Logic in High Definition: Trends in Logical Semantics [1st ed.] 9783030534868, 9783030534875

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Trends in Logic 56

Alessandro Giordani Jacek Malinowski   Editors

Logic in High Definition Trends in Logical Semantics

Trends in Logic Volume 56

TRENDS IN LOGIC Studia Logica Library VOLUME 56 Editor-in-Chief Heinrich Wansing, Department of Philosophy, Ruhr University Bochum, Bochum, Germany Editorial Board Arnon Avron, Department of Computer Science, University of Tel Aviv, Tel Aviv, Israel Katalin Bimbó, Department of Philosophy, University of Alberta, Edmonton, AB, Canada Giovanna Corsi, Department of Philosophy, University of Bologna, Bologna, Italy Janusz Czelakowski, Institute of Mathematics and Informatics, University of Opole, Opole, Poland Roberto Giuntini, Department of Philosophy, University of Cagliari, Cagliari, Italy Rajeev Goré, Australian National University, Canberra, ACT, Australia Andreas Herzig, IRIT, University of Toulouse, Toulouse, France Wesley Holliday, UC Berkeley, Lafayette, CA, USA Andrzej Indrzejczak, Department of Logic, University of Lódz, Lódz, Poland Daniele Mundici, Mathematics and Computer Science, University of Florence, Firenze, Italy Sergei Odintsov, Sobolev Institute of Mathematics, Novosibirsk, Russia Ewa Orlowska, Institute of Telecommunications, Warsaw, Poland Peter Schroeder-Heister, Wilhelm-Schickard-Institut, Universität Tübingen, Tübingen Baden-Württemberg, Germany Yde Venema, ILLC, Universiteit van Amsterdam, Amsterdam, Noord-Holland, The Netherlands Andreas Weiermann, Vakgroep Zuivere Wiskunde en Computeralgebra, University of Ghent, Ghent, Belgium Frank Wolter, Department of Computing, University of Liverpool, Liverpool, UK Ming Xu, Department of Philosophy, Wuhan University, Wuhan, China Jacek Malinowski, Institute of Philosophy and Sociology, Polish Academy of Sciences, Warszawa, Poland Assistant Editor Daniel Skurt, Ruhr-University Bochum, Bochum, Germany Founding Editor Ryszard Wojcicki, Institute of Philosophy and Sociology, Polish Academy of Sciences, Warsaw, Poland The book series Trends in Logic covers essentially the same areas as the journal Studia Logica, that is, contemporary formal logic and its applications and relations to other disciplines. The series aims at publishing monographs and thematically coherent volumes dealing with important developments in logic and presenting significant contributions to logical research. Volumes of Trends in Logic may range from highly focused studies to presentations that make a subject accessible to a broader scientific community or offer new perspectives for research. The series is open to contributions devoted to topics ranging from algebraic logic, model theory, proof theory, philosophical logic, non-classical logic, and logic in computer science to mathematical linguistics and formal epistemology. This thematic spectrum is also reflected in the editorial board of Trends in Logic. Volumes may be devoted to specific logical systems, particular methods and techniques, fundamental concepts, challenging open problems, different approaches to logical consequence, combinations of logics, classes of algebras or other structures, or interconnections between various logic-related domains. Authors interested in proposing a completed book or a manuscript in progress or in conception can contact either [email protected] or one of the Editors of the Series.

More information about this series at http://www.springer.com/series/6645

Alessandro Giordani Jacek Malinowski •

Editors

Logic in High Definition Trends in Logical Semantics

123

Editors Alessandro Giordani Department of Philosophy Università Cattolica del Sacro Cuore Milano, Italy

Jacek Malinowski Institute of Philosophy and Sociology Polish Polish Academy of Sciences Warszawa, Poland

ISSN 1572-6126 ISSN 2212-7313 (electronic) Trends in Logic ISBN 978-3-030-53486-8 ISBN 978-3-030-53487-5 (eBook) https://doi.org/10.1007/978-3-030-53487-5 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Contents

Logic in High Definition—Trends in Logical Semantics . . . . . . . . . . . . . Alessandro Giordani and Jacek Malinowski

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Relating Semantics as Fine-Grained Semantics for Intensional Logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tomasz Jarmużek

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Some Intensional Logics Defined by Relating Semantics and Tableau Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tomasz Jarmużek and Mateusz Klonowski

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Relating Semantics for Connexive Logic . . . . . . . . . . . . . . . . . . . . . . . . Jacek Malinowski and Rafał Palczewski Exact Truthmaking as Inexact Truthmaking by Minimal Totality Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hannes Leitgeb Hyperintensionality in Imagination . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pierre Saint-Germier

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Deontic Logic with Action Types and Tokens . . . . . . . . . . . . . . . . . . . . 117 Alessandro Giordani Causal Agency and Responsibility: A Refinement of STIT Logic . . . . . . 149 Alexandru Baltag, Ilaria Canavotto, and Sonja Smets Impossible Individuals as Necessarily Empty Individual Concepts . . . . . 177 Marie Duží, Bjørn Jespersen, and Daniela Glavaničová Metalinguistic Focus in P-HYPE Semantics . . . . . . . . . . . . . . . . . . . . . . 203 Luke Edward Burke Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

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Logic in High Definition—Trends in Logical Semantics Alessandro Giordani and Jacek Malinowski

Abstract This introductory chapter has two main aims: first, we provide a characterization of the basic notions of logical semantics; second, we give a short description of the remaining chapters of this book.

1 Introduction Sentences, that is sequences of signs constructed on the basis of the rules of a specific language, reflect countless diversity of objects and states of affairs in the world around us. How is it possible? A common intuition is that sentences, like other sequences of signs, allow us to describe the world through their content. Still, to say that the content of expressions is what allows us to describe the world is an unsatisfactory answer as long as we do not specify what kind of entity a content is, or when two expressions are identical as to their content. A classical solution to these problems has it that the content of an expression is its intension, conceived of as a function from possible worlds to the semantic value of the expression, and that two expressions are identical as to their content precisely when they are co-intensional, i.e. when their semantic value coincides across all possible worlds. This solution has led to the development of possible worlds semantics, rightly acknowledged as one of the greatest achievements in contemporary logic, due to its terrific success in modeling a vast variety of modalities, agent-related attitudes, and relational structures. This notwithstanding, possible worlds semantics suffers from crucial limitations, being unable to correctly classify significant invalidities and to capture fine-grained contentrelated distinctions in general. Hyperintensional semantics is intended to overcome such limitations by introducing a richer notion of content and content-equivalence. A. Giordani (B) Università Cattolica di Milano, L. Gemelli 1, 20123 Milano, Italy e-mail: [email protected] J. Malinowski Institute of Philosophy and Sociology, Polish Academy of Sciences, Warszawa, Poland e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Giordani and J. Malinowski (eds.), Logic in High Definition, Trends in Logic 56, https://doi.org/10.1007/978-3-030-53487-5_1

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This volume collects papers aimed to develop and apply fine-grained semantics to a number of open problems, thus contributing to these fascinating new trends. The rest of this introduction will provide some background to better understand the reasons why hyperintensional semantics is desirable.

1.1 Intensional Semantics On a standard conception of truth conditional semantics, to know the content of an expression is to know its semantic value in all possible cases. Thus, to know the content of a sentence is to know its truth conditions, that is the truth value of the sentence in all possible cases. Accordingly, the content of an expression is uniquely determined by its semantic value, so that expressions having the same semantic values in all possible cases have the same content. In more detail, standard truth conditional semantics is based on intuitions akin to the following. 1. An expression has a content: – the content of a name is an individual role; – the content of a sentence is a proposition. 2. A content has an extension: – the extension of an individual role is an individual object; – the extension of a proposition is a truth value. 3. An expression has a referent, which is the extension of its content: – the referent of a name is an individual object; – the referent of a sentence is a truth value. 4. The content of a composite expression is uniquely determined by the content of its components and the form of composition. In particular, the content of a sentence is uniquely determined by the content of its components and its form. 5. A proposition entails another proposition when it includes the content of the second proposition. Therefore, the relation of entailment is at least a reflexive and transitive relation. 6. A proposition is equivalent to another proposition when it includes and is included by the second proposition. Therefore, the relation of equivalence is an equivalence relation. 7. A proposition entails the truth of another proposition when, in all cases, that proposition is true only if the second proposition is true. Therefore, the relation of truth-entailment is at least a reflexive and transitive relation. 8. A proposition is equivalent in truth to another proposition when, in all cases, that proposition is true if and only if the second proposition is true. Therefore, the relation of truth-equivalence is an equivalence relation.

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Intensional semantics builds on these intuitions by exploiting the notion of possible world, introduced to make precise the notion of case, and the notion of intension, introduced to make precise the notion of content. In particular, the intension of an expression is a function that assigns to the expression an extension at each possible world. Intensional semantics is then characterized by a simple assumption: The content of an expression is its intension: – the content of a name is a function from possible worlds to objects; – the content of a sentence is a function from possible worlds to truth values. Two consequences of this basic assumption are noteworthy: Consequence 1: the referent of a sentence at a world is a truth value; Consequence 2: content-equivalence is truth-equivalence. This depends on the fact that the identity conditions of a function are extensional: functions are identical, in a set-theoretical context, provided that they have the same domain and the same values for each element of the domain. Introducing intensions allows for a twofold classification of contexts, viewed as sentences in which other sentences occur. Let  be a relation of logical consequence, ψ C a context and Cφ the context obtained by substituting ψ for φ in C. Then, in a system of logic characterized by , we say that ψ (i) C is extensional iff  (φ ↔ ψ) → (C ↔ Cφ ) for all φ, ψ; ψ (ii) C is intensional iff  (φ ↔ ψ) → (C ↔ Cφ ) for all φ, ψ. Hence, C is said to be extensional with respect to a sentence φ occurring in it when its extension is a function of the extension of φ, so that, the substitution of an equivalent sentence for φ gives rise to an equivalent context, and it is said to be intensional with respect to a sentence φ occurring in it when its intension is a function of the intension of φ, so that, the substitution of a necessarily equivalent sentence for φ gives rise to a necessarily equivalent context. It is then evident that, if contents are intensions, then the existence of non-intensional contexts is to be excluded, since content-equivalent sentences should be substitutable in all contexts.

1.2 Problems The second of the afore-mentioned consequences provides us with a criterion of content-equivalence: two sentences are content-equivalent precisely when they have the same intension. Still, such a criterion gives rise to well-known difficulties, due to the fact that it is not sufficiently fine-grained to capture important content-related distinctions. Specifically, it can be argued that having the same referent at each possible world is not sufficient for two sentences to be content-equivalent. Let us assume that the content of an expression is its intension and that sentences composed by expressions having the same content have the same content. Then we conclude that: sentences having the same intension have the same content; sentences composed

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by expressions having the same intension have the same content. However, an account consistent with these conclusions suffers from the following well-known problems, essentially due to Frege. Frege’s case I: under the assumption that individual names are rigid, the content of a sentence can differ from the content of a sentence having the same intension. “Hesperus” and “Phosphorus” have the same extension at the actual world “Hesperus” and “Phosphorus” have the same intension “Hesperus is Hesperus” and “Hesperus is Phosphorus” have the same intension “Hesperus is Hesperus” and “Hesperus is Phosphorus” have the same content Still, “Hesperus is Hesperus” and “Hesperus is Phosphorus” do not seem to have the same content. Frege’s case II: someone can understand the content of a sentence without understanding the content of a sentence having the same intension. Alice understands the content of “1 + 1 = 2” “1 + 1 = 2” and “2 · sin(π/2) = 2” have the same intension “1 + 1 = 2” and “2 · sin(π/2) = 2” have the same content Alice understands the content of “2 · sin(π/2) = 2” Still, Alice is six years old and does not understand the content of “2 · sin(π/2) = 2”, since she does not understand the content of “sin(π/2)”. Frege’s case III: someone can understand the content of a sentence without understanding the content of a sentence constituted by expressions having the same intension of the expressions constituting the first one. Alice understands the content of “1 + 1 = 2” “1” and “sin(π/2)” have the same intension “1 + 1 = 2” and “sin(π/2) + sin(π/2) = 2” have the same content Alice understands the content of “sin(π/2) + sin(π/2) = 2” Again, Alice is six years old and does not understand the content of “sin(π/2) + sin(π/2) = 2”, since she does not understand the content of “sin(π/2)”. Arguments like these give rise to the problem of granularity: a purely intensional framework is not sufficiently fine-grained to account for crucial content-related distinctions and crucial aspects of modal reasoning.

1.3 Addressing the Problems The previous problems derive from the assumption that sentences having the same intension have the same content and that sentences sharing the same form of composition and composed by expressions having the same intension have the same content. Let W be a set of possible worlds and L a language, identified with the set of sentences that can be generated in it. Since intensions are functions from possible worlds to extensions, truth values in case of sentences, we have 1. int : L → 2W ; 2. given φ ∈ L , intφ : W → 2; 3. given w ∈ W : intφ (w) ∈ 2.

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Since intφ ∈ 2W , there is a one-to-one correspondence between intensions and sets of possible worlds in W . As a consequence, given that contents are intensions, sentences having the same content have the same truth value at each possible worlds. In addition, since being different in content is equivalent to being true in different possible worlds, discernibility relative to content just amounts to discernibility in terms of sets of possible world. Hence, the problem of granularity takes the following form: discernibility in terms of sets of possible world is not sufficiently fine-grained to account for crucial content-related distinctions and aspects of modal reasoning. The present formulation of the problem suggests two immediate solutions. A first solution consists in changing W so as to allow for different kinds of elements, like situations, structured propositions or procedures, thus granting the possibility of making more distinctions in terms of these elements. This solution gives rise to situation semantics [1, 2], state semantics [14, 19], truthmaker semantics [7, 8], impossible world semantics [4, 18], possibility semantics [11, 12], proposition semantics [9, 16], and transparent intensional logic [5, 17], among others. All these frameworks, even those that are not hyperintensional as they stand, see [15], provide us with appropriate tools for capturing more content-related distinctions in modal reasoning. A second solution consists in changing the notion of content, so as to introduce a novel element that, together with the intension, contributes to specify the sense of a sentence. This solution gives rise to topic-based semantics [3, 10], which incorporates substantial elements from the first solution, and, in its general form, relatedness semantics [6] and, finally, to relating semantics [13], which abstracts from the specific way of identifying the new element.

1.4 Hyperintensional Semantics In logical semantics, the phenomenon of hyperintensionality primarily concerns the identity conditions of entities that constitutes the referents of expressions such as individual constants and sentences. Thus, an account of the identity conditions of the referents of some kinds of expressions is hyperintensional provided that necessary indiscernibility of those referents is not sufficient for ensuring their identity: e.g., an account of the identity conditions of propositions is hyperintensional provided that necessary indiscernibility of truth value is not sufficient for ensuring propositional identity. In light of this trait, an hyperintensional perspective on the world is more fine-grained than an intensional one, thus providing us with high discriminative power and allowing us to see the world in higher definition. Introducing hyperintensional identity conditions is crucial to clarify why different kinds of inference based on the substitution of co-referential expressions or of necessarily co-referential expressions, i.e. intensionally identical expressions, are invalid. As paradigmatic cases consider the following inference schemas: 1. a is imagining that φ (φ ↔ ψ) a is imagining that ψ

2. a knows that φ (φ ↔ ψ) a knows that ψ

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3. a sees to it that φ (φ ↔ ψ) a sees to it that ψ

4. a ought to do φ (φ ↔ ψ) a ought to do ψ

It is widely acknowledged that schemas like these do not preserve truth. So, as to 1 and 2, it seems to be possible for someone to imagine, or to know, that the sky be cloudless without imagining, or knowing, that the sky be cloudless and 210 = 1024. Similarly, as to 3 and 4, it seems to be possible for someone to see to it that, or to have an obligation to the effect that, the window be closed without seeing to it that, or having an obligation to the effect that, the window be closed and either destroyed or not destroyed. In order to provide an account of the invalidity of such schemas, one typically assumes that verbs expressing acts of imagination, states of knowledge, actions and obligations generate hyperintensional contexts, i.e., contexts in which it is not legitimate to substitute an expression for an intensionally equivalent expression. Hence, hyperintensional contents, i.e., contents that are different though necessarily equivalent, are to be introduced. The argument is then as follows: (i) there exist hyperintensional operators, such as the ones associated with imagining, knowing, acting, and it can be shown that the contexts generated by using such operators are hyperintensional; (ii) it is possible for an agent to know that φ without knowing that ψ, where ψ is necessarily equivalent to φ, so that the content of “a knows that φ” is different from the content of “a knows that ψ”; (iii) the content of composite expressions is a function of the content of its component, so that the content of a sentence like “a knows that φ” is a function of the contents of the operator “a knows that” and the sentence “φ”; (iv) therefore, the content of φ is different from the content of ψ, even though ψ is necessarily equivalent to φ, so that contents are hyperintensional. In focusing on hyperintensional phenomena, two main lines of research are open. The first one is aimed at providing a positive definition of hyperintensional equivalence, typically developing semantic devices that go beyond possible world semantics. The main problem in this case is to provide a principled account capable of avoiding both the identification of content-equivalence with intensional equivalence, so as to exclude that the content of φ be identical with the content of (φ ∧ ψ) ∨ φ, and the identification of content-equivalence with syntactical equivalence, so as to exclude that the content of (φ ∧ ψ) be different from the content of (ψ ∧ φ). The second line assumes that a general characterization of hyperintensional equivalence is given and aims at devising hyperintensional systems sufficiently strong to enable us to cope with specific problems that arise in the application of modal logic. In turn, in following the second line, two main options are available. On the one hand, we can opt for providing a new interpretation of existent modalities so as to account for the fact that they give rise to hyperintensional contexts. This option leads to an improvement of standard modal semantics. On the other hand, we can opt for further analyzing existent modalities in terms of more fine-grained modalities.

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1.5 Contributions This volume collects contributions that mainly follow the second line of research mentioned above, introducing hyperintensional systems aimed at solving some open philosophical problems. Specifically, the first three studies focus on relating semantics, while the following ones discuss fundamental issues related to hyperintensional semantics or develop hyperintensional frameworks to address issues in modal, in epistemic, deontic, and action logic.

2 Tomasz Jarmu˙zek: “Relating Semantics as Fine-Grained Semantics for Intensional Logics” This text has a programmatic and introductory character. In the paper, we outline a fine-grained semantics for intensional logics. The fundamental idea of the semantics is that the logical value of a given complex proposition is the result of two things: a valuation of propositional variables supplemented with a valuation of relation between the main components of this complex proposition. The latter thing is a formal representation of intensionality that emerges from the connection of several simpler propositions into one more complex proposition. In the first part of the paper, we present some linguistic motivations for the semantics. Later, we propose a very general, multi-valued view on relating semantics, and, in a more detailed way, we consider its two-valued specification, referring also to its historical applications and origin. A further generalization is made when we combine relating semantics with possible world semantics in the subsequent part. The paper concludes with a proposal of defining intensional operators as secondary notions that are based on relating connectives. By dint of the proposal, we can control the behavior of the operators by changing properties of semantic structures for the relating connectives that we use in the definitions.

3 Tomasz Jarmu˙zek and Mateusz Klonowski: “Some Intensional Logics Defined by Relating Semantics and Tableau Systems” The phenomenon of hyperintensionality indicates the problem of propositional identity condition. If the agent knows that (resp. sees to it that, brings it about that, ought to do) φ, then what kind of equivalence ensures that the agent knows that (resp. sees to it that, brings it about that, ought to do) ψ? Such an equivalence should be based on some kind of a relationship between φ and ψ that would allow ψ to inherit the φ’s property imposed by the propositional operator. One of the possible solutions is offered by relating logics. Relating logics are examples of non-classical logics that

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make it possible to seriously consider various non-logical relationships that occur between sentences. The main idea behind such logics is that logical values is not the only thing that matters when we consider truth conditions for formulas built by propositional connectives. What should be considered as another important factor is a content relationship or some other relations, such as causality and temporal successions which hold between sentences or between the states of affairs that are usually expressed by means of those sentences. Hence, the factor is in fact intensional. In the article, we analyze relating logics and make a philosophical introduction to this subject; we introduce a relating language and its general semantic framework. It is a short introduction to the field of relating logics. Next, we describe some special cases of relating logics from a semantic point of view, and finally, as a decision procedure we introduce adequate tableau systems for those logics.

4 Jacek Malinowski and Rafał Palczewski: “Relating Semantics for Connexive Logic” By a connexive logic we mean any logical system, i.e. any set of sentences closed under substitutions and modus ponens rule, containing well known Aritotle’s and Boethian Theses. In this paper, we construct a relating semantics for connexive logic. We show an expressive power of this semantics applying it to Barbershop paradox noted by Lewis Carrol in 1894. This paradox shows unexpected features of conditionals. First we show that connexive interpretation shed new light on Barbershop paradox. We show that relating semantics allow to elucidate unexpected properties of a conditional considered in Barbershop paradox. This way this paper serves as an illustration of first and second paper in this volume applying the method developed there to analysis of connexive implication.

5 Hannes Leitgeb: “Exact Truthmaking as Inexact Truthmaking by Minimal Totality Facts” This article sketches a proposal for how to interpret exact truthmaker semantics within inexact truthmaker semantics: exact truthmaking might be viewed as inexact truthmaking by minimal totality facts. While the philosophical idea is explained by reference to an example, the logical details are left to follow-up work.

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6 Pierre Saint-Germier: “Hyperintensionality in Imagination” Franz Berto has recently proposed two distinct semantics for the logic of ceteris paribus imagination, one which combines a theory of topics with a standard possible worlds semantics, another based on impossible worlds. An important motivation for using these tools is to handle the hyperintensionality of imagination reports. I argue however that both semantics prove inadequate for different reasons: the former fails to draw some intuitive hyperintensional distinctions, while the latter draws counterintuitive hyperintensional distinctions. I propose an alternative truthmaker semantics, guided by an independently motivated philosophical analysis of the content of imagination acts, that preserves the attractive features of both approaches, while avoiding their symmetrical defects in the treatment of hyperintensionality.

7 Alessandro Giordani: “Deontic Logic with Action Types and Tokens” A new characterization of the deontic operators of permission and prohibition is introduced based on a distinction between action types and action tokens. The resulting deontic action logic constitutes a hyperintensional system providing resources for a fine-grained study of the basic deontic notions. The logic is proved to be complete with respect to an appropriate semantics, where models include both possible worlds and action tokens, and the philosophical significance of the distinction is demonstrated by showing that a number of puzzles afflicting current accounts of the deontic operators find intuitive solutions in the new framework.

8 Alexandru Baltag, Ilaria Canavotto and Sonja Smets: “Causal Agency and Responsibility: A Refinement of STIT Logic” This paper proposes a refinement of STIT logic to make it suitable to model causal agency and responsibility in basic multi-agent scenarios in which agents can interfere with one another. We do this by supplementing STIT semantics, first, with action types and, second, with a relation of opposing between action types. We exploit these novel elements to represent a test for potential causation, based on an intuitive notion of expected result of an action, and two tests for actual causation from the legal literature, i.e. the but-for and the NESS tests. We then introduce three new STIT operators modeling corresponding notions of causal responsibility, which we call potential, strong, and plain responsibility, and use them to provide a fine-grained analysis of a number of case studies involving both individual agents and groups.

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9 Marie Du˘zí, Daniela Glavani˘cová and Bjørn Jespersen: “Impossible Individuals as Necessarily Empty Individual Concepts” We talk about ‘impossible objects’ in many areas, ranging from empirical and nonempirical theories to the realm of fiction, myth and folklore: a mathematical pendulum, a perfect market, the set of all sets that are not members of themselves, Kafka’s Gregor Samsa, Pegasus, The Puss in the Boots, and so forth. This paper proposes a hyperintensional account of a special case of impossible objects, so-called ‘impossible individuals’. Our (broadly Fregean) proposal is to identify ‘impossible individuals’ with necessarily empty individual concepts. The main goal of the paper is to develop a method that enables us to discover inconsistencies in specifications of individual concepts and thus prove that such concepts could not possibly be matched by an extension (an individual). Furthermore, this approach allows for a fine-grained individuation of impossible individuals. Fine-graining will not be explored in the present paper, but its very possibility adds to the overall plausibility of the present account.

10 Luke Burke: “Metalinguistic Focus in P-HYPE Semantics” P-HYPE is a hyperintensional situation semantics in which hyperintensionality is modelled via the notion of a side effect. The concept of a side effect comes from the discussion of monads in functional programming languages and from the computational lambda calculus, and it has recently been found a useful way of conceptualising certain features of natural language semantics, such as intensionality and scope. We combine a perspective-sensitive semantic theory with Hannes Leitgeb’s logic, HYPE, using monads from category theory in order to upgrade an ordinary intensional semantics to a possible hyperintensional counterpart. In order to gradually extend the types of phenomena that P-HYPE can capture, we have elsewhere expanded our semantic theory to capture cases of intra-sentential intensional anaphora. Here we increase the coverage of P-HYPE further, by using the pointed power set monad to account for certain cases of hyperintensionality involving metalinguistic focus which seem to resist treatment in P-HYPE as it stands. In so doing, we provide a mini case-study in how to combine monads together in order to integrate different side effects in one language, thereby exemplifying one of the advantages of monads as a tool in compositional natural language semantics.

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11 The Conference The work on this volume began during the Studia Logica conference “Trends in Logic XVIII: Fine-Grained Semantics for Modal Logic: Formal and Foundational Issues” which took place on September 24–27 at the Catholic University of Milan—Italy. At the conference, 24 contributions on the current developments of the standard intensional semantic were presented and discussed. Some of the papers collected in this volume were presented on the conference. Others were initiated by the discussions that took place. This volume is not so much the aftermath of that conference, as it is more the result of the work that was initiated there. Acknowledgements At this point, we would like to thank everyone who contributed to the organization of this conference. We would also like to thank those who contributed to the event and to the development of this volume. Special words of gratitude are due to the reviewers. They have significantly influenced the final form of the works contained herein. Thank you wholeheartedly.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

Barwise, J., & Perry, J. (1983). Situations and attitudes. Cambridge: MIT Press. Barwise, J. (1989). The situation in logic. Center for the Study of Language (CSLI). Berto, F. (2018). Aboutness in imagination. Philosophical Studies, 175, 1871–1886. Berto, F., & Jago, M. (2019). Impossible worlds. Oxford: Oxford University Press. Du˘zí, M., Jespersen, B., Materna, P. (2010). Procedural semantics for hyperintensional logic: Foundations and applications of transparent intensional logic (p. 2010). Dordrecht: Springer. Epstein, R. L. (1990). The semantic foundations of logic. Vol. 1: Propositional logics. Dordrecht: Springer. Fine, K. (2014). Truth-maker semantics for intuitionistic logic. Journal of Philosophical Logic, 43, 549–577. Fine, K. (2017). A theory of truthmaker content i: Conjunction, disjunction and negation. Journal of Philosophical Logic, 46, 625–674. Fox, C., & Lappin, S. (2005). Formal foundations of intensional semantics. Oxford: Blackwell. Giordani, A. (2019). Axiomatizing the logic of imagination. Studia Logica, 107(4), 639–657. Holliday, W. H. (2014). Partiality and adjointness in modal logic. Advances in Modal Logic (pp. 313–332). London: College Publications. Humberstone, L. I. (1981). From worlds to possibilities. Journal of Philosophical Logic, 10, 313–339. Jarmu˙zek, T., Kaczkowski, B. (2014). On some logic with a relation imposed on formulae: Tableau system F. Bulletin of the Section of Logic, 43(1/2), 53–72. Leitgeb, H. (2019). HYPE: A system of hyperintensional logic. Journal of Philosophical Logic, 48, 305–405. Odintsov, S., Wansing, H. (2020). Routley star and hyperintensionality. Journal of Philosophical Logic, first online 2020. Pollard, C. (2008). Hyperintensions. Journal of Logic and Computation, 18, 257–282. Tichý, P. (1988). The Foundations of Frege’s Logic. Berlin: de Gruyter. Wansing, H. (1990). A general possible worlds framework for reasoning about knowledge and belief. Studia Logica, 49, 523–539. Wolniewicz, B. (1982). A Formal Ontology of Situations. Studia Logica, 41, 381–413.

Relating Semantics as Fine-Grained Semantics for Intensional Logics Tomasz Jarmu˙zek

Abstract This text has a programmatic and introductory character. In the paper, we outline a fine-grained semantics for intensional logics. The fundamental idea of the semantics is that the logical value of a given complex proposition is the result of two things: a valuation of propositional variables supplemented with a valuation of relation between the main components of this complex proposition. The latter thing is a formal representation of intensionality that emerges from the connection of several simpler propositions into one more complex proposition. In the first part of the paper, we present some linguistic motivations for the semantics. Later, we propose a very general, multi-valued view on relating semantics, and, in a more detailed way, we consider its two-valued specification, referring also to its historical applications and origin. A further generalization is made when we combine relating semantics with possible world semantics in the subsequent part. The paper concludes with a proposal of defining intensional operators as secondary notions that are based on relating connectives. By dint of the proposal, we can control the behavior of the operators by changing properties of semantic structures for the relating connectives that we use in the definitions.

1 Omnipresence of Intensionality In philosophical logic ([18, 19] etc.) there are many examples of schemas of argumentation which are classically valid (i.e. valid in Classical Propositional Logic, in short: CPL), but invalid or strange from a common sense point of view. For example, the following arguments are classically valid: p→q q →r p→r

(a)

T. Jarmu˙zek (B) Department of Logic, Nicolaus Copernicus University in Toru´n, Torun, Poland e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Giordani and J. Malinowski (eds.), Logic in High Definition, Trends in Logic 56, https://doi.org/10.1007/978-3-030-53487-5_2

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p∧q →r ( p → r ) ∨ (q → r )

(b)

p∧q q∧p

(c)

However, when we replace the parameters with real sentences and the classical connectives with some natural language connectives, we obtain quite bizarre inferences: If Ann dies, then Mark will be in despair. If Mark is in despair, then Ann will call for a doctor. (a1) If Ann dies, then Ann will call for a doctor. If Mark loves Ann and Ann loves Mark, then Mark and Ann love each other. If Mark loves Ann, then Mark and Ann love each other or if Ann loves Mark, then Mark and Ann love each other. Mark goes home and (he) makes a meal for himself. Mark makes a meal for himself and (he) goes home.

(b1)

(c1)

The problem appears because, in the contexts of making the arguments, we consider not only the logical values of sentences, but we also expect some relationships between the content of the sentences involved in inferences (a1), (b1), and (c1). For example, in the case of (c1), we expect that Mark firstly goes home and then he makes a meal for himself, not the other way round. So, what happens, happens in some temporal order which is asymmetric. But CPL is an extensional logic and Boolean connectives capture only truthfunctional aspects. We cannot, therefore, expect the additional (non-logical) connections we see between the sentences contained in particular examples to be expressed in CPL. This lack is, presumably, one of the several factors that gave rise to nonclassical logics. For example, arguments (a) and (b) are invalid in the case of conditional logic C, while argument (b) is invalid in intuitionistic logic, modal logic S2, or basic relevant logic B. On the other hand, argument (c) is always valid if the conjunction is understood merely extensionally. Let us concentrate on classical implication and conjunction for a while. The material implication A → B states that sentence A is false or sentence B is true, while the classical conjunction A ∧ B states the truth of A as well as B, and nothing more. Nonetheless, when we use the natural language, we express additional relationships (e.g. analytical, causal, temporal, in some sense relevant etc.), so in a formal language

Relating Semantics as Fine-Grained Semantics for Intensional Logics

15

we need to have an implication, a conjunction etc., that state more than merely the extensional aspect. In general, classical connectives were treated primarily as reflecting the meaning of certain natural language connectives, but in fact they reflect only their extensional aspect, which is usually defined as follows. From the extensional perspective, the logical value of a complex proposition that is built of an extensional connective depends only on two factors: 1. logical values of propositions that are parts of the complex proposition 2. logical structure of the complex proposition. In many contexts, however, the logical value of simpler sentences is not a unique determinant of logical values of more complex propositions, and, as we know, there exists a lot of counter-examples to the extensional view. Let us consider the conditional sentence: If George Boole was born in Lincoln, then 2 + 3 = 5.

(1)

Since the sentences: George Boole was born in Lincoln.

(2)

2+3=5.

(3)

George Boole was born in Lincoln → 2 + 3 = 5

(4)

are true, the material implication:

is true as well. However sentence (1) is not true from a common sense point of view. In order for this sentence to be true, some kind of content-related relationship is required between the precedent and the succedent that are the sentence’s grammatical components. No one can say what (1) and (2) have in common from this point of view, because this fusion seems just nonsensical when considered as an actually uttered proposition. Hence, sentence (4) is true as a material implication, but not true, awkward, from a common sense point of view, or at least false. Very similar problems appear in respect to other classical connectives and their counterparts in natural language. For example, the sentences: George Boole was born in Lincoln.

(2)

George Boole moved from Lincoln to Cork.

(5)

are true, so their classical conjunction is also true: George Boole moved from Lincoln to Cork ∧ George Boole was born in Lincoln. (6)

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However, the complex sentence: George Boole moved from Lincoln to Cork and was born in Lincoln.

(7)

seems to state a temporal relationship between the components and could be properly rewritten as: George Boole moved from Lincoln to Cork and then was born in Lincoln.

(8)

which is undoubtedly false, since the fact described by (2) took place first, and then the fact described by sentence (5). In most cases, a conjunction in natural language requires not only the truth of simpler sentences, but also some relationship. In this example, it is a relationship of temporal precedence. Summing up, the “real sentences” we use in non-formal languages contain something more then logical values and grammatical structure. This “something” is an intensional aspect. It can be emphasized by some stylizations, like in the examples of relationships listed below: causal, temporal, preferential or analytical ones. If you turned off the light, it was pretty dark in the room.

(9)

If proposition (9) is true as a conditional, then the antecedent denotes the cause of what is denoted by the succedent. Therefore, the components are related by some causal aspects. A causal relationship on the level of language can be expressed by a conditional or conjunction: • if A happens, then it causes B • A happened and it caused B. If Jan arrived home, then his wife arrived, too.

(10)

If proposition (10) is true as a conditional, either the precedent is false or the consequence true, although, additionally, they must be related by temporal precedence. A temporal relationship on the level of language can be expressed by a conditional or conjunction: • if A happens, then B happens next • A happened and B happened next. Basia would like to go to the cinema or the theatre.

(11)

If proposition (11) is true as a disjunction, at least one of the components has to be true, although Basia probably prefers the cinema to the theatre. So they are additionally related by preference relation. On the level of language, a preferential relationship can be expressed by some kinds of disjunction or conjunction: • A or B (but rather A)

Relating Semantics as Fine-Grained Semantics for Intensional Logics

17

• A and B (but rather A). One of the interesting relationships from a philosophical point of view is an analytic relationship: If Mark is a parent, then he has got at least one child.

(12)

If proposition (12) is true, then either the precedent is false, or the consequence true, and, additionally, the property of being a parent (a) requires the property of having at least one child (b) (in the same aspect: biological or adoptive). But (b) happens, if (a) happens, since (b) analytically follows from (a). On the level of language, an analitycal relationship can be expressed by some kinds of conditional or conjunction: • if A is true, then by that fact it analytically follows that B is true • A is true and by that fact it analytically follows that B is true. It seems that anytime we consider a more complex sentence, we come across the phenomenon of intensionality. If we do not treat it seriously, the problems outlined on the level of sentences are automatically transported to the level of argumentation, like in examples (a1), (b1), (c1) presented at the beginning. In the case of conditionals, those problems are called paradoxes of material implication. All of the examples of relationships we have given so far show that connectives require something more than merely a valuation of atomic sentences. For example, the conditional: If A is true, then by that fact it analytically follows that B is true can be interpreted as: 1. A is false or B is true 2. sentence A is related to sentence B by an analytic relationship. As we see, regular condition 1. that defines material implication has become enhanced, since by 2., additionally, sentence A must be analytically related to sentence B. This way we obtain a new kind of implication that works in intensional contexts of argumentation, while material implication may still be usable in extensional contexts. Of course, the second condition (of being in some relation) might be also imposed on other classical connectives (or, more generally on all extensional connectives), which results in the emergence of new connectives that are called relating connectives (since they require the condition of being related). Moreover, by imposing some additional conditions on the mentioned relation between sentences, it is possible to model special and interesting interpretations of relating connectives, suitable for some philosophical and technical applications, like in the examples above. Finally, we can define logical systems based on those interpretations. The logics under examination we call relating logics, since they cover ways of making arguments in a language of formulas built of some relating connectives (the idea is similar to modal logics, which are logics defined in a language with some modal operators).

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But the starting point is an elaboration of the appropriate relating semantics. The semantics we would like to examine takes into account not only logical values of atomic and complex sentences, but also it valuates in an independent way (or dependent, if we wish) whether two (or more sentences) are somehow related. Hence, now, the logical value of a complex sentence can be not only the result of the valuation of simpler sentences. This semantic mechanism allows us to distinguish two situations: (a) a complex sentence is true from the extensional point of view, since its components satisfy some semantic conditions; (b) a complex sentence is true from the extensional point of view, but not from the intensional one, since a connection of its components does not satisfy some extra semantic condition. By dint of this, we can operate on two sorts of logical connectives: classical (or Boolean) and non-classical, intensional ones, and, finally, we can get intensional logics that in an innovative way propose a solution to the problem of non-extensional relationships between sentences in an argumentation. But we must start with some intensional semantics. Such a solution seems to be quite natural, since when two (or more) propositions in natural language are connected by a connective belonging to this language, some sort of emergence occurs (as in the examples). In fact, the key feature of intensionality is that adding a new connective results in the emergence of a new quality, which itself does not belong to the components of a given complex proposition built by means of the same connective. An additional valuation function determines precisely this quality. The use of the concept of emergence is justified here, because the quality that arises as a result of the connection of a few simpler propositions is not reducible to the properties of the initial propositions. Consequently, if this emergence is to be determined, we need additional valuations in a model, which is of key-importance in the relating semantics that we propose.

2 Propositional Language and Non-modal Relating Semantics In order to discuss the issue of relating semantics, we need some formal language. As the primitive symbols of that language we assume the non-empty set of propositional variables Var. We also need a set of propositional connectives. Let N be a set of natural numbers. We assume that: I = {1, 2, . . . , n : n ∈ N} and K ⊆ N. Intuitively, i ∈ I is a number of a given propositional connective, while k ∈ K is an arity of the connective. So, we assume a non-empty set of propositional connectives: Con = {cik : i ∈ I, k ∈ K }. The set of formulas For is the least set X that satisfies the conditions: • Var ⊆ X • cik (A1 , . . . , Ak ) ∈ X , where cik ∈ Con and A1 , . . . , Ak ∈ X .

Relating Semantics as Fine-Grained Semantics for Intensional Logics

19

Now, we can propose a simple relating semantics to the language For. Let LV be a set of logical values — it contains at least two values — while DLV be a non-empty proper subset of LV of designated values. Both sets will serve to valuate the extensional aspect of formulas. Analogous sets are needed for a valuation of connections between sentences. So we have LVr , a set of values for connections that contains at least two values, while DLVr will be a non-empty proper subset of LVr of designated values. Both sets will serve to valuate the intensional aspect of formulas. Hence, in a model we will have at least two functions: (1) a valuation of variables, and (2) a valuation of connections between formulas from For. The first function is just: v : Var → LV. However, any connective from the set Con should possess its own valuation of being related. That is, instead of one function (2) in a semantic structure, there is a family of valuations: {vi }i∈I where for any cik ∈ Con, vi is a function vi : For × · · · × For → LVr . So, if there k

is a connective cik with arity k in the language, then there exists a valuation of all k-tuples that belong to the Cartesian product on the set of formulas (of arity k) into the set of values for connections LVr . Now, we can say that a model (or relating model) is a structure v, {vi }i∈I , where the first element is a valuation of variables, while the second one is a family of valuations of connections between sentences from For, exactly one per each connective from Con. Models of this kind are basic (non-modal) tools of relating semantics. If a family {vi }i∈I in the models is of cardinality bigger than one, we call them multi-relating. If a family {vi }i∈I contains only one valuation, we call the model mono-relating. An interesting thing is that mono-relating semantics can be considered as a special case of multi-relating semantics. It happens when for all i, j ∈ I , vi = v j . Then, for simplicity’s sake, {vi }i∈I can be treated as a single valuation. Of course, in order to extend a valuation of propositional parameters to the set of all formulas, we need to know how the extension of function v works in the case of more complex formulas in respect to a given connective. So, we need the set { f cik : cik ∈ Con} (similarly, as in the case of many-valued logics that can be treated as a structure LV, DLV, { f c : c ∈ Con} ). Now, we can uniquely extend the function v : Var → LV to function V : For → LV in a model, by induction, for all formulas A ∈ For:  v(A), if A ∈ Var V (A) = k f ci (V (B1 ), . . . V (Bk )), if A := cik (B1 , . . . , Bk ) and cik ∈ Con. The question arises: how do the models work? Let us consider some formula, for example cik (A1 , . . . , Ak ) and some model M = v, {vi }i∈I . When is the formula satisfied in the model: M |= cik (A1 , . . . , Ak )? To explain that, we use some standard notions. Assume V is the extension of v onto For. Let V (A1 ), …V (Ak ) be values

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of formulas A1 , . . . , Ak in model M. We say that formula cik (A1 , . . . , Ak ) has got a designated value (is satisfied) in the model trough the condition: M |= cik (A1 , . . . , Ak ) ⇐⇒ V (cik (A1 , . . . , Ak )) ∈ DLV & vi (A1 , . . . , Ak ) ∈ DLVr , which means that the formula has got in the model a designated value (V (cik (A1 , . . . , Ak )) ∈ DLV) — the extensional condition, and the relation between formulas A1 , . . . , Ak takes also a designated value (vi (A1 , . . . , Ak ) ∈ DLVr ) — the intensional condition of being related in a sufficient way. Clearly, by negation, we also know when the formula is not satisfied (has not got a designated value) in a model: / DLV or vi (A1 , . . . , Ak ) ∈ / DLVr , M |= cik (A1 , . . . , Ak ) ⇐⇒ V (cik (A1 , . . . , Ak )) ∈ which means that either the extensional condition, or the intensional one is not fulfilled (the components are not related enough). Some readers can almost immediately ask about the reasons why we introduce such a subtle and sophisticated semantics with many-valuedness of being related. Look at the example. Let us examine some kind of conjunction ∧s . If two sentences A and B are connected by conjunction A ∧s B, this compound sentence reads as follows: A happens and soon B happens. So, the conjunction states some kind of temporal relationship. First A happens, then B happens. But B must happen soon, not too late! What does it mean? We can assume that in the set LVr there is some order of temporal distances between the facts that are denoted by the formulas in the given model, and soon for a pair (A, B) means that v∧s (A, B) ∈ DLVr . What if we assume that in a given model A ∧s B and B ∧s C? Is A ∧s C also satisfied? So, A happens and soon C happens as well? When we assume that the distances in LVr are closed under some kind of operation of addition +, then the answer is: A happens and soon C happens iff v∧s (A, B) + v∧s (B, C) ∈ DLVr , since v∧s (A, B) + v∧s (B, C) = v∧s (A, C). Of course, on the function v∧s there could be imposed other important and interesting constraints that reflect our semantic intuitions on time, temporal order and similar issues. The set LVr can be, for instance, a metric space, or other kind of space under additional operations. We should remember that it is merely an example. Very similar frames can be defined for spatial or other kinds of relations or properties that are graduated (like probability). Summing up this part, a relating model (non-modal, basic) is a pair: M = v, {vi }i∈I

(1∗)

provided we assume the sets: LV, DLV, { f cik : cik ∈ Con}, LVr , and DLVr . Later we will be discussing a special, limited case of models (1∗), and, finally, some extensions to the modal, possible worlds context.

Relating Semantics as Fine-Grained Semantics for Intensional Logics

21

3 Two-Valued Relating Semantics For further considerations, we must try to simplify our semantic context a bit. So now, let us assume two things on models (1∗). The assumptions we will make show that relating models can be two-valued in two different senses. First, let LVr = {0, 1} and so DLVr = {1}. Therefore, the valuations in the family {vi }i∈I are just characteristic functions. Instead of writing vi (A1 , . . . , Ak ) = 1, we can write: Ri (A1 , . . . , Ak ) (which states that formulas A1 , . . . , Ak are related), as Ri (A1 , . . . , Ak ) (which states well as instead of vi (A1 , . . . , Ak ) = 0, we can write:  Ri is the complement of R). Note that formulas A1 , . . . , Ak are not related, because  that model: (2∗) M = v, {Ri }i∈I contains then a family of relations {Ri }i∈I , instead of a family of functions {vi }i∈I . In these models, a valuation v can still be not binary. For example, we can take LV = {1, 21 , 0} and DLV = {1}. Now, considering the implication p → q as defined by Łukasiewicz, we can expect that not only the extensional condition should be satisfied. So, even if v( p) = v(q) = 21 , the implication is true in a model, when R→ ( p, q). We see that in this way it is possible to define interesting many-valued relating logics. Second, independently, we can make another assumption: we work in a twovalued logic, so v assigns to the variables 0 or 1 (where 1 is designated, obviously), hence LV = {0, 1}, while DLV = {1}. Both assumptions can be combined as long as we have any need to model some semantic structures. Now, we consider the models with two values for variables and with relations between formulas, so we use both assumptions. Moreover, let Con consist of: (a) one unary connective ¬, and (b) eight binary connectives: ∧, ∨, →, ↔, ∧w , ∨w , →w , ↔w . Theoretically, we should operate with multi-relating models of the form: v, {R¬ , R∧ , R∨ , R→ , R↔ , R∧ , R∨ , R→ , R↔ } , w

w

w

w

where v : Var −→ {1, 0} and {R¬ , R∧ , R∨ , R→ , R↔ , R∧ , R∨ , R→ , R↔ } is a family of relations — exactly one for each of the connectives (as in (2∗)). But since we would like to interpret the connectives ¬, ∧, ∨, →, ↔ classically, we assume that: R¬ = For and R∧ = R∨ = R→ = R↔ = For × For. Hence, for the classical connectives, we have two relations and they both are universal relations: For and For × For. But this means that when we valuate formulas compounded of the connectives in the superscripts (i.e. ¬, ∧, ∨, →, ↔), only the extensional, classical conditions will play a role. In this way we can treat CPL as a very special instance (and trivial, in fact) of relating logics, where all of the formulas are related to each other, and only valuations of variables determine the logical value of any formula. So, because the universal relations are redundant, we may omit them in the models. As a consequence, our structures have been reduced to the following form: w

w

w

w

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v, {R∧ , R∨ , R→ , R↔ } . w

w

w

w

But we would like to go forward. Let also R∧ = R∨ = R→ = R↔ , so in a model we can have in fact only one relation that is used for the interpretation of all the non-classical connectives, and then models are ordered pairs: v, R . The models are minimal models that can be defined in the relating semantics. Assuming that the classical connectives do not need relations in the models, for our language we have the following truth-conditions for all of the connectives1 : w

w

w

w

Definition 1 Let M be a model and A ∈ For. A is true in M (in symb.: M |= A; and M |= A, if false) iff for any B, C ∈ For: • for propositional variables and formulas built by classical connectives: vM (A) = 1,

if A ∈ Var

M |= B, M |= B and M |= C,

if A := ¬B if A := B ∧ C

M |= B or M |= C, M |= B or M |= C,

if A := B ∨ C if A := B → C

M |= B iff M |= C,

if A := B ↔ C

• for formulas built by strictly relating connectives: [M |= B and M |= C] and RM (B, C), [M |= B or M |= C] and RM (B, C),

if A := B ∧w C if A := B ∨w C

[M |= B or M |= C] and RM (B, C), [M |= B iff M |= C] and RM (B, C),

if A := B →w C if A := B ↔w C.

Let  ⊆ For, we will write M |=  instead of ∀ A∈ M |= A. As we can see (in the first part of the definition), in the case of classical connectives we have the standard extensional interpretation. Of course, the connectives are also relating, but not strictly, since all formulas are related from their point of view. In the case of strictly relating connectives, two conditions must be satisfied. The first one is the standard truth-condition for a classical counterpart of a given relating connective. The second one says that both formulas must be related. We can also define a notion of truth with respect to a relation. Let R = {R : R ⊆ For × For} — the set of all binary relations on For. Definition 2 Let R ∈ R and A ∈ For. A is true with respect to R (in symb.: R |= A) iff ∀v∈{1,0}Var v, R |= A. Let  ⊆ For, we will write R |=  instead of ∀ A∈ R |= A. 1 For

the more extensive explanation and introduction of the notions given below see [7].

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23

Having at disposal Definitions 1 and 2, it is possible to define two notions of semantic consequence. Definition 3 Let Q ⊆ R and  ∪ {A} ⊆ For. Let MQ = {v, R : v is a valuation of variables, R ∈ Q}. A is a semantic consequence of  over MQ (in symb.:  |=MQ A) iff ∀M∈MQ (M |=  =⇒ M |= A). Definition 4 Let Q ⊆ R and  ∪ {A} ⊆ For. A is a semantic consequence of  over Q (in symb.:  |=Q A) iff ∀R∈Q (R |=  =⇒ R |= A). The difference between the above definitions resembles the difference between the local and global semantic consequence relation in modal logic. The notion in Definition 3 is a notion of “local” relation. In this case, the relation is defined with a preservation of truth in the model for any valuation and any relation. The relation of semantic consequence in the sense of the Definition 4 is a “global” relation, since it is defined with a preservation of truth in respect to relations from some set.2 A large amount of mono-relating, two valued logics that are based on models v, R arise when we impose some constraints on the relations, thus taking some special subsets of R. We distinguish horizontal, vertical and diagonal relations determined this way (see details in [7]). From a historical point of view, the idea of an additional relation in models (resulting in models similar to v, R ) was proposed for the first time in [20]. Its application to a philosophical problem of relevance was probably proposed for the first time in [3] (see also [4, 11, 12, 16, 17]).3 In those studies, the authors analyzed a special kind of relating logics: relatedness logic. There, a binary relation was considered; this relation was needed to define an extraordinary kind of content-related implication. A relatedness relation concerns some kind of similarity of arguments of relations. The relating logics initiated in [6] are more general. In the paper, two relating connectives were analyzed, in the models without any constraints imposed on R (so, it was the weakest two-valued mono-relating logic). It was suggested there that considering any formal conditions defining the classes of relations, one could determine the subclasses of relations, and in consequence, define a multitude of specific relating logics equipped with a large number of philosophical and not only philosophical motivations and interpretations. Hence, in principle, the relating semantics might be applied to any logic. The relatedness logic (and dependence logic; see [7]) can be found as a part of a much wider class of the relating logics with the multi-domain of applications. In the relating logics, formulas are related in models, but not necessarily by relatedness relation, since they do not have to be similar or content-related in any way, and clearly, not only two-valued. Another example of application of relating semantics to some philosophical logic is Boolean connexive logics and modal Boolean connexive logics. In the papers [9, 10, 14], a relating semantics approach to the connexive implication was presented 2 More 3 In

about the issue of connections between both consequence relations can be found in [7]. [4] multi-relating models similar to models (2∗), but only two-valued, were suggested.

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(for more details see: [15]). We find this as a part of the wider program of developing the relating semantics for intensional logics that we propose here. Now, we come back to our initial reasoning from the beginning of Sect. 1. Let us assume that the implication that appears in all places in (a), (b) is the relating implication →w , while the conjunction in (c) is the relating conjunction ∧w , both with the interpretations from Definition 1 and at the same time let the rest of the connectives in the arguments (i.e. ∧ and ∨ in (b)) be interpreted in the classical way. So, we need some two-valued relating models v, R . It is easy to observe (by Definitions 3, 1, as well as by Definitions 4, 2) that if models v, R contain: • transitive relations R, then schema (a) is valid • relations R which satisfy the condition: R(A ∧ B, C) ⇒ R(A, C) or R(B, C), for all A, B, C ∈ For, then schema (b) is valid • symmetric relations R, then schema (c) is valid. Therefore, the three conditions are the intensional properties that may have the conditional occurring in (a1), (b2) and the conjunction occurring in (c1), to accept all of the three arguments. And partially inversely, if for parameters p, q, r these conditions are not satisfied, the schemas (a), (b), (c) are not valid, and the arguments can be rejected.4

4 Modal Enhancement So far, we have defined the most general semantic structure (1∗) v, {vi }i∈I in Sect. 2. As we said, however, it is a very general, but non-modal structure. But what happens if we combine the structure with the possible world semantics? In fact, we already may say that structure (1∗) is a one-world model, indeed. The world is a valuation of variables it contains. But naturally, we would like to also have models with more worlds and a non-empty relation of accessibility. In this way, we could have more general semantic structures at our disposal. Let us recall the notion of a possible world model. A possible world model is a structure W, Q, v where: • W • Q ⊆W ×W • v : Var × W −→ {0, 1} 4 The

is a non-empty set of possible worlds is an accessibility relation is a valuation at possible worlds.

question about necessary conditions is a crucial part of the research on the foundations of relating semantics that we already conduct. Both problems — sufficient and necessary conditions — are two sides of the correspondence theory we propose. However, these issues need further, less programmatic articles. And we have already some solutions to the problems.

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25

Now, to each w ∈ W , we attach a structure of the kind (1∗) v, {vi }i∈I without a valuation of variables, and hence we get a family of these structures {{{vi }i∈I }w }w∈W , one for each world w ∈ W . Then we extend a model W, Q, v to: W, Q, {{{vi }i∈I }w }w∈W , v .

(3∗)

Structures (3∗) are very general, especially since all valuations in the models can be many-valued. To convince the reader that the models have a strong power of expression,5 we will limit these models to the model in which, instead of families of valuations {{{vi }i∈I }w }w∈W , we add to the model families of binary relations {{{Ri }i∈I }w }w∈W indexed by worlds w ∈ W (as in (2∗)), but with one relation per a world, so just {Rw }w∈W . We will therefore consider the models: W, Q, {Rw }w∈W , v .

(4∗)

Lastly, let us add to Con the unary, modal operator , understood as it is neccessary that …, and let us assume that the set of formulas For is closed under it. Note that now in our language we have the relating implication with , i.e.: (A →w B). Surely, it is a relating counterpart of the good, old strict implication, since for  we assume its standard truth condition, but adjusted to the new models: W, Q, {Rw }w∈W , v , w |= A iff ∀u∈W (wQu ⇒ W, Q, {Rw }w∈W , v , u |= A). The strict relating implication appears to be a very strong tool. It says that in all accessible worlds a relating implication A →w B should be true, so at every accessible world not only A is false or B is true, but also at every accessible world A should be related to B. As opposed to the strict implication: (A → B), the relating strict implication does not preserve some undesirable arguments. For example, for the strict implication we have so-called paradoxes of strict implication: q |= ( p → q) ¬ p |= ( p → q).

(1) (2)

They are not valid if we replace the material implication with the relating one: q |= ( p →w q) ¬ p |= ( p →w q), 5 It

(R1) (R2)

may sound strange to talk about power of expression in the context of semantics, because this term is usually used in reference to a syntax, to some formal language. Here, we intensionally refer it to semantics, as we think that also logical semantics can be compared in respect with which logical systems can be determined by those logical semantics. Maybe a better term would be power of determining.

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since we have straight-forward counter-examples. For (R1) the counter-model can be: W = {w, u}; Q = {w, u }; Rw = Ru = ∅; v(x, w) = v(x, u) = 1, for all x ∈ Var. In turn, for (R2) the counter-model is: W = {w, u}; Q = {w, u }; Rw = Ru = ∅; v(x, w) = v(x, u) = 0, for all x ∈ Var. The examples show that the proposed semantics is more fine-grained than usual one, also in the case of modal issues, since when we interpret the conditional as →w , we have much more control over the conditional’s behavior, as we could even make inferences (R1) and (R2) valid in some class of models, when we assumed appropriate conditions for relating relations in the models. It is worth mentioning that the models of the form (4∗) have been used at least two times so far. In order to define modal Boolean connexive logics, so to model some connexive implication in the language with  and ♦, in [10] a possible world semantics with the structures (4∗) was applied. We also used such structures in the field of deontic logic. In that case we wanted to weaken deontic logic D, and by interpreting relation Rw as a kind of deontic relevance in a world w, we were able to omit most (if not all) of so-called deontic paradoxes that are present in standard deontic logic [8].

5 Defining Modal Operators in Relating Semantics Let us finish with some exemplary proposals for defining several unary modal operators as secondary notions, reducible to relating connectives. We start with some reference to epistemic logic, so, in a narrow sense, a logic of knowledge operator K and belief operator B. We assume that our relating implication A →w B will be understood as an epistemic implication. (We assume additionally the classical connectives from Sect. 3.) So, A →w B will read as: proposition A epistemically implies proposition B. Of course, in the model v, R it has the logical meaning of the relating implication according Definition 1. But the philosophical interpretation is: • proposition B has not got a smaller logical value than proposition A has • for proposition B exists a justification that is not worse than the best justification for A that exists. In the proposed approach, the central notion is relation R interpreted as a comparison of justifications in a cognitive system of an agent. Some propositional constants are added to the language: 1 , …, i , for some i ≥ 1. We can assume that 1 , written just as , is true at any of considered models: M |= . Hence,  represents the standard of Platonic knowledge — the best justified and true sentence of the knowledge system, some kind of epistemic tautology. The remaining 2 , …, i may be true at some classes of models in respect of what is needed to be modeled. We define the

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27

knowledge operator as the secondary notion to the relating epistemic implication6 : KA :=  →w A. This definition says that the agent knows A iff sentence  epistemically implies sentence A. (Naturally, for various i we may have different notions of knowledge.) Currently, the Gödel’s rule (GR) is not valid without any assumptions, because: |− A |− KA

means now:

|− A |−  →w A

But, obviously, it does not have to be that R(, A). If we want (GR) to be valid, we may, for example, consider models restricted by the condition: R(, A), if A is a tautology of the new epistemic logic. Such models should be defined by induction on the complexity of formulas. The distribution of knowledge operator over implication also is not valid any more. The axiom (K): KA ∧ K(A → B) → KB means now: ( →w A) ∧ ( →w (A → B)) → ( →w B). However, as we can see R(, A), R(, (A → B)) do not guarantee R(, B). If we want (K) to be a tautology, we may need to impose on models the condition: R(, A) & R(, (A → B)) ⇒ R(, B). Hence, our epistemic logic is free from the omniscience paradox (and some other epistemic logic paradoxes7 ), since an agent does not have to know all logical consequences of his knowledge. Instead of (K) we could accept the epistemic implication with the careful distribution: KA ∧ (A →w B) → KB. It seems to be better than (K): an agent does not have to know all his epistemic capabilities. (So, he can have a sufficiently good justification for B, but he does not have to know that A →w B.) For this reason, it is sufficient to take the transitivity of relating relation R (or some restricted transitivity). On the other hand, the axiom (T): KA → A, which is accepted in the standard epistemic logic, is a tautology of the relating epistemic logic — it does not require any restrictions imposed on models, provided that  is true in any model. Also other traditional laws of epistemic logic need some constraints. For the positive introspection: KA → KKA, we can assume the condition on models: R(, A) ⇒ R(, ( →w A)). When we impose the condition: R(, ¬( →w A)), if  R(, A) or R |= A, we have then negative introspection: ¬KA → K¬KA. In this approach, each agent is restricted to his epistemic relation, so more epistemic implications in a language: →w 1 , →w 2 , . . . and more relations in the model

6 Our

approach is similar to some interpretations of obligation operator in deontic logic, where O is treated as a secondary logical notion defined by an implication and a specific constant (see [1, 2, 5, 13]). However, here we use the relating implication. 7 For example, non-validity of K-distribution might be helpful in avoiding Fitch’s paradox.

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v, {Ri }i∈N can be added as personal justification processes. Finally, group knowledge, common sense knowledge and etc. can be also defined in this paradigm. What if we want to define belief operator as the secondary notion to the relating epistemic implication? Surely, we repeat the similar maneuver. Some propositional constans may be added to the language: ⊥1 , …, ⊥i , for some i ≥ 1. We can assume that ⊥1 , written just as ⊥, is false at any model M |=⊥. Hence, ⊥ represents the anti-standard of Platonic knowledge — the worst justified and false sentence of the system, some kind of epistemic counter-tautology. The remaining ⊥2 , …, ⊥i may be false at some classes of models in respect of what is needed to be modeled. Clearly, now BA :=⊥→w A. It means that an agent has got some justification of proposition A that is not worse than the worst one in the system of beliefs. Another interesting things can be some bridge laws between K and B operators and combining (fibring) a logic of knowledge with a logic of belief, and then to move on to a multi-relating logic with models v, RK , RB . It seems that also other non-extensional operators can be defined similarly. For example, an intensional negation: ¬i could be defined as ¬i A := ¬( →w A), so A is not justified or false. Another proposal is ¬i A :=  →w ¬A, so it is justified that A is false, while A itself might be expected after a translation to be equivalent to the formula  →w A. As we can see, there are many options for reducing intensional negation. In this way (or similar way, by using ⊥ or the opposite relating implications) we could try to reconstruct intuitionistic, intermediate or other non-classical logics. At the end: also the remaining relating connectives may provide a base for other modal operators (and intensional connectives) to be defined.

6 Conclusions: Essence and Perspectives of Relating Semantics In the work, I have demonstrated a remarkably flexible and intuitive logical semantics for intensional propositional logics. Its fundamental feature, apart from valuations of propositional variables (in one or many possible worlds), is that it also requires additional valuations of relations between propositions, relations that must occur when these propositions are connected by non-extensional connectives. Such an additional valuation determines whether or not a relevant intensional relation occurs between these propositions. It is quite natural to adopt such a solution, as also in natural language a connection of two (or more) propositions by a connective that belongs to this language produces some sort of emergence (see examples in Sect. 1). In fact, this is precisely what intensionality consists in: the addition of a new connective results in the emergence of a new quality, which itself does not belong to the components of a given complex proposition built by means of the same connective. It is this quality that is determined by an additional valuation function. The notion of emergence is used here for a reason, as it seems the quality that arises as a result of the connection of a few

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29

simpler propositions, especially in a particular context, cannot be reduced to the properties of those initial propositions. Therefore, in order to be able to determine this emergence, additional valuations are necessary in a model, and that is the essence of the relating semantics that we put forward. Note that all of the proposals for relating semantics that we have described here (including the ones that we have omitted so as not to overburden the article) can be reproduced for predicate logic. Then, however, the problem of relation will concern predicates, variables, quantifiers, and, possibly, other symbols.8 This would nevertheless require further, more thorough investigations. There are many indications that each existing logical semantics can be extended to a relating semantics. It can be then assumed that relating semantics is, in a sense, the most general because every semantics in which we do not valuate connections between propositions in a model can be considered as a special case of relating semantics, in which relations between these propositions are treated as the universal relation, and, therefore, we can skip those relations in this class of models. Acknowledgements The research presented in the following article was financed by the National Science Centre, Poland, grant No.: UMO-2015/19/B/HS1/02478. However, the author would also like to express his gratitude to the doctoral students under his supervision, especially to Mateusz Klonowski, for the discussions, inspirations and for the joint exploration of the field of relating logics.

References 1. Anderson, A. (1958). A reduction of deontic logic to alethic modal logic. Mind, 57, 100–103. 2. Anderson, A. (1968). A new square of opposition: Eubouliatic logic. In Akten des XIV. Internationalen Kongresses für Philosophie (Vol. 2, pp. 271–284). Vienna: Herder 3. Epstein, R. L. (1979). Relatedness and Implication. Philosophical Studies, 36, 137–173. 4. Epstein, R. L. (with the assistance and collaboration of: W. A. Carnielli, I. M. L. D’Ottaviano, S. Krajewski, R. D. Maddux). (1990). The semantic foundations of logic. Volume 1: Propositional logics. Dordrecht: Springer Science+Business Media. 5. Hilpinen, R. Deontic logic. In L. Goble (Ed.) The Blackwell guide to philosopohical logic (pp. 159–182). Oxford: Blackwell Publishers Ltd. 6. Jarmu˙zek, T., & Kaczkowski, B. (2014). On some logic with a relation imposed on formulae: Tableau system F . Bulletin of the Section of Logic, 43(1/2), 53–72. 7. Jarmu˙zek, T., Klonowski, M. (2020). Some intensional logics defined by relating semantics and tableau systems. In Giordani, A., Malinowski, J. (Eds.) Logic in High Definition. Trends in Logical Semantics (pp. 31–48). Berlin: Springer. 8. Jarmu˙zek, T., Klonowski, M. (2020). On Logic of Strictly-deontic Modalities. Semantic and tableau approach. Logic and Logical Philosophy, 29(3), 335–380. 9. Jarmu˙zek, T., & Malinowski, J. (2019). Boolean connexive logics: Semantics and tableau approach. Logic and Logical Philosophy, 28(3), 427–448. 8 It has to be mentioned that some activities in this scope have been performed in an attempt to adapt

relatedness logic to first-order logic [12]. However, as in the context of propositional logic, this is an interesting, but a special case of relating logic only with regard to the connective of implication and content-related problem.

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10. Jarmu˙zek, T., Malinowski, J. (2019). Modal Boolean connexive logics: Semantics and tableau approach. Bulletin of the Section of Logic, 48(3), 213–243. 11. Klonowski, M. (2018). Post’s completeness theorem for symmetric relatedness logic S. Bulletin of the Section of Logic, 47(3), 201–215. 12. Krajewski, S. (1982). On Relatedness Logic of Richard L. Epstein. Bulletin of the Section of Logic 11(1/2), 24–30. 13. Lokhorst, G.-J. (2006). Andersonian deontic logic, propositional quantification, and Mally. Notre Dame Journal of Formal Logic, 47(3), 385–395. 14. Malinowski, J. (2019). Barbershop paradox and connexive implication. Ruch Filozoficzny (Philosophical Movement), 75(2), 107–114. 15. Malinowski, J., & Palczewski, R. (2020). Relating semantics for connexive logic. In A. Giordani, J. Malinowski (Eds.) Logic in High Definition. Trends in Logical Semantics (pp. 49–65). Berlin: Springer. 16. Paoli, F. (1993). Semantics for first degree relatedness logic. Reports on Mathematical Logic, 27, 81–94. 17. Paoli, F. (1996). S is Constructively Complete. Reports on Mathematical Logic, 30, 31–47. 18. Priest, G. (2008). An introduction to non-classical logic. From If to Is. Cambridge: Cambridge Univeristy Press. 19. Sainsbury, M. (1991). Logical forms: An introduction to philosophical logic. Oxford: Basil Blackwell. 20. Walton, D. (1979). Philosophical basis of relatedness logic. Philosophical Studies, 36, 115–136.

Some Intensional Logics Defined by Relating Semantics and Tableau Systems Tomasz Jarmu˙zek and Mateusz Klonowski

Abstract Relating logics are examples of non-classical logics that make it possible to seriously consider various non-logical relationships that occur between sentences. The main idea behind such logics is that logical values is not the only thing that matters when we consider truth conditions for formulas built by propositional connectives. What should be considered as another important factor is a content relationship or some other relations, such as causality and temporal successions which hold between sentences or between the states of affairs that are usually expressed by means of those sentences. Hence, the factor is in fact intensional. In the article, we analyze relating logics and make a philosophical introduction to this subject; we introduce a relating language and its general semantic framework. It is a short introduction to the field of relating logics. Next, we describe some special cases of relating logics from a semantic point of view, and finally, as a decision procedure we introduce adequate tableau systems for those logics.

1 Introduction The logics we examine in the paper are relating logics. Their main and distinctive feature is that they cover ways of making formulas related by some special connectives. Relating logics are in some very special sense relevant logics, where the phenomenon of relevance appears on the level of non-Boolean functors. The main idea behind them is that not only logical values matter when we consider truth conditions for formulas built by propositional connectives, but also other factors, mainly intensional ones. The classical interpretation of propositional connectives includes only one aspect of propositions – their logical values (even in possible worlds – we still discuss T. Jarmu˙zek (B) · M. Klonowski Department of Logic, Nicolaus Copernicus University in Toru´n, Toru´n, Poland e-mail: [email protected] M. Klonowski e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Giordani and J. Malinowski (eds.), Logic in High Definition, Trends in Logic 56, https://doi.org/10.1007/978-3-030-53487-5_3

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logical values). Our standpoint is that such an approach disables one from analysing the usage of propositional connectives in intensional contexts—the contexts in which a logical value of component propositions is not the only thing that affects the logical value of the complex proposition. In order to be closer to some linguistic practice, let us look at an example where the temporal or causal context plays a very important role: John met Anna and he fell in love with her. John bought an engagement ring and he proposed to her.

(1) (2)

If Anna smiles to John then he is happy. If John loves Anna then he wants to impress her.

(3) (4)

The connective . . . and . . . in propositions (1) and (2) expresses not only the truth of component propositions, but also the temporal order between the events that are referred to by these components. Proposition (1) states not only that John met Anna and fell in love with her, but also that he met Anna first, and then he fell in love with her. Similarly in (2), it is said that firstly John bought the engagement ring, and then he proposed. In propositions (3) and (4), the connective if . . . then . . . expresses not only that the truth of the antecedent is sufficient for the truth of the consequent, but also the causal order between the events that are referred to by component propositions. In (3), it said that John’s happiness is caused by the fact that Anna smiles to him. In (4), the cause of the fact that John is willing to impress Anna is the fact that he loves her. The causal and temporal contexts are exemplary dependencies that form the basis of non-classical philosophical logics. However, it is easy to indicate many other dependencies, which can also lead to an interesting interpretation of connectives, based on a similar idea to the one outlined above. For example, we have a structural relationship regarding a geographical or political location. We express such a dependence in the following sentence: If Toru´n is located near the Vistula river then it is located in Poland.

(5)

As before, by using (5) we can express something more than the material implication of the component sentences. Proposition (5) lets us state a connection between the fact that Toru´n lies on the Vistula and that Toru´n is in Poland. Another interesting relationship can be a relation of preferences. We have a proposition: Anna would like to have gingerbread or biscuits.

(6)

If (6) is true, at least one of the constituent propositions has to be true. But (6) can also express that Anna, for instance, prefers gingerbread to biscuits. So we additionally can state that the components are related by a preference relation. Of course, in relating logics we may also state a content relationship or analytical relationship:

Some Intensional Logics Defined by Relating Semantics …

Venus is the morning star if and only if it is the evening star.

33

(7)

John is a bachelor if and only if he has no wife and has not been married so far. (8) The consideration of various dependencies, examples of which we have given above, leads to the introduction of new interpretations of the connectives …and …, …or…, if…then…, as well as …if and only if …. To perform an analysis of propositions built by such connectives, one needs to take into account different types of relations that can occur between constituent propositions due to different types of connections. It is quite evident that in the case of such expressions, not only the logical value of component propositions, but also the relation between them plays a significant role. The connectives of which interpretation relies on the two of aforementioned aspects, i.e. logical value and relation between component propositions, are called relating connectives, since in their scope two (or more) sentences are to be related somehow (cf. [9]). A propositional logic that contains in its language some relating connectives is called relating logic (for short: RL). It is possible to take into account various ways of connecting propositions by diverse relationships and generally the fact that propositions are somehow related. As we see, the idea of such connectives and logics reflects intensional aspects of everyday language, so they are intensional in spirit. From a historical point of view, the content relationship aspect was the first investigated relationship in the context of relating logics (done by Richard Epstein, see: [3, pp. 61–84, 115–143], cf. [11, 18]). However, Epstein considered the propositional language with negation ¬, conjunction ∧ and implication →, where negation and conjunction were interpreted in the extensional, classical way. The implication was supposed to preserve a content-relationship. So, in order to define a truth condition for formulas built by implication, a binary relation based on a set of formulas with some additional constraints was introduced (see [2] and [3, pp. 61–145], cf. [10, 11, 14–16]).1 This kinds of logics with Epstein’s variants of implication devoted to a content-relationship problem are called dependence or relatedness logic (although Epstein’s dependence logics are determined by special functions, the functions can be reduced to relations). A wider approach to using an additional relation in semantics was proposed by Jarmu˙zek and Kaczkowski in [5] (cf. [7, pp. 432–433], [10, pp. 201–203]). First, they decided to maintain in the language all the basic Boolean connectives with the extensional interpretation and add to such a language a new kind of implication and conjunction: relating implication and relating conjunction. Second, they thought of their connectives as a convenient tool for expressing various kinds of relationships, not only a content one. Third, their aim was to present the least logic defined by models based on just one binary relation without any extra constraints. Then, they introduced a tool which, by using certain relational conditions, enables one to define various relationships on the formal ground. This approach can be regarded as the proper 1 Epstein in [3, pp. 61–84, 115–143] considers relating connectives on the grounds of the following logics: relatedness logics S and R, dependence logics D, dD, Eq and DPC.

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beginning of studies devoted to relating logics in general. This logic is considered as a proper fragment of the least RL. Here, we develop and continue work of Jarmu˙zek and Kaczkowski presented in [5]. However, we also go further than that. We use all relating counterparts of binary classical connectives, so that we can work with a full language. Moreover, we impose some conditions on relating semantics to show how semantics can be used for modelling interesting examples of non-classical logics. This is possible because relating semantics is a typical fine-grained semantics. As we will show, it enables us to define logics in a simple, intuitive way. To supplement the paper with some proof theory, we add an outline of tableau approach to the systems we introduce.2

2 Language and Semantics of Relating Logic 2.1 Language The language of RL (Boolean-relating language) consists of countably many propositional variables: p, q, r, . . .; Boolean connectives: ¬, ∧, ∨, →, ↔, relating counterparts of binary Boolean connectives (relating connectives): ∧w , ∨w , →w , ↔w and parentheses ), (.3 The set of propositional variables is denoted by Var. The set of connectives is denoted by Con. A set of formulas of RL is defined in the standard way and denoted by For. By F we denote the set of formulas of Classical Propositional Logic and by CPL a set of tautologies of the classical logic. Obviously F ⊂ For. We assume the following conventions and notions: • In the case of any formula from For we will omit the outermost parentheses. We use parentheses ], [ in order to make metalogical expressions more readable. • By the complexity of a given formula we understand an output of function c : For −→ N defined in the standard way, i.e. c(A) = 0, if A ∈ Var; c(A) = c(B) + 1, if A = ¬B; c(A) = c(B) + c(C) + 1, if A = B ◦ C, where ◦ ∈ Con \ {¬}.

2.2 Semantics A semantics for RL is based on the concept of so-called a relating relation. Definition 1 A relating relation is a relation R ⊆ Formn , where n ≥ 1 and Form is a fixed set of formulas of some formal language. 2 More

about the ideas of relating semantics (generalizations, historical issues, applications etc.) can be read in [9]. 3 The letter w in notation of relating connectives comes from the Polish words ‘wia˛za´ c’ (verb) and ‘wia˛˙za˛cy’ (adjective) which might be translated as ‘relate’ and ‘relating’.

Some Intensional Logics Defined by Relating Semantics …

35

In the case of RL, we limit the notion to binary relations on For. However, for some philosophical reasons, we might need to introduce modalities or other intensional connectives and consider relations on a set of formulas built by such connectives. For example, in [6], by a relating relation some notion of deontic relation was expressed, where the considered propositional language contained modalities. But we also might be interested only in some fragment of a given relating logic, for we would like to consider whether some connexive or causal implication might be expressed by means of a relating implication (cf. [7, 8]). That is why we suggest a possibility of defining the notion of relating relation in a more general way. The class of all relating relations on For is denoted by R. If a relation does not hold for some formulas A, B, in the metalanguage we write: ∼R(A, B). Definition 2 A model of RL (a model of RL over R) is an ordered pair v, R such that: • v ∈ {1, 0}Var is a valuation of propositional variables ({1, 0}Var is the set of all valuations) • R ∈ R is a binary relating relation. A valuation v (resp. a relation R) of a model M ∈ M is denoted by vM (resp. RM ). Let Q ⊆ R. A class of models of RL over Q is a class of models of RL over any R from Q. Such class is denoted by MQ. The class of all models of RL is MR. Definition 3 Let M ∈ MR and A ∈ For. A is true in M (in symb.: M |= A; and M |= A, if false) iff for any B, C ∈ For: • for propositional variables and formulas built by Boolean connectives: vM (A) = 1,

if A ∈ Var

M |= B, M |= B and M |= C,

if A := ¬B if A := B ∧ C

M |= B or M |= C, M |= B or M |= C,

if A := B ∨ C if A := B → C

M |= B iff M |= C,

if A := B ↔ C

• for formulas built by relating connectives: [M |= B and M |= C] and RM (B, C), [M |= B or M |= C] and RM (B, C),

if A := B ∧w C if A := B ∨w C

[M |= B or M |= C] and RM (B, C), [M |= B iff M |= C] and RM (B, C),

if A := B →w C if A := B ↔w C.

Let Σ ⊆ For. We will write M |= Σ instead of ∀ A∈Σ M |= A.4 4 It

is possible to express in the considered language purely intensional functors. For instance, we can examine a functor defined in the following way A  B := (A ∨w B) ∨ (A →w B). We can

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As we can see, in the case of Boolean connectives, we have the standard extensional interpretation. In the case of relating connectives, two conditions must be satisfied. The first one is the standard truth-condition for a Boolean counterpart of a given relating connective. In accordance with the second one, both formulas must be related. We can also define a notion of truth with respect to a relation. Definition 4 Let R ∈ R and A ∈ For. A is true with respect to R (in symb.: R |= A) iff ∀v∈{1,0}Var v, R |= A. Let Σ ⊆ For. We will write R |= Σ instead of ∀ A∈Σ R |= A. Below we define two notions of semantic consequence and provide two ways of defining the notion of tautology. Definition 5 Let ∅ = Q ⊆ R and Σ ∪ {A} ⊆ For. • A is a tautology (of RL) over MQ (in symb.: |=MQ A) iff ∀M∈MQ M |=MQ A. • A is a semantic consequence (in RL) of Σ over MQ (in symb.: Σ |=MQ A) iff ∀M∈MQ (M |= Σ =⇒ M |= A). Definition 6 Let ∅ = Q ⊆ R and Σ ∪ {A} ⊆ For. • A is a tautology (of RL) over Q (in symb.: |=Q A) iff ∀R∈Q R |=Q A. • A is a semantic consequence (in RL) of Σ over Q (in symb.: Σ |=Q A) iff ∀R∈Q (R |= Σ =⇒ R |= A).5 We have the following proposition: Proposition 1 Let ∅ = Q ⊆ R and A ∈ For. Then: (1) |=MQ A iff ∅ |=MQ A. (2) |=Q A iff ∅ |=Q A. Let us note that any formula is a semantic consequence, over some class of relations, from any set that contains at least one element of F \ CPL. Proposition 2 Let ∅ = Q ⊆ R, Σ ⊆ For and A ∈ For. Then, if there is B ∈ Σ such that B ∈ F \ CPL, then Σ |=Q A. Proof Assume all the hypotheses. Let R ∈ Q. There is a valuation V such that V (B) = 0. Let v ∈ {1, 0}Var be the reduction of V to Var. Therefore, for valuation v, v, R |= Σ. Hence, by Definition 6, Σ |=Q A.

easily check that v, R |= A  B iff R(A, B). Let us also notice that it is possible to consider unary relating functors. For instance, a kind of a relating negation (the relating counterpart of Boolean negation) could be defined as A →w ⊥, where ⊥ is interpreted in the standard way. These issues, however, require a separate analysis that goes beyond the scope of this article. 5 Surely, if Q = ∅, then |= A and Σ |= A, for all Σ ∪ {A} ⊆ For. But we would not like to Q Q discuss the trivial logic here. The very similar situation is in Definition 5.

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In modal logic, the “local” relation of semantic consequence over some class of models is defined with the preservation of truth in a model, in a world. In turn, the “global” one is defined with the preservation of truth in a model (cf. [1, pp. 31–32]). We can use a similar naming convention and call the relation of semantic consequence in the sense of Definition 5, a “local” one. In this case, the relation is defined with the preservation of truth in the model for any valuation and any relation. In turn, the relation of semantic consequence in the sense of the Definition 6 can be called a “global” one. In this case, the relation is defined with a preservation of truth with respect to relations from some set. The next proposition determines the dependence between the “local” and “global” relations. In accordance with this proposition, the notion of tautology is definable in the “local” way as well as the “global” one. Proposition 3 Let ∅ = Q ⊆ R and A ∈ For. Then: (1) |=MQ A iff |=Q A. (2) |=MQ ⊂ |=Q . Proof Assume all the hypothesis. For (1). „=⇒” Suppose |=MQ A and let R ∈ Q. Then, by Definition 5, ∀M∈MQ M |= A. Let v ∈ {1, 0}Var . We have v, R |= A, so ∀v∈{1,0}Var v, R |= A. Hence, by Definition 4, R |= A, so ∀R∈Q R |= A. Therefore, by Definition 6, |=Q A. „⇐=” Suppose |=Q A and let M ∈ MQ. Hence, by Definition 6, ∀R∈Q R |= A. And so, by Definition 4, ∀R∈Q ∀v∈{1,0}Var v, R |= A. Hence, especially for RM and vM , vM , RM |= A. Thus ∀M∈MQ M |= A. Therefore, by Definition 5, |=MQ A. For (2). Let Σ ∈ For. Suppose Σ |=MQ A. Thus, by Definition 5, ∀R∈Q ∀v∈{1,0}Var ( v, R |= Σ =⇒ v, R |= A). Hence, by distribution of universal quantifier, ∀R∈Q ( ∀v∈{1,0}Var v, R |= Σ =⇒ ∀v∈{1,0}Var v, R |= A). Therefore, by Definition 6, Σ |=Q A. However, the inverse implication does not hold. By Proposition 2 { p} |=Q q. Let M be a model defined in the following way: • for any a ∈ Var we put:  vM (a) =

1, 0,

if a = p if a ∈ Var \ { p}

• RM = ∅. Therefore M |= p and M |= q. Hence, by Definition 5, { p} |=MQ q.

3 Examples of Relating Logics By PL we denote a set of all formulas which are instances of classical tautologies in the language of RL. More precisely, let At := {A ∈ For : A ∈ Var or A := B ◦ C, where ◦ ∈ Conw }. By the valuation of At we mean a function w : At −→ {1, 0}.

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As in the case of classical valuations and formulas from F, we extend the valuation w of At on For to the function V w : For −→ {1, 0} in the standard way, preserving the meaning of classical connectives (for example: V w (A ∧ B) = 1 iff V w (A) = 1 and V w (B) = 1, and similarly for the remaining classical connectives.) Then the set of all instances of classical tautologies is the set PL = {A ∈ For : ∀w∈{1,0}At V w (A) = 1}. Notice that any model M determines a valuation wM : At −→ {1, 0}. For any A ∈ At we put: wM (A) = 1 iff M |= A. Therefore, for any model M and any formula A ∈ For we have M |= A iff V wM (A) = 1, where V wM is the extension of wM determined by M. Clearly, we have PL ⊂ {A ∈ For | ∀R∈R R |= A}. So, relating logics can be considered as sets of tautologies containing PL. The least RL is determined by R and shall be called logic W.6 It is easy to see that logic W is a conservative extension of logic F introduced by Jarmu˙zek and Kaczkowski [5] and of Classical Propositional Logic. We distinguish between three types of logic W extensions according to the classification of relational conditions that determine the subclasses of the class R. We can point out at three types of conditions that can be imposed on relations from R and determine its subsets Q ⊆ R. Here, we describe them heuristically 7 : • Horizontal conditions refer neither to the complexity nor to ways of constructing formulas, e.g. reflexivity, symmetry, irreflexivity, transitivity etc. • Vertical conditions refer to the complexity or ways of constructing formulas. • Diagonal conditions are combinations, or more specifically conjunctions in the cases with finitely many conditions, of former two kinds of conditions (a couple of them was the subject of the analysis performed by Epstein for his subject matter implication (see [2, pp. 138–146], [3, pp. 65–68, 116–123])). Since we consider three types of relations, we obtain three types of subclasses of R: horizontal, vertical and diagonal, and respectively, we distinguish between three types of logics that extend logic W, which are named after the conditions: horizontal, vertical and diagonal. Now we propose some examples of conditions that will be investigated in the later part of our paper. These conditions are some part of the properties of causality presented by Scheffler (except transitivity, which, according to Scheffler, is optional) [17, pp. 139–142]. Here we present some examples of the first type of the conditions that we focus on: ∼R(A, A) R(A, B) =⇒ ∼R(B, A)

(Ir) (Asym)

6 The name of logic W comes from the Polish words ‘wia˛za´ c’ (verb) and ‘wia˛˙za˛cy’ (adjective) which

might be translated as ‘relate’ and ‘relating’ (cf. footnote 3). write ‘heuristically’, since precise definitions of these distinctions have not been worked out yet.

7 We

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[R(A, B) and R(B, C)] =⇒ R(A, C),

(Tran)

for any A, B, C ∈ For. By means of the conditions for causality, we can model a behaviour of →w understood as a causal implication and ∧w understood as a causal conjunction. For example, the semantic conditions imposed on models give the following tautologies as causal laws of the proper logics determined by the conditions: ¬(A →w A)

(Ir∗1 )

¬(A ∧w A)

(Ir∗2 )

(A →w B) → ¬(B →w A)

(Asym∗1 )

(A ∧w B) → ¬(B ∧w A)

(Asym∗2 )

[(A →w B) ∧ (B →w C)] → (A →w C)

(Tran∗1 )

[(A ∧w B) ∧ (B ∧w C)] → (A ∧w C),

(Tran∗2 )

for any A, B, C ∈ For. The first parts of labels inform under what condition a formula is a tautology. Let us also consider the following examples of the second type: ∼R(A, ¬A)

(a1)

∼R(¬A, A)

(a2)

[R(A, B) =⇒ ∼R(A, ¬B)] and R(A →w B, ¬(A →w ¬B))

(b1)

[R(A, ¬B) =⇒ ∼R(A, B)] and R(A →w ¬B, ¬(A →w B)),

(b2)

for any A, B ∈ For. They were utilized in [7] (cf. [8, 12, 13]) to regain semantics for Boolean-connexive logics which contain as theses formulas characteristic of connexive implication. The following theses are of course in relation to the suitable conditions given above: ¬(A →w ¬A)

(a1∗ )

¬(¬A →w A)

(a2∗ )

(A →w B) →w ¬(A →w ¬B)

(b1∗ )

(A →w ¬B) →w ¬(A →w B),

(b2∗ )

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for any A, B ∈ For. Since in the paper we refer to, a semantic analysis and tableau systems for the conditions were given, here we concentrate on other vertical conditions: R(¬A, B) =⇒ ∼R(A, B) (N1) R(A, ¬B) =⇒ ∼R(A, B)

(N2)

R(A ∧ B, C) =⇒ [R(A, C) or R(B, C)]

(C1)

R(A, B ∧ C) =⇒ [R(A, B) or R(A, C)],

(C2)

for any A, B, C ∈ For. They come from [17, pp. 139–142] as properties of a causal relationship. However, here we can show how they can model preference disjunction by our relating disjunction ∨w . The particular properties correspond to the following formulas containing ∨w : (¬A ∨w B) → ¬(A ∨w B)

(N1∗ )

(A ∨w ¬B) → ¬(A ∨w B)

(N2∗ )

[(A ∧ B) ∨w C] → [(A ∨w C) ∨ (B ∨w C)]

(C1∗ )

[A ∨w (B ∧ C)] → [(A ∨w B) ∨ (A ∨w C)],

(C2∗ )

for any A, B, C ∈ For. For the sake of further considerations, we will now introduce some notations. Let RC be the powerset of the set of mentioned conditions {(Ir), (Asym), (Tran), (N1), (N2), (C1), (C2)}. By L we shall denote the set of logics determined by sets from RC. Let us observe that various sets from RC may determine the same logic Λ (for example sets {(Ir), (Asym) } and { (Asym) } provide the same logic, since (Asym) just implies (Ir)). So, there exists more than one function that to any logic from L assigns a unique set of conditions from RC that determines that logic. Let us take one of such functions and denote it by rc. For any Λ ∈ L , by rc(Λ) we denote a set of relational conditions from RC that determines Λ and by m(Λ) a class of Λ’s models determined by conditions from rc(Λ).

4 Tableau Systems for Relating Logics In this section we present tableau systems for all relating logics determined by sets from RC. The analysis we provide is a continuation of the research presented in [5–7] and in many aspects based on the metatheory of tableau systems provided by Jarmu˙zek in [4], for modal logics.

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4.1 Tableau Expressions The language of tableau system is the language of RL extended by auxiliary expressions. Definition 7 The set of auxiliary expressions (in symb.: Ae) is the least set Σ such that: if A, B ∈ For, then A r B, A r B ∈ Σ. Expressions of the form A r B and A r B are supposed to represent relations in the tableau language and state that A, B are or are not related respectively. Definition 8 The set of tableau expressions (for short: the set of t-expressions) is set Ex = For ∪ Ae. Elements of Ex will be called tableau expressions (for short: t-expressions). For any Σ ⊆ Ex by Ae(Σ) we shall denote the following set of expressions {A ∈ Ae : A ∈ Σ}. A tableau inconsistent set is defined in the standard way: Definition 9 Let Σ ⊆ Ex. • Σ is tableau inconsistent (for short: t-inconsistent) iff at least one of the following conditions is satisfied: – there is A ∈ For such that A, ¬A ∈ Σ – there are A, B ∈ For such that A r B, A r B ∈ Σ. • Σ is tableau consistent (for short: t-consistent) iff Σ is not t-inconsistent.

4.2 Tableau Rules We assume the standard elimination rules for Boolean connectives, see Fig. 1. In the case of relating connectives we follow the idea of rules for Boolean connectives, see Fig. 1 (cf. [5, pp. 58–59]). Any formula built by a relating connective is decomposed in such a way as to get (a) elements that we would receive by decomposition of a Boolean-counterpart of the given relating connective and (b) an expression that determines whether a relating relation holds or not. In Fig. 2 we deal with special rules for relational conditions. The rule (RIr ) for irreflexivity allows us to finish proof if we get an expression A r A. The rules (RAsym ) and (RTran ) for asymmetry and transitivity respectively seem to be self-explanatory. Similarly, the rules (RN1 ), (RN2 ) corresponding to the conditions of negation elimination (N1), (N2) as well as the rules (RC1 ) and (RC2 ) for the condition of distribution of conjunction (C1), (C2) respectively. The set of all tableau rules shall be denoted as TR while the set of all elimination rules for connectives as eTR. For any rules from TR expressions in the numerator will be called input, while expressions from the denominator will be called output. Let us take as an example the rule (R∧ ). One of its inputs is { p ∧ q} and then the

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Fig. 1 Elimination rules for logical constants

Fig. 2 Rules for relational conditions

corresponding output is set { p, q}. Notice that this rule is a non-branching one, i.e. for any input, it has got only one output (one set of formulas). On the other hand, (R¬∧w ) is a branching rule and in this case we have three outputs, for example: {¬ p}, {¬q} and { p r q}, if the input is {¬( p ∧w q)}. Once we have the notion of input we can define the notion of applicability of a rule.

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Definition 10 Let (r) ∈ TR and Σ ⊆ Ex. (r) is applicable to Σ iff an input of the (r) is a subset of Σ. We assign to logics from L sets of rules in the following way:  tr(Λ) =

eTR, eTR ∪ {(RX ) : (X) ∈ rc(Λ)},

if Λ = W if Λ = W.

We define the relation of tableau consequence by referring to the concept of closure under tableau rules. Definition 11 Let X ⊆ TR and Σ, Γ ⊆ Ex. Γ is a closure of Σ under tableau rules from X (for short: X-closure of Σ) iff there exists such a subset of natural numbers K that: • K = N or K = {1, . . . n} for some n ∈ N • there exists such an injective string f : K −→ {Y : Y ⊆ Ex} that: – Y1 = Σ – for all i, i + 1 ∈ K there exists such a tableau rule (r) ∈ X that its input is included in Yi , while one of its corresponding outputs is equal to Yi+1 \ Yi – for all i, i + 1 ∈ K for any tableau rule (r) ∈ X if (r)’ input is included in Yi and one of (r)’ corresponding outputs is equal to Yi+1 \ Yi , then there are no such j, j + 1 ∈ K that j > i and one of the remaining outputs of (r) is equal to Y j+1 \ Y j  – for any tableau rule (r) ∈ X if (r)’ input is included in i∈K Yi , then one of the (r)’ corresponding outputs is in i∈K Yi  • Γ = i∈K Yi . In practice, we can treat the notion of closure given in Definition 11 as the notion of a complete branch. In fact, it is a union of all elements that are on a complete branch. A tableau consequence relation in logic Λ ⊇ W is defined relative to a tableau rule set which is assigned by function tr to relational conditions that determine Λ. Definition 12 Let Λ ∈ L and Σ ∪ {A} ⊆ For. • A is a tableau consequence of Σ in tr(Λ) (in symb.: Σ tr(Λ) A) iff there is a finite set Γ ⊆ Σ such that any tr(Λ)-closure of Γ ∪ {¬A} is t-inconsistent. • A is a thesis (of RL) in Λ (in symb.: tr(Λ) A) iff ∅ tr(Λ) A.

4.3 Soundness and Completeness The analysis of soundness for the defined systems we shall begin with a definition of a model suitable for a set of t-expressions.

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Definition 13 Let M ∈ MR and Σ ⊆ Ex. M is suitable for Σ iff for all A, B ∈ For: • if A ∈ Σ then M |= A • if A r B ∈ Σ then RM (A, B) • if A r B ∈ Σ then ∼RM (A, B). The following lemma enables us to prove that any formula received by applications of the introduced tableau rules will also be a tautology over a proper class of relations. Lemma 1 Let Λ ∈ L , Σ ⊆ Ex and M ∈ m(Λ) be suitable for Σ. For any (r) ∈ tr(Λ), if (r) has been applied to Σ, then M is suitable for the union of Σ and at least one output obtained by the application of rule (r). Proof Assume all the hypotheses. For the cases in which the elimination rules of Boolean connectives are applied, the proof is obvious. For the cases in which the elimination rules of relating connectives (Fig. 1) are applied, we have: • Suppose (R∧w ) has been applied to Σ. Then, by Definition 10, A ∧w B ∈ Σ and we get output {A, B, A r B}. Because model M is suitable for Σ, so by Definition 13 M |= A ∧w B. Thus, by Definition 3, M |= A, M |= B and RM (A, B). Hence, by Definition 13, M is suitable for the set Σ ∪ {A, B, A r B}. • Suppose (R¬∧w ) has been applied to Σ. Then, by Definition 10, ¬(A ∧w B) ∈ Σ and we get outputs {¬A}, {¬B}, and {A r B}. Because model M is suitable for Σ, so by Definition 13 M |= ¬(A ∧w B). Thus, by Definition 3, either M |= ¬A or M |= ¬B, or ∼RM (A, B). Hence, by Definition 13, M is suitable for at least one of the following sets: Σ ∪ {¬A}, Σ ∪ {¬B} or Σ ∪ {A r B}. • For the remaining elimination rules of relating connectives we reason in a similar way. For the cases in which the rules for relational conditions of the first type (Fig. 2) are applied, we have: • Let (RIr ) ∈ tr(Λ), so RM satisfies (Ir). Hence ∼RM (A, A), for any A ∈ For. But M is suitable for the set Σ, so A r A ∈ / Σ. • Let (RAsym ) ∈ tr(Λ), so RM satisfies (Asym). Suppose (RAsym ) has been applied to Σ. Then, by Definition 10, A r B ∈ Σ and we get output {B r A}. Because model M is suitable for Σ, then by Definition 13 RM (A, B). Thus ∼RM (B, A), since RM satisfies (Asym). Hence, by Definition 13, M is suitable for the set Σ ∪ {B r A}. • Let (RTran ) ∈ tr(Λ), so RM satisfies (Tran). Suppose (RTran ) has been applied to Σ. Then, by Definition 10, A r B, B r C ∈ Σ and we get output {A r C}. Because model M is suitable for Σ, then by Definition 13 RM (A, B) and RM (B, C). Thus RM (A, C), since RM satisfies (Tran). Hence, by Definition 13, M is suitable for the set Σ ∪ {A r C}. For the cases in which the rules for relational conditions of the second type (see Fig. 2) is applied, we have:

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• Let (RN1 ) ∈ tr(Λ), so RM satisfies (N1). Suppose (RN1 ) has been applied to Σ. Then, by Definition 10, ¬A r B ∈ Σ and we get output {A r B}. Because model M is suitable for Σ, then by Definition 13 RM (¬A, B). Thus ∼RM (A, B), since RM satisfies (N1). Hence, by Definition 13, M is suitable for set Σ ∪ {A r B}. • Let (RC1 ) ∈ tr(Λ), so RM satisfies (C1). Suppose (RC1 ) has been applied to Σ. Then, by Definition 10, A ∧ B r C ∈ Σ and we get outputs {A r C}, {B r C}. Because model M is suitable for Σ, then by Definition 13 RM (A ∧ B, C). Thus either RM (A, C) or RM (B, C), since RM satisfies (C1). Hence, by Definition 13, M is suitable for at least one of the following sets Σ ∪ {A r C}, Σ ∪ {B r C}. • For the remaining cases we reason in the similar way. Theorem 1 Let Λ ∈ L and Σ ∪ {A} ⊆ For. Then: Σ tr(Λ) A =⇒ Σ |=Λ A. Proof Assume all the hypotheses. Suppose Σ tr(Λ) A. Hence, by Definition 12, there is a finite Γ ⊆ Σ such that any tr(Λ)-closure of Γ ∪ {¬A} is t-inconsistent. Suppose that there is a model M ∈ m(Λ) such that M |= Σ and M |= ¬A. By Definition 13 M is suitable for Γ ∪ {¬A}. By Lemma 1 there is tr(Λ)-closure Γ ∪ {¬A}, for which M is suitable. However, such a closure must be t-inconsistent. Thus in the closure either there is B ∈ For such that M |= B and M |= B or there are B, C ∈ For such that RM (B, C) and ∼RM (B, C). Hence, for any model M ∈ m(Λ), M |= Σ implies M |= A. Therefore Σ |=Λ A. Next we will introduce the notion of a model generated by a t-consistent closure under some set of tableau rules from TR. Definition 14 Let X ⊆ TR and Σ ⊆ Ex be a t-consistent closure under X. A model generated by Σ (for short: Σ-model) is a model vΣ , RΣ such that: • for any A ∈ Var we put:  vΣ (A) = • for any A, B ∈ For:

1, 0,

if A ∈ Σ if A ∈ /Σ

RΣ (A, B) iff A r B ∈ Σ.

The next fact is the completeness lemma: Lemma 2 Let Λ ∈ L and Σ be a t-consistent closure under tr(Λ). Then, there is a model M ∈ MR such that: (1) M ∈ m(Λ) (2) for any A ∈ For, if A ∈ Σ then M |= A. Proof Assume all the hypotheses. As a model in the thesis we take Σ-model M = vΣ , RΣ (Definition 14). For (1):

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• Suppose (RIr ) ∈ tr(Λ). Let A ∈ For and assume RΣ (A, A), toward a contradiction. By Definition 14 A r A ∈ Σ. Thus, by application of (RIr ), A r A ∈ Σ. Hence ∼RΣ (A, A). Therefore RΣ satisfies (Ir). • Suppose (RAsym ) ∈ tr(Λ). Let A, B ∈ For and assume RΣ (A, B). By Definition 14 A r B ∈ Σ. Thus, by application of (RAsym ), B r A ∈ Σ. Hence, by Definition 14, ∼RΣ (B, A). Therefore RΣ satisfies (Asym). • Suppose (RTran ) ∈ tr(Λ). Let A, B, C ∈ For and assume RΣ (A, B) and RΣ (B, C). By Definition 14 A r B, B r C ∈ Σ. Thus, by application of (RTran ), A r C ∈ Σ. Hence, by Definition 14, RΣ (A, C). Therefore RΣ satisfies (Tran). • Suppose (RN1 ) ∈ tr(Λ). Let A, B ∈ For and assume RΣ (¬A, B). By Definition 14 ¬A r B ∈ Σ. Thus, by application of (RN1 ), A r B ∈ Σ. Hence ∼RΣ (A, B), because the closure is t-consistent. Therefore RΣ satisfies (N1). • Suppose (RC1 ) ∈ tr(Λ). Let A, B, C ∈ For and assume RΣ (A ∧ B, C). By Definition 14 A ∧ B r C ∈ Σ. Thus, by application of (RC1 ), either A r C ∈ Γ or B r C ∈ Σ. Hence, by Definition 14, either RΣ (A, C) or RΣ (B, C). Therefore RΣ satisfies (C1). • For the remaining cases we conduct the proof similarly. For (2): Base case. Let A ∈ For and c(A) = 0. Thus A ∈ Var. Suppose A ∈ Σ. Then, by Definition 11, M |= A. Inductive hypothesis. Let n ∈ N. Suppose that for any A ∈ For such that c(A) ≤ n, if A ∈ Σ then M |= A. Inductive step. Let A ∈ For and c(A) = n + 1. The cases for formulas built by means of Boolean connectives and their negations follow from the inductive hypothesis. Let us consider the following cases; the other ones (for relating connectives) are provable in a similar way. Let A := B ∧w C. Suppose A ∈ Σ, so B ∧w C ∈ Σ. Hence, by Definition 11 and application of the rule (R∧w ), B, C ∈ Σ and B r C. Hence, by inductive hypothesis and Definition 14, M |= B, M |= C and RM (B, C). Therefore, by Definition 3, M |= B ∧w C, so M |= A. Let A := ¬(B ∧w C). Suppose A ∈ Σ, so ¬(B ∧w C) ∈ Σ. Hence, by Definition 11 and application of the rule (R¬∧w ), either ¬B ∈ Σ or ¬C ∈ Σ, or B r C ∈ Σ. Thus, by inductive hypothesis, either M |= ¬B or M |= ¬C, or B r C ∈ Σ. Therefore, in the first two cases by Definition 3, M |= ¬(B ∧w C), so M |= A. In the third case B r C ∈ / Σ, since Σ is a t-consistent closure under tr(Λ). Thus, by Definition 14, ∼RM (B, C). Therefore, by Definition 3, M |= ¬(B ∧w C), so M |= A. Now we can easily obtain the completeness of our tableau systems. Theorem 2 Let Λ ∈ L and Σ ∪ {A} ⊆ For. Then: Σ |=Λ A =⇒ Σ tr(Λ) A.

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Proof Assume all the hypotheses. Suppose Σ   tr(Λ) A. So, for any finite Γ ⊆ Σ there is a t-consistent tr(Λ)-closure Δ of Γ ∪ {¬A}, such that Γ ∪ {¬A} ⊆ Δ. Hence, there is a t-consistent tr(Λ)-closure Δ such that Σ ∪ {¬A} ⊆ Δ . Otherwise, any of such a closure would consist some t-inconsistency. But by Definition 11 this would mean that for some finite Γ ⊆ Σ no tr(Λ)-closure of Γ ∪ {¬A} is tconsistent. As a consequence, by Lemma 2, there is a Δ -model M ∈ m(Λ) such that M |= Σ ∪ {¬A}. Therefore, by Definition 5, Σ |=Λ A. Acknowledgements The research of Tomasz Jarmu˙zek presented in the following article was financed by the National Science Centre, Poland, grant No.: UMO-2015/19/B/HS1/02478. While the research of Mateusz Klonowski presented in the following article was financed by the National Science Centre, Poland, grant No.: UMO-2015/19/N/HS1/02401. The authors also would like to thank the anonymous referee for all his remarks and interesting suggestions.

References 1. Blackburn, P., de Rijke, M., & Venema, Y. (2001). Modal logic. Cambridge: Cambridge University Press. 2. Epstein, R. L. (1979). Relatedness and implication. Philosophical Studies, 36, 137–173. 3. Epstein, R. L. (with the assistance and collaboration of: W. A. Carnielli, I. M. L. D’Ottaviano, S. Krajewski, R. D. Maddux). (1990). The semantic foundations of logic. Volume 1: Propositional logics. Dordrecht: Springer Science+Business Media. 4. Jarmu˙zek, T. (2013). Tableau metatheorem for modal logics. In R. Ciuni, H. Wansing, C. Willkommen (Eds.), Recent trends in philosphical logic (Vol. 41, pp. 103–126). Trends in logic. Berlin: Springer. 5. Jarmu˙zek, T., & Kaczkowski, B. (2014). On some logic with a relation imposed on formulae: Tableau system F . Bulletin of the Section of Logic, 43(1/2), 53–72. 6. Jarmu˙zek, T., & Klonowski, M. (2020). On logic of strictly-deontic modalities. Logic and Logical Philosophy, 29(3), 335–380. 7. Jarmu˙zek, T., & Malinowski, J. (2019). Boolean connexive logics: Semantics and tableau approach. Logic and Logical Philosophy, 28(3), 427–448. 8. Jarmu˙zek, T., & Malinowski, J. (2019). Modal Boolean connexive logics: Semantics and tableau approach. Bulletin of the Section of Logic, 48(3), 213–243. 9. Jarmu˙zek, T. (2020). Relating semantics as fine-grained semantics for intensional logics. In A. Giordani, J. Malinowski (Eds.), Logic in High Definition. Trends in Logical Semantics (this volume). Berlin: Springer, (pp. 13–30) (2020), 335–380. 10. Klonowski, M. (2018). Post’s completeness theorem for symmetric relatedness logic S. Bulletin of the Section of Logic, 47(3), 201–215. 11. Krajewski, S. (1982). On relatedness logic of Richard L. Epstein. Bulletin of the Section of Logic, 11(1/2), 24–30. 12. Malinowski, J. (2019). Barbershop paradox and connexive implication. Ruch Filozoficzny (Philosophical Movement), 75(2), 107–114. 13. Malinowski, J., & Palczewski R. (2020). Relating semantics for connexive logics. In A. Giordani, J. Malinowski (Eds.), Logic in High Definition. Trends in Logical Semantics (pp. 49–65). Berlin: Springer. 14. Paoli, F. (1993). Semantics for first degree relatedness logic. Reports on Mathematical Logic, 27, 81–94. 15. Paoli, F. (1996). S is constructively complete. Reports on Mathematical Logic, 30, 31–47.

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16. Paoli, F. (2007). Tautological entailments and their rivals. In J. Y. Béziau, W. Carnielli, & D. Gabbay (Eds.), Handbook of paraconsistency (pp. 153–175). London: College Publications. 17. Scheffler, U. (1993). On the logic of event causation. Logic and Logical Philosophy, 1, 129–155. 18. Walton, D. (1979). Philosophical basis of relatedness logic. Philosophical Studies, 36, 115–136.

Relating Semantics for Connexive Logic Jacek Malinowski and Rafał Palczewski

Abstract In this paper, we present a relating semantics in order to investigate the connexive logic. We remind Barbershop paradox noted by Lewis Carroll in 1894. Next we apply the relating semantics to investigate this paradox in detail and to show its relation to the connexive logic. Keywords Boolean connexive logics · Connexive implication · Lewis Carroll’s barbershop paradox · Relatedness · Relating semantics

1 Introduction Connexive logic occupies a very special place among non-classical logics. The classical logic and connexive logic are mutually orthogonal in the sense that the smallest logic containing both of them is the full inconsistent logic and the greatest logic included in both of them is the trivial logic. The mainstream non-classical logics have their roots in paradoxes of material implication. Intuitionistic logic, constructive logic, Johansson logic are typical examples here. They share the main feature: all their theses are also classical tautologies. They all are constructed, in a sense, by a negative selection: we start from classical logic and delete some of their theses which do not fit to a given intuitions. One could tell that, in a sense, connexive logic is really non-classical while mainstream non-classical logics are just modifications of classical one. The implication occuring in a connexive logic expresses very different intuitions than material implication. In Sect. 2, we discuss in detail the motivations of connexivity occurring in texts of ancient and medieval philosophers. We show there that J. Malinowski (B) Institute of Philosophy and Sociology, Polish Academy of Sciences, Warszawa, Poland e-mail: [email protected] R. Palczewski Department of Logic, Nicolaus Copernicus University in Toru´n, Toru´n, Poland e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Giordani and J. Malinowski (eds.), Logic in High Definition, Trends in Logic 56, https://doi.org/10.1007/978-3-030-53487-5_4

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although we still do not know the full shape of connexive logic, there is a common agreement that each connexive logic should satisfy the following Aristotle’s and Boethian Theses: (A1) (A2) (B1) (B2)

∼(∼A → A) (First Aristotle Thesis) ∼(A → ∼A) (Second Aristotle Thesis) (A → B) → ∼(A → ∼B) (First Boethian Thesis) (A →∼B) → ∼(A → B) (Second Boethian Thesis)

On the other hand, the logical system based solely on (A1), (A2), (B1), (B2) appears too weak to mirror the desired features of connexivity. This system should be essentially strengthened, however, we do not know how to do it. The final shape of connexive logic is the matter of future research. For this reason, in this paper, by the connexive logic we mean any set of formulas closed under the rules of substitution and modus ponens containing formulas (A1), (A2), (B1), (B2). Theses (A1), (A2), (B1), (B2) could be satisfied by material implication provided we admit strange interpretation of negation operator like: negation of any sentence is true. We would like to exclude such cases, focusing on Boolean connexive logics— connexive logics where negation and other Boolean connectives behave in the same way as in the classical logics. The aim of this paper is to invesitgate the connexive logics by means of relating semantics—a general semantic tool proposed by Jarmu˙zek and Kaczkowski in [15] and having its roots in Epstein [10]. We present here the results from paper [17]. In paper [18], the authors extended them onto the domain of modal logics. Relating semantics is flexible enough to make, apparently, each connexive logic possess a natural relating semantics. In this paper, we recall the barbershop paradox from Lewis Carroll’s paper [6]. It concerns some conditionals, which seem to lead to paradoxical results. The general idea of relating semantics is based on the relating relation R. The fact that A R B—sentences A and B relate with respect to R—could be interpreted in many ways depending on the motivations of a given logical system. For example, A and B could be related causally, they could be related in the sense of time order, they could have a common content. Generally, two sentences are related because they have something in common. The former chapter in this volume, i.e. papers [14, 16] present relating semantics in detail. Just for the sake of completeness, in section three of this paper, we discuss the main ideas of relating semantics. We focus there on the results and notions used further in this paper as a semantics for the scenario related to barbershop paradox. Usually, the barbershop paradox is associated with a barber from Sevilla—a selfreference paradox related to the Russell’s antinomy. The barbershop paradox we present here after Lewis Carroll is a different paradox published a decade before Russell’s antinomy. It concerns a simple argument leading to paradoxical results provided A → ∼B and A → B are incompatible. Those sentences are incompatible under the connexive interpretation of implication connective, they are compatible if

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we interpret → as a material implication. In section four, we discuss this paradox in detail interpreting → as a connexive implication. We show that under the natural connexive meaning, barbershop conditional does not lead to a paradox.

2 Connexive Logic Non-classical interpretation of implication functor plays the main role in connexive logic. The root of such an interpretation comes back to ancient times. Non-classically interpreted implication appeared especially in the discussion concerning the right meaning of conditional, led among others in the Stoic School. Let us trace in short selected strains of that debate. We will find there fundamental justifications of the connexive meaning of implication as well as the theses defining connexive logic called by names of the major participants of this debate. Sextus Empiricus in widely known fragment “Outline of Pyrrhonism” (II, 111, cf. “Against the Logicians” II, 112–115; see [8, 9]) distinguished and characterized four approaches to the conditionals considered in that time. He ordered them from the weakest to the strongest one (see [28], p. 47) as: material, strict, connexive and emphatic.1 Non-classical interpretation of implication in connexive logics could be found in the third approach to conditionals. As Sextus wrote: [T]hose who introduce ‘connection’ or ‘coherence’ say that a conditional is sound when the denial of its consequent is inconsistent with the antecedent.2

Two concepts are most important for our presentation: , translated as , translated as „incompatibility”, „incon„connexion” („connection”), and sistency” or „contradiction”. In that terms, a conditional with antecedent A and consequent B is true, if it expresses a connection that A and non-B are incompatible, and so, they lead to inconsistency. Let us note that if we take non-A as B, then we obtain as false the conditional where A is an antecedent and non-A is a consequent 1 Sextus

attributed the first interpretation to Filon, and second one to Diodorus. Historians recognized them as Filon from Megara and his master Diodorus Cronos; see [13], fn. 4—about interrelations between the two interpretations, see e.g. [28], ch. 4, Sect. 1, [20], ch. 3, Sect. 3. Sextus left unattributed connexive and emphatic interpretations, however, one can figure out that the third approach comes from Chrysippus and the fourth one was defended by the Peripatetic School, see [20], p. 129. We employ the term “emphatic” as Sextus uses the word in the sense that “in that implication we emphasise in consequent what is expressed in antecedent”. Let us add that “Outline of Pyrrhonism” successively lived until two full English translations: Robert Gregg Bury’s dated 1933 (see [8]) and Beson Mates’ dated 1996 (see [29]). Moreover, a long passage devoted to the four conditionals interpretation appears in English in Kneales, [20], p. 129. 2 Translation: Beson Mates 1996, see. previous footnote. Earlier in the paper dated 1961 (see [28], pp. 47–48) Mates uses “holds” instead of “sound” and “incompatible” instead of “inconsistent”. In logical literature one could find Kneales translation: “And those who introduce the notion of connection say that a conditional is sound when the contradictory of its consequent is incompatible with its antecedent”. In Bury’s translation it looks as follows: “And those who introduce ‘connexion’ or ‘coherence’ assert that it is a valid hypothetical syllogism whenever the opposite of its consequent contradicts its antecedent clause”.

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(and otherwise), as A and non-non-A (i.e. A) are not incompatible. So, as Sextus points out, a conditional “If atomic elements of things do not exist, then atomic elements of things do exist”, is false and not true, as it is true only if we interpret the conditional in a material or strict way, where the falsity of antecedent makes the conditional true. As a consequence, under material interpretation we get: ∼(∼A → A), as well as ∼(A → ∼A). According to McCall ([30], p. 415) it shows that in the third interpretation disis a relational notion: an implication antecedent and cussed by Sextus, consequent cannot be considered separately but only in respect to inconsistency relation.3 A similar characterization of implication could be found earlier in Diogenes Laertios, in the passage where he describes stoic, i.e. among others Chrysippian approach : to “nonsimple propositions”. However he does not use explicite the term A conditional proposition is true if the contradictory of its consequent conflicts with its antecedent, for example, ‘If it is day, it is light.’ This is true; for the opposite of the conclusion, namely ‘It is not light,” conflicts with ‘It is day.’ A conditional proposition is false if the contradictory of its consequent does not conflict with its antecedent, for example, ‘If it is day, Dion is walking.’ For the proposition ‘Dion is not walking’ does not conflict with ‘It is day.’4

Let us note, that in the above examples Diogenes emphasises the content relation between the antecedent and consequent and although he writes explicitly on “a conditional proposition”, using a contemporary knowledge we could tell that we have here to do with a kind of semantic (analytic) entailment: it is impossible that the first sentence is true and the second is false, because the falsity of the later remains in the content contradiction with the true of the former. Similar doubts concern in principle all passages considered above and are shared by contemporary researchers, see e.g. [4], p. 177. 3 One should emphasise that in the subject literature Philo’s approach evokes a some interpretational

problems, whilst Diodorean and even more Chrysippian approaches are interpreted in many different ways. For example, Daniel Bonevac and Josh Dever ([4], p. 179) interpret the first three approaches in the following way, considering the fourth one as unclear. Philo: A → B iff ∼(A ∧ ∼B), Diodorus: A → B iff it is always the case that ∼(A ∧ ∼B), Chrysippus: A → B iff it is necessary that ∼(A ∧ ∼B). Accordingly, one can recognize that the strict implication is expressed by Chrysippian, and not Diodoren approach. However Bonevac and Josh interpretation pass over an important relation between the antecedent and consequent ( ) which allows us to consider it doubtful. 4 “Lives of the Eminent Philosophers” VII, 73, trans. Pamela Mensch, [24]. Classical XX-century translation by Robert Drew Hicks in 1925 (see [23]): “A hypothetical proposition is therefore true, if the contradictory of its conclusion is incompatible with its premiss, e.g. ‘If it is day, it is light.’ This is true. For the statement ‘It is not light,’ contradicting the conclusion, is incompatible with the premiss ‘It is day.’ On the other hand, a hypothetical proposition is false, if the contradictory of its conclusion does not conflict with the premiss, e.g. ‘If it is day, Dion is walking.’ For the statement ‘Dion is not walking’ does not conflict with the premiss ‘It is day.”’ On whether Diogenes really characterizes here the third type of conditional discussed by Sextus, see [28], pp. 48–49. Let us also note, that the very sentences employed by Diogenes as example of true conditional occur in writings by many ancient authors discussing implication and argumentation.

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More complex argument having to lead to connexive interpretation of conditional could be found in Aristotle’s “Prior Analytics”: [I]t is impossible for the same thing to be of necessity both when a certain thing is and when the same thing is not (I mean, for example, for B to be large of necessity when A is white, and for B to be large of necessity when A is not white). For whenever it is the case that if this thing, A, is white, then that thing, B, is necessarily large, and that if B is large then C is not white, then, if A is white, it is necessary for C not to be white. And when there are two things such that if one of them is, it is necessary for the other to be, then if the latter is not, the first necessarily is not. So if B is not large, then A cannot be white. But if it is necessary for B to be large when A is not white, then it results of necessity that when B is not large then this very thing B is large: but that is impossible. For if B is not large, then A will of necessity not be white; therefore, if B will also be large when this is not white, then it results that if B is not large it is large, just as if by means of three terms.5

Assuming that by “of necessity” Aristotle means a kind of implication, his argument could be reconstructed as follows (cf. [4], p. 176): 1. A → B 2. ∼A → B 3. (A → B) → (∼B → ∼A) 4. ∼B → ∼A 5. ∼B → B 6. ∼(∼B → B) 7. ∼((A → B) ∧ (∼A → B))

(assumption) (assumption) (contraposition) (modus ponens, 1, 3) (transitivity, 4, 2) (A1) (1–6)

It is hard to find why Aristotle assumes that it is not true that negation of B implies B (step 6; more strictly: that it is impossible, gr. )—perhaps he refers implicite to Chrysippian interpretation—assuming this fact as a fundamental principle, at least as a law stronger than the other assumptions. As a consequence, in this step, ∼(∼B → B) leads us to the acceptation of ∼((A → B) ∧ (∼A → B)). Both were called in the literature Aristotle Theses, although the first one is attributed to Chrysippus. Many centuries later Boethius provides a similar argument. According to Rahman ([34], p. 289), Boethius distinguishes two types of if-then propositions—in his terminology conditionals or connected: “secundum accidens, e.g. ‘If the fire is warm, the sky is spheroidal’, which expresses only a temporal causal connection, and secundum naturam, such propositions express a strong, i.e. a necessary, intensional and causal, connection”. In De hypotheticis syllogismis one can find the following inference scheme related to strongly connected propositions: Si est A, cum sit B, est C [...] atqui cum sit B, non est C; non est igitur A.6 5 Prior Analytics 57b3–14; trans. Robin Smith, [2]. In studies concerning connexive logics this citation occurs rather in abridged version and in Łukasiewicz’ translation, see [26], pp. 49–50. There are also other translations but the distinctions between them are not essential for our purposes. 6 [3], p. 286 (bk II, chap. vi, Sect. 1, lines 1–3). The notation adapted to the present paper.

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In translation: If A, then if B, then C [...] (and) if B, then not-C; Therefore not-A.

A → (B → C) B → ∼C ∼A

The argument is not sound if we interpret the above conditional as material or strict implication.7 As we noted above, Boethius meant here if-then propositions as secundum naturam, hence as connexive implication. Let us trace the argument, which could justify such an inference scheme (cf. [34], p. 290, [30], p. 416): 1. A → (B → C) (assumption) 2. B → ∼C (assumption) 3. C → ∼B (contraposition, 2, double negation) 4. A → ((B → C) ∧ (C → ∼B)) (1, 3, antecedent law, composition) 5. ((B → C) ∧ (C → ∼B)) → (B → ∼B) (transitivity) 6. ∼(B → ∼B) (A1) 7. ∼((B → C) ∧ (C → ∼B)) (modus tollens, 5, 6) 8. ∼A (modus tollens, 4, 7) As we can see, connexively interpreted formula ∼(B → ∼B), makes the above argument sound. It is easy to observe that the above argument is similar to Aristotle’s argument presented earlier. Aristotle shows ∼((A → B) ∧ (∼A → B)) assuming ∼(∼B → B) while Boethius (adopting notations) shows: ∼((A → B) ∧ (A → ∼B)) assuming ∼(B → ∼B). The conclusion ∼((A → B) ∧ (A → ∼B)) will be called, according to commonly accepted use, the Boethian Thesis. Instead of the former Boethian Thesis we distinguish two Boethian implicational theses (B1), (B2).8 The question of right conditionals interpretation was the subject of considerations in later times. It was especially discussed in times when the contemporary form of logic has emerged, see [34]. In general, the third conditional interpretation pointed out by Sextus and other ancient authors could be meant either in the truth-conditional (extensional) or content (intensional) way. Truth-conditional approach is related to the connexive logic discussed later, while the content approach was proposed by Everett Nelson ([33]). He defined an “entailment” in the following way ([33], pp. 444–445; notation adopted):

7 In our times, the propositional forms and inference schemes postulated by Boethius were analysed

by Karl Dürr (see [7]). This author tries to find the conditional being sound independently if interpreted as material or strict implication. The author’s conclusion is that in both interpretations the formula corresponding to the above inference schema is not a tautology—more strictly, it is not provable neither in the system of Principia Mathematica nor in S2. He did not take account of the possibility of connexive interpretation. 8 Boethian Thesis in the form of negation of conjunction is sometimes called Abelard’s Principle or Strawson’s Thesis, in turn connexive theses used by Aristotle as well as by Boethius are called Aristotelian, see [12], pp. 346–347.

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The propositional function ‘A entails B’ means that A is inconsistent with the propositional function that is the proper contradictory of B. Entailment, not being defined in terms of truth-values, is a necessary connexion between meanings.

According to Nelson such a notion “comes much closer to expressing what ‘implies’ means in ordinary discourse than does either material or strict implication”, [33], p. 445. Therefore Nelson seems to be close to Sextus view, although he does not refer to Sextus argument. It is interesting that Nelson intensional approach is expressed by similar theses as the discussed above extensional approach, see also: [30], pp. 420–421. Truth-conditional approach is intended to formalize the intuitions of the connexive conditionals interpretation in terms of the propositional logics. There appeared many constructions of connexive logics. The most known of them are the Angell’s logic CC1 (see [1]), Mortensen’s M3V (see [32]) and the logic CN by Cantwell (see [5]). There is a relatively broad agreement among researchers that each connexive logic should satisfy formulas (A1), (A2), (B1) and (B2) known as Aristotle and Boethian Theses introduced in the Sect. 1.9 Furthermore, implication in minimal connexive logic is non-symmetric (so it is not a kind of a biconditional relationship, see [40]): (NI)

 (A → B) → (B → A)

(Non-Symmetry of Implication)

Let us note that none of the formulas (A1), (A2), (B1), (B2), is a thesis of classical logics. If we add any of them to the classical logic, we will receive the full inconsistent logic. It shows that the negation or implication involved in them must be interpreted in a non-classical, connexive way. Defined this way minimal connexive logic appears to be so weak that it possesses hard to accept peculiar properties.10 Let us note, for example, that formulas (A1), (A2), (B1), (B2) are true in two-element matrix {1, 0} with unit element 1 designated, material implication and negation defined as: ∼1 = ∼ 0 = 1, as well as in a two-element matrix, with classical negation and implication defined as: x → y = 1 iff x = y. Such examples are obviously counterintuitive. They show the necessity of strengthening the minimal connexive logic—the proper connexive logic should be stronger than minimal one. Let us define, after [12], five other types of connexive logics by assuming some natural conditions additional to Aristotle’s and Boethian Theses. Abelardian logic is a logic satisfying two additional theses—reverse to Aristotle’s and Boethian ones: (A3) (B3) 9 See

∼((A → B) ∧ (∼A → B)) ∼((A → B) ∧ (A → ∼B))

(Third Aristotle’s Thesis) (Third Boethian Thesis)

e.g. [12], p. 346, [40]. We follow the commonly accepted terminology although we shown that A2 could be equally called a Boethian Thesis.. 10 One distinguishes as well a class of so called subminimal connexive logics or quasi-connexive logics: consisting of all logics satisfying at least some of (A1), (A2), (B1), (B2), and (NI), but not all of them, see [12, 17].

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Anti-paradox logic is a logic where we additionally assume that material implication paradoxes do not hold: (PP) (NP) (PN)

 A → (B → A)  A → (∼A → B)  A → (B → C)11

(Positive Paradox of Implication) (Negative Paradox of Implication) (Paradox of Necessity)

Simplificative logic is a logic containing additionally the following two theses: (S1) (S2)

(A ∧ B) → A (A → B) → B

(Simplification 1) (Simplification 2)

Conjunction-idempotent logic is a logic containing additionally the following two theses: (I1) (I2)

(A ∧ A) → A A → (A ∧ A)

(Idempotence 1) (Idempotence 2)

Kapsner-strong logic is a logic satisfying at the same time the following two conditions (see [19]): (K1) (K2)

A → ∼A is unsatisfiable, A → B and A → ∼B are not simultaneously satisfiable.

One could define many systems related to the notion of connexivity. It is not easy task to determine the final shape of a logical systems expressing connexivity in an exact way. Such a task requires further research. For this reason, we do not prejudge now how the proper connexive logic looks like and we consider this term in a general way: thus there exists many connexive logics. For this reason, further in this paper we will follow: Definition 1 By a connexive logic we mean any logical system, i.e. any set of sentences closed under substitutions and modus ponens rule, containing Aristotle’s and Boethian formulas (A1), (A2), (B1), (B2).

3 Relating Semantics Relating semantics is a fruitful combination of extensional and intentional conditions. Apart from the extensional truth-value interpretation, relating satisfaction conditions contain an additional requirement: the formulas have to be related with respect to a given intentional binary relation R on formulas. Extensional truth-value part of relating satisfaction conditions are defined in the same way as in the classical logic.12 There are many possible philosophical motivations of relation R. Among others R(A, B) could be understood (cf. [21], p. 24; [22], p. 7; [15], p. 54): A is a contingent truth and B → C a logical truth, see [30], p. 427. conditions could be here replaced by non-classical interpretations i.e. intuitionistic, paraconsistent etc., see [21], pp. 27–28. However, this remains out of the scope of this paper. 11 Where

12 Classical

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• syntactically: A and B relate in syntactical way, if, for example, each (or respectively some) syntactical unit from A occurs in B (or inversely); • semantically: A and B relate in semantic way, if, for example, any (respectively some) referent of a name occurring in A is a referent of a name occurring in B; • pragmatically: A and B are related by pragmatic content, for example each presupposition of A is a presupposition of B (or inversely); • contextually: A and B contextually relate, if, for example, A and B are related in spatial or thematic way or they both belong to a common knowledge; • temporally: A and B relate in a temporal way, if, for example B is a case immediately after A is the case; • causally: A and B relate causally, if, for example A is a direct cause of B.13 A relation R could interpret one or many binary functors. One can also introduce many relations linking them to different functors. For example, the relation of time order is a natural interpretation of conjunction and relation of a cause is one of possible interpretations of implication. Relation R has then a very simple motivation and at a very same time it has a huge philosophical potential because it allows a broad spectrum of interpretations. The roots of relating logics come back to 70s, when the concept of relatedness logic was founded. The main ides behind this logic were born in 1976 in a discussion concerning the logic of action, which took place on a Logic Seminar at Victoria University of Wellington.14 Richard Epstein and Douglas Walton [10, 36, 37] played the main roles there. In the following years, Epstein developed many formal details of logic of relatedness in cooperation with other authors. The aftermath of those investigations form chapter three of [11]. Other authors continued those investigations, e.g. [21, 22]. Walton looked for practical applications of logic of relatedness in the argumentation theory (see [38, 39]). Logic of relatedness is motivated by problems with proper interpretation of conditionals. Originally relation R was introduced for the implication functor. The very title of Epstein’s paper [10]—“Relatedness and implication”—shows this fact well. Whereas Walton writes: Our philosophical beginning point is the principle that A implies B only if A is related to B. A → B (relatedness implication) is the primitive notion characterized by the matrix (truthtable): A → B is true in all cases except where (i) either A is not related to B or (ii) A is true

13 This classification is not disjoint. A lot depends on theoretical settles. For example, indexicals come within a contextual interpretation. However it could also come under pragmatic or semantic interpretation depending on how to make a demarcation line between semantics and pragmatics. In turn, temporal or causal interpretation could also mean some type of contextual interpretation. 14 See e.g. [37], p. 176 and: Preface to special issue of Philosophical Studies: 36(2), 1979. In Preface one can find a list of seminar participants. One emphasizes there, among others, Robert Goldblatt contribution to constructing proofs of metatheorems of this logic, David Lewis’ proposal to give ‘A is related to B’ a meaning ‘A and B share some common subject-matter’, see [38], p. 39. In 80s. Lewis developes the issue of a “subject-matter” independent from relating logic, see e.g. [25].

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It shows that Epstein’s and Walton’s relatedness implications combine two approaches mentioned above in this paper: classical extensional one, expressed in (ii) of the above quote, and non-classical intentional one, expressed in (i). Recently, Tomasz Jarmu˙zek and Bartosz Kaczkowski (see [15]) proposed a more general approach. In their approach, relation R only interprets conjunction and implication, however no specific properties of relation R are assumed. This approach could be easily generalized to the relating semantics in a subject languages where connectives of ∼, ∧, ∨, →, ↔ are duplicated. Each of them occurs twice, one time with classical meaning and second time as a relating connective. Let ForCPL be a set of formulas of Classical Propositional Logic (CPL), made up in a standard manner from: variables Var = { p, q, r, p1 , q1 , r1 , . . . }, connectives: Con = {∼, ∧, ∨, →, ↔}, and brackets: ), (. Let |=CPL be consequence relation of CPL defined on ForCPL by set of all classical valuations of formulas from ForCPL . By contrast, the set of formulas of Relating Logic (RL) ForRL is generated with Var, Con, four relating connectives that are relating counterparts of classical binary connectives: ∧w , ∨w , →w , ↔w , and brackets: ), (. Thus ForCPL ⊂ For RL . A model for relating formulas is pair v, R , where v : Var → {0, 1} and R ⊆ ForRL × ForRL . Relation R is called relating relation. We have the following, general truth conditions for relating formulas: v, R |= A iff v(A) = 1, if A ∈ Var v, R |= ∼A iff v, R |= A v, R |= A ∧w B iff v, R |= A & v, R |= B & R(A, B) v, R |= A ∨w B iff [v, R |= A or v, R |= B] & R(A, B) v, R |= A →w B iff [v, R |= A or v, R |= B] & R(A, B) v, R |= A ↔w B iff [v, R |= A iff v, R |= B] & R(A, B). The set of all models for RL we denote by MRL . By taking any subset M of MRL in the standard way we define a relating logic |=M : for any X ∪ {A} ⊆ ForRL X |=M A iff for all M ∈ M, if M |= X , then M |= A. Let us note that typical conditions of relation R like, for example, reflexivity or symmetry, reflect properties of interpreted domain. For example reflexivity condition, assumed in logic of relatedness, is satisfied for analytic connection but fail for causal connection because nothing is the cause of itself. For this reason we consider relating relation a primitive notion and do not assume any of its properties but that it is a binary relation on formulas. In this paper, we focus on a logic with only one relating connective—relating implication. 15 See [36], p. 116. In another paper (see [37], p. 176) Walton writes: „The relatedness approach gives us a much more flexible way of dealing with conditionals that allows us to deal better especially with practical conditionals concerned with time and events”.

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We will construct a relating semantics for connexive logic (for more details see [17]). The language of this logic contains only one relating connective— connexive implication. Thus the formulas are built up by means of: variables Var = { p, q, r, p1 , q1 , r1 , . . . }, three Boolean connectives: ∼, ∧, ∨, relating implication →w , and brackets ). ( In this way we obtain set of formulas ForCF , where CF is an abbreviation for combined formulas. Set For CF was on purpose designed so as to include formulas made up from a single non-classical connective that behaves intensionally (the relating implication: →w ), whereas the other preserve their classical Boolean nature. Of course, the set of formulas For CF might be simply treated as a subset of formulas of relating logic ForRL : ForCF ⊂ ForRL . A model for the combined formulas is a regular model for the relating logic: pair v, R , where the relation was reduced to set ForCF : R ⊆ ForCF × For CF , while the valuation remained unchanged: v : Var → {0, 1}. Let us denote the set of all models for the combined formulas as MCF . Let us adopt some abbreviations. To simplify, instead of ForCF × For CF , we will shortly write For2CF . Moreover, for convenience, here and there instead of R(A, B), we shall write ARB. Additionally we shall introduce some notation for the nonRB iff occurrence of relation R. Let R ⊆ For2CF . For all A, B ∈ ForCF we put A ARB does not hold. The truth conditions for the propositional letters and developed formulas remain standard, i.e. as in the classical logic. In turn, our relating implication →w can have an intensional nature (where v, R ∈ MCF and A, B ∈ ForCF ): • v, R |= A →w B iff [v, R |= A or v, R |= B] & R(A, B) Therefore a proposition A →w B is true in a model iff not only the truth of proposition A must guarantee the truth of proposition B; proposition A must also be connected with proposition B. The following conditions on relation R will be needed to determine a connexive logic. R ∼A • R is (a1) iff for all A ∈ ForCF , A RA • R is (a2) iff for all A ∈ ForCF , ∼A • R is (b1) iff for all A, B ∈ ForCF : – if ARB, then A R ∼B – (A →w B)R ∼(A →w ∼B) • R is (b2) iff for all A, B ∈ ForCF : – if ARB, then A R ∼B – (A →w ∼B)R ∼(A →w B). • R is (c1) iff for all A, B ∈ ForCF , if ARB then ∼AR ∼B. Relation R satisfying the above conditions (a1), (a2), (b1), (b2) will be called a connexive relation. The above conditions allow us do define the relating semantics

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adequate for the connexive logic. The following theorem shows that if R ⊆ For2CF satisfies (c1), then Aristotle and Boethian theses are satisfied in all relating models with relation R satisfying conditions (a1), (a2), (b1), (b2). As a consequence, if relation R satisfies (c1), then: R is (a1) ⇔ R |= ∼(A →w ∼A) R is (a2) ⇔ R |= ∼(∼A →w A) R is (b1) ⇔ R |= (A →w B) →w ∼(A →w ∼B) R is (b2) ⇔ R |= (A →w ∼ B) →w ∼(A →w B). The following theorem shows that any connexive relation R determine a connexive implication. Theorem 3.1 If relation R ⊆ For2CF satisfies conditions (a1), (a2), (b1), (b2) then in any model v, R with R: v, R |= ∼ (A →w ∼A) v, R |= ∼ (∼A →w A) v, R |= (A →w B) →w ∼ (A →w ∼B) v, R |= (A →w ∼B) →w ∼ (A →w B). It is easy to observe that there is no connexive relation R such that AR ∼ A, as it contradicts to (a1). As a consequence, not every set of pair of formulas of ForCF could be extended onto a connexive relation. However if R0 satisfies some natural conditions then it is easy to show that it could be extended onto connexive relation. The following theorem provides a useful tool to construct relating semantics for connexive implication. Theorem 3.2 Suppose that R0 ⊆ For2CF . The following conditions are equivalent: (i) For any A, B ∈ ForCF , AR0 ∼A, ∼AR0 A, if AR0 B, then AR0 ∼B. (ii) There exists smallest connexive relation R such that R0 ⊆ R. In the next section, we extent the results from [27] in order to interpret the conditional occuring in the Lewis Carroll Barbershop paradox as a connexive implication. To simplify notations let us assume the following conventions: Suppose v, R ∈ MCF is a model, X is a set of formulas and A is a formula. We say that X and A are consistent iff there exists a model v, R such that v, R |= B for any B ∈ X ∪ {A}. If X and ∼A are consistent then we say that X and A are independent. Otherwise we say that A follows from X . If X is a singleton, then we will skip brackets. We will say that the set of formulas X is independent iff for any A ∈ X the set X − {A} and the formula A are independent. We call the formulas A1 , . . . , Ak independent iff the set {A1 , . . . , Ak } is independent.

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4 Barbershop Paradox in Connexive Interpretation In 1894 Lewis Carroll presented a paradox concerning properties of conditionals (see [6]). It is based on a story which could be summarized as follows: Allen, Brown and Carr run a barbershop. Somebody must be in to take care of the customers. Moreover, Allen and Brown are always together. Carroll formulated the issue in the following two hypotheticals: First. If Carr is out, it follows that if Allen is out Brown must be in. Second. Allen always takes Brown with him. Then Lewis Carroll writes: “If Carr is out, these two Hypotheticals are true together. And we know that they cannot be true together. Which is absurd. Therefore Carr cannot be out. There’s a nice Reductio ad Absurdum for you!”. This way we proved in a logical way that Carr must be always in. It is obviously paradoxical because one can easily see that Allen and Brown might be in and then Carr might be out. So, if Allen and Brown are in and Carr is out then both hypotheticals are true. To recognize all the details of this argumentation we have to formalize it. Let propositions A, B, C denote respectively ‘Allen is out’, ‘Brown is out’, ‘Car is out’. This way Carroll’s hypotheticals take the form: (1) C → (A → ∼B); (2) A → B. Lewis Carroll claims that: (3) A → ∼B and A → B are incompatible. If (1), (2) and (3) are together true then we get paradoxical result that C is false. Interpretation—material implication. Suppose that we stay within the classical logic and → means material implication. Then (3) does not work as A → ∼B and A → B are both true provided A is false. The paradox disappears.16 Moreover, since A is false which means that Allen is in, then C, (1) and (2) are both true only when Allen is in. So, this apparent paradox is based on a mistake. This is not a real paradox like for example liar paradox but just a paralogism—unexpected result based on a mistake in reasoning. However, it could be a bad idea to blame Lewis Carroll. He did not write that he interpreted the conditional as material implication (cf. [31], p. 492–493). 40 years earlier, in 1854 George Boole published his famous book ‘The Laws of Thoughts’. 16 Bertrand

Russell in the second edition of The Principles of Mathematics wrote: “The principle that false propositions imply all propositions solves Lewis Carroll’s paradox [...] The assertion made in that paradox is that, if p, q, r be propositions, and q implies r , while p implies that q implies not-r , then p must be false, on the supposed ground that ‘q implies r ’ and ‘q implies not-r ’ are incompatible. But in virtue of our definition of negation, if q be false, both these implications will hold: the two together, in fact, whatever proposition r may be, are equivalent to not-q. Thus the only inference warranted by Lewis Carroll’s premisses is that if p be true, q must be false, i.e. that p implies not-q; and this is the conclusion, oddly enough, which common sense would have drawn in the particular case which he discusses”, [35], fn. on p. 18.

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Probably we might recognize this book as a first publication where classical propositional logic was precisely defined. We can take for granted that when Lewis Carroll wrote his paper all major concepts of the classical logic with the notion of material implication had already been well established. Nevertheless, the classical propositional logic in general and material implication in particular were not considered as a default logical system. The big part of the history of contemporary logic is a history of criticism of material implication. There are many natural examples of conditionals with false assumptions which considered as material implication bring us to a counterintuitive true proposition. This is why we should consider the conditional in the barbershop paradox in a non-classical way. Let us note that within the classical logic each of the following conditions is equivalent to (1): (1.1) (1.2) (1.3)

∼ C ∨ ∼B ∨ ∼A; (C ∧ A) → ∼B; A → (C → ∼B).

For this reason, within the classical logic, we can replace (1) by any condition from among (1.1), (1.2), (1.3) without any change in the reasoning about the barbershop. It will be very different if we interpret the conditional in a non-classical way. We will come back to it later. Interpretation—connexive implication. If we interpret barbershop conditional as connexive implication, then from Boethian Theses (B1) and (B2) it is easy to deduce that Lewis Carroll’s claim (3) is true. However conditions (1) and (2) are more problematic than in the classical case. For (2) we need that ARB—A relates to B with respect to R. It sounds quite reasonable as according to the story Allen and Brown are always together. For (1) we need that C relates to connexive implication A → ∼B. One could hardly understand what such a relation means. It is hard to justify such a connection. As a consequence, under natural connexive interpretations one could incline to accept (2) and reject (1). Propositions (1), (1.1), (1.2), (1.3) express the same condition—somebody must be in to run a shop. They are mutually equivalent with respect to classical logic. However, for any two of those propositions it is easy to construct a connexive logic such that only one of the propositions is true. To accomplish this goal it is sufficient to define relation R satisfying appropriate conditions. Let us note however that some of such relations could not fit the scenario of barbershop paradox. One can argue that (1) is a very sophisticated form of expressing a simple idea. It seems (1.1) expresses it in the simplest way. (1.2) looks natural also. However (1.3) is as sophisticated as (1). One could easily observe that with respect to connexive implication each of them expresses a different thought. Theorem 4.1 (i) There is no connexive model for the set {A → ∼B, A → B}, hence (3) is valid in any connexive model;

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(ii) The formula (1.1) ∼C ∨ ∼B ∨ ∼A follows from any of formulas (1.2) (C ∧ A) → ∼B, (1.3) A → (C → ∼B), (1) C → (A → ∼B); (iii) The formulas (1) C → (A → ∼B), (2) A → B, (1.2) (C ∧ A) → ∼B, (1.3) A → (C → ∼B) are independent. (iv) The formulas (1.1) ∼ C ∨ ∼B ∨ ∼A and (2) A → B are independent. Proof (i) is immediate consequence of Boethian Theses and 3.1. (ii) Formulas (1.1), (1.2), (1.3) and (1) are valid under the same valuations. In (1.1) no implication occurs while it occurs in the remaining formulas. They have just an additional validity condition: an implication antecedent and consequent relate with respect to R. The conclusion follows directly from satisfaction definition for relating implication. (iii) To show it, one needs to construct—for any three formulas among (1), (2), (1.2), (1.3)—a connexive relation R such that the three formulas are satisfied under R while the fourth is not. The set R0 = {(A, ∼ B), (C, A → ∼B), (C ∧ B, ∼ B), (A, C → ∼ B)} satisfies conditions of Theorem 3.2, let R be the smallest connexive relation containing R0 . It is easy to observe that for any model v, R we have v, R |= A → B while for valuation v such that v(A) = v(C) = 0 (1), (1.2), (1.3) are satisfied in model v, R . It shows that (2) is independent from (1), (1.2), (1.3). Proof of the remaining conditions and of (iv) is similar. It is based on an appropriate definition of a set R0 .  The theorem above provides us with a full picture of logical interrelations between the considered conditions. (2), (1), (1.2), (1.3) are independent. (1), (1.2), (1.3) infer (1.1) while (2) and (1.1) are independent and finally (3) as a connexive thesis follows from all remaining formulas. As one can see, connexive relations defined in a proof of the above theorem are very formal. In general, they do not reflect any desirable interrelations between formulas involved in the barbershop scenario. Let us try to look for relation R being connexive and at the same time reflecting the barbershop scenario. Let us consider a simple example of connexive logic defined by relation R on some language containing sentences A, B, C. Let us take a spatial intuitions that there are two possible locations “in the shop” and “out of the shop”. Let relation R bonds formulas that determine the same location of a barber. Thus ARB, but A R ∼B. This way each two of A, B, C are related, i.e. ARB, ARC, BRC. Let us close this relation under reflexivity—A is in the same place as A, symmetry—if A is in the same place as B then B is in the same place as A. Let us note that this relation is closed under (c1)—if A and B are in the same place then ∼A and ∼B are in the same place as there are only two places. It is easy to observe that relation R0 containing all the pairs mentioned below: ARA, BRB, CRC, BRA, CR A, CRB, ∼AR ∼B, ∼AR ∼C, ∼CR ∼A, ∼AR ∼A, etc. satisfies assumption of Theorem 3.2. As a consequence, there exists connective relation R extended R0 . This way barbershop paradox disappears. In the connexive logic defined by R (2) is satisfiable by some valuation, (3) is always true. Although (1) is false, its classical

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equivalent (1.1) is satisfiable by some valuation. This way we have a right description of the barbershop scenario without falling into contradiction. Conditions (1.2), (1.3) are false. In general, propositions with implication connective have more severe truth conditions than other ones. Theorem 4.1 shows that it is possible to define such a relation that also (1.2) is satisfied, however such a relation would unlikely be as natural as the relation above. As a consequence the fact that connexive logic might be analysed by means of relating models, shows a huge expressive power of relating semantics.

References 1. Angell, R. B. (1962). A propositional logic with subjunctive conditionals. Journal of Symbolic Logic, 27(3), 327–343. 2. Aristotle (1989). Prior analytics. Robert Smith (trans.). Indianapolis, Cambridge: Hackett Publishing Company. 3. Boethius, A. M. S. (1969). De hypotheticis syllogismis. Edited, commented and translated by Luca Libertello, Brescia: Paideia. 4. Bonevac, D., & Dever, J. (2012). A history of the connectives. In D. M. Gabbay et al. (Eds.), Handbook of the history of logic. Volume 11: Logic: A history of its central concepts (pp. 175– 233). Amsterdam: Elsevier. 5. Cantwell, J. (2008). The logic of conditional negation. Notre Dame Journal of Formal Logic, 49(3), 245–260. 6. Carroll, L. (1894). A logical paradox. Mind, 3(11), 436–438. 7. Dürr, K. (1951). The propositional logic of Boethius. Amsterdam: North-Holland. 8. Empiricus, S. (1933). Outline of pyrrhonism. Robert G. Bury (trans.). Cambridge, MA: Harvard University Press. 9. Empiricus, S. (1935). Against the logicians. Robert G. Bury (trans.). Cambridge, MA: Harvard University Press. 10. Epstein, R. L. (1979). Relatedness and implication. Philosophical Studies, 36, 137–173. 11. Epstein, R. L. (1990). The semantic foundations of logic. Vol. 1: Propositional logics. Dordrecht: Kluwer Academic Publishers. 12. Estrada-González, L., & Ramírez-Cámara, E. (2016). A comparison of connexive logics. The IfCoLog Journal of Logics and their Applications, 3(3), 341–355. 13. Hurst, M. (1935). Implication in the fourth century B.C. Mind, 176(44), 484–495. 14. Jarmu˙zek, T. (2020). Relating semantics as fine-grained semantics for intensional propositional logics. In A. Giordani, J. Malinowski (Eds.), Logic in high definition, trends in logical semantics (this volume). Berlin: Springer. 15. Jarmu˙zek, T., & Kaczkowski, B. J. (2014). On some logic with a relation imposed on formulae: Tableau system F. Bulletin of the Section of Logic, 43(1/2), 53–72. 16. Jarmu˙zek, T., & Klonowski, M. Some intensional logics defined by relating semantics and tableau systems. In A. Giordani, J. Malinowski (Eds.), Logic in high definition, trends in logical semantics (this volume). Berlin: Springer. 17. Jarmu˙zek, T., & Malinowski, J. (2019). Boolean connexive logics: Semantics and tableau approach. Logic and Logical Philosophy, 28(3), 427–448. 18. Jarmu˙zek, T. & Malinowski, J. (2019). Modal Boolean connexive logics: Semantics and tableau approach. Bulletin of the Section of Logic, 48(3), 215–245. 19. Kapsner, A. (2012). Strong connexivity. Thought: A Journal of Philosophy 1(2): 141–145. 20. Kneale, W., & Kneale, M. (1962). The development of logic. London: Duckworth. 21. Krajewski, S. (1982). On relatedness logic of Richard L. Epstein. Bulletin of the Section of Logic, 11(1–2), 24–28.

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22. Krajewski, S. (1986). Relatedness logic. Reports on Mathematical Logic, 20, 7–14. 23. Laertius, D. (1925). Lives of eminent philosophers. Robert D. Hicks (trans.). Cambridge, MA: Harvard University Press, 24. Laertius, D. (2018). Lives of the eminent philosophers. Pamela Mensch (trans.). In J. Miller (Ed.). Oxford: Oxford University Press. 25. Lewis, D. (1988). Relevant implication. Theoria, 54(3), 161–174. 26. Łukasiewicz, J. (1957). Aristotle’s syllogistic from the standpoint of modern formal logic (2nd ed.). Oxford: Clarendon Press. 27. Malinowski, J. (2019). Barbershop paradox and connexive implication. Ruch Filozoficzny (Philosophical Movement), 75(2), 109–115. 28. Mates, B. (1961). Stoic logic. Berkeley and Los Angeles: University of California Press. 29. Mates, B. (1996). The skeptic way: Sextus empiricus’s outlines of pyrrhonism. Oxford: Oxford University Press. 30. McCall, S. (2012). A history of connexivity. In D. M. Gabbay et al. (Eds.), Handbook of the history of logic. Volume 11: Logic: A history of its central concepts (pp. 415–449). Amsterdam: Elsevier. 31. Moktefi, A. (2008). Lewis Carroll’s logic. In D. M. Gabbay, J. Woods (Eds.), Handbook of the history of logic. Volume 4: British logic in the nineteenth century (pp. 457–505). Amsterdam: North-Holland. 32. Mortensen, C. (1984). Aristotle’s thesis in consistent and inconsistent logics. Studia Logica, 43(1–2), 107–116. 33. Nelson, E. J. (1930). Intensional relations. Mind, 156(39), 440–453. 34. Rahman, S. (2012). Hugh MacColl and George Boole on hypotheticals. In J. Gasser (Ed.), A Boole anthology. Recent and classical studies in the logic of George Boole (pp. 287–310). Amsterdam: Elsevier. 35. Russell, B. (1938). The principles of mathematics (2nd ed.). New York: W.W. Norton & Company. 36. Walton, D. N. (1979). Philosophical basis of relatedness logic. Philosophical Studies, 36(2), 115–136. 37. Walton, D. N. (1979). Relatedness in intensional action chains. Philosophical Studies, 36(2), 175–223. 38. Walton, D. N. (1982). Topical relevance in argumentation. Philadelphia: John Benjamins. 39. Walton, D. N. (2004). Relevance in argumentation. London: Lawrence Erlbaum Associates. 40. Wansing, H. (2016). Connexive logic. In E. N. Zalta (Ed.), The stanford encyclopedia of philosophy. https://plato.stanford.edu/archives/spr2016/entries/logic-connexive/.

Exact Truthmaking as Inexact Truthmaking by Minimal Totality Facts Hannes Leitgeb

Abstract This article sketches a proposal for how to interpret exact truthmaker semantics within inexact truthmaker semantics: exact truthmaking might be viewed as inexact truthmaking by minimal totality facts. While the philosophical idea is explained by reference to an example, the logical details are left to follow-up work.

1 Introduction: Exact Versus Inexact Truthmaking In a sequence of stimulating publications, Kit Fine has argued that exact truthmaker semantics may be used to address various salient problems and questions in logic and semantics. (See e.g. Fine [2–12]). Just like models of inexact truthmaker semantics, models for exact truthmaking come equipped with a partial parthood ordering ≤ of the states s at which formulas are evaluated; roughly, the higher up in ≤, the more “content” such states have. The satisfaction/verification relation of inexact truthmaker semantics tracks this partial order by being monotonic or hereditary: if s inexactly satisfies the formula A (formally, s |=in A), and s ≤ s  , then also s  inexactly satisfies A. In contrast, the verification relation of exact truthmaker semantics is supposed to express that a state is “wholly relevant to” a formula, to the effect that monotonicity is not presupposed: when s exactly verifies A (formally, s |=vex A), and s ≤ s  , it may well be the case that s  does not exactly satisfy A; when s  ≤ s  , it may be that that s  exactly satisfies A again; and so forth. (Exact truthmaker semanf tics also involves a separate falsification relation |=ex for which monotonicity is not presupposed either.) Fine takes one of the selling points of exact truthmaker semantics to be that it contains, in a suitable sense, inexact truthmaker semantics: “it is possible to interpret the inexact semantics within the exact semantics. For we may take a state to inexactly verify a given statement just in case it contains a state that exactly verifies H. Leitgeb (B) Munich Center for Mathematical Philosophy, Ludwig-Maximilians-University Munich, Geschwister-Scholl-Platz 1, D-80539 Munich, Germany e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Giordani and J. Malinowski (eds.), Logic in High Definition, Trends in Logic 56, https://doi.org/10.1007/978-3-030-53487-5_5

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the statement; the inexact verifiers of a statement are simply those that contain exact verifiers” (Fine [12]). Filling in more of the details, the claim is twofold: (A) Every model of exact truthmaking with an exact verification relation |=vex can be “turned into” a model for inexact truthmaking, such that, when the satisfaction relation of inexact truthmaking gets defined by Def. 1: s |=in A iff there is an s  ≤ s, such that s  |=vex A, all of “the” inexact truth conditions for complex formulas are going to follow. And: (B) “Every” model for inexact truthmaking can be determined in that way from a model for exact truthmaking. While Fine’s [5] own interpretation of Kripkean inexact truthmaker semantics for intuitionistic logic within a novel exact truthmaker semantics for intuitionistic logic may serve as a paradigm case instance of what (A) and (B) should amount to, more would have to said still to make them sufficiently clear and determinate in the general case. In particular, it remains to be explained how the scare quotes in the formulations of (A) and (B) above ought to be eliminated: What exactly does “turn into” mean? (Define explicitly? On the basis of what conceptual resources exactly? What changes are allowed to be made?) What are “the” truth conditions of inexact truthmaker semantics, and is it always possible to reconstruct them in exact terms? (E.g., the truth conditions of negation formulas in Leitgeb’s [14] inexact truthmaker semantics involve an incompatibility relation of states: what is the exact counterpart thereof?) And does “every” model of inexact truthmaking really mean every such model? (Or rather every model of inexact semantics up to some kind of structure-preserving map? If so: what kind?) Even if inexact truthmaker semantics turns out to be generally interpretable within exact truthmaker semantics, one might wonder whether the direction of interpretation could also be reversed: is it possible to interpret, in a suitable sense, the exact semantics within the inexact semantics? As Fine himself points out at various places, the obvious attempt at doing so would involve the notion of a minimal inexact truthmaker: “There has been a persistent tendency in the literature (we might call it ‘minimalitis’) to start off with a hereditary notion of verification and then attempt to get the corresponding notion of minimal verification, or some variant of it, to do the work of exact verification· · · But if I am correct, all such attempts are doomed to failure” (Fine [10], p. 564). Fine supports this claim by pointing to cases of exact truthmakers that are not at the same time minimal inexact truthmakers: e.g., he regards an exact pq-truthmaker to exactly verify p ∨ ( p ∧ q) by each part of that truthmaker playing an “active role” (Fine [8], p. 564) in the exact verification of p ∧ q and thus of p ∨ ( p ∧ q), even though the exact p-part of the same truthmaker would already be sufficient to verify p and hence p ∨ ( p ∧ q). And in certain cases a formula might have a sequence of exact truthmakers descending infinitely along the ≤-relation (see Fine [9]), while the same formula would seem to have no minimal inexact truthmakers at all (in the terminology of Leitgeb [14], Sect. 1, the relevant inexact truthmaker model would fail to be “smooth”). Recently, Deigan [1] proposed to circumvent Fine’s argument by defining exact truthmaking in terms of inexact truthmaking without referring to minimal states: first,

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a Kratzerian notion of exemplification gets defined on the basis of ≤ and |=in (cf. e.g. Kratzer [13]); then the exact verification of atomic formulas is defined by means of exemplification; and finally the definition of exact verification is extended to complex formulas by recursive clauses that mimick those of exact truthmaker semantics. In a similar spirit but on different grounds, I want to argue in the following section that exact truthmaker semantics might even be interpreted within inexact truthmaker semantics, such that minimality is still crucially involved: exact truthmaking is minimal inexact truthmaking by a special type of facts—totality facts. The ultimate goal is to support claims that are analogous to (A) and (B) above except for reversing the roles of inexact truthmaker semantics and exact truthmaker semantics. While the philosophical idea will be explained and motivated, the formal/logical details will have to be deferred to follow-up work, and questions similar to those left open by Fine’s project will also remain open in what follows.1

2 An Interpretation of Exact Truthmaking Within Inexact Truthmaking In what follows, I am going to suggest that exact truthmaking may be interpreted as inexact truthmaking by minimal totality facts. I will explain how the interpretation is meant to work with the help of a concrete example. Consider the following model from exact truthmaker semantics: The model from Fig. 1 consists of five states, s1 through s5 . By the valuation function of the model, s1 is an exact verifier of p, s2 is an exact falsifier of q and thus an exact verifier of q (= ¬q), each of s3 and s5 is an exact verifier of r , and s4 is neither an exact verifier nor falsifier of any propositional variable. Finally, the lines in Fig. 1 depict the parthood relation in the model (presented in the style of Hasse diagrams in lattice theory): e.g., it holds that s1 ≤ s3 , s3 ≤ s3 (by the reflexivity of ≤), s3 ≤ s4 , and s1 ≤ s4 (by the transitivity of ≤). See Fine [8] for more on the formal details of such models for exact truthmaker semantics. Clearly, the model above can be turned into a (preliminary) model of inexact truthmaker semantics by applying Fine’s definition of inexact truthmaking in terms of exact truthmaking: For instance, in the model depicted by Fig. 2, s3 inexactly satisfies p, q, r , since: s3 contains a state which exactly satisfies p in the original model (s1 ≤ s3 ); s3 contains a state which exactly satisfies q in the original model (s2 ≤ s3 ); and s3 also contains a state which exactly satisfies r in the original model (s3 ≤ s3 ). One can also see that s3 is the only minimal r -state in the model, while the original model involved two states (s3 and s5 ) that exactly verified r —confirming Fine’s criticism of exactnessas-minimality.

1 Acknowledgements: I am grateful to Ilaria Canavotto, Alessandro Giordani, and Jacek Malinowski,

without whom this discussion note would not exist.

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Fig. 1 A model of exact truthmaker semantics

s5 |=vex r

s5

s4

s3 |=vex r

s1 |=vex p

Fig. 2 A (preliminary) model of inexact truthmaker semantics

s3

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But now suppose that we are not quite done transforming the exact truthmaker model from Fig. 1 into one for inexact truthmaker semantics: once we will have taken five additional steps, which I am going to describe now one after the other (and one of which is optional), exact truthmaking will become definable in terms of a form of minimal inexact truthmaking after all. Step (i): Adding totality facts. When the valuation mapping of the inexact truthmaker model from Fig. 2 assigned p to s1 , this was meant to represent that the state

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s1 contains the fact that p: at s1 it is a fact that p. Over and above facts like that, let us now consider also facts of another kind which I am going to call ‘totality facts’: s1 does not just contain the fact that p but even the total fact that p; so, at s1 , it is a total fact that p. Or in other words: s1 contains a fact that is “wholly relevant to” p (using Finean terminology) or that is a total way of it being the case that p. Exact truthmaker semanticists could not complain about any such commitment to totality facts, as they share the same commitment, whether implicitly or explicitly: for when they say that s1 exactly verifies p in Fig. 1, this may be regarded as just a different way of saying that it is a total fact that p at s1 . And in his favorite approach to the exact truthmaker semantics for quantifiers, Fine ([8] , p. 568) himself supposes that “for each subdomain B of possible individuals, there is a totality state. . . to the effect that the individuals of B are exactly the individuals that there are”, by which he commits himself to a variant of totality facts that is solely concerned with the existence of individuals. Inexact truthmaking by totality facts is just an instance of inexact truthmaking by facts of a particular kind, and the kind of facts in question should not be regarded as any more outlandish than exact truthmakers. Step (ii): Describing totality facts by total-fact operators. One might restrict talk of such totality facts to the very metalanguage in which one refers to states, formulas, and how they relate to each other. Or—optionally, though conveniently—one may introduce some means of describing totality facts into one’s object language: for now, let us assume that there is some sentential operator Tot in the object language, such that Tot(A) expresses that it is a total fact that A; hence, s |=in Tot(A) just in case s inexactly satisfies a total way of it being the case that A. (If s |=in Tot(A) then a fortiori s |=in A.) The introduction of operators of such kind is not unheard of either: e.g., sentential ‘all I know’ or total-knowledge operators that express the total knowledge of a person are well-known from the semantics of autoepistemic logic (see Levesque [15]) and have been applied successfully in nonmonotonic reasoning. The present approach is going to use similar total-facts operators to reconstruct the nonmonotonicity of exact truthmaking within (monotonic) inexact truthmaking. (In what follows, we will only need to apply total-fact operators to sentences A that do not themselves include total-facts operators.) Step (iii): Indexing totality facts and total-fact operators. Reconsidering the model of exact truthmaking from Fig. 1, it becomes clear that there may be more than one way of being a total fact that A: for s3 and s5 , which are numerically distinct states, are both exact truthmakers of r . So let us assume that there may be numerically distinct totality-facts-that-A, e.g., a total1 fact that A and a total2 fact that A. Accordingly, there should be a sequence of Tot i -operators over an index set I of sufficient cardinality by which the different ways of being total facts that A can be expressed in the object language: s |=in Tot i (A) just in case s inexactly satisfies the total way #i of it being the case that A. The underlying index set I is just some arbitrary set; but for instance, and most conveniently, there might be a uniquely determined index i s in I for each state s, such that the corresponding operator Tot is expresses the s-way of being a total fact. E.g., in the case of the simple models from Figs. 1 and 2, the index for s1 might be 1, the index for s2 might be 2, and so forth; Tot 1 may then be

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used to describe the s1 -way of being a total fact, Tot 2 to describe the s2 -way of being a total fact, and the like.2 Step (iv): Adding closure conditions for totality facts. Once totality facts for atomic formulas and their negations have been introduced, the penultimate step is to reinterpret the semantic rules of exact truthmaker semantics as ontological closure conditions for “complex” totality facts; thus, the valuation function of the type of model that we are interested in will capture that the existence of certain totality facts at certain states may depend on the existence of certain totality facts at other states. For instance, employing the state-indices that were mentioned before, the following should hold for every state s: s |=in Tot is (A ∧ B) just in case there are s  and s  , such that s = s  ◦ s  , s  |=in Tot is (A), and s  |=in Tot is (B). (s  ◦ s  is the fusion of the states s  and s  , which may be defined as the least upper bound of s  and s  with respect to ≤; see Fine [8], p. 560.) But actually all that we are going to need in what follows is just the existence of certain kinds of indexed totality facts: there is an index i, such that s |=in Tot i (A ∧ B) and for all u < s it holds that u |=in Tot i (A ∧ B) (that is, s is minimal in satisfying Tot i (A ∧ B)), if and only if there are states s  and s  , such that s = s  ◦ s  , there is an index j with s  |=in Tot j (A) and for all u < s  it holds that u |=in Tot j (A) (that is, s  is minimal in satisfying Tot j (A)), and there is an index k with s  |=in Tot k (B) and for all u < s  it holds that u |=in Tot k (B) (that is, s  is minimal in satisfying Tot k (B)). (< is the strict order that is defined from ≤ by: u < s if and only if u ≤ s but not s ≤ u.) The resulting closure condition on “conjunctive totality facts” should be no less plausible than the corresponding semantic rule for the exact verification of conjunctions that constitutes one part of exact truthmaker semantics. Similar closure conditions might be introduced for “negative totality facts” and “disjunctive totality facts”, the details of which might be lined up with the ways in which the semantic rules for negation and disjunction are formulated in exact truthmaker semantics. (This said, it is actually not so clear how the details should be filled in, since Fine formulates the semantic rules for the falsification of conjunctions and for the verification of disjunctions differently at different places.) As already hinted at before, so far as literals A (that is, propositional variables or their negations) are concerned, one may also view the following as a closure condition on facts: if s |=in Tot i (A), then s |=in A. The model of inexact truthmaker semantics that results from applying steps (i)– (iv) to the model of exact truthmaker semantics that was given in Fig. 1 looks like this: The model from Fig. 3 strictly extends that of Fig. 2 by totality facts, only some of which are described explicitly in the diagram. For example: s1 |=in Tot 1 ( p) since it is a total fact that p at s1 (as determined by the model’s valuation function), and that particular total fact has been indexed by 1. For analogous reasons, it holds that s3 |=in Tot 3 (r ) and s5 |=in Tot 5 (r ). Moreover, by the closure of totality facts in step 2 In the case of an uncountable state space, the resulting object language would be uncountable; but

never mind.

Exact Truthmaking as Inexact Truthmaking by Minimal Totality Facts s5 |=in p, q, r, Tot1 (p), Tot2 (q), Tot3 (r), Tot3 (p ∧ q), Tot5 (r)

s5

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Fig. 3 The resulting model of inexact truthmaker semantics

(iv) from above, it must be the case that s3 |=in Tot 3 ( p ∧ q), since s3 is the fusion of s1 and s2 , and s1 |=in Tot 1 ( p) and s2 |=in Tot 2 (q). Last but not least, since the model in Fig. 3 is one of inexact truthmaker semantics, monotonicity is presupposed: e.g., since s1 |=in Tot 1 ( p), every state above s1 in the ≤-order inexactly satisfies Tot 1 ( p), too; e.g.: s4 |=in Tot 1 ( p), that is, s4 inexactly satisfies the total way #1 of it being the case that A. Step (v): Defining exact truthmaking as inexact truthmaking by minimal totality facts. Finally, we can interpret exact truthmaking within inexact truthmaking by considering minimal totality facts, just as planned: Def. 2: s |=vex A iff ∃i: s |=in Tot i (A) and for all s  < s, s  |=in Tot i (A), that is, s exactly verifies A just in case there is an index i, such that s is minimal in inexactly satisfying Tot i (A). For example, in Fig. 3, s3 is minimal in inexactly satisfying Tot 3 (r ), s5 is minimal in inexactly satisfying Tot 5 (r ), and these are the only states that are minimal in satisfying a totality state of the Tot i (r )-kind. E.g., while s4 also inexactly satisfies Tot 3 (r ), it is not minimal in doing so, as s3 < s4 . One might think that the existence of such minimal totality facts would only be guaranteed in finite models, such as the one from Fig. 3. But actually, even if the procedure described in (i)–(v) were applied to an infinite model from exact truthmaker semantics, it would still manage to translate exact truthmaker states into suitable minimal totality facts that would be assigned to inexact truthmaker states: e.g., an infinitely descending sequence of exact truthmakers of A would be translated into an infinitely descending sequence of inexact truthmaker states s and corresponding totality facts indexed by i s , such that each such s would be the minimal Tot is (A)-state. And, by the definition

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of exact truthmaking in terms of inexact truthmaking (see Def. 2) and the closure conditions for totality facts (recall (iv)), the semantic rules for exact truthmaking would still follow as intended. In particular, for conjunctions and conjunctive total facts it follows: s |=vex A ∧ B if and only if (by Def. 2) there is an index i, such that s |=in Tot i (A ∧ B) and for all u < s it holds that u |=in Tot i (A ∧ B), if and only if (by step (iv)) there are states s  and s  , such that s = s  ◦ s  , there is an index j with s  |=in Tot j (A) and for all u < s  it holds that u |=in Tot j (A), and there is an index k with s  |=in Tot k (B) and for all u < s  it holds that u |=in Tot k (B), if and only if (by Def. 2) there are states s  , s  , such that s = s  ◦ s  , s  |=vex A and s  |=vex B. In the terminology of Sect. 1, steps (i)–(v) from above execute Part (B) of the project of interpreting exact truthmaker semantics within inexact truthmaker semantics: Every model for exact truthmaking can be determined in a particular manner from a model for inexact truthmaking with totality facts, where totality facts constitute but a particular type of facts assigned to states in the model. Carrying out part (A) of the project would then consist in showing that every model of inexact truthmaking with totality facts can be turned into a model for exact truthmaking, such that, when the verification relation of exact truthmaking gets defined by Def. 2 from above, all of the truth conditions for complex formulas in exact truthmaker semantics are going to follow. (An analogous claim would have to be shown for exact falsification.) Similar questions would need to be answered as those alluded to in the description of Fine’s project in Sect. 1: What exactly does “turn into” mean? What are “the” truth conditions of exact truthmaker semantics? Does “every” model of exact truthmaking really mean every such model? I will not try to fill in the details here, but at least the rough outlines of part (A) of the project should be clear enough: all relevant assumptions on inexact truthmaker models with totality facts that were involved in (i)–(v) above would be used to determine the very class of inexact truthmaker models with totality facts that are meant to be translatable into models for exact truthmaking.

3 Conclusions and Open Questions I have argued that exact truthmaking might be interpretable as minimal inexact truthmaking after all, and hence that exact truthmaker semantics might be interpretable within inexact truthmaker semantics. The key idea is to assume that some of the facts that are assigned to states in inexact truthmaker semantics are totality facts: total facts that something is the case, describable with the help of sentential total-fact operators. It seems neither implausible ontologically to postulate the existence of such totality facts, nor are their corresponding totality operators semantically problematic. If the

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interpretation is successful, every model of inexact truthmaking might be turned into a model for exact truthmaking with totality facts in which exact verification may be defined as inexact truthmaking by minimal totality facts; and every model for exact truthmaking might be determined in that way from a model for inexact truthmaking with totality facts. However, the exact formal details of the interpretation still need to be filled in, which is left to future work. Once this has been carried out, interesting new logical questions might emerge, such as: what is the sound and complete system of logic of the corresponding total-fact operators? Would it be sensible to allow for nested occurrences of these operators, and could such nestings be usefully applied, e.g., in the logical study of the grounding of grounding-facts? And independently of the questions of mutual interpretability between exact and inexact truthmaker semantics: how useful—philosophically and logically—is an inexact truthmaker semantics in which the fact that A and the total fact that A can be described and investigated at the same time?

References 1. Deigan, M. (forthcoming). A please for inexact truthmaking, forthcoming in Linguistics and Philosophy. 2. Fine, K. (2012a). A guide to ground. In F. Correia & B. Schnieder (Eds.), Metaphysical grounding (pp. 37–80). Cambridge: Cambridge University Press. 3. Fine, K. (2012b). Counterfactuals without possible worlds. Journal of Philosophy, 109, 221–46. 4. Fine, K. (2012c). The pure logic of ground. Review of Symbolic Logic, 25, 1–25. 5. Fine, K. (2014a). Truth-maker semantics for intuitionistic logic. Journal of Philosophical Logic, 43, 549–577. 6. Fine, K. (2014b). Permission and possible worlds. dialectica 68, 317–36. 7. Fine, K. (2016). Angellic content. Journal of Philosophical Logic, 45, 199–226. 8. Fine, K. (2017a). Truthmaker semantics. In B. Hale, C. Wright, & A. Miller (Eds.) A Companion to the Philosophy of Language (2nd Edn., pp. 556–77). Chichester: Wiley. 9. Fine, K. (2017b). A theory of truth-conditional content I: Conjunction, disjunction and negation. Journal of Philosophical Logic, 46(6), 625–74. 10. Fine, K. (2017c). A theory of truth-conditional content II: Subject-matter, common content, remainder and ground. Journal of Philosophical Logic, 46(6), 675–702. 11. Fine, K. (forthcoming). Verisimilitude and truthmaking, forthcoming in Erkenntnis. 12. Fine, K. (unpublished). Truthmaker semantics, unpublished manuscript, www.academia.edu/ 8980719. 13. Kratzer, A. (2002). Facts: Particulars or information units? Linguistics and Philosophy, 25, 655–70. 14. Leitgeb, H. (2019). HYPE: A system of hyperintensional logic (with an application to semantic paradoxes). Journal of Philosophical Logic, 48(2), 305–405. 15. Levesque, H. J. (1990). All I know: A study in autoepistemic logic. Artificial Intelligence, 42(2–3), 263–309.

Hyperintensionality in Imagination Pierre Saint-Germier

Abstract Franz Berto has recently proposed two distinct semantics for the logic of ceteris paribus imagination, one which combines a theory of topics with a standard possible worlds semantics, another based on impossible worlds. An important motivation for using these tools is to handle the hyperintensionality of imagination reports. I argue however that both semantics prove inadequate for different reasons: the former fails to draw some intuitive hyperintensional distinctions, while the latter draws counterintuitive hyperintensional distinctions. I propose an alternative truthmaker semantics, guided by an independently motivated philosophical analysis of the content of imagination acts, that preserves the attractive features of both approaches, while avoiding their symmetrical defects in the treatment of hyperintensionality.

1 Introduction Even though imagination has the reputation to defy logic, our discourse about acts of imagination seems to allow for non-trivial logical inferences. For example, of the two following arguments, the first seems intuitively valid, while the second seems not: (1)

a. Alice imagines at t that Sherlock Holmes is a left-handed smart detective. Therefore Alice imagines at t that Sherlock Holmes is left-handed. b. Bob imagines at t that Sherlock Holmes is either left-handed or righthanded. Therefore, either Bob imagines at t that Sherlock Holmes is lefthanded, or Bob imagines at t that Sherlock Holmes is right-handed.

P. Saint-Germier (B) Centre for Science Studies, Aarhus University and IRCAM, Aarhus, Denmark e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Giordani and J. Malinowski (eds.), Logic in High Definition, Trends in Logic 56, https://doi.org/10.1007/978-3-030-53487-5_6

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For Bob’s act of imagination can leave it open whether Sherlock is left- or righthanded, without determinately representing Sherlock as left-handed or as righthanded. Now, it is widely acknowledged that our reports about the contents of intentional states such as beliefs behave hyperintensionally in the sense that they violate the law of substitution of necessary equivalents: (SNE)

A ⇔ A , B  B AA





where “⇔” symbolizes necessary equivalence and “B AA ” is the result of uniformly substituting all occurrences of “A” in “B” by an occurrence of “A ”.1 Imagination seems to be no exception. Consider the case of Carol whose mind wanders on the way back from her Ancient Philosophy class. Of the following pair of sentences, the first member is true, while the second is false: (2)

a. When Carol imagines that Socrates is talking with Plato, she imagines that Socrates talks and Plato listens respectfully. b. When Carol imagines that Socrates is talking with Plato, she imagines that Socrates talks, Plato listens respectfully and both {Socrates} and {Plato} exist.

From the fact that Carol imagines that Socrates and Plato are talking, it hardly follows logically that she imagines anything about the singleton sets having respectively Socrates and Plato as their only members. However, assuming that, necessarily, a singleton set {a} exists just in case its unique member a exists, (2-b) differs from (2-a) only by the substitution of necessary equivalents, so that we have a clear violation of (SNE). Thus the grain of imagination seems to be cut finer than the grain of necessary equivalence. The logic of imagination reports should therefore be hyperintensional. It follows that traditional possible-worlds semantics alone will hardly do full justice to the logic of imagination. Various alternative semantic frameworks have been proposed to enable the fine-grained discrimination of necessarily equivalent contents in the context of propositional attitudes. One may add impossible worlds, e.g. [5, 16, 18], or partial situations, e.g. [1], or work with structured propositions [7], or take propositions as primitive entities [19]. In the case of imagination reports, Berto, who has recently done much innovative work, has relied on impossible worlds [2, 5], as well as on the combination of a theory of topics with a standard possible worlds semantics [3, 4]. I take it that a good treatment of hyperintensionality for a logic of imagination should meet two complementary desiderata. On the one hand, the semantics should allow to draw all the hyperintensional distinctions that are needed to account for the logic of imagination reports, e.g. the distinction between (2-a) and (2-b). On the other hand, the semantics should not draw unnecessary hyperintensional distinctions, i.e.

1 We use this nonstandard notation for uniform substitution, instead of the more familiar “B[A /A]”

because we will need the same brackets to symbolize the imagination operator.

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it should not draw spurious logical distinctions between contents that are intuitively equivalent in imagination. Consider: (3)

a. When Carol imagines that Socrates is talking with Plato, she imagines that Socrates talks and Plato listens respectfully. b. When Carol imagines that Socrates is talking with Plato, she imagines that Plato listens respectfully and Socrates talks.

Assuming that “and” here expresses nothing more than logical conjunction, it seems unnecessary and even inadequate to make room for the logical possibility that the first sentence be true and the second false. How could anyone imagine that p ∧ q without imagining that q ∧ p? A logic of imagination should therefore be fine-grained enough to account for hyperintensional distinctions as in (2), but not too fine-grained, so that the intuitive equivalence of (3-a) and (3-b) is accounted for. While there is much to be said in favour of both the topics semantics and the impossible worlds semantics proposed by Berto, in that they nicely reflect in their own way some characteristic properties of imagination, I argue that neither meets both of our desiderata regarding hyperintensionality. I therefore propose an alternative semantics, based on the framework of truthmaker semantics as developed recently by Kit Fine in [10–12], and directly inspired by a philosophical view of the contents of imagination popularized recently by Yablo [20] and Chalmers [6]. I argue that it preserves all the attractive features of Berto’s semantics while meeting both of our desiderata regarding hyperintensionality. After introducing the target notion of ceteris paribus imagination in Sect.2, I describe in Sect.3 Berto’s possible worlds semantics with topics and show that it fails to draw some intuitive hyperintensional distinctions. In Sect.4, I turn to Berto’s impossible worlds semantics and argue that it draws counterintuitive hyperintensional distinctions. I then introduce the truthmaker semantics that I favor in Sect.5 and argue that it provides a more satisfactory treatment of hyperintensionality in Sect.6, before drawing some general conclusions in Sect.7.

2 Ceteris Paribus Imagination One way to regiment our discourse about acts of imagination is to focus on conditionals such as: (4)

As agent i decides to imagine that A, she also imagines, ceteris paribus, that B.

The sentential variable “A” stands for the “input” of the imagination act, i.e. the content with which one explicitly and voluntarily initiates an exercise of imagination. The sentential variable B stands for an “output” of the imagination act, i.e. a content that results from imagining the input. The input of the imagination act is the part which is under our voluntary control, whereas the output is not directly controlled by

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our will, but follows implicitly from background beliefs and contextual assumptions. For example, when I decide to imagine that Sherlock Holmes smokes in his living room, I thereby imagine a number of other facts about Holmes: that he lives in Baker Street, wears glasses, that his pipe is made of wood and contains tobacco, and that he does nothing illegal. The enriched content of this act of imagination depends on background beliefs that I hold about Holmes and about pipe-smoking. These beliefs are imported in the content of my imagination provided they remain plausible, given the input “Sherlock smokes in his living room”. If some of those background beliefs had been different, the output may have been different. Also, if the input is enriched, some outputs may be canceled. For example, as I decide to imagine that Sherlock Holmes smokes crack in his living room, I still imagine that his pipe is made of wood, but I do not imagine he is doing something legal anymore. In contrast with simpler imagination reports of the form (5)

Agent i imagines that A.

imagination conditionals have the advantage of making explicit a number of important distinctions between aspects of the content of imaginative acts (input vs output, explicit vs implicit, under voluntary control vs automatic) that would otherwise be blurred. So, following Berto, we will take these conditional constructions as the canonical way to formulate imagination reports. Given a set of propositional variables V, a set of agents I and for each i ∈ I a set of entertainable formulas, K i , the language L I for our logic of imagination is defined by the following grammar: A := V | ¬A | (A ∧ B) | (A ∨ B) | (A ⇒ B) | [A]i B if A ∈ K i , for all i ∈ I Outermost brackets will be omitted, following usual conventions. Note that the symbol “⇒” will be interpreted as a strict conditional. Thus “A ⇒ B” says that B is true in all the possible worlds in which A is true. The symbol of strict equivalence “⇔” is introduced by the usual definition and will be a natural means to express the necessary equivalence of two formulas. Formulas of the form [A]i B are meant to formalize imagination conditionals of the form (4). Thus [·]i can be seen as a conditional operator, taking an antecedent A and a consequent B to form the complex formula [A]i B. Our agents are assumed to be real-world, rationally bounded and cognitively finite beings. Thus we assume that they are not able to take any input whatsoever as a starting point for an exercise of imagination. That is why we specify for each agent i a set K i of entertainable formulas that constitute admissible inputs for the conditional operator. Thus whenever A ∈ / K i , the string of symbols [A]i B does not belong to the language L I . Since we will only focus on a single arbitrary agent in what follows, the subscript indicating the identity of the agent will be dropped. We will sometimes refer to the boolean fragment of L I , by which we mean the set of formulas of L I containing only propositional variables and the boolean connectives “¬”, “∧” and “∨”. The modal fragment of L I is the set of formulas of

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L I containing only propositional variables, the boolean connectives, and the strict conditional and biconditional “⇒” and “⇔”. Now that we have fixed our formal language and offered a brief sketch of its intended interpretation, the next task is to endow it with an adequate semantics.

3 Possible Worlds with Topics A sensible diagnosis of the failure of (SNE) in example (2) is that the act of imagination reported in (2-a) is only about Socrates and Plato, and not about {Socrates} or {Plato}. If so, then one might hope to provide a good account of the hyperintensionality of imagination reports by giving a decisive role to what sentences are about, namely topics. Berto’s approach in [3, 4] is to treat the operator [·] as a variably strict conditional on a space of possible worlds with the additional constraint that the consequent should preserve the topic of the antecedent. Informally, [A]B is true in a possible world w just in case B is true in all the possible worlds A-accessible to w (or equivalently, in all the possible worlds selected on basis of A and w) and the topic of B is part of the topic of A. This approach has the feature of being relatively conservative with respect to traditional possible worlds semantics. If it works, it shows that it is not necessary to depart radically from the framework of possible world semantics to account for hyperintensionality, but only to enrich it with additional components such as topics. In order to make this idea of topic preservation precise, Berto treats topics as entities that can be associated with sentences. Since two necessarily equivalent sentences may not have the same topic, this is, prima facie, a promising resource to deal with hyperintensionality. Berto endows topics with mereological structure: a topic can be part of a wider topic and have another topic as a proper part. To model these relations, one can start with a set AT of atomic topics and an operation ⊕ of fusion governed by the following laws: • x = x ⊕ x (idempotence) • x ⊕ y = y ⊕ x (commutativity) • x ⊕ (y ⊕ z) = (x ⊕ y) ⊕ z (associativity). Then the set of topics T can be characterized as the closure of AT under ⊕. Thus for all x, y ∈ T , there is a fusion x ⊕ y ∈ T and more generally for all finite set of topics S ⊆ T , the fusion of the members of S, i.e. ⊕S, is also a member of T . From this operation of fusion, one can define a binary relation ≤ on T as follows: x ≤ y := x ⊕ y = y. It is then easy to see that ≤ is reflexive, anti-symmetric and transitive so that it partially orders the set T . (T , ≤) is easily seen to be a join semi-lattice, i.e. for all finite subset S ∈ T , S has a least upper bound, namely ⊕S. The partial order ≤ on T will then be understood as a relation of topic inclusion or parthood.

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In order to assign topics to formulas of L I , one introduces a mapping t from V to AT , so that for any propositional variable p, t ( p) represents the topic of p. This mapping is extended to complex formulas by the following rule: t (A) = ⊕{x ∈ T : ∃ p ( p ∈ V(A) & x = t ( p))} = t ( p1 ) ⊕ · · · ⊕ t ( pn ) where V(A) = { p1 , . . . , pn } is the set of all propositional variables occurring in A. It follows immediately, for example, that t (A) = t (¬A) and that t (A ∨ B) = t (A ∧ B), which is in line with recent accounts of topics [11, 21]. More generally, on this approach, V(A) ⊆ V(B) implies t (A) ≤ t (B). A topic frame, or T -frame for short, is then a tuple W, {R A : A ∈ K }, T , ⊕, t where: • W is the set of possible worlds, • {R A : A ∈ K } is a set of accessibility relations between worlds: each entertainable formula A has its own accessibility relation R A ⊆ W × W , • T is the set of topics, • ⊕ is the operation of topic fusion, • t is a mapping from propositional variables to topics. A topic model, or T -model for short, is a tuple W, {R A : A ∈ K }, T , ⊕, t,  where W, {R A : A ∈ K }, T , ⊕, t is a T -frame and  is a relation of relative truth between possible worlds and propositional variables.  can be extended to arbitrary formulas of L I , according to the following recursive rules: (T¬) w  ¬A iff w  A (T∧) w  A ∧ B iff w  A and w  B (T∨) w  A ∨ B iff w  A or w  B (T⇒) w  A ⇒ B iff, for all w ∈ W such that w  A, w  B (T[·]) w  [A]B iff, for all w ∈ W such that w R A w , w  B and t (B) ≤ t (A). The first four rules are standard. The last one corresponds to the informal truthconditions stated above, namely that “[A]B” is true at a possible world w just in case B is true at all possible worlds A-accessible to w and the topic of B is part of the topic of A. What one says in terms of accessibility relations can also be said in terms of a selection function f from K × W to P(W ), according to the following definition: f (A, w) := {w ∈ W : w R A w } If one also denotes by |A| the set of possible worlds at which A is true, one can rephrase the rule for the [·] operator as follows: (T[·]’)

w  [A]B iff f (A, w) ⊆ |B| and t (B) ≤ t (A).

Following Berto, we will alternate between describing the semantics of the [·] operator in terms of accessibilities and in terms of a selection function, since in some contexts, the latter is handier than the former.

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Two more constraints need to be added to the definition of a T -model. First, we need the following “Basic Constraint”: f (A, w) ⊆ |A|

(BC)

This ensures that the content of the input will always be present in the output, which seems to be required by the intended interpretation of the operator: whenever I imagine that A, I imagine ceteris paribus that A, if I imagine anything at all. The second constraint, called the “Principle of Imaginative Equivalents” specifies the conditions under which two inputs are imaginatively equivalent in the sense that they systematically yield the same outputs in the same possible worlds. (PIE)

If f (A, w) ⊆ |B| and f (B, w) ⊆ |A|, then f (A, w) = f (B, w).

With this definition of a T -model in hand, one can define the notions of validity in a T -model, T -validity tout court, and T -logical consequence in the usual way: • for all T -model M = W, {R A : A ∈ K }, T , ⊕, t, , M T A iff w  A for all w ∈ W. • T A iff M T A for all T -model M. •  T A iff for all T -model M = W, {R A : A ∈ K }, T , ⊕, t,  and all w ∈ W , if w  B for all B ∈ , then w  A. We will call IT the logic determined by the class of T -models, i.e. IT = {A ∈ L I : T A}.2 Here is a list of schemata that come out valid on the present semantics3 : 1. 2. 3. 4.

T [A]A for all A ∈ K (success) [A](B ∧ C) T [A]B (left simplification) [A](B ∧ C) T [A]C (right simplification) [A]B, [A]C T [A](B ∧ C) (adjunction)

These validities seem indeed to be principles that any logic of ceteris paribus imagination should contain. Success expresses the fact that when one initiates an act of imagination with content A, one also imagines ceteris paribus A, which seems right. The content of the input is always preserved in the output. Simplification and Adjunction characterize what Berto calls the “mereology” of imagination, namely the fact that imagining a whole entails imagining the parts, and imagining all the parts entail imagining the whole. There are also schemata that the present semantics does not validate: 1. 2. 3. 4.

[A]B T [A ∧ C]B (non-monotonicity) [A](B ∨ C) T [A]B ∨ [A]C (indeterminacy) [A]B T [A](B ∨ C) (relevance) It is not the case that: if T B, then T [A]B (no logical omniscience)

2 Giordani

offers in [15] a sound and complete axiomatization of the logic determined by the class of models that satisfy (BC), without assuming (PIE). 3 We provide proofs only for the results not already proved in Berto’s papers. The curious reader is kindly referred to [3, 4] or invited to find the proofs as an exercise.

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5. A ⇒ B T [A]B (no imaginative entailment) 6. T [A ∧ ¬A]B (non-vacuity) Indeed, those principles should be invalid for any logic of ceteris paribus imagination. Non-monotonicity is adequate, because some information C added to an input A, might turn out to be inconsistent with the output B imagined on the basis of input A alone. We already observed this phenomenon with the example of Sherlock Holmes smoking simpliciter, versus smoking crack. Indeterminacy captures the way imagination represents indeterminate states of affairs. It sometimes happens that we imagine something indeterminately, in the sense that we imagine explicitly that an object has a determinable or disjunctive property F, without imagining explicitly or implicitly a particular determinate or a particular disjunct of F. For example, I can imagine that a zebra has a number of stripes on his back without imagining a definite number of stripes. Relevance expresses the fact that the output of an imaginative act is bounded by its topical relationship with the input. Although the disjunction B ∨ C is true whenever B is true, the topic of C might not be part of the topic of A, even if the topic of B is. No imaginative entailment, no logical omniscience and non vacuity are also consequences of the topical constraints that bear on the connection between inputs and outputs. Even if B is a logical consequence of A, or just an arbitrary logical truth, it does not follow that B will automatically be imagined on the basis of input A. For that, the output should be relevant to the input, that is, on the present proposal, topic-preserving. For the same reason, it is not the case that anything whatsoever will be imagined on the basis of an inconsistent input. So far so good. Let us now examine in more detail how this semantics deals with hyperintensionality. A logic is said to be hyperintensional just in case it is not closed under (SNE). It is easily verified that IT is hyperintensional. Proposition 1 A ⇔ A , B T B AA Proof Consider a T -model M = W, {R A | A ∈ K }, T , ⊕, t,  such that W = {w}, R A = ∅ for all A ∈ K , t ( p) = t (q). By (T¬), (T∨) and (T⇒), we have w  ( p ∨ ¬ p) ⇔ (q ∨ ¬q). By (T[·]), w  [ p]( p ∨ ¬ p) since f ( p, w) ⊆ | p ∨ ¬ p| and t ( p ∨ ¬ p) ≤ t ( p). Since t (q ∨ ¬q) = t (q)  t ( p), it follows by (T[·]) that  w  [ p](q ∨ ¬q). Hyperintensional logics such as IT are thus able to make fine-grained hyperintensional distinctions between necessarily equivalent formulas. But stating the mere fact that some hyperintensional distinctions are available does not tell us which hyperintensional distinctions are available. We need to ask ourselves whether it makes the right hyperintensional distinctions. Now for all its virtues, Berto’s topics-based semantics does not offer a completely satisfactory treatment of the hyperintensionality of imagination. As we shall see shortly, in some specific instances, it still treats as equivalent some contents that should intuitively be distinguished when we reason about ceteris paribus imagination reports. Before we consider these instances in detail, I would like first to discuss a feature of IT that could be taken to raise a difficulty with regard to hyperintensionality, but

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really does not, or so I shall argue. We saw above that IT has the feature of no logical omniscience, in the sense that it is not the case that any logical truth whatsoever will be part of the output of any act of imagination. However, IT validates the following schema, which may be taken to reintroduce a form of logical omniscience: Proposition 2 If T B, and V(B) ⊆ V(A), then T [A]B. Proof Let w be an arbitrary possible world from an arbitrary T -model M =

W, {R A : A ∈ K }, T , ⊕, t, . By assumption T B, so that |B| = W . Thus f (A, w) ⊆ |B|. Since V(B) ⊆ V(A), t (B) ≤ t (A). Therefore, by (T[·]), w  [A]B.  Is it really the case that I imagine all the logical validities involving p and q whenever I imagine explicitly that p and q? Giving a positive answer to this question seems to either allow for too easy a way to acquire logical knowledge via the imagination or endow agents with unrealistic logical knowledge from the start. Such interpretations of Proposition 2 however forget that the content represented by the consequent is meant to remain implicit. It follows that the particular syntactic expression of this content is irrelevant. What matters is the information it carries, i.e. the set of possible worlds it rules out. Because of the Basic Constraint, the overall content of the output, modeled by f (A, w) for input A in possible world w, should rule out at least as many possibilities as the input A does. But this is compatible with making true reports of the form [A]B where |A| ⊂ |B|, because B does not have to express the whole content of the output for this report to be true, but only a relevant aspect thereof. Even though I implicitly imagine much more, it is true to say that as I imagine explicitly that Sherlock Holmes is a man, I imagine implicitly that he is a human being. For the same reason, it is true to say, although of dubious relevance in most ordinary contexts, that as I imagine that Sherlock Holmes is a man, I imagine implicitly that he is either a man or not a man. Accepting this is compatible with denying that I thereby imagine implicitly that either Moriarty is a human being or Moriarty is not a human being, for the humanity of Moriarty is completely off-topic in that case. So, even though Berto [3, p. 1883] presents this feature of his logic as a potential cause for worry, I do not think that this points towards an inadequacy of IT as a logic of ceteris paribus imagination, provided one takes seriously the idea that the content expressed by the consequent of imagination conditionals is implicit. Let us turn to more problematic aspects in the treatment of hyperintensionality offered by IT . (PIE) plays an important role in the limitation of hyperintensionality. This enables the logic, on the one hand, to account for intuitive equivalences in imagination, such as the following: Proposition 3 [A ∧ B]C T [B ∧ A]C. Proof Let w be an arbitrary possible world from an arbitrary T -model M =

W, {R A | A ∈ K }, T , ⊕, t,  such that w  [A ∧ B]C. By (T[·]), f (A ∧ B, w) ⊆ |C| and t (C) ≤ t (A ∧ B). By (BC), f (A ∧ B, w) ⊆ |A ∧ B| = |B ∧ A| ⊇ f (B ∧

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A, w). By (PIE), then, f (A ∧ B, w) = f (B ∧ A, w). Now f (B ∧ A, w) = f (A ∧ B, w) ⊆ |C| and t (C) ≤ t (A ∧ B) = t (B ∧ A). Thus, by (T[·]), w  [B ∧ A]C.  But (PIE) also has more problematic consequences. Let us call two formulas A and A topically strictly equivalent, or T S-equivalent for short, i.e A ≈T S A , whenever T A ⇔ A and for all T -model, t (A) = t (A ). The first difficulty stems from the observation that IT licenses the substitution of T S-equivalents in all contexts: 

Proposition 4 If A ≈T S A , then T B ⇔ B AA . Proof By induction on the construction of B. Since A and A , by assumption, are true in the same possible worlds, the property is easily seen to hold whenever B is a propositional variable, a negation, a conjunction, a disjunction and a strict condi tional. The only remaining case is when B is of the form [C]D. Let us rewrite [C]D AA   as [C  ]D  . Then C  = C AA and D  = D AA . By the induction hypothesis, |C| = |C  | and |D| = |D  |. Let w be an arbitrary possible world from an arbitrary T -model M = W, {R A | A ∈ K }, T , ⊕, t, . By (BC), f (C, w) ⊆ |C| = |C  | ⊇ f (C  , w). By (PIE), then, f (C, w) = f (C  , w). Then f (C  , w) = f (C, w) ⊆ |D| = |D  |. By assumption, t (A) = t (A ). Therefore, t (D  ) = t (D) ≤ t (C) = t (C  ). Then by (T[·])   and (T⇒), w  [C]D ⇔ C  [D  ], i.e. w  [C]D ⇔ [C]D AA . In other words, the granularity of imagination, according to IT , is never finer than T S-equivalence. As the proof shows, (PIE) plays a decisive role in securing this substitution. The important question, then, is whether this relation of T S-equivalence provides the right measure of the grain of imagination. Now it is easy to see that (A ∨ B) ∧ (A ∨ C) ≈T S (A ∨ (B ∧ C)). It follows immediately from Proposition 4 that contents of this form are substitutable within the scope of a ceteris paribus imagination operator. Consider however the following case: There are rumors that Albert is in love, but no one really knows with whom. Close friends of his have at best partial information to offer. Bob claims that Albert is in love with Patricia or with Quiana. Carl contends that Albert is in love with Patricia or with Roberta. Zeno does not know how reliable Bob and Carl are. Although he does not rule out in principle the possibility that some people may be in love with several people at the same time, he assumes by default that people typically are in love with one person, and only one, at a time. He also believes, for independent reasons, that Patricia loves Albert. Thus, as Zeno imagines that Bob and Carl are both right, he imagines ceteris paribus that Albert loves Patricia and that his love will be requited. Now Daniel claims that Albert loves Patricia or that he loves both Quiana and Roberta. Furthermore, Zeno believes that Quiana and Roberta both hate Albert. Thus, as Zeno imagines that Daniel is right, he does not imagine ceteris paribus that Albert’s love will be requited. For it is consistent with what Zeno imagines that Albert may love both Quiana and Roberta, neither of whom will love him in return.

I take this little story to be logically consistent and Zeno’s exercises of imagination to be coherent and plausible within the context provided by the story.

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Let us define: • p := Albert loves Patricia, • q := Albert loves Quiana, • r := Albert loves Roberta, and • s := Albert’s love will be requited. Zeno’s imagination reports can then be formalized as follows: (6)

a. [( p ∨ q) ∧ ( p ∨ r )]s b. ¬[( p ∨ (q ∧ r )]s

Since the antecedents of the two imagination conditionals are T S-equivalent, (6-a) and (6-b) offer together a prima facie counterexample to Proposition 4. Perhaps the significance of the example could be disputed along the following lines. It seems that in (6-a), Zeno only considers possible worlds where Albert loves exactly one person, whereas in (6-b), he somehow also considers possible worlds where Albert loves two people. Are we not surreptitiously shifting the context here, having the assumption that Albert loves only one person constrain the way the input will be enriched in the former case, while removing that assumption in the latter case? It may be tempting to think that those two ways of enriching the input should be represented by two distinct selection functions, and thus by two distinct models. If this is so, then the intuitive data offered above cannot be used to build a counterexample to Proposition 4: it simply fails to show that there is a model where (6-a) and (6-b) are both true in the same possible world. This objection correctly points to an important feature of the example, i.e. that the respective evaluations of (6-a) and (6-b) presuppose that Zeno develops the two inputs in different directions, relying on different background assumptions. For sure, it would be illegitimate to import a background assumption p into the output of an act of imagination based on input A and block the very same assumption p when given input A if inputs A and A are the same, or equivalent as inputs. But whether inputs ( p ∨ q) ∧ ( p ∨ r ) and p ∨ (q ∧ r ), in the present case, are equivalent is precisely the point at issue. If the move consisting in distinguishing two models to account for the intuitive data simply assumes that those two contents are equivalent, or is based on the view that they should be taken to be equivalent because they are T S-equivalent, it simply begs the question. Furthermore, it seems independently compelling to treat ( p ∨ q) ∧ ( p ∨ r ) and p ∨ (q ∧ r ) as distinct inputs, each having, for good reason, different effects on the imaginative enrichment process. Remember that Zeno assumes by default that Albert loves at most one person, i.e. unless the possibility that he loves more than one person is made particularly salient. In the antecedent of (6-a), the possibility that Albert loves more than one person, although logically possible, is not explicitly presented as a live hypothesis. The default assumption is thus imported and constrains the imaginative enrichment, ruling out possible worlds where Albert loves more than one woman. In contrast, by explicitly having the conjunction of q and r as a subformula, the input p ∨ (q ∧ r ) makes it salient that Albert

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may love both Quiana and Roberta, and thus cancels the default assumption. Possible worlds where Albert loves more than one woman are then ruled in, which explains the difference between (6-a) and (6-b). At this point, it might still be objected that if the difference between (6-a) and (6-b) is really explained by the sudden salience of the possibility that Zeno loves two women, then the formalization of the imagination report described by (6-b) should rather be: (6b’)

¬[ p  ∧ ( p ∨ (q ∧ r ))]s

where p  :=Albert may love two women. Now, according to Berto himself, “an act of imagination (in a given context) is individuated by its explicit content” [2, p. 1292]. It is one thing for Zeno to explicitly imagine that Daniel is right, another to explicitly imagine that Daniel is right and that Albert may love more than one woman, since the second conjunct is not part of what Daniel says.4 Now it is stipulated in the scenario above that Zeno engages explicitly with the former content, not the latter. So (6b’) formalizes an imagination conditional describing a different act of imagination than the one reported in the scenario, and (6b) is the correct formalization of the conditional describing that act of imagination. On a more charitable reading, the objection might be taken to question the very plausibility of the report expressed by (6-b), as long as p  is not added to its antecedent. To this version of the objection, it may be replied that it is not necessary that a content be explicitly part of the input of an act of imagination in order to constrain the enrichment process. On the contrary, it is part of the very idea of ceteris paribus imagination, as introduced by Berto, that an input, in a particular context, may implicitly trigger the importation of relevant background information into the content of the imaginative act.5 So there is no necessity to insert p  within the antecedent of (6-b), if there is sufficient reason to expect, in the context provided by the scenario, that the assumption p  will be triggered by the explicit content of the input, so that the same output will be reached as a result. Now, in the context of the scenario, the initial assumption that Albert loves exactly one person, which explains (6a), has the status of a default assumption. The input p ∨ (q ∧ r ) makes the possibility that Albert may love both Quiana and Roberta salient and thus cancels the default assumption. It triggers instead the assumption that Albert may love more than one woman. Thus there is no need to insert p  in the antecedent of (6-b), since we have good reason to assume that p  will be imported as a result of the input of (6-b) in the context of the scenario. might point out that p  is implicit in what Daniel says, and I am prepared to agree. But what is merely implied or suggested by what someone says is not part of what is said. So when Zeno imagines that Daniel is right, he takes as input of his act of imagination that what is said by Daniel is true, which does not contain p  . 5 For instance, Berto emphasizes that “different acts of imagining the same explicit content can trigger the import of different background information depending on contexts (the time and place at which the cognitive agent performs the act, the status of its background information, etc.)” [2, p. 1287]. 4 One

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Although the complexity of the story and of the imagination reports it contains invite to consider the counterexample offered by (6-a) and (6-b) with some prudence, I think this case, under the analysis just proposed, at least indicates how topically strictly equivalent contents could play non-equivalent roles in ceteris paribus imagination reports. Even though this hardly counts as a knock-down argument, it already suggests a potential difficulty for Berto’s topics-based semantics. There are however other, and perhaps clearer, counter-examples. One of them relates to another feature of IT , which is also a consequence of (PIE): Proposition 5 If T A ⇔ A and V(A) ⊆ V(A ), then [A]B T [A ]B. Proof Suppose T A ⇔ A and V(A) ⊆ V(A ). Let w be a possible world from a T -model M = W, {R A : A ∈ K }, T , ⊕, t,  such that w  [A]B. By (BC), f (A, w) ⊆ |A| = |A | ⊇ f (A , w). By (PIE), f (A, w) = f (A , w). By (T[·]), we have f (A, w) ⊆ |B| and t (B) ≤ t (A). It follows that f (A , w) ⊆ |B|. Since V(A) ⊆ V(A ), t (A) ≤ t (A ). Then t (B) ≤ t (A) ≤ t (A ) and by (T[·]) we have w  [A ]B.  Now consider the following example: Ted is a person who eats well, goes to the gym twice a week and sleeps eight hours every night. The only thing that prevents his lifestyle from being really healthy is that he drinks a whole bottle of Cognac every day for breakfast. As I imagine that Ted does not drink alcohol, I imagine ceteris paribus that he lives a healthy life. But as I imagine that Ted does not drink alcohol, or does not drink alcohol but takes heroin, I do not imagine ceteris paribus that he lives a healthy life. For it is consistent with what I imagine that he may take heroin and this is not healthy at all.

Again, this little story seems logically consistent and the two imagination reports seem coherent and plausible within the context of the scenario. Now let: • p := Ted does not drink alcohol, • q := Ted lives a healthy life, • r := Ted takes heroin The imagination reports in the scenario above can be formalized, respectively, as: (7)

a. [ p]q. b. ¬[ p ∨ ( p ∧ r )]q.

Since p and p ∨ ( p ∧ r ) are strictly equivalent, and V( p) ⊆ V( p ∨ ( p ∧ r )), it follows from Proposition 5 that the argument from (7-a) to (8)

[ p ∨ ( p ∧ r )]q

is valid. However the act of imagination described by (8) is highly counterintuitive given the background facts. Furthermore, if that inference were logically valid, it would make the story above logically inconsistent, which hardly seems to be the case.

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Let us briefly pause and try to understand what has gone wrong with IT . Since (PIE) has the function of reducing the hyperintensionality of the logic and plays a crucial role in the derivation of the problematic Propositions 4 and 5, it is natural to suspect (PIE) to be the main source of these hyperintensionality issues, and to conjecture that they can be solved by removing (PIE) and replacing it with a weaker principle. As a matter of fact, Berto has independently identified another potentially problematic consequence of (PIE), which forces the following schema: Proposition 6 [A]B, [A ∧ B]C T [A]C. Berto, however, has some doubts about its adequacy. …there may be counterexamples around. The issue has to do with cases where C easily pops to mind given B alone, but is only dimly related to A—for then [Proposition 6] (acting a bit like a Cut rule in a logical calculus) washes the bridging B away in the conclusion. Here’s a situation suggested by Claudio Calosi, that may do. [A]B: given the input that I am wearing a red shirt in Pamplona, I imagine that I am being chased by bulls. [A ∧ B]C: given the input that I am being chased by bulls on the streets of Pamplona while wearing a red shirt, I imagine that I die on the street. But it’s not the case that [A]C: Given that I am wearing a red shirt in Pamplona, I don’t imagine that I die on its streets. [4, p. 13]

Looking for a weaker “principle of imaginative equivalents” therefore may look like a promising strategy to solve the issues we have identified about hyperintensionality, as well as other issues identified independently by Berto himself. However, there are reasons to think that such a strategy will not succeed. For Berto’s semantics faces yet another difficulty with hyperintensionality, which is however independent of (PIE). This difficulty shows up in the treatment of counterpossible imagination. Berto’s semantics validates the following schema: Proposition 7 T [A ∧ ¬A]B ⇔ [A ∧ ¬A]¬B Proof Let w and w be two arbitrary possible worlds from an arbitrary T -model M = W, {R A : A ∈ K }, T , ⊕, t, . Suppose that w  [A ∧ ¬A]B. By (T[·]) it follows that (i) f (A ∧ ¬A, w ) ⊆ |B| and (ii) t (B) ≤ t (A ∧ ¬A). Now, by (BC) f (A ∧ ¬A, w ) ⊆ |A ∧ ¬A| = ∅. It follows that f (A ∧ ¬A, w ) = ∅. Then, vacuously, f (A ∧ ¬A, w ) ⊆ |¬B|. Now t (¬B) = t (B) ≤ t (A ∧ ¬A). It follows by (T[·]) that w  [A ∧ ¬A]¬B. By (T⇒), w  [A ∧ ¬A]B ⇒ [A ∧ ¬A]¬B. Thus  [A ∧ ¬A]B ⇒ [A ∧ ¬A]¬B. A symmetrical reasoning shows that the converse strict implication holds.  But this seems like a wrong result. Suppose I am reading the story about Sylvan’s Box as told in [17]. Sylvan’s Box is there described as a box belonging to the late Richard Sylvan and having the outstanding feature of being both empty and nonempty. Now consider the following imagination report: (9)

As I imagine that Sylvan’s Box is both empty and nonempty, I imagine, ceteris paribus, that it belongs to Richard Sylvan.

That report seems coherent and plausible, given the context. According to Proposition 7, however, it is logically equivalent to:

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As I imagine that Sylvan’s Box is both empty and nonempty, I imagine ceteris paribus that it does not belong to Richard Sylvan.

But this seems patently absurd. Why should the inconsistency in the description of Sylvan’s box force us to imagine only contradictory things about it? Why can’t I simply imagine that it belongs to Richard Sylvan simpliciter? The only reason to accept such an equivalence would be to endorse the view that counterpossible imagination is trivial. This view, however has already been rejected and rightly so. It forces us to treat all inconsistent inputs as equivalent. Then anything would be implicitly imagined on the basis of an impossible input, regardless of any topical connection. But when I imagine that Terence Tao squares the circle, I certainly do not imagine, ceteris paribus, that Donald Trump is a transgender woman. The constraint of topic preservation between the input and the output however does not offer the right solution to this problem. The properties of topics, in particular the fact that t (A) = t (¬A), make this constraint insufficient in this case, as the intuitive non-equivalence of (9) and (10) clearly show. Given that (PIE) plays no role in the proof of Proposition 7, it follows that weakening (PIE) or even removing it altogether from the definition of a T -model, will not clear all the difficulties pertaining to hyperintensionality. Some of these difficulties, at least, should be attributed to a deeper cause. Either topic preservation is insufficient to capture the hyperintensional link between antecedents and consequents of imagination conditionals, or the very idea of treating imagination conditionals as variably strict conditionals on a space of possible worlds is misguided. Either way, some drastic changes have to be made if one wants to have an adequate treatment of the hyperintensionality of imagination.

4 Impossible Worlds Berto has offered an alternative impossible worlds semantics for the language L I [2, 5]. While this semantics preserves the attractive features of IT , Berto insists that it is more “flexible” [3, p. 1884] than the topics-based semantics, in a sense that we will clarify shortly. It is then natural to expect that the introduction of impossible worlds will allow to account for the hyperintensional distinctions we identified in the previous section. An impossible worlds frame, or I W -frame for short, is a triple P, I, {R A : A ∈ K } where P is a non-empty set of possible worlds, I is a (possibly empty) set of impossible worlds, P ∩ I = ∅, and {R A : A ∈ K } is a set of accessibility relations on the set of all possible and impossible worlds W = P ∪ I , i.e. for all A ∈ K , RA ⊆ W × W . An impossible worlds model, or I W -model for short, is a quintuple P, I, {R A : A ∈ K }, , - where P, I, {R A : A ∈ K } is an I W -frame, while  and - are relations between worlds and formulas, corresponding respectively to verification and falsification, such that for all w ∈ P:

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(IW¬+ ) w  ¬A iff w - A (IW¬− ) w - ¬A iff w  A (IW∧+ ) w  A ∧ B iff w  A and w  B (IW∧− ) w - A ∧ B iff w - A or w - B (IW∨+ ) w  A ∨ B iff w  A or w  B (IW∨− ) w - A ∨ B iff w - A and w - B (IW⇒+ ) w  A ⇒ B iff w  B, for all w ∈ P such that w  A (IW⇒− ) w - A ⇒ B iff w - B for some w ∈ P such that w  A (IW[·]+ ) w  [A]B iff w  B for all w ∈ W such that w R A w (IW[·]− ) w - [A]B iff w - B for some w ∈ W such that w R A w (CC1) w  A or w - A (CC2) it is not the case that w  A and w - A. Verification and falsification are distinguished in order to have “gappy” worlds, where a formula can be neither true nor false, as well as “glutty” worlds, where a formula can be both true and false. For the same reason they are modeled as relations between worlds and formulas, rather than mappings from worlds and formulas to truth-values. In virtue of conditions (CC1) and (CC2), possible worlds behave classically from a logical point of view. All the tautologies of classical logic are true at all possible worlds, and the set of formulas that are true in a possible world is closed under classical logical consequence. Impossible worlds, in contrast, are completely anarchic. Each formula is treated there as an atom, with no recursive condition whatsoever on its verification or falsification by a world. It follows that for any formula A, there are impossible worlds w and w such that w  A and w - A. Thus, one will find impossible worlds where (classical) logical truths are falsified, e.g. A ∨ ¬A and impossible worlds where contradictions are true. Also, for each set  of formulas and each formula A, there is an impossible world such that w  B for all B ∈  and w  A. There are impossible worlds, thus, where A ∧ B is verified even though A is not. Even worse, for each set of formulas  and formula A, there is an impossible world such that w  B for all B ∈  and w - A. There are impossible worlds where A ∧ B is true while A is false. Such impossible worlds are sometimes called open worlds [5, 18]. The main motivation for this feature is that it allows to account for the logical imperfection of non-ideal cognitive agents. This logical imperfection is indeed one source of the hyperintensionality of imagination reports. When one imagines A to be the case, while not imagining B to be the case, while B is logically equivalent to A, one imagines an impossible world where A is true but B is false. This is precisely what the semantical clause for the imagination operator allows to model, since it quantifies on all worlds, and not just on possible ones. As in the previous semantics, we can rephrase the clause for the imagination operator in terms of a selection function f (A, w) := {w ∈ W : w R A w }

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except that W now includes impossible as well as possible worlds. We now denote by ||A|| the set of worlds, possible and impossible, where A is verified. Thus: (IW[·]+ ’) (IW[·]− ’)

w  [A]B iff f (A, w) ⊆ ||B||. w - [A]B iff f (A, w) ∩ ||¬B|| = ∅.

For the ceteris paribus operator to behave properly, Berto has to impose a number of additional conditions: (IW1) (IW2) (IW3) (IW4)

f (A, w) ⊆ ||A|| if f (A, w) ⊆ ||B ∧ C||, then f (A, w) ⊆ ||B|| and f (A, w) ⊆ ||C|| if f (A, w) ⊆ ||B|| and f (A, w) ⊆ ||C||, then f (A, w) ⊆ ||B ∧ C|| if f (A, w) ⊆ ||B|| and f (B, w) ⊆ ||A||, then f (A, w) = f (B, w).

(IW1) plays the same role as the Basic Constraint in the previous semantics, while (IW4) plays the same role as the Principle of Imaginative Equivalents. The main difference is that (IW4) is formulated in terms of possible and impossible worlds, while (PIE) only considered possible worlds. This will turn out to make a big difference on the treatment of hyperintensionality. (IW2) and (IW3) need to be added to ensure that the resulting logic has the validities expressing the mereology of imagination. Validity in a model, validity and logical consequence are defined as follows: • For all IW -model M = P, I, {R A : A ∈ K }, , -, M  I W A iff w  A for all w ∈ P. •  I W A iff M  I W A for all I W -model M. •   I W A iff for all IW -model M = P, I, {R A : A ∈ K }, , -, and all w ∈ P, w  A if w  B for all B ∈ . We call IIW the logic determined by the class of I W -models, i.e. IIW = {A ∈ L I :  I W A}. This semantics is designed to provide the same attractive features as the topicsbased semantics. It secures the following validities: 1. 2. 3. 4.

 I W [A]A for all A ∈ K (success) [A](B ∧ C)  I W [A]B (left simplification) [A](B ∧ C)  I W [A]C (right simplification) [A]B, [A]C  I W [A](B ∧ C) (adjunction)

and invalidities: 1. 2. 3. 4. 5. 6.

[A]B  I W [A ∧ C]B (non-monotonicity) [A](B ∨ C)  I W [A]B ∨ [A]C (indeterminacy) [A]B  I W [A](B ∨ C) (relevance) It is not the case that: if  I W B, then  I W [A]B (no logical omniscience) A ⇒ B  I W [A]B (no imaginative entailment)  I W [A ∧ ¬A]B (non-vacuity).6

6 As

in the previous section, we do not reproduce the proofs that are already given by Berto in [2], unless they are particularly informative.

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So far so good. The sense in which the impossible worlds framework is more “flexible” is that the conditions on models that are responsible for particular validities can be manipulated independently. This is especially true for the mereology of imagination. For example, left simplification and right simplification follow directly from the condition (IW2) on models. Similarly, adjunction follows from (IW3). Thus if, for some reason, one comes to be dissatisfied with adjunction, one can selectively remove (IW3) from the definition of an I W -model, and keep left and right simplification untouched. Like in the topics-based semantics, the key element governing hyperintensionality is the principle of imaginative equivalents expressed by (IW4). Since it is defined in terms of sets of possible and impossible worlds, the conditions required for f (A, w) to be the same as f (B, w) are stronger, and so the principle as a whole is weaker. As a matter of fact, this weaker principle does allow to draw the hyperintensional distinctions that the topics-based semantics could not: Proposition 8 (i)  I W [A ∨ (B ∧ C)]D ⇔ [(A ∨ B) ∧ (A ∨ C)]D (ii) [A]B  I W [A ∨ (A ∧ C)]B (iii)  I W [A ∧ ¬A]B ⇔ [A ∧ ¬A]¬B Proof We provide a countermodel for each invalidity.7 (i)

We have w1  [( p ∨ q) ∧ ( p ∨ r )] p and w1  [ p ∨ (q ∧ r )] p. A fortiori, w1  [ p ∨ (q ∧ r )] p ⇔ [( p ∨ q) ∧ ( p ∨ r )] p. (ii)

We have w1  [ p]q and w1  [ p ∨ ( p ∧ r )]q. 7 Each node represents a world identified by its name. Names of impossible worlds are followed by a

star. Under each possible world, we indicate the propositional variables that are verified and falsified respectively. All the other formulas that are verified and falsified, respectively, can be deduced with the help of the recursive conditions given in the definition of an I W -model. For impossible worlds, we also indicate verified and falsified non atomic formulas when they cannot be obtained from propositional variables by condition (IW3) and leave implicit those that can be so obtained. An arrow labeled with formula A between worlds w and w indicates that w R A w . It can be checked that the represented models satisfy all the conditions in the definition of an I W -model.

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(iii)

We have w1  [ p ∧ ¬ p]q and w1  [ p ∧ ¬ p]¬q. A fortiori, w1  [ p ∧ ¬ p]q ⇔ [ p ∧ ¬ p]¬q.  Thus the logic of I W -models is fine-grained enough to allow for the hyperintensional distinctions that the previous semantics could not draw. However, it still admits the validity deemed problematic in light of Calosi’s example: Proposition 9 [A]B, [A ∧ B]C  I W [A]C Proof Let M = P, I, {R A : A ∈ K }, , - be an I W -model and let w1 ∈ P be such that w1  [A]B and w1  [A ∧ B]C. Now let w2 be any world such that w1 R A w2 . By (IW1), w2  A and by (IW[·]+ ), w2  B. By (IW3), w2  A ∧ B. Now f (A, w1 ) ⊆ ||A ∧ B|| and, by (IW2), f (A ∧ B, w1 ) ⊆ ||A||. By (IW4), f (A, w1 ) = f (A ∧ B, w1 ). By assumption and (IW[·]+ ), f (A ∧ B, w1 ) ⊆ ||C||. It follows, again  by (IW[·]+ ), that w1  [A]C. The proof is instructive as it shows that all four conditions (IW1)–(IW4) are necessary to get the problematic result. This is where the flexibility of the impossible worlds framework pays off. In order to avoid this result, one is free to choose which condition to remove. While it seems out of the question to give up on (IW1), given the interpretation of ceteris paribus imagination that we have in mind, one may consider removing (IW2) or (IW3) instead of the principle of imaginative equivalents (IW4). Those at least are options considered by Berto [4, p. 13]. However, it is hard to find any independent reason, based on the notion of ceteris paribus imagination, to remove any of these two conditions. Furthermore, the resulting logic would lose either left and right simplification or adjunction. In any case, the “mereology” of imagination, which independently appears as an attractive feature of the logic, is lost. For this reason, the impossible worlds semantics is less than fully satisfactory, even though it correctly accounts for the hyperintensional distinctions that the topicsbased semantics could not draw. There is however another difficulty. While the topics-based semantics failed to draw some intuitive hyperintensional distinctions, this impossible worlds semantics now draws some problematic hyperintensional distinctions. We argued above that we should expect inputs of the form A ∧ B and B ∧ A to be equivalent. For exactly the same reason, one should expect A ∨ B and B ∨ A to be equivalent as inputs, since the order of the disjuncts in the input should not matter for what we imagine as an output. However, the impossible worlds semantics draws a hyperintensional distinction in the latter case, but not in the first:

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Proposition 10 (i)  I W [A ∧ B]C ⇔ [B ∧ A]C (ii)  I W [A ∨ B]C ⇔ [B ∨ A]C. Proof (i) Let M = P, I, {R A : A ∈ K }, , - be an I W -model and let w ∈ P be a possible world. By (IW2), f (A ∧ B, w) ⊆ ||A|| and f (A ∧ B, w) ⊆ ||B||. By (IW3) f (A ∧ B, w) ⊆ ||B ∧ A||. By a symmetrical reasoning, f (B ∧ A, w) ⊆ ||A ∧ B||. By (IW4), f (A ∧ B, w) = f (B ∧ A, w). Then, by (IW[·]+ ) and (IW⇒+ ), w  [A ∧ B]C ⇔ [B ∧ A]C. (ii) Consider the following countermodel:

We have w1  [ p ∨ q] p and w1  [q ∨ p] p. A fortiori, w1  [ p ∨ q] p ⇔ [q ∨ p] p.  Now this situation is rather puzzling. It seems that whatever reason we have to treat A ∧ B and B ∧ A as equivalent inputs applies equally to A ∨ B and B ∨ A, and symmetrically, that whatever reason we have to distinguish A ∨ B from B ∨ A as inputs applies to A ∧ B and B ∧ A. Why should the order of disjuncts matter and the order of conjuncts not? Thanks to the flexibility of the impossible worlds semantics, one may fix this by adding the following condition to the definition of an I W -model: (IW5)

f (A ∨ B, w) = f (B ∨ A, w).8

While this addition is enough to make inputs of the form A ∨ B and B ∨ A, equivalent, it still allows for further unintuitive hyperintensional distinctions. For this modified semantics still treats inputs of the form A and A ∧ A as equivalent, while it does not treat A and A ∨ A as equivalent inputs. Proposition 11 For all I W -model M satisfying (IW5): (i) M  I W [A]B ⇔ [A ∧ A]B (ii) M  I W [A]B ⇔ [A ∨ A]B Proof (i) Let M = P, I, {R A : A ∈ K }, , - be an I W -model satisfying (IW5) and let w ∈ P be a possible world. By (IW1), f (A, w) ⊆ ||A||. By (IW3), f (A, w) ⊆ ||A ∧ A||. By (IW2), f (A ∧ A, w) ⊆ ||A||. By (IW4), f (A, w) = f (A ∧ A, w). Then, by (IW[·]+ ) and (IW⇒+ ), w  [A]B ⇔ [A ∧ A]B. 8I

am indebted to an anonymous reviewer for bringing this solution to my attention.

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(ii) Consider the following counter-model, which satisfies (IW5):

We have w1  [ p]q and w1  [ p ∨ p]q. A fortiori, w1  [ p]q ⇔ [ p ∨ p]q.



Once again, such an asymmetry between conjunction and disjunction seems prima facie unmotivated and unwelcome. Once again, thanks to the flexibility of the impossible worlds semantics it can easily be fixed by the addition of a further condition on IW -models: (IW6)

f (A, w) = f (A ∨ A, w)

Is this enough to remove all the problematic hyperintensional distinctions predicted by the official impossible worlds semantics of [2] and [5]? Let us assume, for the sake of discussion that it does or that it is possible to arrive at an adequate treatment of the hyperintensionality of imagination in this way, by the further addition of appropriate constraints on I W -models in response to new problematic hyperintensional distinctions. In either case, it is hard to see the resulting treatment as anything more than a series of ad hoc adjustments. In the official story told in [2] and [5], (IW4) is the condition on I W -models that is essentially responsible for the limitation of hyperintensionality. The rationale behind it is that if I imagine B on the basis of input A and B on the basis of input A, then A and B are equivalent as inputs. What Propositions 10 and 11 show, then, is that (IW4) does not account for all the required limitation of hyperintensionality. This is a consequence of the fact that the space of impossible worlds contains strange worlds where A ∨ B is true but not B ∨ A, or where A is true but not A ∨ A, and that (IW1)–(IW4) do not rule out the possibility that such worlds be selected on the basis of some inputs. This leaves the further limitations of hyperintensionality achieved by (IW5) and (IW6) without a principled motivation. It does not follow, of course, that it is absolutely impossible to define a well-motivated set of constraints on I W -models that entails (IW1)–(IW6) and provides an adequate account of hyperintensionality in imagination reports. But I think it is fair to point out that we find little indication of how to do this in the official impossible world semantics of [2] and [5]. Let us take stock. We saw that the logic of ceteris paribus imagination determined by the topics-based semantics of [3] is not hyperintensional enough. Now the official impossible worlds semantics of [2, 5] appears to be too hyperintensional. While the flexibility of this approach may allow to remove those excesses of hyperintensionality by the addition of further ad hoc constraints on models, it is not clear how to do this in a principled way.

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5 A Truthmaker Semantics In this section, I describe an alternative semantics for the language L I that preserves the attractive features of Berto’s logics of ceteris paribus imagination, while providing, in a principled way, the right amount of hyperintensionality–or so I shall argue. To do so I will leave aside possible and impossible worlds semantics and work instead within the framework of truthmaker semantics, as recently elaborated by Fine in [10–12]. There are two main motivations for this, one technical, one philosophical. On the technical side, truthmaker semantics proposes to evaluate sentences with respect to partial situations, rather than complete worlds. Those situations have a mereological structure, which enables one to say that a situation s verifies a sentence A exactly, if the whole of s is relevant to the truth of A, or inexactly if at least some part of s is so relevant. On the one hand, the relation of exact verification allows to make some fine-grained hyperintensional distinctions, for example between A and A ∨ (A ∧ B), but also between (A ∨ B) ∧ (A ∨ C) and A ∨ (B ∧ C). On the other hand, truthmaker semantics, at least on the version considered here, allows us to introduce impossible situations. But unlike Berto’s impossible worlds, impossible situations are not logically anarchic and comply with uniform principles governing the logical connectives. For instance, any situation, possible or otherwise, that exactly verifies A ∨ B also exactly verifies B ∨ A. Thus it looks like truthmaker semantics offers promising resources to deal with the problematic cases that caused trouble for Berto’s topics-based and impossible-world-based approaches. The second motivation is more philosophical. Some philosophers trying to get clear on the nature of imagination have indeed proposed the view that when I imagine that p, I imagine a situation s that verifies p [6, 20]. Here is Chalmers’ statement of the view: When one imagines a situation and reasons about it, the object of one’s imagination is often revealed as a situation in which S is this case, for some S. When this is so, we can say that the imagined situation verifies S, and that one has imagined that S. [6, p. 150]

A situation, here, is meant to be “(roughly) a configuration of objects and properties within a world” [6, p. 151]. Here it is made clear that situations, as opposed to worlds, are partial. Moreover, it is natural to think that those situations stand in inclusion or parthood relations with one another. For example, I can imagine a whole situation and then decide to focus only on a part of it. Some sentences verified by the larger situation may not be verified by the smaller one. Truthmaker semantics then looks like a hospitable environment where to develop those ideas rigorously. This way of thinking about imagination is independently attractive for a number of reasons. First, it fits the phenomenology of imagination, which relates us to partial situations, or imaginary scenes, rather than entire possible worlds. Second, it gives an elegant picture of the intentionality of imagination: acts of imagination have both an intentional object, a situation, and an intentional content, a proposition, and both are connected by the relation of verification. Third, the mereological properties of imag-

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ination identified above can be readily explained by reference to the mereological properties of the imagined situation itself. This philosophical view of imagination can easily be adapted to the framework of ceteris paribus imagination. I imagine ceteris paribus that B, as I imagine explicitly that A, if the situation that I come up with when I try to imagine that A also verifies B. So there is no difficulty in using the insights of Chalmers and Yablo to model ceteris paribus imagination. Let us describe in more detail the particular truthmaker-semantical framework we will use. First of all, we will work within a modalized space of situations, containing possible as well as impossible situations, since impossible situations will be useful to model counterpossible imagination. A modalized situation space can be represented by a triple S, S ♦ , , where S represents the set of all situations, S ♦ ⊆ S represents the set of possible situations and  is a partial order on S, representing the relation of parthood between situations, and the following conditions hold: (MSS1) For any X ⊆ S, there is a least upper bound X ∈ S of X , i.e. each set of situations has a mereological fusion. (MSS2) If s ∈ S ♦ and t  s, then t ∈ S ♦ , i.e. each part of a possible situation is possible. Although the fusion of two arbitrarily chosen situations always exists, it does not follow that the fusion of two possible situations is always possible. Parts of possible situations, however, are always possible. Once such a modalized situation space is given, a relation of compatibility between situations can be defined as follows: (Comp) if S is a set of situations, the situations in S are compatible iff there is a possible situation u such that s  u for all s ∈ S. Otherwise they are incompatible. The notion of possible world can also be defined as follows: (PW) a situation w is a possible world iff for all situation s, if s is compatible with w, then s  w. It is then natural to add the following condition to the definition of a modalized situation space: (MSS3)

For all s ∈ S ♦ , there is a possible world w such that s  w.

Thus given a modalized situation space S, S ♦ ,  we can define a set W containing all the possible worlds in that space. Second, there is indeed a huge difference, for a partial situation, between not verifying a sentence and falsifying it, just like there is a huge difference between not falsifying a sentence and verifying it. For example a situation where Alice is sitting alone, neither verifies nor falsifies the sentence: “Bob is standing”. From this point of view, it is plausible to take a partial situation s to verify the negation of a sentence A just in case it falsifies A. If so, we will need sentences to be related to situations by two independent semantical notions of verification and falsification. The proposition expressed by a sentence will then be taken to be the pair of its verifiers and falsifiers.

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Third, the verification and falsification relations between situations and sentences can be understood in various ways. Following [12], I will separate exact, inexact and classical notions of verification and falsification respectively. A situation exactly verifies a sentence A, which we note s  A, just in case it is wholly relevant to the truth of A. In other words, it has no part that is not relevant to the truth of A. For example a situation t where the temperature is hot and the sun shines exactly verifies the sentence (11)

It is hot and sunny

but it does not exactly verify the sentence (12)

It is hot

for the sunny part of t is irrelevant to the truth of (12). Symmetrically, a situation s exactly falsifies a sentence A just in case it is wholly relevant to the falsity of A: it has no part that is not relevant to the falsity of A. Hence t above exactly falsifies (13)

It is neither hot nor sunny

but not (14)

It is not sunny.

We denote by ||A||+ the set of exact verifiers of A and by ||A||− the set of its exact falsifiers. Exact verification and falsification are importantly non persistent, in the sense that if s  s  , then s  A does not entail s   A, and s - A does not entail s  - A. They are also compositional in the sense that the exact verifiers and falsifiers of a complex sentence can be computed from the exact verifiers and falsifiers of its parts: (¬+ ) (¬− ) (∧+ ) (∧− ) t(∨+ ) t (∨− )

s  ¬A iff s - A s - ¬A iff s  A s  A ∧ B iff there are t, u ∈ S such that s = t  u and t  A and u  B s - A ∧ B iff s - A or s - B or there are t, u ∈ S such that s = t  u and A and u - B s  A ∨ B iff s  A or s  B or there are t, u ∈ S such that s = t  u and A and u  B s - A ∨ B iff there are t, u ∈ S such that s = t  u and t - A and u - B.

A situation s inexactly verifies a sentence A, i.e. s ∣> A, just in case it is at least partly relevant to the truth of A. In other words, s is enough to make A true, but it may be more than enough. For example t above inexactly verifies (12). Symmetrically, ∣ A, just in case it makes A false and is a situation s inexactly falsifies A, i.e. s A:= there is a situation t such that t  s and t  A ∣ A := there is a situation t such that t  s and t - A. • s A iff all possible situation inexactly verifying all the members of  inexactly verifies A. • A is a classical consequence of , i.e.   A iff all possible situation classically verifying all the members of  classically verifies A. Similarly, one can define binary relations of exact, inexact and classical equivalence as, respectively, exact, inexact and classical consequence in both directions. It is then not difficult to see, for example that A and A ∨ (A ∧ B) are not in general exactly equivalent, although they are classically equivalent. For the following exact verifier s of p ∨ ( p ∧ q) does not exactly verify p 10 :

9 This definition characterizes a distributive notion of exact consequence which requires that all possible situation exactly verifying all the premises individually also verifies the conclusion. One may also consider a collective understanding of exact consequence which requires that all possible situations exactly verifying the conjunction of all the premises also verifies it conclusion. Unless otherwise specified, by exact consequence we will understand hereafter exact consequence in the distributive sense. For a recent study of exact consequence in the distributive sense see [14]. 10 In the two diagrams below, labeled nodes represent situations. Downward straight lines between situations represent the relation of parthood. Atomic formulas exactly verified by situations are indicated below or above the nodes representing them. Unless they are particularly relevant to the example, we do not indicate the complex formulas exactly verified or falsified by a situation when they can be computed from the atomic formulas it exactly verifies or falsifies, via the recursive clauses given above.

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Similarly, (A ∨ B) ∧ (A ∨ C) and A ∨ (B ∧ C) are not in general exactly equivalent. For the following exact verifier s of ( p ∨ q) ∧ ( p ∨ r ) does not exactly verify p ∨ (q ∧ r ):

In such a framework, it is also possible to define the topic τ (A) of a sentence A as the fusion of its exact verifiers and exact falsifiers, i.e. τ (A) = (||A||+ ∪ ||A||− ). 11 Just like in Berto’s approach to topics, it naturally follows from this definition that τ (A) = τ (¬A) and that τ (A ∧ B) = τ (A ∨ B) = τ (A)  τ (B). We can then express topic inclusion or preservation directly with the relation , without adding a new primitive notion. Thus the topic τ (A) of A is included in the topic τ (B) of B just in case τ (A)  τ (B). It is then easy to see, e.g., that τ (A)  τ (A ∧ B). We now have all the ingredients we need to propose a fine-grained modeling of ceteris paribus imagination. It should be made clear, before we dive into the details, that partial situations will be used only to analyze the contents of acts of imagination. The truth of sentences will be evaluated in possible worlds, so that the overall propositional logic remains classical. It is only when we look inside the scope of the imagination operator that partial situations, and the relations of exact and inexact verification and falsification start to play a role. Acts of imagination will then be modeled with the help of a selection function that takes as argument a formula A and a possible world w and yields a set of situations f (A, w). As before, A corresponds to the explicit input of the exercise of imagination while w, the possible world in which the act of imagination is evaluated, provides the context for the act. The set f (A, w) encodes the implicit content of the act of imagination performed on the basis of A in the context of world w. 11 Defending

this particular definition of topic goes way beyond the purpose of this paper. See however [13] for a detailed defense of this account.

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Since the content of the input has to be preserved in the content of the output, it is required that all the members of f (A, s) inexactly verify A. The relevant notion here is the inexact sort of verification, since those situations may add further components that are not directly relevant to the truth of A itself, but are relevant to the ways A can be made true, given some background beliefs or contextually salient assumptions. For example it may not be itself relevant for the truth of (15)

Sherlock Holmes is smoking alone at home

that he smokes tobacco rather than opium or crack, or that he lives in a small apartment, rather than a big mansion. Yet those extra components may be part of the situation one spontaneously imagines when one imagines Sherlock Holmes smoking alone at home. The basic idea of the proposed truthmaker semantics of imagination is then that a possible world w classically verifies [A]B only if (i) B is inexactly verified by all the members of f (A, s). Since a disjunction B ∨ C is always inexactly verified by an inexact verifier of B, this necessary condition is not sufficient to ensure that the resulting logic will have the relevance property observed above. So we need to add the additional condition that (ii) the topic of B is part of the topic of A. Thus we end up with a semantical clause for the imagination conditional operator that closely mirrors the clause in Berto’s topics-based semantics, but importantly quantifies on partial situations rather than possible worlds. It might be wondered why we have a set of situations rather than a unique situation as a value for our selection function. For Yablo and Chalmers seem to talk as if the object of an act of imagination is a singular situation rather than a set of situations. The reason for this set-theoretic treatment is related to the issue of determinacy. As Yablo himselfs notes, the contents of our imaginings are “more or less determinate” [20, p. 27]. On the one hand, the situation that is the object of my explicit imagining is usually not represented in full specificity. I can imagine that I am very rich, without necessarily imagining exactly how many euros I have on my bank account. Yet, it does not follow that I take this situation to be metaphysically indeterminate with respect to that amount. Assuming the contrary would be making the content of my imagination much weirder that it is. What I explicitly imagine is not fully determinate, but I implicitly imagine it as fully determinate. In this kind of cases, to the indeterminately imagined situation will correspond a number of fully determinate situations. The job of the selection function is then to collect such situations within a set, as a function of the input and the context. Then it will be true that as I imagine that I am very rich, I imagine that I can afford to buy a Steinway grand piano, if the determinate amount on my bank account in all the determinate situations collected by the selection function allow me to buy a Steinway grand. In order to account for this particular mixture of determinacy and indeterminacy in the content of our imaginings, a set-valued selection function is exactly what we need. The framework of truthmaker semantics and the general idea behind the semantics for the imagination operator should now be clear enough.

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A T M-frame F is a quadruple S, S ♦ , , f  where S, S ♦ ,  is a modalized situation space and f : K × W → P(S) is a function mapping ordered pairs A, w, consisting of an entertainable formula taken from K and a possible world, to sets of situations. A T M-model M is a triple F, || · ||+ , || · ||−  where: 1. F is a T M-frame, 2. || · ||+ is a mapping from V to P(S) such that for all p ∈ V, || p||+ is closed under fusion, 3. || · ||− is a mapping from V to P(S) such that for all p ∈ V, || p||− is closed under fusion, (Exc) for all propositional variable p, no member of || p||+ is compatible with a member of || p||− , (Exh) for all propositional variable p, any possible situation is compatible with a member of || p||+ or with a member of || p||− . Given a T M-model M, the notion of exact verification extends to all formulas of the boolean fragment of the language according to the recursive clauses (¬+ ), (¬− ), (∧+ ), (∧− ), (∨+ ) and (∨− ) stated above. It is not difficult to prove that (Exc), (Exh) and the closure under fusion of exact verifiers and falsifiers, respectively, extend to all formulas of the boolean fragment of the language. (Exc) ensures that possible worlds are consistent, (Exh) makes sure that possible worlds are complete in the sense that for any possible world w and any sentence A, A will be either true or false in w. The closure under fusion of exact verifiers and falsifiers yields for that fragment of the language an inclusive truthmaker semantics [12, p. 563], so that for example ||A||+ = ||A ∧ A||+ , which would not be guaranteed otherwise. Given a T M-model M = F, || · ||+ , || · ||− , we can also define recursively the classical truth conditions of a formula in a given world w: (TMp) (TM¬) (TM∧) (TM∨) (TM⇒) (TM[·])

w  p iff s  w for some s ∈ || p||+ w  ¬A iff w  A w  A ∧ B iff w  A and w  B w  A ∨ B iff w  A or w  B w  A ⇒ B iff for all w ∈ W , if w  A, then w  B w  [A]B iff f (A, w) ⊆ |B|+ and τ (B)  τ (A).

The logic of the boolean fragment of L I will thus be classical propositional logic and the logic of the modal fragment of L I will be S5. The reader will have noticed that the strict conditional and the imagination operator only receive classical verification conditions, but no exact verification conditions. For sure, it would have been simpler and more elegant to give exact verification conditions for all the connectives and operators and then derive their classical verification conditions from the bottom up. The problem is that assigning exact verification conditions to the strict conditional is a rather difficult matter. For on the one hand, the exact verification relation is only sensitive to the intrinsic features of situations. Yet, on the other hand, the truth of a strict conditional statement A ⇒ B seems to say

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something about what happens in all other possible situations classically verifying A, namely that they also classically verify B. So it is unclear what properties of a situation one can exploit to say when it exactly verifies a strict conditional, if it can be done at all.12 Similarly, it is easier to define classical verification condition for the imagination operator, than working out its exact verification conditions. Working with two series of verification conditions allows us to leave those problems aside, at least for the time being, in order to focus on the topic of hyperintensionality.13 It can easily be checked that, for any formula A of the boolean fragment of the language, w  A iff w is incompatible with an exact falsifier of A, and w  A iff w is incompatible with an exact verifier of A, so that the two series of semantical clauses are in harmony with each other. The drawback of this maneuver is that we will not be able to evaluate imagination conditionals that involve strict conditionals or embedded imagination conditionals. So we need to restrict the grammar of the language so that a string of the form [A]B is well-formed only if A and B are taken from the boolean fragment of L I . We will call L I  this sub-language of L I . This loss in expressivity, however, has little consequence for the main goal of the present study, since the difficulties pertaining to hyperintensionality that we identified in Berto’s approach only involved imagination conditionals containing booleans connectives and can be expressed all the same in L I  . For the selection function to model imagination correctly, we add the following two conditions to the definition of a T M-model: (I) f (A, w) ⊆ |A|+ (IE) if ||A||+ = ||B||+ and ||A||− = ||B||− , then f (A, w) = f (B, w). The principle of identity (I) ensures, just like (BC) in Berto’s framework, that what is explicitly imagined will always be implicitly imagined. The principle of Imaginative Equivalence (IE) sets the conditions under which two inputs are equivalent: they need to have the same exact verifiers and the same exact falsifiers. Its full justification will have to wait until the next section, which is specifically dedicated to the justification of the treatment of hyperintensionality in the present truthmaker semantics. But the intuitive idea seems clear enough: A ∧ B and B ∧ A will always be treated as equivalent inputs because they have the same sets of exact verifiers and exact falsifiers, respectively. It is thus impossible to imagine a situation verifying A without imagining a situation verifying B and vice versa. Validity and consequence are then defined as follows: • for all T M-model M =

S, S ♦ , , f , || · ||+ , || · ||− , M  A iff w  A for all possible world w in S ♦ , • T M A iff M  A for all T M-model M, 12 A similar problem is addressed by Fine in [9] as part of the project of providing an exact semantics

for intuitionnistic logic. similar strategy is followed by Fine in the exposition of his truthmaker semantics for counterfactuals [8], where he only gives classical verification conditions for counterfactuals, themselves defined in terms of exact and inexact verification. 13 A

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•  T M A iff for all T M-model M =

S, S ♦ , , f , || · ||+ , || · ||− , and all possible world w ∈ S ♦ , if w  B for all B ∈ , then w T M A. Let us call ITM the logic determined by the class of T M-models, i.e ITM = {A ∈ L I  : T M B}. The appropriate validities generated by Berto’s semantics are also valid under the present semantics: Proposition 12 (a) (b) (c) (d)

T M [A]A, for all A ∈ K (success) [A](B ∧ C) T M [A]B (left simplification) [A](B ∧ C) T M [A]C (right simplification) [A]B, [A]C T M [A](B ∧ C) (adjunction)

Proof (a) Let w be an arbitrary possible world taken from an arbitrary T M-model M =

S, S ♦ , , f , || · ||+ , || · ||− . By (I) f (A, w) ⊆ |A|+ . Since τ (A)  τ (A), we have, by (TM[·]), w  [A]A. (b) Let w be an arbitrary possible world taken from an arbitrary T M-model M =

S, S ♦ , , f , || · ||+ , || · ||−  such that w  [A](B ∧ C). Then by (TM[·]), (i) f (A, w) ⊆ |B ∧ C|+ and (ii) τ (B ∧ C)  τ (A). Now |B ∧ C|+ ⊆ |B|+ . It follows from (i) that f (A, w) ⊆ |B|+ . In addition, τ (B)  τ (B ∧ C)  τ (A). Therefore, by (TM[·]), we have w  [A]B. (c) By the same reasoning as immediately above applied to C instead of B in the conjunctive consequent B ∧ C. (d) Let w be an arbitrary possible world taken from an arbitrary T M-model M =

S, S ♦ , , f , || · ||+ , || · ||−  such that w  [A]B and w  [A]C. Then by (TM[·]), we have (i) f (A, w) ⊆ |B|+ and f (A, w) ⊆ |C|+ , while (ii) τ (B)  τ (A) and τ (C)  τ (A). Let s be any member of f (A, w). It follows from (i) that s has a part t1 such that t1  B and a part t2 such that t2  C. Then, by (∧+ ), t = t1  t2  B ∧ C. Since t  s, we have s ∣> B ∧ C. Thus f (A, w) ⊆ |B ∧ C|+ . Since τ (B ∧ C) = τ (B)  τ (C), it follows from (ii) that τ (B ∧ C)  τ (A). By (TM[·]) we conclude that w  [A](B ∧ C).  The unwelcome invalidities avoided by Berto’s semantics are also avoided here: Proposition 13 (a) (b) (c) (d) (e) (f)

[A]B T M [A ∧ C]B (non-monotonicity) [A](B ∨ C) T M [A]B ∨ [A]C (indeterminacy) [A]B T M [A](B ∨ C) (relevance) It is not the case that: if T M B, then T M [A]B (no logical omniscience) A ⇒ B T M [A]B (no imaginative entailment) T M [A ∧ ¬A]B (non-vacuity)

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Proof For each invalidity, we present a countermodel.14 (a)

We have f ( p, w1 ) ⊆ |q|+ and τ (q) = s = τ ( p). By (TM[·]), w1  [ p]q. Now f ( p ∧ r, w1 )  |q|+ . By (TM[·]), w1  [ p ∧ r ]q. (b)

We observe that f ( p, w1 ) ⊆ |q ∨ r |+ . Besides, τ (q ∨ r ) = s = τ ( p). By (TM[·]), w1  [ p](q ∨ r ). Now f ( p, w1 )  |q|+ and f ( p, w1 )  |r |+ . By (TM[·]), w1  [ p]q and w1  [ p]r .

14 Labeled

nodes represent situations. Labels of the form wi indicate situations that are possible worlds. Downward straight lines between situations represent the relation of parthood. All and only possible situations are part of a possible world. Labeled bent lines represent accessibility relations, so that wi is connected to s j by a bent line labeled by A just in case s j ∈ f (A, wi ). For each situation, the propositional variables that it exactly verifies or falsifies according to the model are indicated above or below the node representing that situation. The formulas verified exactly, inexactly and classically by each situation are not represented, but can be easily computed with the help of the recursive semantical clauses given earlier. It can be checked that each diagram satisfies the definition of a T M-model.

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(c)

We have f ( p, w) ⊆ |q|+ and τ (q) = s = τ ( p). By (TM[·]) w  [ p]q. Now τ (q ∨ r ) = w  τ ( p). By (TM[·]), w  [ p](q ∨ r ). (d)

We know by classical logic that T M q ∨ ¬q. Now f ( p, w)  |q ∨ ¬q|+ . By (TM[·]) w  [ p](q ∨ ¬q). (e) By the same countermodel as immediately above. We know by modal logic that T M p ⇒ (q ∨ ¬q). A fortiori w  p ⇒ (q ∨ ¬q). Yet for the same reason as immediately above, w T M [ p](q ∨ ¬q). (f)

We have f ( p ∧ ¬ p, w)  |q|+ . By (TM[·]), w  [ p ∧ ¬ p]q.



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Thus the present truthmaker semantics shares the virtues of Berto’s semantics by validating the same appropriate schemata and invalidating the same inappropriate schemata.

6 The Grain of Imagination Let us now turn to the issue of hyperintensionality. Berto’s topics-based semantics allowed the derivation of principles that were shown to be problematic in light of counterintuitive instances. None of those principles hold in the present semantics. Proposition 14 (a) T M [A ∨ (B ∧ C)]D ⇔ [(A ∨ B) ∧ (A ∨ C)]D (b) [A]B T M [A ∨ (A ∧ C)]B (c) T M [A ∧ ¬A]B ⇔ [A ∧ ¬A]¬B Proof

(a)

We have f (( p ∨ q) ∧ ( p ∧ r ), w1 ) ⊆ |¬q|+ and τ (¬q) = τ (q) = s = τ (( p ∨ q) ∧ ( p ∨ r )). By (TM[·]), w1  [( p ∨ q) ∧ ( p ∨ r )]¬q. Now f ( p ∨ (q ∧ r ), w1 )  |¬q|+ . It follows, by (TM[·]), that w1  [ p ∨ (q ∧ r )]¬q. A fortiori, w1  [( p ∨ (q ∧ r ))]¬q ⇔ [( p ∨ q) ∧ ( p ∨ r )]¬q.

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(b)

We have f ( p, w1 ) ⊆ |q|+ and τ (q) = s = τ ( p). So, by (TM[·]), we have w1  [ p]q. Now f ( p ∨ ( p ∧ r ), w1 )  |q|+ . It follows, by (TM[·]), that w1  [ p ∨ ( p ∧ r )]q. (c)

We have f ( p ∧ ¬ p, w) ⊆ |q|+ and τ (q) = s = τ ( p). By (TM[·]), it follows that w  [ p ∧ ¬ p]q. However, f ( p ∧ ¬ p, w)  |¬q|+ . By (TM[·]), it follows that w  [ p ∧ ¬ p]¬q. A fortiori, w  [ p ∧ ¬ p]q ⇔ [ p ∧ ¬ p]¬q.  In fact, the present semantics also avoids the other difficulty that worried Berto in connection with both (PIE) and (IW4): Proposition 15 [A]B, [A ∧ B]C T M [A]C.

Proof

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We have f ( p, w1 ) ⊆ |q|+ and τ (q) = s = τ ( p). By (TM[·]) w1  [ p]q. Besides, f ( p ∧ q, w1 ) ⊆ |r |+ and τ (r ) = s = τ ( p ∧ q). By (TM[·]), w1  [ p ∧ q]r . Yet f ( p, w1 )  |r |+ . By (TM[·]), w1  [ p]r . Note that || p||+ = || p ∧ q||+ so that (IE) is respected.  The fact that the present semantics seems hyperintensional enough to avoid the problems of the topics-based semantics does not mean that it is not too hyperintensional. In the present framework, the grain of imagination is never finer than bilateral exact equivalence, i.e. the relation that holds between two formulas A and A when ||A||+ = ||A ||+ and ||A||− = ||A ||− , which we symbolize by A ≈ B E A . 

Proposition 16 If A ≈ B E A , then B ≈ B E B AA for all B that belongs to the boolean fragment of the language. Proof An easy induction on the construction of B.





Proposition 17 If A ≈ B E A , then B T M B AA . Proof By induction on the construction of B. Since bilateral exact equivalence entails classical equivalence, i.e. truth in the same class of possible worlds, the property is easily seen to hold whenever B is a propositional variable, a boolean compound or a strict conditional. The only remaining case is when B is of the form    [C]D. Let us write [C]D AA as C  [D  ]. Then C  = C AA and D  = D AA . Let w be an arbitrary possible world from an arbitrary T M-model such that w  [C]D. By Proposition 16, C ≈ B E C  and D ≈ B E D  . By (IE), f (C, w) = f (C  , w). Then, by (TM[·]), f (C  , w) = f (C, w) ⊆ |D|+ = |D  |+ and τ (D  ) = τ (D)  τ (C) =  τ (C  ). Thus, by (TM[·]) w  [C  ]D  . There is a principled reason for this. Our semantics was guided by the independently motivated philosophical view that acts of imagination have situations as their intentional objects while their intentional contents are what those situations verify. If this view is right, then contents of imagination that are exactly verified by the same situations and exactly falsified by the same situations cannot differ. In particular, unlike the refined impossible world semantics described in Sect. 4, we have a principled reason for the acceptance of: Proposition 18 (i) (ii) (iii) (iv)

T M T M T M T M

[A ∧ B]C ⇔ [B ∧ A]C [A ∨ B]C ⇔ [B ∨ A]C [A]B ⇔ [A ∧ A]B [A]B ⇔ [A ∨ A]B

Proof It follows from the exact verification and falsification clauses, and the closure under fusion of exact verifier and exact falsifier sets that A ∧ B ≈ B E B ∧ A, A ∨ B ≈ B E B ∨ A, A ≈ B E A ∧ A and A ≈ B E A ∨ A. Then (i)–(iv) follow from Proposition 17 and (TM⇒). 

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One might still wonder whether it is so clear that bilateral exact equivalence always gives the right measure of hyperintensionality in the case of imagination. Among the less trivial bilateral exact equivalences, one finds the following: (a) (b) (c) (d)

¬(A ∨ B) ≈ B E ¬A ∧ ¬B ¬(A ∧ B) ≈ B E ¬A ∨ ¬B A ≈ B E ¬¬A A ∧ (B ∨ C) ≈ B E (A ∧ B) ∨ (A ∧ C).

It follows straighforwardly from those equivalences and Proposition 17 that the members of each of those pairs will be substitutable salva veritate everywhere in an imagination conditional. Is that plausible? First of all, we gave reasons above not to worry about such substitutions occurring within the consequent of an imagination conditional and will not repeat the argument. The interesting question is whether substitutions in antecedent position should be considered problematic. An objector might argue that this substitutability result amounts to conferring to our agents the knowledge of those logical equivalences. But for sure, a real agent does not necessarily know the De Morgan laws. If so, why couldn’t she imagine something on the basis of ¬( p ∨ q) and something else on the basis of ¬ p ∧ ¬q? If this logic of imagination is meant to apply to real-world agents, this is a rather inadequate result. Such a worry is misplaced, however. Substitutability results of this kind do not say anything about the amount of logic that the agent knows. They only say that the agent will systematically treat two bilaterally equivalent inputs A and B in the same way, to the extent that they understand A and B in virtue of their linguistic competence. And the reason we should expect agents to treat both inputs in the same way is not that they have some background knowledge of logic, but simply that any situation that exactly verifies the former exactly verifies the latter and conversely. If the broad philosophical picture of imagination drawn by Yablo and Chalmers is correct, then for any two such inputs, there is no way an imagined situation can verify one but not the other. On the assumption that this philosophical view is correct, at least in outline, the substitutability of ¬( p ∨ q) and ¬ p ∧ ¬q, derives from the intentional structure of acts of imagination rather than from the possession of logical knowledge. One might still worry that the last equivalence gives rise to counterexamples, in the same way that the topical strict equivalence of A ∨ (B ∧ C) and (A ∨ B) ∧ (A ∨ C) caused trouble for Berto’s topics semantics, as exemplified by the Albert case above. Our truthmaker semantics predicts indeed that the following imagination conditionals are logically equivalent: (16)

a. As I imagine that Albert loves Patricia and that he loves Quiana or Roberta, I imagine that Albert is polyamorous. b. As I imagine that Albert loves Patricia and Quiana or that he loves Patricia and Roberta, I imagine that Albert is polyamorous.

However in this case, the two inputs do seem intuitively equivalent. We can once again appeal to the philosophical account of imagination that we took as a guide to

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explain why this should be so: any situation that verifies the first input, also verifies the second one, and vice versa. As a consequence, it is impossible to imagine a situation verifying one without imagining a situation verifying the other. Thus we should expect both inputs to yield the same outputs. The treatment of hyperintensionality that emerges from the truthmaker semantics developed in the previous section thus avoids all the problems raised for Berto’s topics-based and impossible worlds semantics. The truthmaker semantics meets both the desideratum of drawing sufficiently fine-grained and that of not drawing excessively fine-grained hyperintensionality. And it does so in virtue of an independently plausible and philosophically motivated criterion of imaginative equivalence.

7 Conclusion While it is important for the adequacy of a logic of ceteris paribus imagination to avoid the substitutability salva veritate of necessarily equivalent propositions, it is no less important for such a logic to draw the right hyperintensional distinctions. Berto’s proposal of a possible worlds semantics combined with a theory of topics achieves to draw some important hyperintensional distinctions, but ends up giving too coarse a measure of the granularity of imagination. Symmetrically, his alternative impossible worlds semantics ends up being excessively fine-grained, and misses some clear cases of imaginative equivalence. While it is possible to avoid such problematic results by imposing further appropriate constraints on models, it is unclear that this kind of solution can provide a principled understanding of hyperintensionality in imagination. Truthmaker semantics, however, gives appropriate resources for a more adequate treatment, when developed under the guidance of an independently attractive philosophical picture of imagination. There are perhaps a few more general lessons to be learned from this study of hyperintensionality in imagination. On the one hand, the difference between Berto’s topics-based and the truthmaker semantics is reminiscent of a methodological difference that can be observed in a similar debate between Yablo [21] and Fine [11, 13] on subject-matter. While Yablo starts with possible worlds and adds truthmakers to draw hyperintensional distinctions regarding subject-matter between necessarily equivalent contents, Fine begins from the start with partial situations. The present work can be seen as providing arguments, both technical and philosophical, for approaching the logic of imagination with a situations-first, rather than a possible-worlds-first, methodology and more generally as illustrating the fruitfulness of the situations-first approach. On the other hand, impossible worlds provide another set of tools that seem particularly suited to the treatment of hyperintensionality. For a world where a hyperintensional distinction holds is by definition an impossible world. This being said, there are many different ways to introduce impossibilities to account for hyperintensionality. As we saw, the kind of open impossible worlds used by Berto end up drawing unwanted hyperintensional distinctions. In contrast, the framework of truth-

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maker semantics, by allowing impossible situations that are still subject to a uniform semantics, avoids such an excess. While I argued for the advantages of truthmaker semantics in the treatment of the hyperintensionality of ceteris paribus imagination reports, I do not mean to imply that truthmaker semantics is necessarily always the best tool to deal with hyperintensionality in general. The reason why it works well for the logic of imagination is that it closely fits a natural and attractive way of thinking about imagining as an intentional act directed towards partial situations capable of verifying or falsifying the contents of the sentences that occur in imagination reports. Whether truthmaker semantics, as employed here, will prove fruitful for explaining other hyperintensional phenomena, is thus an open question, but one lesson to draw from this study of imagination is that a way to make progress in the understanding of hyperintensionality is to closely associate the elaboration of technical tools with philosophical analysis.15

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

Barwise, J., & Perry, J. (1983). Situations and attitudes. Cambridge: MIT Press. Berto, F. (2017). Impossible worlds and the logic of imagination. Erkenntnis, 82(6), 1277–1297. Berto, F. (2018). Aboutness in imagination. Philosophical Studies, 175(1), 1871–1886. Berto, F. (2018). Taming the runabout imagination ticket. Synthese (online first), 1–15. Berto, F., & Jago, M. (2019). Impossible worlds. Oxford: Oxford University Press. Chalmers, D. J. (2002). Does conceivability entail possibility? In T. S. Gendler & J. Hawthorne (Eds.), Conceivability and possibility (pp. 145–200). Oxford: Oxford University Press. Cresswell, M. J. (1975). Hyperintensional logic. Studia Logica, 34(1), 25–38. Fine, K. (2012). Counterfactuals without possible worlds. Journal of Philosophy, 109(3), 221– 246. Fine, K. (2014). Truth-maker semantics for intuitionistic logic. Journal of Philosophical Logic, 43(2–3), 549–577. Fine, K. (2017). A theory of truthmaker content I: Conjunction, disjunction and negation. Journal of Philosophical Logic, 46(6), 625–674. Fine, K. (2017). A theory of truthmaker content II: Subject-matter, common content, remainder and ground. Journal of Philosophical Logic, 46(6), 675–702. Fine, K. (2017). Truthmaker semantics. A companion to the philosophy of language (pp. 556– 577). Chichester: Blackwell. Fine, K. (forthcoming). Yablo on subject-matter. Philosophical Studies. Fine, K., & Jago, M. (2019). Logic for exact entailment. Review of Symbolic Logic, 12(3), 536–556. Giordani, A. (2019). Axiomatizing the logic of imagination. Studia Logica, 107(4), 639–657. Jago, M. (2014). The impossible: An essay on hyperintensionality. Oxford: Oxford University Press. Priest, G. (1997). Sylvan’s box: A short story and ten morals. Notre Dame Journal of Formal Logic, 38(4), 573–582.

15 Preliminary versions of this paper have been presented in Bochum, Louvain-la-Neuve, Milan and Prague. I would like to thank all audiences at these venues for their valuable feedback, as well as an anonymous reviewer for considerably helpful criticism and suggestions. This research was supported by the project “Intuitions in Science and Philosophy” funded by the Danish Council for Independent Research DFF 4180-00071.

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18. Priest, G. (2016). Towards non-being: The logic and metaphysics of intentionality (2nd ed.). Oxford: Oxford University Press. 19. Thomason, R. H. (1980). A model theory for propositional attitudes. Linguistics and Philosophy, 4(1), 47–70. 20. Yablo, S. (1993). Is conceivability a guide to possibility? Philosophy and Phenomenological Research, 53(1), 1–42. 21. Yablo, S. (2014). Aboutness. Princeton: Princeton University Press.

Deontic Logic with Action Types and Tokens Alessandro Giordani

Abstract A new characterization of the deontic operators of permission and prohibition is introduced based on a distinction between action types and action tokens. The resulting deontic action logic constitutes a hyperintensional system providing resources for a fine-grained study of the basic deontic notions. The logic is proved to be complete with respect to an appropriate semantics, where models include both possible worlds and action tokens, and the philosophical significance of the distinction is demonstrated by showing that a number of puzzles afflicting current accounts of the deontic operators find intuitive solutions in the new framework.

1 Introduction In this paper I provide a new system of deontic logic of states and actions based on a semantics where the distinction between action types and action tokens is exploited in order to capture crucial principles concerning permissions, prohibitions and obligations. The system is developed in a framework that includes a set of possible states and a subset of ideal ones which identify what is allowed given a deontic ideal. On top of that, it includes a set of action tokens, intended as aspects of the concrete conducts of an agent, which are appropriately connected to terms for action types in order to specify the conditions under which actions are permitted and prohibited. The paper is structured as follows. In Sect. 2 Segerberg’s basic system D AL of deontic action logic is introduced and discussed with respect to the problems it gives rise as to the characterization of the deontic notions. In Sect. 3 a first development of D AL is put forward. This intermediate system, call it I D AL, for intensional deontic action logic, is of interest in as much as it provides us with a framework where the approaches to action logic based on boolean algebra and on dynamic logic can be unified in pleasant way. In Sect. 4, a final development of D AL is proposed and proved to be complete with respect to a novel semantics. This new system, call it H D AL, for A. Giordani (B) Università Cattolica di Milano, L. Gemelli 1, Milano 20123, Italy e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Giordani and J. Malinowski (eds.), Logic in High Definition, Trends in Logic 56, https://doi.org/10.1007/978-3-030-53487-5_7

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hyperintensional deontic action logic, is related to a semantic framework exploiting a distinction between action types and tokens and involving crucial elements of truthmaker semantics. It is then shown that in H D AL the problems affecting the previous systems can be solved, together with a number of puzzles deriving from the classical accounts of permission and prohibition.

2 Introduction to Deontic Action Logic The system advanced in the final section is a development of the deontic action logic proposed by Segerberg in [15].1 In this section I present Segerberg’s logic and highlight the main problems we want to solve. In order to be self-contained, a proof of the completeness of that logic with respect to the deontic action models initially introduced by Segerberg is sketched. The proof differs from the ones offered in [6, 18] in that it is not assumed that the set of terms for basic action types is finite, and so that the corresponding term algebra is atomic.

2.1 Language and Semantics Let AV be a countable set of variables for basic action types. The language L D AL of deontic action logic contains a set T m D AL of terms and a set Fm D AL of formulas, respectively defined by the following rules: α := a | 0 | 1 | α | α  β | α  β, where a ∈ AV φ := α = β | P(α) | F(α) | ¬φ | φ ∧ ψ | φ ∨ ψ, where α is a term. The idea underlying the introduction of such language is that actions, intended as types or generic acts [21], can be negated (a), conjoined () and disjoined () in a suitable way. In addition, two primitive types are introduced: the type of impossible actions, denoted by 0, and the type of the trivial action, denoted by 1. This idea can be fleshed out both in terms of execution of generic acts and in terms of conducts exemplifying action types, where a conduct is a complete individual action, that is what is performed by an agent in a certain occasion.2 Thus, the negation of an action can be defined as the action executed by an agent not executing the original action, the conjunction of two actions as the action executed by an agent executing both actions, and the disjunction of two action as the action executed by an agent executing at least one of them [21]. On this interpretation it is impossible for the impossible action to be 1 Segerberg’s basic system and some significant variations have been extensively studied. See [5,

6, 15, 18, 19] for introduction and discussions. See [10] for an introduction to boolean algebras. 2 A further interpretation, the one proposed by Segerberg, is in terms of action outcomes. The intuition is that, since the execution of an action gives rise to different outcomes in different circumstances, then the action itself can be viewed as the set of these outcomes. See [15].

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executed, while the trivial action is always executed. Alternatively, using the notion of a conduct, or individual action, exemplifying an action type, the negation of a type can be defined as the type that is exemplified by a conduct that is not exemplifying the original type, the conjunction of two types as the type exemplified by a conduct that exemplifies both types, and the disjunction of two types as the type exemplified by a conduct that exemplifies at least one of those types [23]. On this interpretation the impossible action has no examples, while the trivial action is exemplified by every conduct. In what follows I will use both interpretations for illustrating the intended semantics of the systems we are going to discuss. Definition 1 Inclusion and specification. 1. α  β := α  β = α; 2. α  β is a specification of α and β. The action type denoted by α is included in the action type denoted by β precisely when it is impossible for the agent to execute α without executing β: if the examples of α are precisely the examples α  β, then every examples of α is also an example of β. Furthermore, since every example of α  β is an example of both α and β, α  β is said to be a specification of α and β. The language of deontic action logic is interpreted with respect to structures involving two ingredients: (i) a set E containing elements representing the possible conducts of an agent; (ii) a code (Legal, I llegal), where Legal is the subset of E containing elements representing conducts that are permissible according to the code and I llegal is the subset of E containing elements representing conducts that are impermissible according to the code. An interpretation of L D AL with respect to those structures assigns to each action term a set of conducts in such a way that the boolean operators of L D AL are associated to set theoretical operations on the boolean algebra of the subsets of E. Definition 2 Model for L D AL . A model for L D AL is a tuple M = (E, Legal, I llegal, I ) where E = ∅ Legal ⊆ E I llegal ⊆ E Legal ∩ I llegal = ∅ I : T m D AL → ℘ (E) where I is required to satisfy the following conditions. Conditions on I : I1 : I (0) = ∅ I2 : I (1) = E I3 : I (α) = E − I (α) I4 : I (α  β) = I (α) ∩ I (β) I5 : I (α  β) = I (α) ∪ I (β)

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As mentioned above, the basic idea is that every action type has a set of examples, the conduct exemplifying it, and that an action term is extensionally interpreted as referring to that set of examples. The conditions on I ensure that the intended interpretation of the connection between action types and their examples is respected. Definition 3 Truth in a deontic action model. The notion of truth is defined by the following recursion. M |= α = β ⇔ I (α) = I (β) M |= P(α) ⇔ I (α) ⊆ Legal M |= F(α) ⇔ I (α) ⊆ I llegal M |= ¬φ ⇔ M |= φ M |= φ ∧ ψ ⇔ M |= φ and M |= ψ M |= φ ∨ ψ ⇔ M |= φ or M |= ψ The truth conditions of permissions and prohibitions are particularly interesting: action α is permitted provided that all its examples are legal and it is prohibited provided that all its examples are illegal. Hence, the notion of permission characterized in this framework is a notion of strong permission according to which an action is permitted only if all its specifications are also permitted [1, 2, 8, 22].

2.2 Axiomatics and Completeness The logic D AL is characterized by the following set of axioms and rules. Group 1: axioms for classical propositional logic and modus ponens. Group 2: identity axioms. α=α α = β → φ = φ[α/β] where φ[α/β] is the result of substituting β for some occurrence of α in φ. Group 3: boolean axioms. α  (β1  β2 ) = (α  β1 )  β2 αβ =β α α  (α  β) = α α  (β1  β2 ) = (α  β1 )  (α  β2 ) αα =0

α  (β1  β2 ) = (α  β1 )  β2 αβ =β α α  (α  β) = α α  (β1  β2 ) = (α  β1 )  (α  β2 ) αα =1

Group 4: deontic axioms P(α  β) ↔ P(α) ∧ P(β) F(α  β) ↔ F(α) ∧ F(β) P(α) ∧ F(α) ↔ α = 0 In particular, axioms of group 4 characterize P and F as ideal-like predicates, i.e. predicates defining ideals in every boolean algebra on which the language of D AL is interpretable. In fact, the following propositions are derivable in D AL.

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Fact 1 Properties of P and F. P1 : P(0) P2 : P(β) ∧ α  β → P(α) P3 : P(α) ∧ P(β) → P(α  β)

F1 : F(0) F2 : F(β) ∧ α  β → F(α) F3 : F(α) ∧ F(β) → F(α  β)

P1 and F1, stating that the impossible action is both permitted and prohibited, seem to be a bit counterintuitive, but they are harmless in so far as the impossible action is never executable. P2 and F2 are to be assessed separately: F2 states that any specification of a prohibited action is prohibited, which is as it should be, since executing a specification of an action implies executing that action; P2 states that any specification of a permitted action is permitted, which is controversial, as we will see shortly. Finally, P3 and F3 are fairly intuitive: if two actions are permitted, then it is no surprise that an agent is allowed to execute either one of them; similarly, if two actions are prohibited, then there is no admissible choice between them. It is not difficult to show that the axioms previously introduced are sound with respect to the intended semantics. The proof of completeness is in two steps, showing that (i) every D AL-consistent set of formulas can be expanded to a maximal D ALconsistent set of formulas; (ii) every maximal D AL-consistent set of formulas can be used to construct a canonical model for L D AL relative to which we can prove a canonicity lemma, stating that the canonical model is indeed a model for L D AL and a truth lemma, stating that a formula is true in the canonical model precisely when it is an element of the maximal D AL-consistent set we start with. The first step is standard, so that we will focus on step (ii). So, let Δ be a maximal D AL-consistent set of formulas and define [α] := {β : (α = β) ∈ Δ}. Lemma 1 AΔ = AΔ , 0Δ , 1Δ , −Δ , Δ , Δ , where (1) (2) (3) (4) (5) (6)

AΔ = {[α] : (α = α) ∈ Δ} 0Δ = [0] 1Δ = [1] −Δ [α] = [α] [α] Δ [β] = [α  β] [α] Δ [β] = [α  β]

is a boolean algebra. In addition, PΔ = {[α] : P(α) ∈ Δ} and FΔ = {[α] : F(α) ∈ Δ} are ideals on AΔ . Proof The first point follows from axioms of group 2 and group 3. Note that both the constants and the operations of AΔ are well-defined since, by virtue of the axioms of group 2, (α = β) ∈ Δ is a congruence on the algebra of terms. The second point is a straightforward consequence of the axioms of group 4. Let us now define U (AΔ ) as the set of all the ultrafilters of AΔ .

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Definition 4 canonical model for L D AL based on Δ. Let Δ be a maximal D AL-consistent set of formulas. The canonical model based on Δ is the tuple MΔ = (E Δ , LegalΔ , I llegalΔ , IΔ ), where E Δ = U (AΔ ) LegalΔ = {U ∈ U (AΔ ) : PΔ ∩ U = ∅} I llegalΔ = {U ∈ U (AΔ ) : FΔ ∩ U = ∅} IΔ is such that IΔ (α) = {U ∈ U (AΔ ) : [α] ∈ U } Lemma 2 (Canonicity Lemma): MΔ is a model for LDAL . By the definitions of E Δ , LegalΔ and I llegalΔ we get that E = ∅, LegalΔ ⊆ E Δ and I llegalΔ ⊆ E Δ . So, we are only required to show that LegalΔ ∩ I llegalΔ = ∅ and that the canonical valuation satisfies I1 – I5. Fact 2 LegalΔ ∩ I llegalΔ = ∅. Proof Suppose that U ∈ LegalΔ ∩ I llegalΔ . Then PΔ ∩ U = ∅, so that [α] ∈ U , for some α such that P(α) ∈ Δ; FΔ ∩ U = ∅, so that [β] ∈ U for some β such that F(β) ∈ Δ. Since U is an ultrafilter, [α] Δ [β] ∈ U , so that [α] Δ [β] = 0Δ . Still P(α  β) ∈ Δ, given that P(α) ∈ Δ; F(α  β) ∈ Δ, given that F(β) ∈ Δ. Therefore (α  β = 0) ∈ Δ, and so [α] Δ [β] = 0Δ . Contradiction. Fact 3 IΔ satisfies I1 – I5. Proof It follows from the properties of ultrafilters. I1 : IΔ (0) = ∅, since, for all U ∈ E Δ , 0Δ = [0] ∈ / U. I2 : IΔ (1) = E Δ , since, for all U ∈ E Δ , 1Δ = [1] ∈ U. I3 : IΔ (α) = E Δ − IΔ (α). U ∈ IΔ (α) ⇔ [α] ∈ U, by the definition ofIΔ U ∈ IΔ (α) ⇔ −[α] ∈ U, by the definition of − [α] / U, since U is an ultrafilter U ∈ IΔ (α) ⇔ [α] ∈ / IΔ (α), again by the definition of IΔ U ∈ IΔ (α) ⇔ U ∈ I4 : IΔ (α  β) = IΔ (α) ∩ IΔ (β). U ∈ IΔ (α  β) ⇔ [α  β] ∈ U, by the definition of IΔ U ∈ IΔ (α  β) ⇔ [α] Δ [β] ∈ U, by the definition of[α] Δ [β] U ∈ IΔ (α  β) ⇔ [α] ∈ U and [β] ∈ U, since U is an ultrafilter U ∈ IΔ (α  β) ⇔ U ∈ IΔ (α) and U ∈ IΔ (β), again by the definition of IΔ

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I5 : IΔ (α  β) = IΔ (α) ∪ IΔ (β). U ∈ IΔ (α  β) ⇔ [α  β] ∈ U, by the definition of IΔ U ∈ IΔ (α  β) ⇔ [α] Δ [β] ∈ U, by the definition of [α] Δ [β] U ∈ IΔ (α  β) ⇔ [α] ∈ U or [β] ∈ U, since U is an ultrafilter U ∈ IΔ (α  β) ⇔ U ∈ IΔ (α) or U ∈ IΔ (β), again by the definition of IΔ Finally, a formula is true in MΔ just in case it is contained in Δ. Lemma 3 (Truth Lemma): MΔ |= ϕ ⇔ ϕ ∈ Δ. Proof The interesting cases are those concerning α = β, P(α) and F(α). Case 1 : MΔ MΔ MΔ MΔ MΔ MΔ

|= α |= α |= α |= α |= α |= α

=β =β =β =β =β =β

⇔ α = β ∈ Δ. ⇔ IΔ (α) = IΔ (β) ⇔ ∀U ∈ E Δ (U ∈ IΔ (α) ⇔ U ∈ IΔ (β)) ⇔ ∀U ∈ E Δ ([α] ∈ U ⇔ [β] ∈ U ) ⇔ [α] = [β], by the properties of ultrafilters ⇔ (α = β) ∈ Δ, by the definition of [α] and [β]

The crucial step is based on the fact that any two different elements of a boolean algebra are separated by an ultrafilter, so that elements contained in the same ultrafilters are identical. Case 2 : MΔ |= P(α) ⇔ P(α) ∈ Δ. MΔ |= P(α) ⇔ IΔ (α) ⊆ LegalΔ MΔ |= P(α) ⇔ ∀U ∈ U (AΔ )(U ∈ IΔ (α) ⇒ U ∈ LegalΔ ) MΔ |= P(α) ⇔ ∀U ∈ U (AΔ )([α] ∈ U ⇒ PΔ ∩ U = ∅) MΔ |= P(α) ⇔ [α] ∈ PΔ , by the properties of ultrafilters MΔ |= P(α) ⇔ P(α) ∈ Δ, by the definition ofPΔ The crucial step is based on the fact that any ideal is the intersection of all the maximal ideals that include it and that there is a one to one correspondence between maximal ideals and ultrafilters, since every maximal ideal is the complement of an ultrafilter. Thus, if all the ultrafilters containing [α] intersect PΔ , then all maximal ideals including PΔ contain [α], so that PΔ contain [α]. Case 3 : MΔ |= F(α) ⇔ F(α) ∈ Δ. Similar to the previous one.

2.3 Problems Concerning Deontic Action Logic The logic D AL, intended as a system of deontic action logic, poses two main problems: (i) the logical principles characterizing the notion of permission give rise to unintuitive consequences; (ii) there seems to be no way to define the notion of obligation in terms of permission and prohibition.3 3 A more in-depth discussion of the problems concerning the characterization of the deontic notions

in D AL is offered in [5, 19].

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Permission As noted above, P and F are ideal-like predicates. In particular, it holds that 1. P(β) ∧ α  β → P(α) 2. F(β) ∧ α  β → F(α) However, while there is no problem in requiring that any specification of a prohibited action is also prohibited, it is not evident why any specification of a permitted action should also be permitted. Indeed, a straightforward consequence of this requirement is that P(β) → P(α  β), since α  β  β. The principle states that, if the execution of β is permitted, then the execution of β together with any other action is also permitted. This consequence becomes intelligible when we interpret P as permission relative to any circumstance, i.e. relative to any further specification of the action which is permitted. Still, such an interpretation implies a too strict limitation of the application of the notion of permission. To be sure, let us consider a case in which P(α) ∧ F(β). Since P(α  β) ∧ F(α  β), by the previous point, and P(α  β) ∧ F(α  β) → α  β = 0, we conclude that α  β = 0. So, it is impossible for two compatible actions to be such that the first one is permitted and the second one is prohibited. Again, such a consequence becomes intelligible when we interpret P as permission relative to any circumstance, but then it seems that very few actions, if any, turn out to be permissible under this interpretation, since no action that is compatible with something prohibited, that is no action that can be executed together with something prohibited, can be permitted. Obligation Traditionally, the notion of obligation is seen as definable either in terms of permission, assuming that an action is obligatory when the execution of its negation is not permitted, or in terms of prohibition, assuming that an action is obligatory when the execution of its negation is prohibited. Still, in the present framework, both definitions generate unintended consequences. Let O be an operator for obligation. 1. Suppose we define O(α) := F(α). Then, O(α) holds provided that every example of α is illegal. As a consequence, O(α) implies O(α  β), since F(α) implies F(α  β), so that the obligation to execute an action implies the obligation to execute that action or any other action. This is the standard definition of obligation, but the previous consequence pushes us to look for a better alternative. 2. Suppose we define O(α) := ¬P(α). Then, O(α) holds provided that some example of α is not legal. Thus, every action whose examples are strictly included in the set Illegal are obligatory, so that every prohibited action is also obligatory, provided that there are at least two prohibited actions with different examples. Still, the idea that every strict specification of a prohibited action is obligatory is far from satisfactory. 3. Suppose we define O(α) := F(α) ∧ P(α). Then, O(α) holds provided that every example of α is illegal and every example of α is legal. Thus, the examples of α are precisely the legal examples, since the disjunction of α and α is always executed, so that O(α) implies that any other obligatory action coincides with α.

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Still, the idea that, if some action is obligatory, then only one action is obligatory is again far from satisfactory.4 In conclusion, it seems that the intuitive definitions of O present significant drawbacks, so that the notion of obligation can be introduced at best via an axiomatic characterization.5 However, if we want to preserve the idea that O is definable in terms of permission and prohibition, an amendment of D AL is in order.

3 Intensional Deontic Action Logic As a first step in the development of a deontic action logic that overcomes the abovementioned drawbacks let us introduce a new system I D AL of intensional deontic action logic. The main difference between D AL and the new system lies in the semantics, specifically in the interpretation of the action terms: while in D AL action terms are related to sets of possible conducts, in I D AL action terms are related to sets of possible states, i.e., the states where the outcomes of the actions are realized. The task of this intermediate system is twofold: first, it provides us with a suitable starting point to improve D AL; second, due to the associated semantic framework, it allows us to unify the approach to deontic logic based on boolean action algebras and the approach to deontic logic based on dynamic action logic.6

3.1 Language and Semantics Let AV be a countable set of variables for basic action types and P V be a countable set of variables for propositions. The language L I D AL of intensional deontic action logic contains a set T m I D AL of terms and a set Fm I D AL of formulas, respectively defined by the following rules: α := a | 0 | 1 | α | α  β | α  β, where a ∈ AV φ := p | done(α) | P(α) | F(α) | ¬φ | φ ∧ ψ | φ ∨ ψ | φ | [ag]φ | I, where p ∈ PV . Here, done(α) says that the action type denoted by α has just been performed, i.e., that the current state has been reached by performing α. The intended interpretation of the modal formulas is as follows: φ is a global modality, stating that φ holds at 4 See

[2, 5, 7, 19] for further discussions about this definition and its consequences. is the option proposed in [19], where the semantics of D AL is extended by introducing a set of required examples of actions. This move gives us the freedom to put forward conditions on the connections between O, P and F that are consistent with our intuitions. 6 See [1, 2, 4, 13, 20] for introduction and discussions of different systems of deontic action logic based on dynamic action logic. See [5, 12, 16, 17] for detailed presentations of the links between the dynamic approach and the algebraic approach. 5 This

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every possible state; [ag]φ is a local modality, stating that φ if determined from the point of view of the agent, i.e. that all the actions the agent is performing can only lead to states where φ holds; I states that the reference state is a state that respects a deontic ideal which is not necessarily captured by the legal code. The other connectives and the dual operators ♦ and ag are defined as usual. The relation of term inclusion is defined as before. In L I D AL we are able to distinguish to-be deontic notions and to-do deontic notions. In more detail, and in accordance with the tradition of dynamic deontic logic, we are able to define the following kinds of prescription. Definition 5 local to-be permission. State permission: Pst (φ) := ag (I ∧ φ) A state of affairs is permitted precisely when its realization is consistent with the realization of the deontic ideal in the circumstances. The corresponding obligation and prohibition are Ost (φ) := [ag](I → φ) and Fst (φ) := [ag](I → ¬φ). Definition 6 local to-do permission. Action permission: P(α) := ag (I ∧ done(α)) Action α is permitted precisely when performing α is consistent with the realization of the deontic ideal in the circumstances. The corresponding obligation and prohibition are O(α) := [ag](I → done(α)), and F(α) := [ag](I → done(α)). Definition 7 local to-do strong permission. Action strong permission: P∗ (α) := [ag](done(α) → I) Action α is strongly permitted precisely when performing α is sufficient for the realization of the deontic ideal in the circumstance, so that there is no possibility of performing α while violating the ideal. As shown before, strong permission has no intuitive corresponding obligation or prohibition. A global version of these notions can be obtained by substituting , respectively its dual, for [ag], respectively its dual, in the previous definitions. Let us now show how L I D AL is interpreted. Definition 8 Model for L I D AL . A model for L I D AL is a tuple M = (W, R, I deal, D, P, F, V ) where W = ∅ and R : W → ℘ (W ) is such that R(w) = ∅, for all w ∈ W I deal ⊆ W is such that I deal ∩ R(w) = ∅, for all w ∈ W D : T m(L I D AL ) → ℘ (W ) P : W → ℘℘ (W ) is such that P(w) is an ideal on W, for all w ∈ W F : W → ℘℘ (W ) is such that P(w) is an ideal on W, for all w ∈ W V : P V → ℘ (W ) where D, P and F are required to satisfy the following conditions.

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Conditions on D: D1 : D2 : D3 : D4 : D5 :

D(0) = ∅ D(1) = W D(α) = W − D(α) D(α  β) = D(α) ∩ D(β) D(α  β) = D(α) ∪ D(β)

Conditions on P and F: P(w) ∩ F(w) = {∅}, for all w ∈ W . Intuitively, R assigns to each w ∈ W the set R(w) of states that are accessible from w via a particular conduct of the agent; I deal is the set of ideal states, i.e. the set of states where the deontic ideal is not violated; D assigns to each action term α the set D(α) of states where the type denoted by α has just been executed, so that D can be viewed as a function that classifies states in terms of the actions just executed in them [4] (Fig. 1). Since action types are represented by sets of states, the properties of being permitted and prohibited are represented by sets of sets of states. In this kind of models P and F can be viewed as classifiers of actions in terms of their deontic value: sets of states in P(w) correspond to actions that are just executed at w and permitted, while sets of states in F(w) correspond to actions that are just executed at w and prohibited (Fig. 2). In a picture, where the union of P(w) is represented in gray:

Intuitive valuation: Ideal D( )

1. P∗ ( ) is true at w, since all -states are ideal 2. P( ) is true at w, since there are ideal -states

w

Fig. 1 Accessible α-states and to-do permissions. 1. P∗ ( ) is true at w1 2. P( ) is true at w1 3. P( ) is true at w1

Ideal

1. P∗ ( ) is true at w2 2. P( ) is true at w2 3. ¬P( ) is true at w2

w1

Fig. 2 Accessible α-states and kinds of permission

Ideal D( )

D( )

w2

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The constraints on D provide a connection between the algebra of actions and the algebra of states. Hence, according to D1 and D2, there is no state in which 0 is executed, while 1 is executed at any state; according to D3 to execute the negation of an action coincides with not executing that action; and according to D4 and D5 to execute the conjunction of two actions coincides with executing both the first and the second action, while to execute the disjunction of two actions coincides with executing either the first or the second action. Finally, note that, given these conditions, D is completely specified by the values assigned to the variables for basic actions. The following definition of truth is then justified. Definition 9 Truth in a model for L I D AL . The notion of truth is defined by the following recursion. M, w M, w M, w M, w M, w M, w M, w M, w M, w M, w

|= p |= done(α) |= P(α) |= F(α) |= ¬φ |= φ ∧ ψ |= φ ∨ ψ |= φ |= [ag]φ |= I

⇔ w ∈ V ( p) ⇔ w ∈ D(α) ⇔ D(α) ∈ P(w) ⇔ D(α) ∈ F(w) ⇔ M |= φ ⇔ M |= φ and M, w |= ψ ⇔ M |= φ or M, w |= ψ ⇔ ∀v(v ∈ W ⇒ M, v |= φ) ⇔ ∀v(v ∈ R(w) ⇒ M, v |= φ) ⇔ w ∈ I deal

The relation between the deontic notions captured by P and F and the notions captured by P and F can now be characterized in terms of the distinction between permissions and prohibitions as implied by the deontic ideal in the current circumstances and permissions and prohibitions as stated by the deontic code that prescribes what is legal and what is illegal again in the current circumstances. • P(α) is to be intended as: α is the content of an abstract permission in the current circumstances. P(α) can be true at a state even though it is not necessarily true that doing α is consistent with the ideal in those circumstances: e.g. it is permitted, i.e., it is legal, to leave a place, when leaving a place is consistent with all the prescriptions of the reference code, but it is not ideally permitted to leave a place when someone in that place needs help. • P(α) is to be intended as: α is implicitly permitted, being consistent with the ideal in the current circumstances. P(α) can be true at a state even though it is not necessarily true that doing α is permitted by the code: e.g. it may be permitted to use something not yours to help someone in danger, but it can be in principle not permitted, i.e. it can be illegal given the prescriptions of the reference code, to use something not yours.

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• F(α) is to be intended as: α is the content of an abstract prohibition in the current circumstances. F(α) can be true at a state even though it is not necessarily true that doing α violates the ideal in all circumstances: e.g. it may be prohibited, i.e. illegal given the prescriptions of the reference code, to leave a certain place at a certain time, but it is not ideally prohibited to leave that place when someone outside needs help. • F(α) is to be intended as: α is implicitly prohibited, being inconsistent with the ideal in the current circumstances. F(α) can be true at a state even though it is not necessarily true that doing α is prohibited by a code: e.g. it may not be prohibited, i.e. illegal, to cross an intersection when the light is green, but it is ideally prohibited to cross an intersection when the light is green and someone is on the crossroad. As just shown, a first benefit of our intermediate system is that it allows us to define diverse useful notions of permission and prohibition. As we are going to see, a second benefit is that it allows us to characterize Segerberg’s deontic action logic without introducing identity axioms and boolean axioms.

3.2 Axiomatics and Completeness The logic I D AL is characterized by the following set of axioms and rules. Group 1: usual K T 5 axioms and rules for . Group 2: usual K D axioms and rules for [ag] plus φ → [ag]φ ag I Group 3: axioms for done. done 1 : done(1) done 2 : ¬done(0) done 3 : done(α) ↔ ¬done(α) done 4 : done(α  β) ↔ done(α) ∧ done(β) done 5 : done(α  β) ↔ done(α) ∨ done(β) Group 4: deontic axioms. P(α  β) ↔ P(α) ∧ P(β) F(α  β) ↔ F(α) ∧ F(β) P(α) ∧ F(α) ↔ ¬done(α) Group 5: indiscernibility axioms. (done(α) ↔ done(β)) → P(α) ↔ P(β) (done(α) ↔ done(β)) → F(α) ↔ F(β)

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System I D AL is sound with respect to its intended semantics. As to completeness, the proof is based on the construction of the following canonical model. Definition 10 canonical model for L I D AL . Let Δ be a maximal I D AL-consistent set of formulas and let w, v vary on maximal I D AL-consistent sets of formulas. The canonical model MΔ based on Δ is the tuple MΔ = (W, R, I deal, D, P, F, V ), where W = {w : {φ : φ ∈ Δ} ⊆ w} R is such that R(w) = {v ∈ W : {φ : [ag]φ ∈ w} ⊆ v}, for all w ∈ W I deal = {w ∈ W : I ∈ w} D is such that D(α) = {w ∈ W : done(α) ∈ w} P is such that P(w) = {D(α) : P(α) ∈ w}, for all w ∈ W F is such that F(w) = {D(α) : F(α) ∈ w}, for all w ∈ W V is such that V ( p) = {w ∈ W : p ∈ w}, for all p ∈ P V Lemma 4 (Canonicity Lemma): MΔ is a model for LIDAL . Proof Conditions D1 – D5 are satisfied by virtue of the axioms of group 3. In addition, P(w) ∩ F(w) = {∅}: indeed, if D(α) ∈ P(w) and D(α) ∈ F(w), then P(α) ∈ w and F(α) ∈ w. Thus, ¬done(α) ∈ w, by the last axiom of group 4, and so, for every v ∈ W , done(α) ∈ / v. Therefore D(α) = ∅, by the definition of D. Lemma 5 (Truth Lemma): MΔ , w |= ϕ ⇔ ϕ ∈ w. Proof The interesting cases are those concerning done(α), P(α) and F(α). Case 1 : MΔ , w |= done(α) ⇔ done(α) ∈ w. MΔ , w |= done(α) ⇔ w ∈ D(α) MΔ , w |= done(α) ⇔ done(α) ∈ w, by the definition of D(α) Case 2 : MΔ , w |= P(α) ⇔ P(α) ∈ w. MΔ , w |= P(α) ⇔ D(α) ∈ P(w) MΔ , w |= P(α) ⇔ P(α) ∈ w, by the definition of P(w) Case 3 : MΔ , w |= F(α) ⇔ F(α) ∈ w. MΔ , w |= F(α) ⇔ D(α) ∈ F(w) MΔ , w |= F(α) ⇔ F(α) ∈ w, by the definition of F(w) This concludes our sketch of the proof. Let us now focus on the connection between I D AL and Segerberg’s D AL.

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3.3 Interpretation of D AL in I D AL The crucial result of this section is that D AL can be embedded in I D AL, given a specific translation of L D AL into L I D AL . Specifically, we are going to prove that a formula of L D AL is derivable in D AL if and only if its translation is derivable in I D AL. The key idea underlying the translation is that identity of actions can be interpreted as intensional identity, so that α and β coincide precisely when the states where α has just been executed coincide with the states where β has just been executed: D(α) = D(β), corresponding to (done(α) ↔ done(β)). Definition 11 translation function ∗ . ∗

: Fm D AL → Fm I D AL is such that

1. (α = β)∗ = (done(α) ↔ done(β)) 2. (P(α))∗ = P(α), (F(α))∗ = F(α) 3. (¬φ)∗ = ¬φ ∗ , (φ ∧ ψ)∗ = φ ∗ ∧ ψ ∗ , (φ ∨ ψ)∗ = φ ∗ ∨ ψ ∗ . We now show that  D AL φ ⇒ I D AL (φ)∗ , by induction on the length of a derivation, and then that  D AL φ ⇒ I D AL (φ)∗ , by a semantic argument. Theorem 1  D AL φ ⇒ I D AL (φ)∗ , for every φ ∈ L D AL . It suffices to show that the translations of the axioms of D AL are theorems of I D AL. This trivially holds for the D AL axioms of groups 1. As to the D AL axioms of group 2, the translation of every boolean identity is derivable in I D AL, due to the axioms on done. As an example, consider -commutativity: α  β = β  α.  I D AL done(α) ∧ done(β) ↔ done(β) ∧ done(α), by classical logic  I D AL done(α  β) ↔ done(β  α), by done 4 and classical logic  I D AL (done(α  β) ↔ done(β  α)), by the logic of   I D AL (α  β = β  α)∗ , by the definition of ( )∗ As to the D AL axioms of group 3, a straightforward induction on the length of a formula shows that  I D AL (done(α) ↔ done(β)) → φ ↔ φ[α/β]. The induction steps relative to the deontic cases follow from the indiscernibility axioms. Finally, D AL axioms of group 4 coincide with the corresponding I D AL axioms, so that their translation is immediately derivable. Theorem 2  D AL φ ⇒ I D AL (φ)∗ , for every φ ∈ L D AL . Both D AL and I D AL are sound and complete with respect to their intended semantics. Thus, it suffices to show that any model for L D AL can be transformed into a model for L I D AL validating the same translated formulas at any world.

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Definition 12 induced model for I D AL. Let M = (E, Legal, I llegal, I ) be a model for L D AL . The model for L I D AL induced by M is M I = (W, R, I deal, D, P, F, V ), where W =E R =W ×W I deal = W D=I P is such thatP(w) = ℘ (Legal), for every w ∈ W F is such that F(w) = ℘ (I llegal), for every w ∈ W V is such that V ( p) = W, for every p ∈ P V It is evident that M I is a model for L I D AL . Hence, we are only required to show that, for every φ ∈ L D AL , M |= φ ⇔ ∀w ∈ W (M I , w |= (φ)∗ ). Fact 4 M |= φ ⇔ ∀w ∈ W (M1 , w |= (φ)∗ ), for every φ ∈ L D AL . Proof The interesting cases are the non-boolean ones. (1) M |= α = β ⇔ I (α) = I (β) ⇔ ∀w ∈ W (w ∈ D(α) ⇔ w ∈ D(β)), by the definition of D ⇔ ∀w ∈ W (M I , w |= done(α) ⇔ M I , w |= done(β)) ⇔ ∀w ∈ W (M I , w |= (done(α) ↔ done(β))) ⇔ ∀w ∈ W (M I , w |= (α = β)∗ (2) M |= P(α) ⇔ I (α) ⊆ Legal ⇔ D(α) ∈ ℘ (Legal) ⇔ ∀w ∈ W (D(α) ⊆ P(w)) ⇔ ∀w ∈ W (M I , w |= P(α) ⇔ ∀w ∈ W (M I , w |= (P(α))∗ (3) M |= F(α) ⇔ I (α) ⊆ I llegal ⇔ D(α) ∈ ℘ (I llegal) ⇔ ∀w ∈ W (D(α) ⊆ F(w)) ⇔ ∀w ∈ W (M I , w |= F(α) ⇔ ∀w ∈ W (M I , w |= (F(α))∗ This concludes the proof. In conclusion, system I D AL enables us to obtain both an interpretation of D AL and a representation of some key elements of dynamic action logic. Specifically, in systems of action logic based on dynamic logic, action types are represented by sets of transitions, where a set of α-transitions represents all the ways in which action type α can be dynamically accomplished in the current state. This idea can be captured here as follows: action types are represented by sets of worlds, where α-states represent all the ways in which action type α is currently performed. Hence, α-transitions can be

Deontic Logic with Action Types and Tokens Fig. 3 Representation of transitions in I D AL

133 done( )

transitions R

done( )

R R

done( )

R done( )

defined in terms of R-transitions leading to α-states, thus preserving the descriptive power of dynamic action logic (Fig. 3). Still, I D AL presents the same drawbacks as D AL as to the characterization of permissions and obligations, and this brings us to a further development.

4 Hyperintentional Deontic Action Logic Let us now introduce our final system H D AL of hyperintensional deontic action logic. In H D AL we will be able to provide (i) a suitable characterization of the notions of permission and prohibition, (ii) a straightforward definition of the notion of obligation, and (iii) a simple solution to some crucial paradoxes affecting deontic logic.7 As to the first point, we noted that, while in D AL the notion of prohibition respects our intuitive conception concerning the logical relations between prohibited actions, the notion of strong permission turns out to be not so standard, since not every specification of a permitted action seems to be permitted. Strong permission is typically introduced in order to account for the idea of free choice permission, i.e., the idea that, if we are allowed to do α  β, then we are allowed to do both the first and the second action [2, 15, 22]. Still, strong permission, even though successful in providing such an account, seems to be too strong to be understood as an ordinary deontic notion. Hence, our first aim is to introduce a deontic notion consistent with the idea of free choice permission. As to the second point, we noted that in D AL the notion of obligation seems to resist our tentative definitions in terms of prohibition or permission. Hence, our second aim is to introduce notions of prohibition and permission allowing for the definition of obligation. As to the final point, it is wellknown that a variety of systems of deontic logic are subjected to deontic paradoxes [11]. Hence, our final aim is to introduce deontic notions that avoid paradoxes. As we are going to see, H D AL seems to enable us to achieve these three aims in one shot.

7 See

[7, 14] for a presentation of other approaches that allow for a hyperintensional development.

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4.1 Language and Semantics The language L H D AL of hyperintensional deontic action logic coincides with L I D AL , but the intended interpretation of its formulas, specifically the deontic formulas, is different. So, let us start by introducing the notion of model for L H D AL . Definition 13 Model for L H D AL . A model for L H D AL is a tuple M = (W, R, I deal, D, T, P, F, V ) where: W = ∅ R : W → ℘ (W ) is such that R(w) = ∅, for all w ∈ W I deal ⊆ W is such that I deal ∩ R(w) = ∅, for all w ∈ W D : T m(L I D AL ) → ℘ (W ); T = (T, +, tr ) is such that 1. T = ∅ 2. + : T × T → T 3. tr : T m(L DLT ) → ℘ (T ) P : W → ℘ (T ) − {∅} F : W → ℘ (T ) − {∅} where D satisfies the conditions stated in Definition 8 and tr , P and F are required to satisfy specific sets of conditions to be introduced. The new element is given by T = (T, +, tr ), which is a frame for instances of action types, intended as action tropes. T contains three elements: 1. a set T of action tropes; 2. an operation + of trope fusion; 3. an operation tr of type interpretation. The fusion x + y of two tropes is a trope coinciding with that part of the conduct of an agent constituted by trope x and trope y. The operation tr assigns to each term for action types the set tr (α) of tropes instantiating it, so that tr can also be viewed as a classification of tropes in terms of action types. In order to fully understand the significance of this new frame it is crucial to clarify two distinctions: the distinction between action types and action tokens, or tropes, and the distinction between examples and instances of an action types. Let us focus on the conduct that an agent can currently adopt. It is possible for an agent’s conduct to exemplify several action types: thus, it is possible for the conduct of a runner to exemplify, among others, the action of taking a step, the action of taking a breath, and the action of stopping her MP4; moreover, the same runner could exemplify these actions in a huge number of different ways. These ways are the action tropes that can be classified under the corresponding types: a particular taking a step, a particular taking a breath, a particular act of stopping the MP4.

Deontic Logic with Action Types and Tokens

Action type α : taking a step β : taking a breath γ : stopping the MP4

Action tropes classified by tr x1 , x2 , . . . ∈ tr (α) : ways of taking a step y1 , y2 , . . . ∈ tr (β) : ways of taking a breath z 1 , z 2 , . . . ∈ tr (γ ) : ways of stopping an MP4

Conduct type αβ γ αβ γ .. .

Possible conducts current conduct:x1 + y1 + z 1 possible conduct:x2 + y2 + z 2 .. .

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Together, these tropes constitute the particular conduct of the agent. In light of this, we say that an action trope is a particular way in which an action type can be executed by an agent. Thus, if the conduct of the agent is x1 + y1 + z 1 , then the particular way in which α is executed by the agent is x1 . In addition, to distinguish the relation between a conduct and a type α and the relation between a trope and α, we say that the conduct exemplify the action type, while the trope instantiates it, or is an instance of it. Hence, a trope, e.g. this particular taking a step, is an instance of the action of taking a step, which is an action type, and is that in virtue of which a conduct constituted by it is an example of that type. Therefore, a conduct exemplifies action type α precisely when it has as a part a trope that instantiates α.8 The main difference between conducts and action tropes is that it is possible for the same conduct to exemplify, in virtue of its constituting tropes, different action types, while it is impossible for an action trope to instantiate more than one type. This implies that an action type is fully specified in terms of its tropes, thus justifying the introduction of an operation like tr .9 Conditions on tr . T1 : tr (α) = tr (α) T2 : tr (α  β) = tr (α) ∪ tr (β) T3 : tr (α  β) = {x + y : x ∈ tr (α), y ∈ tr (β)} T4 : tr (α  β) = {x + y : x ∈ tr (α), y ∈ tr (β)} T5 : tr (α  β) = tr (α) ∪ tr (β) Given conditions T1 – T5, tr is completely specified by the values assigned to the variables for basic actions and their negations. Let us briefly comment on T1 – T5. The first fact to note is that the negation of an action is not necessarily instantiated by all the tropes that do not instantiate that action, so that we need two different definition for the set of tropes instantiating an 8 I say that a conduct is constituted by a certain number of tropes. The relation of constitution is not

further analyzed, but it could be characterized in terms of the part-whole relation, by assuming that the agent’s conduct is a particular event whose parts are action tropes. 9 In [23] what we intend as an agent’s conduct is presented as an individual action. Tropes, intended as particular actions, and individual actions in von Wright’s sense are not to be confused.

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action and the set of tropes instantiating its negation. In this respect, our semantics is akin to a truth-maker semantics.10 T1 states that α and α share the same tropes, so that α and α are to be considered as the same action type, being indistinguishable in terms of tropes. T2 and T4 characterize the disjunction of two actions. Suppose that a conduct c exemplifies α  β; then that in virtue of which c is such an example is either an instance α or an instance of β. Suppose now that c exemplifies the negation of α  β; then that in virtue of which c is such an example is the fusion of an instance of the negation of α, which prevents c from exemplifying α, and of an instance of the negation of β, which prevents c from exemplifying β. T3 and T5 characterize the conjunction of two actions. Suppose that a conduct c exemplifies α  β; then that in virtue of which c is such an example is the fusion of an instance of α, which ensures that c exemplifies α, and of an instance of β, which ensures that c exemplifies β. Suppose now that c exemplifies the negation of α  β; then that in virtue of which c is such an example is either an instance of the negation of α or an instance of the negation of β. The deontic notions are now interpreted in terms of action tropes: P assigns to each w ∈ W the set P(w) of tropes that are permitted at w, while F assigns to each w ∈ W the set F(w) of tropes that are prohibited at w. Intuitively, P and F can be viewed as a classification of tropes in terms of what is permitted and prohibited according to a the deontic code. Conditions on P and F. C1 : x + y ∈ P(w) ⇒ x, y ∈ P(w) C2 : x, y ∈ F(w) ⇒ x + y ∈ F(w) C3 : P(w) ∩ F(w) = ∅ These conditions are rationality requirements on permissions and prohibitions. Indeed, since a trope x can be viewed as a part of a trope y whenever x + y = y, the first condition implies that, if a composed trope is permitted, then all its parts are also permitted, while the second one implies that, if a trope is prohibited, then all the tropes including it as a part are also prohibited. The idea is simply that no part of a particular legal action is non-legal, while a particular illegal action makes illegal every particular action having it as a part. A further condition to assume could be P(w) ∩ F(w) = ∅, stating that no trope is both permitted and prohibited. However, in what follows I will be more liberal, and so I will not require that this condition is to be satisfied by our models. As a next task, let us define a relation of indiscernibility for action types. Two definitions immediately present themselves: an intensional one and its hyperintensional variant. According to the first one, two action types are indiscernible precisely when 10 The

following exposition will be very sketchy. See [9] for a more detailed presentation of the basics of a truth-maker semantics and [3] for a different application of a truth-maker semantics to deontic logic leading to results partly analogous to the ones obtained here.

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they are performed in the same states. According to the hyperintensional one, two action types are indiscernible precisely when they are performed in the same states and they are denoted by terms built out of the same basic terms. Here we opt for the second definition, which will allow us to solve a number of problems deriving from assuming the intensional option. Definition 14 Indiscernibility. Let var (α) be the set of variables for basic actions occurring in α. I nd(α, β) := D(α) = D(β) and var (α) = var (β) where var (α) is the set of action variables and negated variables occurring in α. Thus, two terms for action types are indiscernible precisely when they are identical both relative to the states in which their referents are executed and relative to the variables and negated variables occurring in them. Conditions on I nd. Ind1 : Ind2 :

I nd(α, β) and tr (α) ⊆ P(w) ⇒ tr (β) ⊆ P(w) I nd(α, β) and tr (α) ⊆ F(w) ⇒ tr (β) ⊆ F(w)

Hence, if action α is permitted or prohibited, then all the corresponding indiscernible actions are permitted or prohibited as well. However, given the hyperintensionality of the notion of indiscernibility, it is not necessary for actions that are intensionally identical to be permitted or prohibited together. The justification of Ind1 and Ind2 is based on a consistency assumption: if the action type corresponding to α is permitted and it is impossible for an agent to discern α from β, then the action type corresponding to β is also to be permitted, otherwise the permission of α would be void, since we could always say that, in the current circumstances, an agent performing α has done something which is not permitted, having done β. Similarly, if the action type corresponding to α is prohibited and it is impossible for an agent to discern α from β, then the action type corresponding to β is also to be prohibited, otherwise the prohibition of α would be void, since one could always say that, in the current circumstances, an agent performing α has done nothing prohibited, having done β. Definition 15 Truth in a model for L H D AL . The notion of truth is defined by the following recursion. M, w M, w M, w M, w M, w M, w M, w M, w M, w M, w

|= p |= done(α) |= P(α) |= F(α) |= ¬φ |= φ ∧ ψ |= φ ∨ ψ |= φ |= [ag]φ |= I

⇔ w ∈ V ( p) ⇔ w ∈ D(α) ⇔ tr (α) ⊆ P(w) ⇔ tr (α) ⊆ F(w) ⇔ M |= φ ⇔ M |= φ and M, w |= ψ ⇔ M |= φ or M, w |= ψ ⇔ ∀v(v ∈ W ⇒ M, v |= φ) ⇔ ∀v(v ∈ R(w) ⇒ M, v |= φ) ⇔ w ∈ I deal

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This definition differs from the definition of truth in a model for L I D AL only relative to the truth conditions of the deontic operators. Here, similarly to the definition of truth in a model of D AL, an action is permitted when all its instances are legal according to the code and it is prohibited when all its instances are illegal according to the code. However, in the present framework, it is possible for actions that are intensionally identical to have different deontic values: for example, it is possible for α to be permitted even if α  (β  β) is not permitted.

4.2 Axiomatics and Completeness The logic H D AL is characterized by the following set of axioms and rules. Group 1 – Group 3: as in I D AL. Group 4: deontic axioms. P(α  β) ↔ P(α) ∧ P(β) F(α  β) ↔ F(α) ∧ F(β)

P(α  β) → P(α) ∧ P(β) F(α) ∨ F(β) → F(α  β)

Group 5: indiscernibility axioms. (done(α) ↔ done(β)) → P(α) ↔ P(β), provided var (α) = var (β) (done(α) ↔ done(β)) → F(α) ↔ F(β), provided var (α) = var (β) As to the axiomatic system, H D AL differs from I D AL in two respects. First: in line with the idea of indiscernibility introduced above, axioms of group 5 are now conditioned, so that two action terms are substitutable for each other in a deontic context only if they share the same variables, making the system hyperintensional. Second: as a consequence, axioms of group 4 no longer imply that P and F are ideallike predicates. Thus, in light of the problems stressed in Sect. 2.3, axioms of group 4 are improved: (i) in line with the idea that legal actions make legal any action which is part of them, we assume the that any permitted specification is the conjunction of permitted actions; (ii) in line with the idea that illegal actions make illegal any action of which they are part, we assume that any specification of a prohibited action is itself prohibited; (iii) in line with the idea that legal actions do not make legal any action of which they are part, we do not assume the that any specification of a permitted action is itself permitted. System H D AL is sound with respect to the intended semantics. Let us check the axioms in groups 4 and 5. Axioms on permission M, w |= P(α  β) ⇔ tr (α  β) ⊆ P(w) ⇔ tr (α) ∪ tr (β) ⊆ P(w), by T2 ⇔ tr (α) ⊆ P(w) and tr (β) ⊆ P(w) ⇔ M, w |= P(α) and M, w |= P(β), by the definition of |=

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M, w |= P(α  β) ⇔ tr (α  β) ⊆ P(w) ⇔ {x + y : x ∈ tr (α), y ∈ tr (β)} ⊆ P(w), byT3 ⇒ tr (α) ⊆ P(w) and tr (β) ⊆ P(w), by C1 ⇒ M, w |= P(α) and M, w |= P(β), by the definition of |= Axioms on prohibition M, w |= F(α  β) ⇔ tr (α  β) ⊆ F(w) ⇔ tr (α) ∪ tr (β) ⊆ F(w), by T2 ⇔ tr (α) ⊆ F(w) and tr (β) ⊆ F(w) ⇔ M, w |= F(α) and M, w |= F(β), by the definition of |= M, w |= F(α) ∨ F(β) ⇔ M, w |= F(α) or M, w |= F(β) ⇔ tr (α) ⊆ F(w) or tr (β) ⊆ F(w) ⇒ {x + y : x ∈ tr (α), y ∈ tr (β)} ⊆ F(w), by C2 ⇒ tr (α  β) ⊆ F(w), by T3, by the definition of |= Axioms on indiscernibility Suppose M, w |= (done(α) ↔ done(β)) and var (α) = var (β). Then D(α) = D(β), so that I nd(α, β), by the definition of I nd. Thus: tr (α) ⊆ P(w) ⇔ tr (β) ⊆ P(w), by Ind1, so that M, w |= P(α) ↔ P(β); tr (α) ⊆ F(w) ⇔ tr (β) ⊆ F(w), by Ind2, so that M, w |= F(α) ↔ F(β). As to completeness, the proof follows from the construction of a canonical model. First of all, let Δ be a maximal H D AL-consistent set of formulas and define the following equivalence relation. Definition 16 α ≈Δ β := (done(α) ↔ done(β)) ∈ Δ and var (α) = var (β). It follows from the axioms on done that ≈Δ is such that every action term α  is equivalent to a term i∈I αi in canonical form, i.e., that every action term α is equivalent to a disjunctive term constituted by conjunctions αi of variables and negated variables from α. Note that the set I of indexes is finite. Example 1 Let α = (a1  a2 )  a3 . Then 1. α ≈Δ (a1  a2  a3 )  (a1  a2  a3 )  (a1  a2  a3 ); 2. (a1  a2  a3 )  (a1  a2  a3 )  (a1  a2  a3 ) is in canonical form, since it is a disjunction of conjunctions of variables and negated variables from α. The procedure for finding a canonical form of an action term is the same procedure used for finding the disjunctive normal form of a formula in classical propositional logic. The canonical forms, up to the order of conjuncts and disjuncts, are canonical representatives for ≈Δ -equivalence classes of action terms. In addition, by choosing the standard lexicographic order of variables, we are able to identify a unique canonical form c(α) for each action term α. This is the canonical representative for the ≈Δ -equivalence class of α, so that β ≈Δ α ⇔ c(β) = c(α).

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Definition 17 conjunctive terms and canonical tropes of α. The notion of conjunctive term is inductively defined as follows: 1. every variable and negated variable is a conjunctive term; 2. if α1 and α2 are conjunctive terms, then α1  α2 is a conjunctive term. The notion of canonical trope of α is then defined exploiting the fact that, since α is equivalent to a disjunctive term in canonical form, it is equivalent to a disjunction of conjunctive terms. Hence, the set T r (α) of canonical tropes occurring in α is defined as T r (α) = {αi : αi is a disjunct in c(α)}. Corollary 1 Let T r (α) be the set of canonical tropes of α. Then:  T r (α) = {αi }i∈I , provided c(α) = i∈I αi . Thus α ≈Δ β ⇔ c(α) = c(β) ⇔ T r (α) = T r (β). Lemma 6 Basic identities satisfied by T r . Without loss of generality, we can assume that α ≈Δ

 i∈I

αi and β ≈Δ

 j∈J

βj.

1. T r (α) = T r (α): straightforward, since α ≈Δ α. 2. T r (α  β) = T r (α) ∪ T r (β):   it follows from the fact that α  β ≈Δ ( i∈I αi )  ( j∈J β j ). β j ∈ T r (β)}: 3. T r (α  β) = {αi  β j : αi ∈ T r (α),   it follows from the fact that α  β ≈Δ ( i∈I αi )  ( j∈J β j ) ≈Δ i∈I, j∈J (αi  β j ). 4. T r (α  β) = {αi  β j : αi ∈ T r (α), β j ∈ T r (β)}, since α  β ≈Δ α  β. 5. T r (α  β) = T r (α) ∪ T r (β), since α  β ≈Δ α  β. Definition 18 canonical model for L H D AL . Let Δ be a maximal H D AL-consistent set of formulas and let w, v vary on maximal H D AL-consistent sets of formulas. The canonical model MΔ based on Δ is the tuple MΔ = (W, R, I deal, D, P, F, V ), where W = {w : {φ : φ ∈ Δ} ⊆ w} R is such that R(w) = {v ∈ W : {φ : [ag]φ ∈ w} ⊆ v}, for all w ∈ W I deal = {w ∈ W : I ∈ w} D is such that D(α) = {w ∈ W : done(α) ∈ w} T = (T, +, tr ) is such that 1. T = {α : α conjunctive} 2. + is such that α + β = α  β 3. tr is such that tr (α) = T r (α) P is such that P(w) = {α : α is conjunctive and P(α) ∈ w}, for all w ∈ W F is such that F(w) = {α : α is conjunctive and F(α) ∈ w}, for all w ∈ W V is such that V ( p) = {w ∈ W : p ∈ w}, for all p ∈ P V

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Note that + is well-defined, since α  β is a conjunctive term, provided that α and β are conjunctive, and that tr (α) = ∅, since every action term is associated to its canonical representative. Lemma 7 (Canonicity Lemma): MΔ is a model for LHDAL . Proof Conditions D1 – D5 are satisfied by virtue of the axioms of group 3. Conditions T1 – T5: are satisfied by lemma 6 and the definition of tr . As to C1. Suppose α + β ∈ P(w). Then P(α  β) ∈ w, by the definition of + and the definition of P, and so P(α), P(β) ∈ w, by the P axioms of group 4. Thus, α, β ∈ P(w), again by the definition of P. As to C2. Suppose α, β ∈ F(w). Then F(α), F(β) ∈ w, by the definition of + and the definition of F. Hence F(α  β) ∈ w, by the F axioms of group 4. Thus α, β ∈ F(w), again by the definition of F. Finally, Ind1 and Ind2 are satisfied given that I nd(α, β) ⇔ α ≈Δ β: I nd(α, β) ⇔ D(α) = D(β) and var (α) = var (β) ⇔ ∀w ∈ W (w ∈ D(α) ⇔ w ∈ D(β)) and var (α) = var (β) ⇔ ∀w ∈ W (done(α) ∈ w ⇔ done(β) ∈ w) and var (α) = var (β) ⇔ ∀w ∈ W (done(α) ↔ done(β) ∈ w) and var (α) = var (β) ⇔ (done(α) ↔ done(β)) ∈ Δ and var (α) = var (β) ⇔ α ≈Δ β Suppose now I nd(α, β), so that α ≈Δ β. Then tr (α) = tr (β), by the definition of tr , and so tr (α) ⊆ P(w) ⇒ tr (β) ⊆ P(w) and tr (α) ⊆ F(w) ⇒ tr (β) ⊆ F(w). Lemma 8 (Truth Lemma): MΔ , w |= ϕ ⇔ ϕ ∈ w. Proof The interesting  cases are those concerning P(α) and F(α). Note that α ≈Δ i∈I αi , where all αi are conjunctive and I is finite. Case 1 : MΔ , w MΔ , w MΔ , w MΔ , w MΔ , w MΔ , w

|= P(α) ⇔ P(α) ∈ w. |= P(α) ⇔ tr (α) ⊆ P(w) |= P(α) ⇔ {αi }i∈I ⊆ P(w) |= P(α) ⇔ ∀i(i ∈ I ⇒ P(αi ) ∈ w) |= P(α) ⇔ P( i∈I αi ) ∈ w |= P(α) ⇔ P(α) ∈ w

by the definition of |= by the definition of tr by the definition of P by the P axioms of group 4 by the P axiom of group 5

Case 2 : MΔ , w MΔ , w MΔ , w MΔ , w MΔ , w MΔ , w

|= F(α) ⇔ F(α) ∈ w. |= F(α) ⇔ tr (α) ⊆ F(w) |= F(α) ⇔ {αi }i∈I ⊆ F(w) |= F(α) ⇔ ∀i(i ∈ I ⇒ F(αi ) ∈ w) |= F(α) ⇔ F( i∈I αi ) ∈ w |= F(α) ⇔ F(α) ∈ w

by the definition of |= by the definition of tr by the definition of F by the F axioms of group 4 by the F axiom of group 5

This concludes the proof. We can now test H D AL as a system for deontic action logic.

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5 Interpreting Prescriptions The notions of permission and obligation give rise to a number of well-known difficult issues in deontic logic. On the one hand, we have to account for different kinds of permission that are not interpretable in terms of a comprehensive general notion [11]. Thus, we distinguish between: (i) bilateral permission, according to which an action is permitted when we are at liberty to perform it, i.e., when we are allowed to perform it and to refrain from performing it, and unilateral permission, according to which an action can be permitted even if we are not allowed to refrain from performing it; (ii) explicit permission, according to which we may perform an action when there is an explicit norm allowing us to do so, and implicit permission, according to which we may perform an action when there is no explicit norm permitting us to do so; (iii) strong permission, according to which an action is permitted when its performance is sufficient for no norm to be violated, and basic permission, according to which an action is permitted even if its performance is consistent with the violation of a norm. On the other hand, it is usually conjectured that the notions of permission, prohibition, and obligation are in specific logical relations and respect some basic logical principles, just like the notions of possibility, impossibility, and necessity. Still, assuming standard logical relations seems to lead to problematic consequences and paradoxes. In this section, by adopting the H D AL-notions of permission and prohibition, I provide a definition of obligation and show that under the proposed interpretation most of the paradoxes affecting deontic action logic can be avoided.

5.1 Basic Principles on Permission and Prohibition Let us start by observing that P captures a notion of free choice permission, since P(α  β) implies both P(α) and P(β). This is indeed one of the most interesting traits of this operator, since it allows us to divide the notion of free choice permission from the notion of strong permission, given that, as we will see, P(α  β) is not derivable from P(α) and P(β). In terms of this notion and the notion of prohibition, we define obligation as follows. Definition 19 The notion O(α) of abstract obligation is defined in terms of prohibition and permission as F(α) ∧ P(α). Let us now consider the relations between P, P, F, F, state some important properties of P and F, and show that the notion of obligation just defined is such that it is possible for two different actions to be obligatory. In order to do that, consider a model M characterized as follows. Let T = (T, +, tr ) be such that

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1. T = {x1 , x2 , x3 } 2. xi + x j = xi , for i = j, and = x3 , for i = j 3. tr (a0 ) = tr (a1 ) = {x1 } and tr (ai ) = {x2 , x3 }, for i = 0, 1 tr (a0 ) = tr (a1 ) = {x2 } and tr (ai ) = {x1 }, for i = 0, 1 Then, define M so that W = {w, v1 , v2 } R is such that R(w) = {v1 , v2 } and R(v1 ) = R(v2 ) = {v2 } I deal = {w, v2 } D is such that D(a1 ) = {w, v1 }, D(ai ) = {w, v2 }, for i = 1 T = (T, +, tr ) is as just defined P is such that P(w) = P(v1 ) = P(v2 ) = {x1 } F is such that F(w) = F(v1 ) = F(v2 ) = {x2 , x3 } V is such that V ( p) = ∅, for all p ∈ P V In a picture: v1

done(a1 ), ¬I

R

P(a1 ), ¬P(a1 ) F(a2 ), ¬F(a2 )

w

R R

v2

done(a2 ), I

R

Then, we obtain the following invalidities. 1. P(α) does not imply P(α). M, w |= P(a1 ), since tr (a1 ) = {x1 } ⊆ P(w), but M, w |= P(a1 ), since R(w) ∩ I deal ∩ D(a1 ) = ∅. 2. P(α) does not imply P(α). M, w |= P(a2 ), since tr (a2 ) = {x2 , x3 }  P(w), but M, w |= P(a2 ), since R(w) ∩ I deal ∩ D(a2 ) = ∅. 3. F(α) does not imply F(α). M, w |= F(a2 ), since tr (a2 ) = {x2 , x3 } ⊆ F(w), and M, w |= F(a2 ), since R(w) ∩ I deal ∩ D(a2 ) = ∅. 4. F(α) does not imply F(α). M, w |= F(a1 ), since tr (a1 ) = {x1 }  F(w), but M, w |= F(a1 ), since R(w) ∩ I deal ∩ D(a1 ) = ∅.

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Hence, the deontic notions defined in terms of the ideal and the deontic notions defined in terms of what is legal and illegal according to a code are logically independent in H D AL. 5. P(α) does not imply P(α  β). M, w |= P(a1 ), since tr (a1 ) = {x1 } ⊆ P(w), but M, w |= ¬P(a1  a2 ), since tr (a1  a2 ) = {x1 , x2 }  P(w). 6. P(α) does not imply P(α  β). M, w |= P(a1 ), since tr (a1 ) = {x1 } ⊆ P(w), but M, w |= ¬P(a1  a2 ), since tr (a1  a2 ) = {x3 }  P(w). 7. F(α  β) does not imply F(α). M, w |= F(a1  a2 ), since tr (a1  a2 ) = {x3 } ⊆ F(w), but M, w |= ¬F(a1 ), since tr (a1 ) = {x1 }  F(w). 8. ¬P(α) does not imply F(α), and so ¬P(α) does not imply O(α). M, w |= ¬P(a1  a2 ), since tr (a1  a2 )  P(w), and M, w |= ¬F(a1  a2 ), since tr (a1  a2 )  F(w). In particular, this point is in line with the idea that P and F represent explicit permission and explicit prohibition, given that not everything that is not explicitly permitted is explicitly prohibited. Two actions are intensionally identical provided that there is no world in which one of them is performed while the other one is not. Hence, the action denoted by α is intensionally identical to action denoted by β precisely when D(α) = D(β). 9. Intensional identity does not imply substitutability in deontic contexts. D(a1 ) = D(a1  (a1  a2 )), since D(a1 ) = D(a1 ) ∪ (D(a1 ) ∩ D(a2 )), but M, w |= P(a1 ) and M, w |= P(a1  (a1  a2 )), since tr (a1 ) ⊆ P(w) but tr (a1  (a1  a2 )) = {x1 , x3 } is not included in P(w). In general, D(α) = D(β) allows us to obtain M, w |= P(α) → P(β) only if the variables occurring in α also occur in β. 10. O(α) ∧ O(β) does not imply (done(α) ↔ done(β)). M, w |= O(a0 ) ∧ O(a1 ), since tr (a0 ) = tr (a1 ) = {x1 } ⊆ P(w) and tr (a0 ) = tr (a1 ) = {x2 } ⊆ F(w), but M, w |= (done(α) ↔ done(β)), since D(a0 ) = D(a1 ). Hence, contrary to what happens in D AL, when O is defined as in Definition 19 it is not true that two obligatory actions are intensionally identical. Since this was the main objection to the adoption of such definition, it seems that we are now legitimated in assuming it.

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5.2 Paradoxes The final task we are to accomplish is to consider the main paradoxes concerning unilateral permission, and specifically free choice permission, and see how they can find a solution in H D AL. In particular, what I will do is to show that a number of well-known deontic arguments with paradoxical conclusions, see [11], turn out to be invalid when interpreted in H D AL. • Result 1: O(α) → O(α  β). (1) P(α  β) → P(α) (2) P(α  β) → P(α) (3) ¬P(α) → ¬P(α  β) (4) O(α) → ¬P(α  β) (5) O(α) → O(α  β) The step from (4) to (5) is invalid: The implication from ¬P(α  β) to O(α  β) fails in H D AL, as shown in 8. Hence, result 1 is not derivable. • Result 2: O(α) → P(β). (1) P(α  β) → P(β) (2) O(α  β) → P(β) (3) O(α) → P(β) The step from (2) to (3) is invalid, since O(α) → O(α  β) fails, given that O(α) is defined in terms of P(α) and P(α) → P(α  β) fails, as shown in 5. • Result 3: P(α) → P(β). (1) (2) (3) (4) (5)

O(α  β) → O(α) O(α  β) → O(α) ¬O(α) → ¬O(α  β) P(α) → ¬O(α  β) P(α) → P(α  β), and so P(α) → P(β)

The step from (4) to (5) is invalid. The implication from ¬O(α  β) to P(α  β) fails in H D AL, as shown in 8. Hence, result 3 is not derivable. • Result 4: P(α) → P(α  β). (1) P((α  β)  (α  β)) → P(α  β) ∧ P(α  β) (2) P((α  β)  (α  β)) → P(α  β) (3) P(α) → P(α  β) The step from (2) to (3) is invalid, since P(α) → P((α  β)  (α  β)) fails, given that the variables occurring in (α  β)  (α  β) do not necessarily occur in α.

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Hansson’s Result 5, to the effect that O(α) ∧ P(β) → P(α  β) and Result 6, to the effect that O(α) ∧ O(β) → P(α  β) give problems only if we neglect the distinction between unilateral and bilateral permission. Since we maintain this distinction we can skip them. • Result 7: P(α) → P(α  β). This implication is not derivable in H D AL, as shown in 6. • Result 8: P(α1  β1 ) ∧ P(α2  β2 ) → P(β1  β2 ). (1) P(α1  α2 ) ∧ P(β1  β2 ) → P(α1 ) (2) P(α1  α2 ) ∧ P(β1  β2 ) → P(β1 ) (3) P(α1  α2 ) ∧ P(β1  β2 ) → P(α1  β1 ) This result is valid, but it is not counterintuitive. Hansson gives us a case of this kind: Suppose that you have a free choice permission to jump or not to jump and you also have a free choice permission to buy or not to buy an ice-cream. It would then follow, according to this derivation, that you have a free choice permission to jump or buy an ice-cream. However, these are not two options to choose between and we suppose free choice permissions to concern options that we have a choice between. However, why is this weird? The idea could be that free choice permissions are permissions we find in normative systems and normative systems do not state free choice permissions concerning options we do not have a choice between, since this would be pointless. Still, it seems that there is no genuine problem in assuming that the implication holds. • Result 9: P(β1  β2 ) → P((α  β1 )  (α  β2 )). (1) (2) (3) (4)

P(β1  β2 ) → P(β1 ) ∧ P(β2 ) P(β1  β2 ) → P((α  β1 )  (α  β1 )) ∧ P((α  β2 )  (α  β2 )) P(β1  β2 ) → P(α  β1 ) ∧ P(α  β2 ) P(β1  β2 ) → P((α  β1 )  (α  β2 ))

The first step is invalid. There is no way to obtain P((α  βi )  (α  βi )) from P(βi ), i = 1, 2, since it is not necessary for the variables occurring in the first term to occur in the second one.

6 Conclusion This paper presents a semantic characterization of the deontic operators of permission and prohibition in terms of sets of action tokens. The resulting deontic action logic is a hyperintensional system that allows us to put forward a fine-grained analysis of the basic deontic notions and to solve some well-known problems in deontic logic. The semantic framework here proposed is sufficiently rich to incorporate some

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insights from both boolean action logic and dynamic logic and two interesting lines of research are open due to that. First, the deontic notions that constitute the core of language L I D AL = L H D AL involve no reference to the conditions under which a certain prescription is triggered. The introduction of corresponding conditional notions would constitute an attractive improvement. Second, in the present framework action tropes are connected to action types in terms of operation tr . However, in the dynamic framework, action tokens are typically identified with transitions between states. Exploring the possibility of interpreting the elements of T on transitions is then an inevitable step in the attempt to better understand deontic action logic, and the logic of action in general.

References 1. Anglberger, A. (2008). Dynamic deontic logic and its paradoxes. Studia Logica, 89, 427–435. 2. Anglberger, A., Gratzl, N., & Roy, O. (2015). Obligation, free choice, and the logic of weakest permissions. The Review of Symbolic Logic, 8, 807–827. 3. Anglberger, A., & Korbmacher, J. (2017). Truthmakers and normative conflicts. Studia Logica, 1–35. 4. Canavotto, I., & Giordani, A. (2018). Enriching deontic logic. Journal of Logic and Computation, 29, 241–263. 5. Castro, P. F., & Kulicki, P. (2014). Deontic logics based on boolean algebra. In R. Trypuz (Ed.), Krister Segerberg on Logic of Actions (pp. 85–117). Dordrecht: Springer. 6. Castro, P. F. & Maibaum, T. S. (2009). Deontic action logic, atomic boolean algebras and fault-tolerance. Journal of Applied Logic, 7, 441–466. 7. Czelakowski, J. Action and deontology. In Ejerhed, E. & Sten L. (ed.) Logic, Action, and Cognition. Essays in Philosophical Logic, (pp. 47–88), Amsterdam: Springer. 8. Dignum, F., Meyer, J.-J. C., & Wieringa, R. J. (1996). Free choice and contextually permitted actions. Studia Logica, 57, 193–220. 9. Fine, K. (2017). A theory of truthmaker content I: Conjunction, disjunction and negation. Journal of Philosophical Logic, 46(6), 625–674. 10. Givant, S., & Halmos, P. (2009). Introduction to Boolean Algebras. Heidelberg: Springer. 11. Hansson, S. O. (2013). The varieties of permissions. In D. Gabbay, et al. (Eds.), Handbook of Deontic Logic and Normative Systems (pp. 195–240). London: College Publications. 12. Kulicki, P., & Trypuz, R. (2017). Connecting Actions and States in Deontic Logic. Studia Logica, 105, 915–942. 13. Meyer, J.-J. C. (1988). A different approach to deontic logic: Deontic logic viewed as a variant of dynamic logic. Notre Dame Journal of Formal Logic, 1, 109–136. 14. Peterson, C. (2017). A logic for human actions. In R. Urbaniak, et al. (Eds.), Applications of Formal Philosophy (pp. 73–112). Cham: Springer. 15. Segerberg, K. (1982). A deontic logic of action. Studia Logica, 41, 269–282. 16. Sergot, M. (2014). Some examples formulated in a ‘seeing to it that’ logic: Illustrations, observations, problems. In T. Müller (Ed.), Nuel Belnap on Indeterminism and Free Action (pp. 223–256). Amsterdam: Springer. 17. Sergot, M., & Robert, C. (2006). The deontic component of action language nC+. In International Workshop on Deontic Logic and Artificial Normative Systems, (pp. 222–237). Berlin: Springer. 18. Trypuz, R., & Kulicki, P. (2009). A systematics of deontic action logics based on Boolean algebra. Logic and Logical Philosophy, 18, 253–270.

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19. Trypuz, R., & Kulicki, P. (2013). On deontic action logics based on Boolean algebra. Journal of Logic and Computation, 25, 1241–1260. 20. Der Meyden, V. R. (1996). The dynamic logic of permission. Journal of Logic and Computation, 6, 465–479. 21. von Wright, G. H. (1951). Deontic logic. Mind, 237, 1–15. 22. von Wright, G. H. (1968). An Essay in Deontic Logic and the General Theory of Action. Amsterdam: North-Holland Publishing Company. 23. von Wright, G. H. (1983). On the logic of norm and action. In Practical Reason, (pp. 100–129). Oxford: Blackwell.

Causal Agency and Responsibility: A Refinement of STIT Logic Alexandru Baltag, Ilaria Canavotto, and Sonja Smets

Abstract We propose a refinement of STIT logic to make it suitable to model causal agency and responsibility in basic multi-agent scenarios in which agents can interfere with one another. We do this by supplementing STIT semantics, first, with action types and, second, with a relation of opposing between action types. We exploit these novel elements to represent a test for potential causation, based on an intuitive notion of expected result of an action, and two tests for actual causation from the legal literature, i.e., the but-for and the NESS tests. We then introduce three new STIT operators modeling corresponding notions of causal responsibility, which we call potential, strong, and plain responsibility, and use them to provide a fine-grained analysis of a number of case studies involving both individual agents and groups.

1 Introduction STIT logic [2, 22] is a powerful framework, under various respects connected to game theory (see [13]), to reason about individual and group agency and, specifically, about the individual and collective notion of seeing-to-it-that. As in game theory, in semantics for STIT logic, every agent is endowed at every moment with a set of available choices, corresponding to the concrete actions she can perform at that moment. Agency is then characterized by the following fundamental feature: A. Baltag · I. Canavotto (B) · S. Smets Institute for Logic, Language and Computation, University of Amsterdam, Amsterdam, The Netherlands e-mail: [email protected] A. Baltag e-mail: [email protected] S. Smets e-mail: [email protected] S. Smets The Logic, Information and Interaction Group, Department of Information Science and Media Studies, University of Bergen, Bergen, Norway © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Giordani and J. Malinowski (eds.), Logic in High Definition, Trends in Logic 56, https://doi.org/10.1007/978-3-030-53487-5_8

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• Independence of agency (IoA): An agent can select any of the choices available to her and something will happen, no matter what the other agents choose. According to IoA, an agent can choose to open the window, and engage in such an activity, independently of what any other agent decides to do. But, precisely because of what these other agents do, her choice may, nevertheless, not result in the window’s being open. In this framework, an agent sees to it that ϕ just in case her choice does indeed ensure that ϕ, no matter what the other agents do. This classic characterization of seeing-to-it-that (or variants thereof [2, 8, 12, 23]) is typically taken to capture the fact that an agent brings about, or causes, ϕ. As attested by a number of recent works [9, 10, 14, 26], this has made STIT theory one of the main reference tools among logicians to reason about different notions of legal and moral responsibility. In fact, establishing a causal connection between what an agent did and ϕ is the very first step towards attributing the responsibility for ϕ to that agent, as no one can be held responsible for something she did not cause. In this perspective, simple variants of the STIT operator taking into account the availability of alternative choices, as the deliberative STIT operator [23], can be taken to represent the lowest level of responsibility, namely causal responsibility, or strict liability, thus providing a basis to analyze more complex notions, involving, for instance, the beliefs and intentions of the agent [9, 10, 26]. The fundamental problem with applying STIT to the analysis of causal responsibility is that the notion of bringing about modeled in STIT semantics is considerably narrower than the corresponding intuitive notion: given the STIT-analysis of seeingto-it-that, the mere fact that other agents may prevent the realization of ϕ is sufficient to exclude that an agent brings about ϕ. This has important consequences on the possibility of using standard STIT semantics to represent causal responsibility—hence, more complex kinds of responsibility—in multi-agent scenarios. To better appreciate this point, consider, for instance, the following hypothetical case. Example 1 Alice shoots David dead. Beth, a bystander, could hit Alice thus preventing David’s death, but she remains still, petrified with fear. Does Alice see to it that David is dead in this scenario? Is she causally responsible for it? According to STIT theory, the answer is negative, since Alice’s choice of shooting does not ensure David’s death (if Beth hits Alice, her shot is diverted). Yet, a jury assessing the case would certainly disagree: after all, Alice did pull the trigger and cause David’s death. The aim of this paper is to refine STIT theory in order to make it suitable to model causal agency and analyze causal responsibility in basic multi-agent scenarios of this sort. To do this, we will view the attribution of causal responsibility for ϕ as a task involving three preliminary phases (see also [16]): 1. Selection of the relevant context. 2. Identification of the potential causes of ϕ. 3. Selection of the actual causes of ϕ.

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In phase 1, we answer the question: Who was involved and what actions did they do? In phase 2, we answer the question: Who did something that was expected to result in ϕ? Finally, in phase 3, we answer the question: Who, in addition, did actually contribute to the result? We devote the rest of this section to clarify the content of each phase and to explain how we will enrich STIT theory to account for it. Relevant context: Agents and action types Imagine we are members of a jury assessing the case in Example 1. Given the evidence presented in court, our first task is to figure out what happened: Who was involved, what did they do, and how did they interact? By answering these questions, we select the relevant agents (individuals or groups) and the relevant actions they were doing when the homicide took place. For instance, we may say that, when David died, Alice and Beth were present at the scene, Alice just fired a shot, and Beth was standing close to her. In this way, we select Alice and Beth as the relevant agents and the actions of Alice’s shooting and Beth’s standing as the relevant actions. Agents and actions we do not deem relevant, like Frank’s reading the newspaper in the park, Carl’s entering a shop two streets away, Beth’s smoking a cigarette (while standing), and so on, are simply ignored. The selected agents and actions form what we can call the relevant context.1 To account for this, we work with a STIT semantics supplied with action types.2 The idea is that, when we describe what an agent did, we typically abstract away from the features of her concrete conduct that are irrelevant to our inquiry, i.e., we refer to a certain type of action she was performing, where the degree of specificity of the description determines the degree of specificity of the corresponding type. In particular, since who performs an action is crucial to attribute responsibility, we work with agent-relative action types. Accordingly, we do not consider the overly generic types of, say, shooting and standing, but the individual types of, say, Alice’s shooting and Beth’s standing or the joint type3 of Alice’s shooting and, simultaneously, Beth’s standing.4 In addition, in line with the tradition of Propositional Dynamic 1 The selection of the relevant context is similar to the process of choosing the variables constituting

a causal model (see, e.g., [16]). As it is not always straightforward to decide which variables should be included in a causal model (and which values they should take), in the same way it is not always straightforward to decide which agents and actions should be included in the relevant context. In modeling case studies in Sect. 4 we will presuppose that the relevant context has been identified, without attempting to pinpoint the criteria leading to this identification—but see [15] for more on this issue. 2 The need to introduce action types in STIT logic is already emphasized in [31] and, more recently, in [3]; the same need is expressed by [20] in relation to the problem of modeling uniform strategies in STIT. In the last decade, several authors, including [11, 21, 24, 32, 34, 40, 41], have taken up this challenge. Our semantics is inspired by these proposals and, in particular, by [21]. From a more philosophical perspective, a detailed analysis of ontology of action against the backdrop of STIT theory can be found, e.g., in [35]. 3 As we will make precise below, joint action types are combinations of individual action types, and their execution coincides with the simultaneous execution of all individual types constituting them. Joint action types are thus the most elementary kind of group actions: they require neither a cohesive group nor coordinated actions. 4 The question naturally arises whether there is a limit to the specification of types: can we admit all of, e.g., Alice’s shooting, Alice’s shooting David, Alice’s shooting David with a rifle, Alice’s

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Logic (P DL) [17], we view the execution of a type at a moment as something that contributes to the course of events at that moment. More specifically, here we focus on one-step actions, meaning that the execution of a type at a moment contributes to the course of events from that moment to the next, or to the transition from the current state of the world to the next. In this framework, it is natural to view agents’ choices as choices to perform certain types of actions. Since the actions chosen at a moment have effects in the next moment, our interpretation of the STIT operators will thus be in the spirit of XSTIT [8, 12].5 Potential causes: Expected results and opposing relation Once we have selected the relevant context (involved agents and their actions), our next task is to identify the potential causes of the negative result before us. We typically do this by considering whether some of the involved agents did something, by herself or with others, that was expected to result in it. Let us consider an elaborated version of Example 1. Example 2 Alice and Carl fire at David, aiming at the heart. Diana also fires at David, but aims at the legs. Beth stands close to Alice, petrified with fear. In assessing the case, we will agree that Alice and Carl did something that was expected to result in David’s death (they both aimed at the heart!), while Beth and Diana did not. We will thus include Alice’s and Carl’s actions in the list of the potential causes of David’s death, while excluding Beth’s and Diana’s actions. Our proposal is to analyze this intuitive notion of expected result by enriching STIT semantics with a primitive relation of opposing between agent-relative action types. In the spirit of interventionist approaches to causation [37], the idea is that, in order to check whether there is a potential causal link between the action performed by an agent and a given state of affairs ϕ, we consider scenarios in which nothing opposes the execution of that action, and see if ϕ is produced.6 In modeling this idea, we will assume a basic view on the opposing relation, namely that an agent-relative action type opposes another just in case the execution of the former intrinsically

shooting David with a rifle while stepping back, and so on, as action types? We assume a minimal view on these matters: the only thing we require is that action types, unlike action tokens, are repeatable, i.e., they can be executed in different places and/or times. Given that this requirement is satisfied, the choice to refer to a more or to a less specific type depends on which details of the considered situation we take to be relevant. Of course assuming this minimal view may raise the worry that it gives the modeler too much flexibility, and that some general criteria to be followed in the modeling practice should be laid down. We leave a more in-depth discussion of this issue to another venue. Thanks to an anonymous referee for raising this point. 5 XSTIT is a version of STIT theory where agents’ choices are modeled as having effects in the next moment, in contrast with classic Chellas STIT theory [2, 22, 23] where choices are represented as having effects at the same moment at which they are taken. 6 It is by performing simple tests of this kind that we expect, for instance, that the man’s walking towards the exit results in him being out of the building, that the kid’s throwing a rock against the window results in the window’s being broken, that Alice’s poisoning David’s water results in David’s death, and so on.

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hinders7 the execution of the latter. The following are then paradigmatic examples of opposing. Example 3 (a) Alice’s pushing a trolley opposes Beth’s pushing in the opposite direction. (b) Beth’s hitting Alice opposes Alice’s firing at David. (c) The crowd’s getting out of the building opposes Carl’s getting in. According to this view, no agent-relative action type opposes itself, since no such type is such that its execution hinders its own execution. In addition, when an agent-relative action type opposes another such type taken alone, it also opposes this latter type taken together with other agents’ actions. So, in Example 3(a), the execution of the type “Alice’s pushing a trolley” intrinsically hinders the execution of the type “Beth’s pushing the trolley in the opposite direction”, and so, a fortiori, the execution of this latter type simultaneously with the execution of any other type.8 Finally, Examples 3(b) and 3(c) show that the opposing relation is not symmetric: Alice’s firing at David does not oppose Beth’s hitting her, nor does Carl’s getting in the building oppose the crowd’s getting out, even if the converse relations hold. Actual causes: But-for and NESS tests Let us go back to Example 2. After identifying Alice’s and Carl’s actions as potential causes of David’s death, we have to decide which of them actually caused it: Was it Alice’s shot, Carl’s shot, or both? In the legal literature,9 the standard test for actual causation is the but-for test: what an agent did was an actual cause of a state of affairs ϕ if, but-for that agent’s action, ϕ would not have occurred. This test is appealing for its simplicity and, in many cases, it provides us with intuitive results, but it cannot handle cases of overdetermination. So, in Example 2, if Alice’s shot and Carl’s shot were separately sufficient for David’s death, then neither of them would satisfy the but-for test and hence there would be no actual cause of the death. This limitation is addressed, in the legal literature, by using the more flexible NESS test [38, 39].10 According to this account, what an agent did was an actual cause of a state of affairs ϕ if it was part of a minimal sufficient condition for ϕ that occurred, where a sufficient condition for ϕ is minimal when none of its parts is a sufficient condition for ϕ (we will come back to this). 7 Note

that hindering is weaker than blocking: in our view, it is possible that two actions, one opposing the other, are executed at the same time without the opposed action being blocked. So, in Example 3(a), Alice’s pushing the trolley and Beth’s pushing in the opposite direction can be executed at the same time without Beth necessarily being stopped. 8 Although the types in these (and later) examples are quite specific, note that they are still types: a situation in which Alice pushes a trolley and Beth pushes it in the opposite direction is a type of situation that can occur at different times and in different places (cf. fn. 4 on this.) 9 See [25] for a recent overview and discussion. 10 The acronym “NESS” stands for “necessary element of a sufficient set”. The NESS test is closely related, in philosophy, to Mackie’s INUS account [27, 28] (“INUS” stands for “insufficient but necessary part of an unnecessary but sufficient condition”) and, in the law, to Hart’s and Honoré’s “causally relevant factor” account [18]. This test is implicit in Belnap’s notion of strict joint agency [1] and has recently been given a game-theoretic formulation and used to analyze responsibility in voting contexts in [5–7].

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In order to account for the last preliminary phase in the attribution of causal responsibility, we will implement both the but-for and the NESS tests in our framework. We will then use the notion of potential cause, understood in terms of expected results, and the notion of actual cause, understood in terms of but-for or NESS conditions, to distinguish and study corresponding notions of causal responsibility, which we call potential, strong, and plain causal responsibility. We start in Sect. 2 by presenting our refined STIT semantics with action types and opposing, and by providing an axiomatization. In Sect. 3, we then define the basic notions of expected result, but-for and NESS dependence, and introduce three new XSTIT operators representing the three above mentioned notions of causal responsibility. After comparing the new operators with the deliberative XSTIT operator, in Sect. 4 we use them to provide a fine-grained analysis of a number of key case studies involving both individual agents and groups. We conclude by suggesting further directions in Sect. 5.

2 XSTIT with Action Types and Opposing Relation Standard frames for STIT logic are defined in terms of branching time structures supplied with agent choice functions (see [22]). In XSTIT, only branching time structures where time is discrete are considered. In this section, we modify STIT frames based on discrete branching time structures by replacing agent choice functions with functions defined in terms of action types and by including a relation of opposing between such types. We then introduce the language LALO of the action logic with opposing ALO and present an axiomatization.

2.1 Frames Let us start by introducing the notions of action type and discrete branching time structure (D BT structure for short). Concerning the former, as we mentioned in the introduction, we work with agent-relative action types. Among them, we distinguish individual, global, and joint action types. Definition 1 (Agent-relative action types) Let Ag ⊆ N be a finite set of agents and T a finite set of basic action types, whose elements are denoted with lower case letters a, b, c, and so on. The sets of individual, global, and joint action types are defined as follows. • Individual action types: for any agent i, Ai = T × {i} is the set of individual action types of agent i (i-action types).  For any a ∈ T, we denote the corresponding element in Ai with ai . IA = i∈Ag Ai is the set of all individual action types.

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• Global action types: any function α : Ag → IA such that α(i) ∈ Ai for all i ∈ Ag is a global action type. We denote the set of all global action types with GA and use Greek letters α, β, γ , and so on, for its elements. • Joint action types: for any group of agents I ⊆ Ag and global action α, the restriction α I of α to I is a joint action of group I (an I -action type). The set of all I -action types is denoted with A I . Clearly, for all α ∈ GA, α Ag = α, and so A Ag = GA. In addition, for all α ∈ GA and i ∈ Ag, we can identify the joint action α{i} and the individual action α(i) and, hence, A{i}and Ai . We can thus define the set of all agent-relative action types as Act = I ⊆Ag A I . A relation of sub-action between the elements of Act can be naturally defined as: α I  β J just in case (i) I ⊆ J and (ii) α I = β I . Intuitively, an individual action type ai is instantiated by any execution of a carried out by i (recall the type “Alice’s shooting”). A global action type (or, in game-theoretic terms, an action profile) is any combination of individual action types, one for each agent in Ag, and it is instantiated by the parallel execution of all individual action types constituting it. Similarly, a joint action type of a group I is any combination of individual action types, one for each member of I , and it is instantiated by the parallel execution of all individual types constituting it (recall the type “Alice’s shooting and, simultaneously, Beth’s standing”). Definition 2 (D BT structure) A D BT structure is a tuple Mom, 0) be constructions of order n over B. Then [X X 1 ...X m ] is a construction of order n over B. (iv) Let x 1 , ..., x m , X (m > 0) be constructions of order n over B. Then [λx 1 ...x m X] is a construction of order n over B. (v) Nothing is a construction of order n over B unless it so follows from (i)-(iv). Tn+1 (types of order n + 1) Let ∗n be the collection of all constructions of order n over B. Then (i) ∗n and every type of order n are types of order n + 1. (ii) If α, β 1 ,...,β m (m > 0) are types of order n + 1 over B, then (α β 1 ... β m ) (see T1 (ii)) is a type of order n + 1 over B. (iii) Nothing is a type of order n + 1 over B unless it so follows from (i) and (ii).

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For the purposes of natural-language analysis, we are usually assuming the following base of ground types: o: ι: τ: ω:

the set of truth-values T, F; the set of individuals (the universe of discourse); the set of real numbers (doubling as times); the set of logically possible worlds (the logical space).

We assume that the universe of discourse ι is multivalued, though we leave aside the cardinality of this basic type. Sets and relations are modelled by (and in typetheoretic terms identified with) their characteristic functions. Thus, for instance, (oι) is the type of a set of individuals, while (oιι) is the type of a relation-in-extension between individuals. Empirical expressions denote empirical conditions, which may or may not be satisfied at the world/time pair selected as points of evaluation. These empirical conditions are modelled as (PWS-)intensions. Intensions are entities of type (βω): mappings from possible worlds ω to an arbitrary type β. The type β is frequently the type of the chronology of α-objects, i.e., a mapping of type (ατ ). Thus α-intensions are frequently functions of type ((ατ )ω) abbreviated as ‘ατ ω ’. Extensional entities are entities of a type α where α = (βω) for any type β. Where w ranges over ω and t over τ , the following logical form essentially characterizes the logical syntax of empirical language: λwλt [...w...t...] Examples of frequently used (PWS-)intensions are: propositions of type oτ ω , properties of individuals of type (oι)τ ω , binary relations-in-intension between individuals of type (oιι)τ ω , and individual offices of type ιτ ω . Logical objects like truth-functions and quantifiers are extensional; truth-functions ∧, ∨, ⊃ are of type (ooo), and ¬ of type (oo). The quantifiers ∀α , ∃α are typetheoretically polymorphic total functions of type (o(oα)), for an arbitrary type α, defined as follows. The universal quantifier ∀α is a function that associates a class A of α-elements with T if A contains all elements of the type α, otherwise with F. The existential quantifier ∃α is a function that associates a class A of α-elements with T if A is a non-empty class, otherwise with F. Notational conventions. Below all type indications will be provided outside the formulae in order not to clutter the notation. Moreover, the outermost brackets of Closures will be omitted whenever no confusion can arise. Furthermore, ‘X /α’ means that an object X is (a member) of type α. ‘X → α’ means that X is typed to v-construct an object of type α, regardless of whether X in fact v-constructs anything. Throughout, it holds that the variables w → ω and t → τ . If C → ατ ω then the frequently used Composition [[C w] t], which is the intensional descent (i.e., extensionalization) of the α-intension v-constructed by C, will be encoded as ‘Cwt ’. When no confusion arises, we use the standard infix notation without Trivialization for the application of truth-functions and quantifiers. Moreover, instead of ‘[0∀λx B]’, ‘[0∃λx B]’ we often write ‘∀x B’, ‘∃x B’, for any B → o, to make quantified formulas easier to read.

Impossible Individuals as Necessarily Empty Individual Concepts E

expresses

construction

v-constructs

denotation

191 has a value at w, t

reference

denotes

Fig. 1 General semantic schema

The general semantic schema involving the meaning (i.e., a construction) of an expression E, denotation (i.e., the object, if any, denoted by E) and reference (i.e., the value of an intension, if the denotation is an intension, in the actual world at the present time) is depicted by Fig. 1. Once the meaning of a term or expression has been given, it can be calculated what its denotation is. Provided the denotation is not a trivial (i.e., constant) intension or a mathematical function, the reference cannot be calculated; instead it must be established by extra-logical and extra-semantic means (i.e., empirical inquiry or mathematical calculation) what the reference, if any, is. TIL does not consider reference a semantic notion, so the semantic value of a term or expression cannot be its reference. Nor can it, in all cases, be its denotation, because there are meaningful terms that do not denote any object, like ‘the greatest prime’. However, the semantic value of an empirical term is invariably its denotation, i.e. an intension. Accordingly, an empirical term expresses a construction of an intension as its meaning and denotes this intension as its semantic value. Therefore, if E expresses as its meaning a construction of the impossible individual office then E denotes the impossible office. Any such E qualifies as an empirical term because its meaning constructs an intension of type ιτ ω .

3.1 Requisites Duží et al. in [9, Chap. 4] introduce a logic of intensions that has been developed into an intensional essentialism which spells out how some intensions supervene on other intensions. The key notion is that of requisite. Intuitively, a requisite is a further intension of a particular type that must, as a matter of analytic necessity, be had by any entity that has—by being (in) the extension of—some initial intension. For instance, the further property of being physically extended is a requisite for having the initial property of being coloured. Formally, a requisite is a relationin-extension between intensions of any type, though typically (for philosophical purposes) between individual properties or offices. The type of requisite we need here is defined as follows. Definition 3 Property as a requisite of an office. Let p → (oι)τ ω and q → ιτ ω be constructions v-constructing a property and an office, respectively; Occ/(o ιτ ω )τ ω the property of an office of being occupied at a given world and time; True/(o oτ ω )τ ω the property of a proposition of being true at a given world and time; and Req/ (o (oι)τ ω ιτ ω ) the relation between a property and an office. Then the property v-

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constructed by p is a requisite of the office v-constructed by q iff [0Req p q] = [0∀λw [0∀λt [[0Occwt q] ⊃ [0Truewt λwλt [pwt qwt ]]]]]. Remark This definition applies the properties Occ and True to an office and a proposition, respectively, because the relation obtains necessarily. If we carelessly defined the relation by way of [0∀λw [0∀λt [pwt qwt ]]] the result would be a falsehood. The reason is that, at those worlds and times at which the office goes vacant, the Composition qwt is v-improper, hence so would the Composition [pwt qwt ] be, hence the universal quantifiers would return the truth-value F. Suppose now that the value of q is the impossible office. Then the antecedent Composition [0Occwt q] v-produces F for every valuation v, and thus the implication v-produces T for any valuation v and any value of p. The upshot is that any property is a requisite of the impossible office. This explosion of requisites is a straightforward corollary of Def. 3. One may wonder: does it matter? Narrowly speaking, it does not, for the explosion simply highlights the fact that the impossibility inherent to the impossible office fails to impose any sort of restriction on what must be true of the occupant. Since this extreme office will nowhere and never have an occupant, it is of no material or empirical import what the requisites of the office are. But – no restrictions, no logic.16 There are basically two options. One is to develop a device to prevent explosion from arising in the first place. The other is to leave the explosion as is and instead develop an additional notion of office that brings explosion to a halt. We are pursuing the second option, so we need to restrict the proliferation of requisites to those that are, intuitively speaking, conceptually relevant, instead of alien, to a given construction that produces an impossible office. That is, we want to devise our logic in such a way that some, but certainly not all, properties can be inferred. What we will be doing is cap the explosion as soon as a witness of inconsistency has been identified.

3.2 Modifiers Another notion we need to analyze due to our example of the fake banknote that is a banknote is that of property modifier.17 This example involves a privative modifier, fake. TIL defines modifiers by way of requisites. First, property modifiers are extensional functions (relations-in-extension) of type ((oι)τ ω (oι)τ ω ), which trade the root property for the modified property. Second, we define the essence of a property. Definition 4 (Essence of a property) Let f , g → (oι)τ ω be constructions of properties; let Ess/(o (oι)τ ω (oι)τ ω ) be a function assigning to a given property f the set of its requisites defined thus: 16 Various versions of relevant/relevance logic have been put forward to address this sort of issue. See, for instance, [4]. 17 See [6] for the TIL treatment of modifiers and [17] for privative modifiers in particular.

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Ess = λf λg [0Req g f ]

0

Then the essence of a property f is the set of its requisites: [0Ess f ] = λg [0Req g f ] Now we are in a position to define a modifier privative with respect to a property h. Definition 5 (Privative modifier) Let the types be: h → (oι)τ ω ; M → ((oι)τ ω (oι)τ ω ); p → (oι)τ ω ; x → ι. Then a modifier M is privative with respect to a property h iff [[0Ess h] ∩ [0Ess [M h]]] = ∅ ∧ 0 ∃p [[[ Ess h] p] ∧ [[0Ess [M h]] λwλt [λx ¬[pwt x]]]]. Hence a private modifier deprives the root property constructed by h of some, but not all, of its requisites by trading them for at least one contradictory property. As a result, the respective properties constructed by h and [M h] are contrary rather than contradictory. No individual can co-instantiate both properties; but many individuals instantiate neither of them.

4 Between Explosion and Sterility We want to be able to infer that (colloquially speaking, at first) the fake banknote that is a banknote must be both a banknote and a fake banknote, and cannot be a zebra, a unicorn, a planet, a former smoker, . . . . The hyperoffice subsumes the properties of being a banknote and being a fake banknote, but it does not subsume being a zebra, etc. Since these two subsumed properties are contrary, hence logically mutually exclusive (because fake is privative with respect to banknote), this hyperoffice produces the impossible office.

4.1 Hyperrequisites of a Hyperoffice Above we defined the essence of an office Off /ιτ ω as the set of all the requisite properties of the office. Next, we are going to show how to derive conceptually relevant hyperrequisites of a hyperoffice. Each hyperoffice *Off /∗n → ιτ ω produces an office Off /ιτ ω . The essence of Off being infinite, we cannot derive constructions of all the requisites belonging to the essence of Off. Yet this does not debar us from deriving those that are conceptually relevant. Therefore, the set of the properties produced by the conceptually relevant hyperrequisites of *Off is a proper subset of the essence of Off.

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The definition of hyperrequisite of a hyperoffice is the central definition being put forward in this paper. It must provide a solution to the problem of explosion of requisites of the impossible office, and the solution must not engender new problems of its own. In particular, it must not reinstate the problem of explosion, only now of hyperrequisites. This explains why we do not apply ex falso quodlibet. And this also explains why we define only primary hyperrequisite of a hyperoffice, where a primary hyperrequisite is defined negatively as one that does not demand refinement in order to be derived. Or, if defined positively, primary requisites are those constructions of properties that can be derived directly from the hyperoffice, i.e. that can be, as it were, ‘read off of’ the hyperoffice as it immediately presents itself.18 Definition 6 (Primary hyperrequisite of a hyperoffice) Let *Off /∗n → ιτ ω ; hyperrequisite *Req/∗n → (o (o ι)τ ω ιτ ω ). Then the primary hyperrequisites *Req of the hyperoffice *Off are those property-producing constructions that are provably derivable from *Off without applying ex falso quodlibet. Remark In TIL we have developed several kinds of proof calculus. They include, inter alia, a general resolution method, the sequent calculus and natural deduction.19 Hence by ‘provably derivable’ we intend the application of any of these methods. Since our goal is to track down an inconsistency somewhere in a given definition of the impossible office, in case there is no explicit inconsistency among the primary hyperrequisites, we go on to derive secondary hyperrequisites of a given hyperoffice *Off. Secondary hyperrequisites of *Off are primary hyperrequisites of another hyperoffice *Off r , where *Off r is obtained by refining *Off. Since the refined construction is provably equivalent to the original one in the sense of producing the same office, in case the office in question is impossible we arrive after a finite number of steps at a pair of contradictory hyperrequisites, at which point we terminate the process. Now we illustrate by way of an example the method of deriving hyperrequisites of a hyperoffice by natural deduction. Let the example be: ‘the only bird that is red or blue’. The meaning of this definite description is the hyperoffice identified with the following Closure: λwλt [0I λx [[0Birdwt x] ∧ [[0Redwt x] ∨ [0Bluewt x]]]] Types. I /(ι(oι)): singularizer, i.e., a function that associates a singleton S of individuals with its unique element and is otherwise undefined; Bird, Red, Blue/(oι)τ ω ; x → ι. We want to derive the set ∗ Req0 of primary hyperrequisites of the above Closure (i.e., of the Closure itself rather than of its product of type ιτ ω ). Here is how. Assume 18 Refinement has been described and defined in [9, Sect. 5.4.4]. The idea behind refinement is that it takes us from an atomic construction (typically, a Trivialization) of a property F to a molecular construction of F. Refinement takes us, e.g., from being a bachelor to being an unmarried adult male. For applications, see [14, Sect. 5], which also defines simplification, the dual of refinement. 19 For details, see, e.g. [3, 5, 10].

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that in a world w and time t an individual a happens to be the only bird that is red or blue. Then: (1) a = λwλt [0I λx [[0Bird wt x] ∧ [[0Red wt x] ∨ [0Bluewt x]]]]wt (2) a = [0I λx [[0Bird wt x] ∧ [[0Red wt x] ∨ [0Bluewt x]]]] (3) [[0Bird wt a] ∧ [[0Red wt a] ∨ [0Bluewt a]]] (4) [0Bird wt a] (5) [[0Red wt a] ∨ [0Bluewt a]] (6) [λwλt λx [[0Red wt x] ∨ [0Bluewt x]]wt a]

∅ β-reduction, 1 IE, 2 ∧E, 3 ∧E, 3 β-expansion, 5

Remark Line 3 employs a rule we call IE (‘singularizer elimination’), and which can be stated formally thus: [a = [0I λx H (x)]] H (a) The rule dictates that if a is an entity equal to the only x such that H (x) then it is derivable that H (a). The types involved are these: H (x)/∗n → o: construction with a free variable x; λx H (x)/∗n → (oα); x/∗n → α; a/∗n → α; I /(α(oα)). The proof of the rule follows immediately from the definition of singularizer as taking a singleton to its element and being undefined at empty or multi-element sets. If [0I λx H (x)] is proper then the set produced by [λx H (x)] is a singleton populated by a; therefore, [a = [0I λx H (x)]] = H (a). For the sake of simplicity, in the proofs that follow we will omit the first two steps and start directly at step (3), thus beginning at the condition that any individual a that would occupy the office would have to satisfy. We just proved that any individual a that would be the only bird that is red or blue would have to have the properties of being a bird and being red or blue ∗

Req0 : {0Bird , λwλt λx [[0Redwt x] ∨ [0Bluewt x]]}

(1)

where the constructions 0Bird → (oι)τ ω , λwλt λx [[0Redwt x] ∨ [0Bluewt x]] → (oι)τ ω are provably derivable from the hyperoffice of the only bird that is red or blue without refinement. By iteration we keep deriving constructions of other properties from the constructions obtained at the previous step. If a given construction is of the form 0F, i.e., the Trivialization of a property F, then we apply the method of refinement. It consists in using an equivalent, molecular construction defining the property F instead of 0F. Hence, at step 2 we obtain the set ∗ Req1 of constructions that are provably derivable from the refined elements of ∗ Req0 : Derivation of ∗ Req1

(2)

Since among ∗ Req0 there is a Trivialization of the property of being a bird, we include a definition of this property instead of 0Bird . We rely on the following defini-

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tion20 : a bird is an endothermic vertebrate of the class Aves, having a body covered with feathers, forelimbs modified into wings, scaly legs, a beak, and no teeth, and laying hard-shelled eggs. Formally, where the identity obtains between the property constructed by the Trivialization and the property constructed by the Closure: Bird = λwλt λx [ [[0Endothermic 0V ertebrate]wt x] ∧

0

[0Featherwt x] ∧ [0W ingwt x] ∧ [[0Toothless [0Beaked 0Jaw]]wt x] ∧ ∀y [[0Laywt x y] ⊃ [[0Hardshelled 0Egg]wt y]] ∧ . . . ] Hence, ∗ Req1 are the constructions of properties that are provably derivable from this definition: ∗

Req1 : {[0Endothermic 0V ertebrate], 0Feather, 0W ing, [0Toothless [0Beaked 0Jaw]], λwλt λx ∀y [[0Laywt x y] ⊃ [[0Hardshelled 0Egg]wt y]], . . .}

Types: Vertebrate, Feather, Wing, Jaw, Egg/(oι)τ ω ; Endothermic, Beaked, Toothless, Hardshelled/((oι)τ ω (oι)τ ω ): property modifiers; [0Endothermic 0Vertebrate], [0Beaked 0Jaw], [0Hardschelled 0Egg] → (oι)τ ω ; [0Toothless [0Beaked 0Jaw]] → (oι)τ ω ; Lay/(oιι)τ ω ; x, y → ι; … – and so on. For instance, in step 3 we can derive from [0Endothermic 0Vertebrate] the constructions of the properties of being a vertebrate and being endothermic, because Endothermic is an intersective modifier with respect to Vertebrate.21 Similarly, we can derive constructions of the properties of having a beak, being toothless, etc. Note that we cannot derive constructions of the property being red or of the property being blue, which is as it should be. These are not among the hyperrequisites of our hyperoffice. Rather, the construction of the (‘disjunctive’) property being red or blue is one of its hyperrequisites. We can check that the set of properties produced by the hyperrequisites of the hyperoffice the only bird who is red or blue is a subset of the set of requisites of the office produced by this hyperoffice. For the sake of simplicity, let us denote the respective office as ‘BRB’. Then, since we proved that if for any world w and time t the office of the only bird that is red or blue is occupied by an individual a then a is a bird and a is red or blue at w, t, we can generalize to obtain this: ∀w∀t [[0Occupiedwt 0BRB] ⊃ [0Truewt λwλt [0Birdwt 0BRBwt ]]] = [0Req 0Bird 0BRB]

20 See 21 For

https://www.dictionary.com/browse/bird. a classification and description of three kinds of property modifiers, see [6, 15, 17].

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∀w∀t [[0Occupiedwt 0BRB] ⊃ [0Truewt λwλt [[0Redwt 0BRBwt ] ∨ [0Bluewt 0BRBwt ]]]] = [0Req λwλt λx [[0Redwt x] ∨ [0Bluewt x]] 0BRB]

Anyway, since there is nothing inconsistent in the definition of the office of the only bird that is red or blue, we might derive still other hyperrequistes without arriving at a pair of contradictory properties. Consider the impossible hyperoffice the only fake banknote that is a banknote. The office produced by this hyperoffice is necessarily vacant, because the properties of being a banknote and being a fake banknote are contrary. As explained above, this is due to the modifier fake being privative with respect to banknote. Thus, fake deprives the property of being a banknote of some—yet not all—of its requisites. For instance, a banknote must be issued by a bank and acceptable as tender, while a fake banknote is a fabrication that is parasitic on this institution, though it may in all other respects be both materially and conceptually indistinguishable from banknotes. Hence, though a fake banknote has much in common with a banknote, it is, and must be, a non-banknote. To forestall an explosion of requisites, we again apply our method of deriving conceptually relevant primary and secondary hyperrequisites of the hyperoffice, until we come across a witness of inconsistency. Thus, we prove that any x that might occupy the office produced by the hyperoffice would have to instantiate also the properties produced by the hyperrequisites. The hyperoffice is this Closure: λwλt [0I λx [[0Banknotewt x] ∧ [[0Fake 0Banknote]wt x]]] Step 1. Derivation of primary hyperrequisites Banknote]}

0

∗

Req0 : {0Banknote, [0Fake

(1) [[0Banknotewt x] ∧ [[0Fake 0Banknote]wt x]] (2) [0Banknotewt x] (3) [[0Fake 0Banknote]wt x]

∅ ∧ E, 1 ∧ E, 1

Step 2. First, refinement of 0Banknote. For now, this definition should do: “A banknote is a promissory note issued by a bank payable to the bearer on demand without interest and acceptable as money, a financial instrument to settle debt”. Furthermore, we can apply the rule that no fake banknote is acceptable as money: [[0Fake 0Banknote]wt x] ¬[0Acceptable-aswt 0Money x] This gives us secondary hyperrequisites ∗ Req1 : {0Promissory-note, λwλt λx [0Issued -bywt 0Bank x], λwλt λx [0Acceptable-aswt 0Money x], . . . , λwλt λx ¬[0Acceptable-aswt 0Money x], . . .}

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We might proceed to steps 3, 4, and so on, ad infinitum. But then we would end up with the explosion of requisites that we wanted to prevent. However, since by deriving relevant hyperrequisites of a hyperoffice producing the impossible office we eventually discover some inconsistency that makes the office impossible, we finish at this point. Among the hyperrequisites in ∗ Req1 there is a pair of contradictory hyperrequisites, namely: λwλt λx [0Acceptable-aswt 0Money x] and λwλt λx ¬[0Acceptable-aswt 0Money x] This is the main reason for introducing the notion of hyperrequisite of a hyperoffice. True, since the requisites of the impossible office are any and all properties whatsoever, they must include pairs such that they contradict each other. Only we do not know in advance whether a given hyperoffice under scrutiny produces the impossible office. Our method of deriving hyperrequisites of a hyperoffice makes it possible to eventually discover such conceptually relevant witnesses of impossibility. Hence, in no possible world at no time is there an individual that could hold the office, because this individual would have to be acceptable as money and at the same time not acceptable as money. For another example, we want to derive that 0Man is a hyperrequisite of the hyperoffice the man without properties, thus proving this hyperoffice to be inconsistent without also deriving just any property-producing procedure as yet another of its hyperrequisites. Here is the derivation: (1) λwλt [0I λx [[0Manwt x] ∧ ∀p ¬[pwt x]] = a] (2) [0I λx [[0Manwt x] ∧ ∀p ¬[pwt x]] = a] (3) [[0Manwt a] ∧ ∀p ¬[pwt a]] (4) [0Manwt a] (5) ∀p ¬[pwt a] (6) ¬[0Manwt a]

∅ λE, 1 IE, 2 ∧E, 3 ∧E, 3 ∀E, 0Man/p, 5

In lines (4) and (6) we have derived a pair of contradictory hyperrequisites. So, our derivation terminates in keeping with Def. 6. Had our derivation not terminated, we could have gone on to derive that any property-producing procedure was a hyperrequisite of the hyperoffice in question, including a procedure producing the property of being a woman: (7) ¬[0Manwt a] ∨ [0W omanwt a] (8) [0W omanwt a]

∨I, 6 MTP, 4,7

If we were to allow the derivation that the man without properties is a woman we would be pulling the rug from under our key notion of hyperoffice and as a result could not carry out the advertised hyperintensional exploration of the realm of

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‘impossible individuals’. This goes to show why the ban on quodlibet is central to Def. 6 and far from being ad hoc. For a more elaborate example, we turn to the example of the hyperoffice of the quickest runner (a) who never overtakes the slowest runner (b), though a runs m times faster than b, provided a allows b a head start of n > 0 metres and both run at a constant speed. Our task is to prove that this hyperoffice constructs the impossible office by demonstrating that there is an inconsistency buried in its definition. Let us analyze the definition. We start out with type assignments: variables x, y → ι; u, t → τ ; Runner/(oι)τ ω ; Quickest, Slowest/(ιτ ω (oιτ ω )): a function taking a property to an office; (constant) V(elocity-of)/(τ ι)τ ω ; D/(τ ι)τ ω : the distance covered by an individual. The hyperoffice has this logical structure: λwλt [0I λx ∃y [[x = [0Quickest 0Runner]wt ] ∧ [y = [0Slowest 0Runner]wt ] ∧ [[0Vw x] = [m [0Vw y]]] ∧ [[0Dw x] = 00] ∧ [[0Dw y] = n] ∧ ¬∃u [[u > t] ∧ [[0Dwu x] = [n + [0Dwu y]]]]]] Derivation: (a) λwλt [0I λx ∃y [[x = [0Quickest 0Runner]wt ] ∧ [y = [0Slowest 0Runner]wt ] ∧ [[0Vw x] = [m [0Vw y]]] ∧ [[0Dw x] = 00] ∧ [[0Dw y] = n] ∧ ∅ ¬∃u [[u > t] ∧ [[0Dwu x] = [n + [0Dwu y]]]]]] (b) [0I λx ∃y [[x = [0Quickest 0Runner]wt ] ∧ [y = [0Slowest 0Runner]wt ] ∧ [[0Vw x] = [m [0Vw y]]] ∧ [[0Dw x] = 00] ∧ [[0Dw y] = n] ∧ λE, (a) ¬∃u [[u > t] ∧ [[0Dwu x] = [n + [0Dwu y]]]]]] (c) λx ∃y [[x = [0Quickest 0Runner]wt ] ∧ [y = [0Slowest 0Runner]wt ] ∧ [[0Vw x] = [m [0Vw y]]] ∧ [[0Dw x] = 00] ∧ [[0Dw y] = n] ∧ IE, (b) ¬∃u [[u > t] ∧ [[0Dwu x] = [n + [0Dwu y]]]]] (d) ∃y [[a = [0Quickest 0Runner]wt ] ∧ [y = [0Slowest 0Runner]wt ] ∧ [[0Vw a] = [m [0Vw y]]] ∧ [[0Dw a] = 00] ∧ [[0Dw y] = n] ∧ λE, a/x, (c) ¬∃u [[u > t] ∧ [[0Dwu a] = [n + [0Dwu y]]]]] (e) [[a = [0Quickest 0Runner]wt ] ∧ [b = [0Slowest 0Runner]wt ] ∧ [[0Vw a] = [m [0Vw b]]] ∧ [[0Dw a] = 00] ∧ [[0Dw b] = n] ∧ ∃E, b/y, (d ) ¬∃u [[u > t] ∧ [[0Dwu a] = [n + [0Dwu b]]]]] ∧E, (e) (f) [a = [0Quickest 0Runner]wt ] ∧E, (e) (g) [b = [0Slowest 0Runner]wt ] ∧E, (e) (h) [[0Vw a] = [m [0Vw b]]] ∧E, (e) (i) [[0Dw a] = 00] ∧E, (e) (j) [[0Dw b] = n] ∧E, (e) (k) ¬∃u [[u > t] ∧ [[0Dwu a] = [n + [0Dwu b]]]] In line (k) we have derived that there is no future moment u at which Achilles (a) has caught up with the tortoise (b). With a view to unearthing an inconsistency in this hyperoffice, we have to take physical laws on board, thereby stepping beyond pure logic. Physics tells us that d = v t, where d is distance, v velocity, and t time elapsed. At a constant velocity, the distance that something travels is equal to its

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velocity multiplied by the time spent. Since physical laws are empirical necessities, in TIL notation the law can be formulated like this.22 Let z → ι; t → τ : the time spent moving. Then: λw ∀t ∀z [[0Dw z] = [[0Vw z] t]] Let us calculate. The distance the quickest runner, Achilles (a), covers at time u > t is m times larger than the distance the slowest runner, the tortoise (b), covers at the same time u, because the velocity of a is equal m times the velocity of b: [[0Vw a] = [m [0Vw b]]] ⇔ [[[0Dw u] a] = [m [[0Dw u] b]]] or “[0Dwu a] = [m [0Dwu b]]” for short. Achilles gives the tortoise a much-needed head start n, [[0Dwt a] = 00] ∧ [[0Dwt b] = n]. Should Achilles overtake the tortoise at some time u > t, he must catch up with it first. The distance Achilles must run at this time u is the head start n plus the distance the tortoise runs at the same time u: [0Dwu a] = [n + [0Dwu b]] Since [0Dwu a] = [m [0Dwu b]], i.e. [0Dwu b] = [[0Dwu a] : m], we have the simple algebraic equation: [0Dwu a] = [n + [[0Dwu a] : m]]. Yet this equation has a solution, namely: [0Dwu a] = [[n m] : [m − 1]] Since both n and m are given constant numbers, there is a time u at which Achilles catches up with the tortoise: ∃u [[u > t] ∧ [[0Dwu a] = [n + [0Dwu b]]]] which contradicts the construction at line (k). Thus, we have just proved that there is an inconsistency in the above hyperoffice; hence, there is no quickest runner (provided the laws of physics remain fixed); i.e., the hyperoffice constructs the impossible office.23 22 For the sake of simplicity, we use usual mathematical notation for arithmetic operations. For instance, we write ‘[m [0Vw b]]’, ‘[[0Vw a] t]’ for ‘[0Times m [0Vw b]]’ and ‘[0Times [0Vw a] t]’, respectively, where Times/(τ τ τ ) is the multiplication function. Similarly for the other mathematical functions of addition (+), subtraction (−), and division (:). 23 Note, however, that the condition of both runners maintaining a constant speed is critical here, because otherwise we could not apply the above physical law d = v t. If Achilles and the tortoise would get tired and keep slowing down in the right way, the series of gaps between them would not converge to zero. In Zeno’s paradox the gaps are n, mn , mn2 , mn3 , . . . , which converge, because the series m1 + m12 + m13 + . . . converges to 1. But, for example, the series 21 + 13 + 14 + . . . which is ostensibly convergent is actually divergent. If Achilles had run the first part of the rate at 21 km/h

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5 Conclusion The problem we have addressed above is how to develop a hyperintensional notion of ‘impossible individual’ within Transparent Intensional Logic. The solution we put forward contains the novel concepts of hyperoffice and hyperrequisite. We identified ‘impossible individuals’ with a particular category of hyperoffices. The two novel concepts enabled us to calculate for a given hyperoffice whether or not it constructs the impossible office, which is necessarily vacant because any construction or definition of it is logically inconsistent. Our solution invokes no impossible individualsin-extension and only marginally intensions, namely the impossible office. Rather the focus is on structured hyperintensions—constructions as defined by Transparent Intensional Logic—some of which are typed to construct offices, including the impossible office. Acknowledgements Marie Duží and Bjørn Jespersen were supported by the Grant Agency of the Czech Republic, project No. GA18-23891S, Hyperintensional Reasoning over Natural Language Texts. Daniela Glavaniˇcová was supported by the Slovak Research and Development Agency under the contract no. APVV-17–0057 and by the VEGA grant No. 2/0117/19 Logic, Epistemology and Metaphysics of Fiction. We are grateful to members of the Department of Analytic Philosophy (Slovak Academy of Sciences) for comments on a previous version of this paper, in particular to Miloš Kosterec, Martin Vacek and Marián Zouhar. Versions of this paper were presented by Daniela Glavaniˇcová at Trends in Logic, Milan, 24- 27 September 2018; and Bjørn Jespersen at OZSW 6th Conference, University of Twente, 9–10 November 2018, and Modal Metaphysics: Issues of the (Im)Possible VI, Bratislava, 2–4 August 2018.

References 1. Bach, E. (1986). Natural language metaphysics. In Barcan Marcus et al. (Ed.) Logic, Methodology and Philosophy of Science VII (pp. 573–595). Elsevier. 2. Berto, F. (2017). Impossible worlds and the logic of imagination. Erkenntnis, 82, 1277–1297. ˇ 3. Cíhalová, M., Duží, M., Ciprich, N., & Menšík, M. (2010). Agents’ reasoning using TIL-Script and Prolog. Frontiers in Artificial Intelligence and Applications, 206, 135–154. 4. Dunn, J. M., & Restall, G. (2002). Relevance logic. In F. Guenthner & D. Gabbay (Eds.), Handbook of Philosophical Logic 6 (pp. 1–136). Dordrecht: Kluwer. 5. Duží, M. (2012). Extensional logic of hyperintensions. Lecture Notes in Computer Science, 7260, 268–290. 6. Duží, M. (2017). Property modifiers and intensional essentialism. Computación y Sistemas, 7260, 268–290. 7. Duží, M. (2018). Negation and presupposition, truth and falsity. Studies in Logic, Grammar and Rhetoric, 54, 15–46. 8. Duží, M. (2019). If structured propositions are logical procedures then how are procedures individuated? Synthese, 196, 1249–1283. 9. Duží, M., Jespersen, B., & Materna, P. (2010). Procedural Semantics for Hyperintensional Logic. Foundations and Applications of Transparent Intensional Logic. Berlin: Springer. and the tortoise at 21 km/h, then they would slow to thus the tortoise would always remain ahead.

1 3

km/h and

1 4

km/h, respectively, and so on,

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10. Duží, M., Menšík, M. (2020). Inferring knowledge from textual data by natural deduction. Computación y Sistemas 24(1), pp. 29–48 . 11. Fine, K. Constructing the impossible. In a forthcoming collection of papers for Dorothy Edgington. 12. Glavaniˇcová, D. (2017). Tichý and fictional names. Organon F, 24, 384–404. 13. Glavaniˇcová, D. (2018). Fictional names and semantics: towards a hybrid view. Objects of Inquiry in Philosophy of Language and Literature (pp. 59–73). Studies in Philosophy of Language and Linguistics, Berlin: Peter Lang. 14. Jespersen, B. (2015). Structured lexical concepts, property modifiers, and Transparent Intensional Logic. Philosophical Studies, 172, 321–345. 15. Jespersen, B. (2016). Left subsectivity: how to infer that a round peg is round. dialectica 70, 531–547. 16. Jespersen, B. (2019). Anatomy of a proposition. Synthese, 196, 1285–1324. 17. Jespersen, B., Carrara, M., & Duží, M. (2017). Iterated privation and positive predication. Journal of Applied Logic, 25, 48–71. 18. Kaplan, D. (1975). How to Russell a Frege-Church. Journal of Philosophy, 72, 716–729. 19. Kosterec, M. (2018). On the essence of empty properties. Synthese, https://doi.org/10.1007/ s11229-018-02036-1 20. Priest, G. (2005). Towards Non-Being. Oxford: Oxford University Press. 21. Sedlár, I. (2019). Hyperintensional logic for everyone. Synthese, https://doi.org/10.1007/ s11229-018-02076-7. 22. Slater, B. H. (1994). The epsilon calculus’ problematic. Philosophical Papers, 23, 217–242. 23. Tichý, P. (1975). What do we talk about? Philosophy of Science, 42, 80–93. 24. Tichý, P. (1979). Existence and God. Journal of Philosophy, 76, 403–420. 25. Tichý, P. (1988). The Foundations of Frege’s Logic. De Gruyter. 26. Vacek, M. (2017). Fiction: impossible! Axiomathes, 28, 247–252.

Metalinguistic Focus in P-HYPE Semantics Luke Edward Burke

Abstract P-HYPE is a hyperintensional situation semantics in which hyperintensionality is modelled as a ‘side effect’, as this term has been understood in natural language semantics [8, 42], and in functional programming [21]. In [5], we combine [2]’s perspective-sensitive semantic theory with a hyperintensional situation semantics, HYPE [28], using monads from category theory in order to ‘upgrade’ an ordinary intensional semantics to a possible hyperintensional counterpart. In [6] we expand our semantic theory to capture cases of intra-sentential anaphora. We here supplement the framework with the pointed power set monad so that we can account for certain cases of hyperintensionality involving metalinguistic focus [29] which seem to resist treatment in P-HYPE as it stands. This gives a mini case study in how to combine monads together in order to integrate different side effects in one language, one of the advantages of monads as a tool in compositional semantics.

1 Introduction Hyperintensional semantic theories, as we are understanding them in this paper, are semantic theories that sometimes distinguish the semantic values of sentences that express logical and mathematical truths (or falsities), where by ‘logical truths’ we are referring to the validities of classical logic. Such sentences are often assigned the same semantic value in semantic theories of natural language based on standard possible world semantics [18]. L. E. Burke (B) Department of Philosophy, The University of London, Gower Street, WC1E 6BT London, England University College London, London, England Foundations of Computer Science, The University of Bamberg, An der Weberei 5 (ERBA), 96047 Bamberg, Germany Universität Bamberg, Bamberg, Germany e-mail: [email protected]; [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Giordani and J. Malinowski (eds.), Logic in High Definition, Trends in Logic 56, https://doi.org/10.1007/978-3-030-53487-5_10

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In these frameworks, intensionality is usually modelled via possible worlds, and the semantic value of an expression is its intension, construed as a function from indices to its extension, where an index might include a world, time and speaker and possibly other parameters.1 This gives rise to the prediction that the semantic values of two sentences will be the same if and only if they are intensionally equivalent, in the sense of denoting the same function from indices to truth values (see [17] and references therein).2 But this prediction is highly counterintuitive. Consider the following example, from [11, p.82].3 Let us call a set ‘finite’ iff it cannot be put into a one-one correspondence with a proper subset of itself, and a set ‘inductive’ iff it can be put into a one-one correspondence with a proper initial segment of the natural numbers. Readers may be more familiar with the terms ‘Dedekind finite’ for the former term and ‘finite’ for the latter [7]. If the axioms of Zermelo-Fraenkel set theory with the axiom of Choice (ZFC) are true in all worlds of every standard model, then (1) and (2) have the same intension, as do (3) and (4), since the inductive sets and the finite sets are provably equivalent in ZFC.4 (1)

1 In

Kim proved that the prime numbers are infinite.

(2)

Kim proved that the prime numbers are not inductive.

order to account for context sensitivity, semantic values might be construed as functions from contexts (of utterance) to indices to extensions (Kaplanian characters [25]), where both contexts and indices are construed as tuples of features which may include a world, a location, a speaker, a standard of taste, a variable assignment and various other so-called aspects [50] which are relevant to the interpretation of sentences. For this reason, semantic values are usually taken to be much more complex than functions from worlds to extensions. The question has been posed [36] as to whether such complicated semantic values as functions from indices to truth values can play the role traditionally attributed to propositions or contents, as being the objects of assertion and various speech acts; likewise, one can question whether the contents of sub-sentential words are identical to their semantic values with some theorists ([49] and references therein) retaining that semantic values and contents are in fact distinct. Given a distinction between semantic values and contents, the question of hyperintensionality can thus be posed at two levels: at the level of semantic value and at the level of content. We will not explore this question here, but in this article we intend to present a particular theory of hyperintensional semantic values. 2 Of course, in a framework in which intensions are functions from indices to extensions, various notions of intensional equivalence can be defined, depending on what aspects of the index are relevant for assessing intensional equivalence. It is therefore striking that not much of the literature on hyperintensionality fails to consider the ramifications of this point, resting content with a simplistic understanding of intensional equivalence (expressing the same function from worlds to extensions) which fails to take into account what the concept might mean in a richer semantics in which indices contain a number of so-called aspects in the sense of [50]. 3 This example also brings out the theory relativity of certain mathematical statements. For example, the finite and the inductive sets are not the same in some constructive forms of set theory. We have considered ways of modelling this relativity to a mathematical theory via our semantic theory, but will not discuss this issue here. 4 Throughout this article ‘the prime numbers’ denotes the set of prime numbers. This particular example could, of course, be replaced by another example in which two mathematical predicates have the same intension but differ in meaning, but which does not involve the assumption that the axioms of ZFC are valid in every model.

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(3)

The prime numbers are infinite.

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(4)

The prime numbers are not inductive.

But (1), (2) intuitively differ in meaning, as do (3) and (4). For this reason, finegrained semantic values have been advocated as a way of capturing intuitive distinctions in meaning which intensional semantic values fail to capture. P-HYPE is a hyperintensional semantic theory which extends HYPE [28] to the subsentential level, providing a compositional semantic theory of natural language. The fact that P-HYPE is compositional is a point in its favour in comparison to some of the alternative theories of hyperintensionality in the philosophical literature [15, 23, 48], which do not provide a compositional analysis of sentences at the subsentential level, or at least present a systematic method of pursuing compositional semantics. Another point in its favour is that P-HYPE is also able to capture certain perspective relative phenomena, since it incorporates and develops the semantics of perspective in [2].5 The basic strategy for dealing with hyperintensionality in P-HYPE is this: in P-HYPE we sometimes assign distinct semantic values to logically equivalent sentences because semantic values in P-HYPE vary according to the perspective index they are supplied with.6 Perspective relativity can be banished whenever necessary, by using the enlightened perspective, a special designated perspective which allows expressions to receive their ordinary intensional interpretation. In P-HYPE, following [2](‘AG’ from now on), names are perspective relative, and attitude verbs take complement sentences whose semantic value varies according to a perspective index. But in P-HYPE we make predicates perspective relative, too. We also follow [2] in emphasising the distinction between the perspective of the grammatical subject of a sentence containing an attitude verb, as opposed to the perspective of the utterer of that sentence. It is widely acknowledged in the literature on perspective relativity (see [27] and references therein, [32, 40]) that the perspective of the grammatical subject of a sentence containing an attitude verb and the perspective of the utterer of that sentence play a particularly important role in the semantics of perspectival language, and, as [27] has pointed out, sometimes the relevant perspective is neither the subject nor the utterer.7 We employ monads in P-HYPE because it allows us to include perspective sensitive semantic values without revision of compositional rules. However, monads offer a compelling lens with which to view hyperintensionality itself, as a ‘side-effect’ of semantic computations.

5 We have also considered an extension of HYPE which uses the simply typed λ-calculus but which

does not incorporate perspective relativity. reviewer asks how we would distinguish co-intensional predicates like ‘opthalmologist’ and ‘oculist’ in P-HYPE. We make these predicates perspective relative, though they have the same semantic value relative to the so-called enlightened perspective, which we discuss later. For this reason, we can sometimes distinguish the predicates. 7 We do not have a theory of how perspective indices are chosen in a context and leave the exploration of this topic for another paper. We suspect that our account will have much in common with the hyperintensional semantic theories in [10, 13], in which hyperintensionality is modelled by means of de re interpretations and various pragmatic principles enable us to select an appropriate de re reading. 6A

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In Sect. 2 we propose to analyse hyperintensionality as a side effect, and then discuss [2]’s semantic theory in Sect. 2.1. In Sect. 3 we introduce HYPE and then PHYPE (Sect. 3.1). We then (Sect. 4) discuss what perspectives are in our framework and introduce toy lexical entries (Sect. 4.1) and examples of compositional analyses in P-HYPE involving attitude verbs Sect. 4.2.8 In (Sect. 5) we discuss how we can integrate metalinguistic focus [29] into our account and suggest in closing how it solves a potential problem with HYPE as a hyperintensional theory.

2 Hyperintensionality as a Side Effect Let intensional semantic values be functions from sets of worlds to other entities [18] and focus semantic values be sets of alternatives (see [38]). Different linguistic phenomena require different kinds of semantic value. For example, nondeterministic, intensional and state-changing semantic values are used to capture the effects of focus, modality and anaphora compositionally [8]. Reference [41] had the intuition that we can model many phenomena such as intensionality, focus and anaphora—which might naively seem non-compositional—by using monads. Monads had long been used in functional programming to model so-called ‘side effects’ [21]—programming language features which result in two expressions that normally have the same denotation yielding different results from one another in certain larger expressions. But [42] argues that we can speak of ‘linguistic side effects’. We can see intensionality, focus sensitivity and perspective relativity as side effects: they are features which give rise to linguistic environments in which two expressions cannot be substituted salva veritate, despite ordinarily having the same denotation; and, more generally, we can think of a side effect of an expression as those aspects of it which go beyond its normal denotation. [42] includes amongst so-called linguistic side effects certain types of referential opacity [44] and certain expressions whose meaning and compositional contribution is not pre-theoretically transparent. Hyperintensionality is arguably a good example of a linguistic side effect in Shan’s sense, since it is not pre-theoretically obvious how to model what fine-grained distinctions between logically equivalent statements are necessary and how to characterise their behaviour compositionally. But from both [8] and Shan’s list of linguistic side effects, hyperintensionality is conspicuous by its absence. We propose that hyperintensionality be added to the list of linguistic side-effects, and try to study it from this vantage point.9 Monads have been used to model linguistic side effects (see [1, 3, 4, 8, 9, 19, 43, 45]); and, in particular, to obviate revision of compositional rules whenever new types of semantic value are added to a semantic theory (for motivation along these lines, see [8, 41]). Intuitively, they can be thought of as algebraic structures which generate 8 Sections 3.1

and 4.1 recap some of the material in [5].

9 Of course, it might be the case that there are in fact various different types of hyperintensional side

effect, and the fact that we have to supplement P-HYPE with additional monads later on suggests this may indeed be the case.

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fancy semantic values and integrate them into a semantic theory with minimal disruption to compositional derivations. Monads (which we define below) map operations and values in a given type-space-for example, the type space provided by the simply typed λ-calculus with basic types for individuals and Booleans-with operations and values in an enriched type-space (which might include such exotic semantic values as intensions, focus-sensitive and perspective relative semantic values10 ) whilst preserving ordinary extensional function application as the main compositional principle. Moreover, they do this without generalising to the worst case, where, by ’generalising to the worst case’ we mean the strategy of assigning all expressions of some syntactic category a complicated semantic type, whenever one expression of that syntactic category requires that complicated type to deal with some phenomena.11 Normally, to compositionally combine intensions we need intensional function application [20], in addition to extensional function application. Monads allow us to do forego this additional compositional rule [41]. It is important to emphasise that monads are a technique for implementing a theory of hyperintensionality, as opposed to a theory of hyperintensionality itself. But monads are also attractive specifically for the design and implementation of hyperintensional semantic theories, since they provide a means of combining many side effects in one language [8], and, discussions of hyperintensionality often include many different kinds of phenomena: the question of the semantic treatment of propositional attitudes towards contradictions or impossibilities, the treatment of co-referring names, the role of ‘metalinguistic’ or syntactic information in propositional attitude reports and the treatment of resource-bounded reasoning. Different theories of hyperintensionality have explored these topics in different ways with different methods. Monads, we suggest, may be a natural way of integrating different ‘effects’ that these theories employ in one language. Indeed, in [8] we already see how monads have used to integrate intensionality, focus semantics, dynamic semantics and continuations, and the integration of multiple ‘side-effects’ via monads in functional programming is a well-established topic in theoretical computer science [22]. We therefore expect that many of the traditional hyperintensional semantic

10 By focus-sensitive and perspective-relative semantic values, we mean semantic values which incorporate the truth-conditional effects of focus and of perspective. For discussion of focus sensitive semantic values see [38]. For discussion of perspective relative semantic values, see [27]. The focus relative semantic values in [38] have been incorporated into the monadic fold by [41]. And this article discusses one way of incorporating perspective relativity into the monadic fold. 11 Montague required that every expression of the same syntactic category must be assigned the same semantic type, and so whenever a certain expression of some syntactic category requires a complex semantic type, all expressions of that syntactic category must be given that type. For example, although an extensional transitive verb find expresses a relation between individuals of type e → e → t, since an intensional transitive verb like seek requires a more complex type assignment, the orthodox Montagovian strategy requires that we give find a more complicated type. This is known as ’generalising to the worst case’ and it has been criticised on various grounds. In particular, it has been argued that allowing the semantic types of expressions to be shifted via type shifters better captures certain empirical data, and also allows us to adopt simple lexical entries.

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theories can be formulated in monadic terms.12 Adding monads to the simply typed λ-calculus also provides a more expressive semantic theory than the simply typed λ-calculus alone. It is therefore worthwhile, as a methodological exercise, to see how far we can go just with monads and the simply typed λ-calculus. Since monads have been so useful in capturing other ‘side effects’ in natural language, we might propose a possible methodological hypothesis for determining which hyperintensional semantic theories are preferable; namely, that ceterus paribus, a semantic theory T1 of hyperintensionality is preferable to another semantic theory of hyperintensionality T2 if we can capture T1 by adding monads to the simply typed λ-calculus but we cannot capture T2 .13 The intuitive basis of the hypothesis is that we should try to devise semantic theories which capture the phenomena we want to capture as simply as possibly. The usual practice in science is to identify the weakest theory that captures a phenomeon in order to minimise extraneous assumptions and reduce complexity. If monads have proved a theoretically elegant way in which to capture a diverse array of phenomena in natural language, then we should try to see whether they can capture hyperintensionality. Let us now consider the monad used in P-HYPE and the semantics of AG. This is the reader monad [41] defined on P, the set of perspective indices.14 Monads were originally discussed in Category theory [30], but subsequently became part of the functional programming toolkit. Here we discuss monads as they are usually presented in the functional programming literature [47], but for the relationship between monads as used in functional programming and the equivalent definition usually employed in Category theory, see [31]. A reader monad is a triple (♦, η, ). ♦ : TYPE → TYPE, which takes a type and returns another type. ♦ is a so-called type-constructor, which takes a type as argument and returns another type. ♦ behaves as a special modal operator in Lax logic [14].15 ♦ maps any type τ to p → τ (where p is the type of perspective indices) and, for all a, b, maps a function f : a → b

12 We mentioned above that we can implement HYPE monadically. Our implementation involves a reader monad for impossible worlds, and this monad is employed to implement intensionality in [41]. 13 The hypothesis would require refining. We would need to specify what counts as ‘capturing’ a semantic theory of hyperintensionality. We might discover that virtually all hyperintensional semantic theories can be captured via monads, in which case the hypothesis would not be very useful. Also, the hypothesis could perhaps be weakened, if all the phenomena that Monads have been used to capture can be captured by the use of the applicative functor, which is like a monadic functor but slightly weaker. We (tentatively) suggest (ft.33) that using a monad in our semantic theory is preferable to using an applicative. 14 We will discuss how perspective indices are employed later. For now we can just say that to every agent in a discourse there corresponds a perspective index, and that certain terms which are perspective sensitive are interpreted relative to perspective indices. Given a background theory which includes the simply typed λ-calculus, in which lambda terms are assigned types, if an expression has type α, a perspectivally sensitive expression has type p → α. 15 It is in fact an endofunctor, as this is understood in category theory; that is, a functor that maps a category to itself (in this case the category of types).

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to a function ♦ f : ♦a → ♦b,16 such that (♦ f )(x) = λi. f (xi).17 η : τ → ♦τ is a value-constructor that takes a non-monadic value x : τ and trivially upgrades it to monadic values by forming a constant function from perspective indices to x. It is called the unit of the monad: Definition 1 η(x) =de f λi.x : p → α Finally,  (called bind) is a polymorphic binary infix operator acting as a sort of functional application: (5)

 (‘bind’): ♦τ → (τ → ♦δ) → ♦δ

Definition 2 a f =de f λi. f (a(i))(i)

where a : ♦τ, f : τ → ♦δ

 applies a monadic function f to a monadic argument a, by threading a perspective index i through a and f . In the following sections we assume our reader is familiar with the rules of the simply typed λ-calculus (β-reduction, etc) and we present our semantics via labelled binary trees (see the examples below) in which nodes are labelled with expressions of the form a : α where a is a lambda term and ‘a : α  is read ‘a is of type α’. We follow the general convention of type-driven composition [20, 26] that, in the case of a binary branching node α consisting of daughters β, γ , either β is a function which applies to γ , or γ is a function which applies to β. For this reason, we often omit the lambda term to the left of the colon and label nodes with types only, since we can infer the lambda term given the general convention of type driven interpretation. The main exception to this convention is the rule of predicate abstraction in [20], which we adopt (although our version of predicate abstraction varies from the notation used in op. cit). Predicate abstraction, in our presentation, permits trees such as the tree in (6), in which one of the daughters is labelled with λx : α and the other is labelled as p : β, where α is the type of the variable x. The intuitive interpretation of such trees is that ‘λx  represents the point at which λx binds into the formula p on the adjacent node of the tree, where p might contain a free variable which becomes bound by λx. However, in (6), ‘λx’ is not a well formed term of the simply typed λ-calculus and so does not have a type; consequently, λx cannot combine with p via function application, contradicting the convention of type driven composition. However, if need be, the rule of predicate abstraction can actually be replaced [37] with a similar rule which has the same effect but which preserves the convention of type-driven composition. Since adopting this alternative rule would introduce irrelevant complications, we stick to the rule of predicate abstraction. (6)

16 This

latter property ensures that ♦ is a functor. ‘x : α’ is read ‘x is of type α’

17 Throughout

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We will present  as a type-shifting operation as in (7), which shifts something of type ♦α, to a generalised quantifier of type (α → ♦β) → ♦β which is then able to take a predicate abstract of type α → ♦β as argument. The η operator instead will be presented via a non-branching tree (see (8)): (8)

(7)

2.1 AG’s Semantic Theory We now discuss AG’s semantic theory, with a view to introducing P-HYPE in the next section. AG’s theory concentrates on the behaviour of proper names in attitude reports, but in P-HYPE our aim is much broader: we intend to provide a hyperintensional semantic theory which can deal with other types of noun phrase, and with predicates, and examples below will sketch our approach. The purpose of inventing P-HYPE was not to give an account of names–and it may be that other accounts of names are superior to the account that AG give18 –but to give a hyperintensional semantic framework suitable for linguists to utilise in giving a compositional semantic theory of natural language. Let us consider AG’s theory by way of an example. Consider the sentence (9a), uttered in the scenario (9b): (9)

a. Mary Jane loves Spiderman. b. Scenario: Mary Jane does not know Peter Parker is Spiderman and she loves the man she calls ‘Peter Parker’. A speaker σ who knows or is ‘enlightened’ [51]19 about Peter Parker’s secret identity utters (9a)

18 AG’s account of names was devised in order to deal with the sorts of cases that [39] has written about, in which the substitution of one name for another name seems to be blocked in sentences with no intensional or hyperintensional operators. The examples which [39] discusses involved fictional names, but the substitutivity failure she discusses arises with non-fictional names, too. However, a full treatment of fictional names using the semantics of AG would no doubt involve further refinement of their account, and non-fictional names could easily have been used to illustrate the phenomena discussed in this article. 19 A reviewer asks how the distinction between enlightened and unenlightened speakers relates to [16]’s use of these terms. Our distinction was taken from [51] and [2], who explicitly reference [16]. So as far as we know our distinction is the same as in [16].

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According to AG, there is a sense in which (9a) is true, from the perspective of an enlightened utterer, but false from Mary Jane’s perspective. Reference [2] model this by making certain names perspective relative, so that Mary Jane can associate a distinct denotation with the names ‘Spiderman’ and ‘Peter Parker’, and these denotations are mental representations, which they call perspectives. We can then imagine a private mental lexicon for each person, consisting of the set of perspectives that a given person associates with terms of her language, which we call that person’s perspective or their mental model.20 Throughout this article we use ‘perspective’ ambiguously–both to denote a semantic value in someone’s mental model, and that person’s mental model–with the context serving to disambiguate which notion we have in mind. Consider the lexicon (Table 1) of the enlightened speaker σ of (9a). Plain names are subscripted with σ indicating the denotation of that name for σ . The names which are type ♦e have different denotations, depending on what perspective index they are interpreted relative to. AG suppose that certain names vary in perspective but others do not. Those which do not vary in perspective have something like a default status (see [51] for more discussion), in the following sense: if someone becomes enlightened, and learns, for example that Spiderman and Peter Parker are one and the same thing, then they will, by and large,21 just use plain ‘Peter Parker’, and this name will thence have default status.22 Notice the κ operator in the denotation of believe and love. κ has the following interpretation (where De is the domain of individuals of a model and P is the set of perspective indices): (10)

∀x ∈ De (κ(x) ∈ P)

Perspective sensitive expressions that scope below κ are interpreted relative to the perspective index corresponding to the subject of the attitude report (see Table 1). Expressions that scope above , are interpreted relative to the default perspective of the utterer. Let us consider the two readings of (9a). The false reading of (9a) is represented by (11),which β-reduces to (12), and the true reading is represented by (13), which β-reduces to (14):

20 A reviewer asks whether the notion of perspective we employ is similar to that of [12]. We were not aware of [12] prior to writing this article, but many aspects of his semantics seem to be similar to ours. In particular, he relativises the denotation of expressions to theories, which are sets of indices, and we relativise the interpretation of expressions to perspectives, which are sets of indices. However, we employ no counterpart relations in our semantics, unlike [12]. Also, [12] does not give an account of hyperintensionality; rather, his account is aimed at solving problems regarding intensional identity and the de re. We would eventually like to extend our semantics to cover the cases which [12] discusses. This should be possible, given we have an account [6] of hyperintensional anaphora in P-HYPE. 21 Reference [51] discusses some of the exceptions to this claim. Sometimes, for example, we seem to utter a name like ’Spiderman’ as if we were an unenlightened person who did not know that Spiderman is Peter Parker. 22 The default status of certain names might be clearer with names of places. An utterance of ’I’ve never been to Constantinople’ by an utterer who knows that Constantinople is an old name for the city presently called Istanbul, generally sounds infelicitous or semantically anomalous.

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Table 1 Lexicon of σ , the enlightened speaker Word Denotation Mary Jane Peter Parker believe love

mjσ ppσ λc.λs.B(s, c(κ(s))) λo.λs.love(s, o(κ(s)))  sm(i) if i = κ(m j) λi. pp(i) if i = κ(σ )

Spider-Man



Type e e ♦t → e → t ♦e → e → t ♦e

sm(i) if i = κ(m j) (κ(m j))) pp(i) if i = κ(σ )

(11)

love(m jσ , λi.

(12)

love(m jσ , sm(κ(m j)))    sm(i) if i = κ(m j) : p λi. λz.η (love(mj, z))) (κ(σ )) pp(i) if i = κ(σ ) : p

(13) (14)

love(m jσ , sm(κ(σ )))

Since Spiderman in (3) scopes above  and above κ, it is interpreted relative to the default perspective index, which is the index of the speaker, who they assume in their model to be enlightened. They thus stipulate that the speaker’s perspective is the one fed to an expression of the form a f , which by definition denotes λi. f (a(i))(i), and thus, if the speaker’s index is j, we always evaluate some expression of the form a f at j.23 When, however, a perspective relative expression scopes below the function f in a f , it is caught by the κ operator. One feature of P-HYPE is that we are able to capture formally an aspect of AG’s theory which they do not formalise; namely, that the utterer of (9a) never really understands the word Spiderman from Mary Jane’s perspective, since Mary Jane’s understanding of this word is only really accessible to her. Rather, in this case, according to AG, the denotation of Spiderman for the utterer is something in the perspective of the utterer which represents what the utterer takes to be the denotation of the expression relative to the perspective of Mary Jane. And, in general, according to AG, an expression occurring in a sentence only really ever receives a denotation in the mental model of the utterer of that sentence. In P-HYPE this is ensured by the optional condition (see section 3.2) of Isolation of perspectives, which ensures that the denotation of an expression from the perspective of one individual never overlaps the denotation of an expression from a distinct individual.24 23 The

technical stipulation they make is based on the fact [27] that sentences or expressions which are perspective relative are usually relative to the perspective of the utterer of them, and if they are relative to other perspectives, they are relative either to individuals salient in some group within a given context, or are relative to the perspective of the grammatical subject of the sentence. 24 A reviewer asks whether the condition is consistent with our being able to distinguish correct from incorrect uses of a given name. Since we are not aiming to give a fully fledged theory of names in this article we only make a few suggestions here. When names appear outside the scope of

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3 P-HYPE: A Combination of HYPE and Monads Here we briefly introduce some features of HYPE which are relevant to P-HYPE. The reader should consult [28] for further details. HYPE [28] is a logic which employs states/situations.25 States may be like classical possible worlds, but may also be partial (or gappy)– verifying a formula and its negation. One nice feature of HYPE is that it behaves entirely classically at a subset of states; as such, linguistic analyses couched in classical logics can be transferred to HYPE. But HYPE incorporates special incompatibility ⊥ and fusion operators ◦ in the satisfaction clauses for negation and the conditional, somewhat like [46, pp. 202–7] and truthmaker semantics: 1. ◦ : S × S → S is a partial commutative, associative binary function (called fusion), such that: – Either s ◦ s  is undefined, or s ◦ s  is defined (and hence in S) in which case, where V is a function from states to atomic formulas true at them, it is required that V (s ◦ s  ) ⊇ V (s) ∪ V (s  ). – s ◦ s is defined, and s ◦ s = s. 2. ⊥ is a binary symmetric relation on S (the incompatibility relation), such that: – If there is a v with v ∈ V (s) and v ∈ V (s  ), then s⊥s  . – If s⊥s  and both s ◦ s  and s  ◦ s  are defined, then s ◦ s  ⊥s  ◦ s  . ◦ gives rise to a partial order ≤, such that, for all s, s  ∈ S, s ≤ s  iff s ◦ s  is defined and s ◦ s  = s  . Importantly, truth is monotonic under fusion extension: for all s, if s |= A and s ◦ s  is defined, then s ◦ s  |= A. Variable assignments ρ and their modified variants ρ(d/x) behave as in Classical Predicate logic. Satisfaction of a formula φ is defined relative to a state and a variable attitude verbs or other hyperintensional operators, we suppose that the default interpretation of an expression will be the enlightened interpretation. This ensures that an ordinary utterance of ‘Barack Obama’ in ‘Barack Obama flies to work’ cannot have as its denotation the individual Spiderman, at least relative to the enlightened perspective, given that the name is not being used pragmatically as a secret way of referring to Spiderman, and given that the ordinary use of a name N usually carries a presupposition that the individual referred to is called N . But the details of what perspectives are used to interpret sentences in which no attitude verbs of hyperintensional operators are present will depend on a theory of how the context of utterance determines which perspectives are relevant, just as a detailed theory of what individuals can be used as the referents of indexicals in context of utterance will depend on the details of some theory of what makes certain individuals in a context salient. As regards the correctness of the use of a name when it appears in the scope of certain attitude verbs or hyperintensional operator, according to our theory, the use of a name can only be correct or incorrect relative to the perspective of a given individual, in which case, the use is incorrect if the person uses ‘Spiderman’ to denote the mental representation she associates with Peter Parker, when that individual in fact distinguishes a distinct mental representation with Peter Parker. Certain attitude verbs, such as Prove will presuppose that their complement sentences are true relative to the enlightened perspective, thereby ensuring that it can never be true that someone has proved something which is false. 25 We will use the words ‘states’ and ‘situations’ interchangeably and we steer clear of metaphysical issues relating to states in this article.

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assignment (written: s, ρ |= φ), and the clauses for the logical symbols are as usual, except for ¬ and ⊃, which have a distinctly modal flavour: s, ρ |= ¬A iff for all s  : if s  , ρ |= A then s⊥s  and s, ρ |= A ⊃ B iff for all s  : if s  , ρ |= A and s ◦ s  is defined, then s ◦ s  , ρ |= B. We should like to point out, before moving on, that besides Classical and Intuitionistic logic, whose relation to HYPE is discussed by [28], the fusion relation HYPE employs and the use of this in the clause for the conditional seem similar to some aspects of bunched logic of [35]. We hope to explore this connection in greater depth elsewhere.

3.1 Introducing P-HYPE: A Combinination of HYPE and AG’s Perspective-Sensitive Semantics In P-HYPE, we require the usual hierarchy of typed domains familiar from [18], where elements of different types correspond to different kinds of entities. Let e, t, P be three distinct objects which are not ordered pairs.26 T Y P E be the smallest set such that: • e, t, s, p ∈ T Y P E • α, β ∈ T Y P E implies α → β ∈ T Y P E • α ∈ T Y P E implies ♦α ∈ T Y P E, where ♦α : p → α Let V A Rα , and C O Nα be countably infinite sets of variables and constants, for each α ∈ T Y P E:  • C O N =  τ ∈T Y P E C O Nτ • V A R = τ ∈T Y P E V A Rτ The set of terms is the union of the sets of terms of type τ , for arbitrary τ :  T E R M = τ ∈T Y P E T E R Mτ The set T E R Mτ for every type τ is then defined as follows: • • • • • • • • •

c ∈ C O Nτ implies c ∈ T E R Mτ x ∈ V A Rτ implies x ∈ T E R Mτ τ = α → β, tβ and xα ∈ V A Rα imply (λx.t) ∈ T E R Mτ tα→τ and u α are terms then (t u) ∈ T E R Mτ . At ∈ T E R Mt and xα ∈ V A Rα implies (∀x.A) ∈ T E R Mt A ∈ T E R Mt implies ¬c A ∈ T E R Mt A, B ∈ T E R Mt implies (A ∗ B)t ∈ T E R Mt , for ∗ ∈ {∧, ∨, →} x ∈ T E R Me implies D O X x ∈ T E R Ms→s→t and P R O Vx ∈ T E R Ms→s→t ◦ ∈ T E R Ms→s→s

26 We

note that we use s as a type and s as a variable over states.

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• • • •

215

≤ ∈ T E R Ms→s→t π ∈ T E R M p→s→s→t κ ∈ T E R Me→s→ p ⊥ ∈ T E R Ms→s→t In P-HYPE we have sets S, D, P, where:

• S = Sc  Sn = ∅ splits into classical and non-classical states. • D = ∅ is the domain of individuals • P = ∅ is the set of perspective indices. In P-HYPE we have functions κ, π : •

κ : D → S → P, is a function that associates a unique perspective index κ(d)(s) to each individual d in a state s. • π : P → S → P(S) maps every perspective index κ(d)(s) ∈ P and states s ∈ S to a set of states π(κ(d)(s)) (s) ⊆ S, the perspective set or p-set of d at s. We present π in the term language as a characteristic function of type p → s → s → t. The reader should observe that κ and π are used to denote either the terms discussed above or the functions κ and π just described, which are the model-theoretic interpretation of the terms κ and π . Likewise ≤ and ⊥ are used to denote either the terms discussed above or the relations ≤ and ⊥ that hold between states described above, which are the model-theoretic interpretations of the terms ≤ and ⊥. However, since terms are usually presented with their types, this should present no confusion. In P-HYPE we also have a distinguished perspective index: • E ∈ P is called enlightened and is such that for all s, s  : π(E)(s)(s  ) iff s  ∈ {s}, whence s  = s

• (Isolation of perspectives): For all d1 , d2 ∈ D for which d1 = d2 , and all s ∈ S, π(κ(d1 )(s))(s) ∩ π(κ(d2 )(s), s) = ∅. HYPE negation involves ⊥ and the conditional involves ◦, and can be defined directly in our type-theoretic framework, in the object language: • (¬As→t ) = λs.∀s  ((As→t )(s  ) → s ⊥ s  ) • (As→t ⊃ Bs→t ) = λs.∀s  (s ≤ s  ∧ (As→t )(s  ) → (Bs→t )(s  )) A frame F = S, D, κ, P, E, π, ◦, ⊥ based on D, S and P is a family of sets D = {Dα | α ∈ T Y P E} such that De = D, Dt = {1, 0}, Ds = S and D p = P, and Dα→β ⊆ { f | f : Dα → Dβ } for each type α → β and E, κ, π, ◦, ⊥ are as above. A P-HYPE model is a structure M = F, m where F is as above and m : α.C O Nα → Dα is an interpretation of constants. An assignment over M is a function ρ : α.V A Rα → Dα . Variable assignments ρ and their modified variants ρ(d/x) (i.e x → d) behave as in Classical Predicate logic. An admissible valuation

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V exists, which assigns to each assignment ρ over M and each term Aα a value Va (Aα ) ∈ Dα , provided the conditions below are met in model M: Vρ (xα ) = ρ(xα ) Vρ (cα ) = m(cα) Vρ (∀xα A) =  d∈Dα Vρ(d/x) (A) Vρ (∃xα A) = d∈Dα Vρ(d/x) (A) Vρ (Aα→β Bα ) = Vρ (Aα→β )Vρ (Bα ) Vρ (λxα Aβ ) = the function f on Dα whose value at d ∈ Dα is equal to Vρ  Aβ , where ρ  = ρ(d/x) • Vρ (Aα ≡ Bα ) = {s | Vρ (Aα ) = Vρ (Bα )}

• • • • • •

In P-HYPE we add a set P of perspective indices, relative to which perspective sensitive expressions are interpreted. These are in the image of κ : De × S → P, a function much like AG’s κ function, which assigns perspective indices to agents at states.27 We then provide a function π : P × S → P(S), mapping perspective indexes and states to a set of states which we call the perspective set (p-set) of an agent at that state (this is the perspective or mental model of the agent, in the sense discussed above). Following AG, we can then, if we desire, enforce the distinctness requirement amongst perspectives they impose by requiring that the perspective set of two distinct agents is always distinct (This is the condition Isolation of perspectives above). This models their intuition that perspectives are entirely private. The semantic values of perspective-sensitive expressions then pick out subsets of the perspective set of agents. In this sense a name and predicate denotation interpreted relative to a perspective is something which only inhabits that perspective.

4 What Are Perspectives? Consider the perspective index κ(a)(s) of an arbitrary agent a at state s, and consider a’s p-set, π(κ(a)(s))(s). a’s p-set is supposed to represent a’s perspective or mental model by containing the mental representations which a associates with expressions of her language. Moreover, the perspective of an agent is supposed to be able to distinguish logical and mathematical truths. We need therefore to consider what role the states in π(κ(a)(s))(s) are playing, and how the states in a given perspective are ordered and relate to one another. In HYPE we can fuse states together using ◦, so fusion provides a potentially interesting way of relating states in a perspective, perhaps in such a way as to reflect cognitively significant properties of that person’s perspective. In fact, Leitgeb makes fusion partially defined, and writes [28] that “HYPE-models allow for two states not to be fusable, just as, e.g., mereological fusion might be undefined for certain entities-the fusion of the sun with any of my fingernails may simply not count as a “natural” object.” Fusion might be an operator 27 In

a fuller development it should probably be a function which assigns such indices to agents at states, in certain contexts and at certain times.

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which merges together states in a way which is constrained by principles which model cognitively significant properties of a person’s perspective. In this section, we briefly consider three ways of understanding the states in a person’s p-set and how the fusion operator interacts with the states in someone’s p-set.28 The first is to think of the states in a perspective as like the doxastically or epistemically accessible worlds of epistemic logic, except private to particular individuals, and to think of fusion as a way of constructing scenarios which are epistemically possible according to the agent whose perspective is being considered. The second is to think of the states in a perspective as sorted via subject matter, and to view the fusion operator as a way of combining subject matters according to a particular agent. The final way of viewing the states in a perspective is to think of certain of these states as being minimal verifiers of our beliefs and fusion as a way of creating a state in which all our beliefs are verified. Our suggestions here are highly tentative, and would need significant development in order to provide an adequate theory of what perspectives are. Consider the first way of understanding an agent’s p-set, on which states in the perspective of an agent a are supposed to represent what is epistemically possible according to a. One proposal is to assume that A is epistemically possible for a if, and only if a state in a’s perspective verifies A. If this is the case, then we need to constrain fusion, for the following reason. Suppose that a perspective is closed under fusion, in the sense that, for any two distinct states s, s  in a perspective, s ◦ s  is in that perspective. Suppose a does not consider it epistemically possible that A ∧ ¬A is true but considers A and ¬A both individually epistemically possible. Since a considers each of A and ¬A epistemically possible and A is epistemically possible for a if, and only if a state in a’s perspective verifies A, there is a state s in a’s perspective that verifies A and a distinct state s  in a’s perspective that verifies ¬A. For in HYPE, whenever a state r verifies some formula, r ◦ r  verifies that formula, where r  is any state (this follows from ≤ being monotonic. Therefore, s ◦ s  verifies A ∧ ¬A holds, and so, given closure under fusion, s ◦ s  should be epistemically possible for a, contradicting our assumption. There are various ways of dealing with this problem, but the three most obvious are: (a) abandoning the right to left direction of the assumption that A is epistemically possible for a if, and only if a state in a’s perspective verifies A; (b) hanging on to the assumption that fusion is used to construct epistemic possibilities but constraining the fusion operator in some way; or, (b) abandoning the assumption that fusion constructs epistemic possibilities.29

28 One way to think of the states picked out by π which is orthogonal to the issue of hyperintensionality and which we wish to develop in future work is to think of the states as entities that store information about certain discourse referents [24], or as being stacks of discourse referents as in [8]. Fusion would then enable the addition of a discourse referent to a stack. 29 For example, regarding option (b), we could see fusion as modelling some mental procedure of construction which an agent has attempted, which involves grouping facts together.

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Consider the second second way of understanding an agent’s p-set, on which the states in an agents p-set distinguish the subject matters [48] involved in the interpretation of sentences. perhaps by being partitioned in certain ways [48]. Considerations of fineness of grain would then involve subject matters. For example, we could require that when an agent utters a sentence s, the sentence has a certain subject matter according to that agent, and so, relative to that agent’s perspective index, s denotes a subset of the states in the p-set that concern that particular subject matter, however subject matters are individuated. The fusion relation would then combine together subject matters according to how they are seen by a particular agent. Such a conception of the perspective of an agent might be useful for analysing notions of aboutness. However, it is not immediately clear how it would apply to all cases of hyperintensionality, since many examples of hyperintensionality concern sentences which are intuitively about the same subject matter. Our third and final proposal is to think of the perspective of an agent via the notion of a minimal verifier, a concept which has been discussed in truthmaker semantics [15]. Consider (16a)-(16d) (where is.white : ♦se → ♦st), which represent different denotations that (15) might have in a context c and state s, where speaker (c) and addr essee(c) pick out, respectively, the speaker and addressee of the context of utterance and κ is as specified in P-HYPE: (15)

Snow is white

(16)

a. b. c. d.

is.white(κ(speaker (c))(s))(s)(snow(κ(speaker (c))(s))(s)) is.white(κ(addr essee(c))(s))(s)(snow(κ(addr essee(c))(s))(s)) is.white(κ(speaker (c))(s))(s)(snow(κ(addr essee(c)(s))(s)) is.white(κ(addr essee(c))(s))(s)(snow(κ(speaker (c))(s))(s))

Suppose that, for each formula φ in (16a)-(16d) of (15), there is a unique state in the perspective of the utterer called the minimal verifier of φ. This state verifies φ and only the formulas that φ logically implies in HYPE, and φ is called the distinguished formula of the state. A distinguished subset of the minimal verifiers in π(κ(d)(s))(s) are the states whose distinguished formulas represent things which an agent believes. For example, in this distinguished subset we might have the minimal verifiers of (16a) and (16d), but not (16b) and (16d). In this distinguished set of states, the role of fusion would be to connect states via fusion if an agent endorses the formulas minimally verified by them. We hope that the three ways of viewing p-sets that we have discussed give some idea as to how our theory might be further developed. Since it is not clear at this stage which theory is best for our purposes (modelling hyperintensionality in semantic theory), we remain neutral between them.

Metalinguistic Focus in P-HYPE Semantics Table 2 Simplified lexical entries DENOTATION  sm(i) if ∃s ∈ S. i = κ(h, s) λi. : ♦e pp(i) else ∃s ∈ S. i = E λx, j, s.  I ( j)(s)(x( j)) if ∃s ∈ S. j = κ(h)(s) f inite( j)(s)(x( j)) else ∃s : S. j = E : ♦e → ♦(s → t) λy, x, i, s. . ∀s  [π(i)(s)(s  ) → love(i)(s  )(x)(y)(κ(x)(s  )] : ♦e → e → ♦(s → t) λp, x, i, s.∀s  [s ≤ s  ∧ π(i)(s)(s  )] → ∀s  [D O X x (s)(s  ) → p(κ(x)(s  ))(s  )] : ♦(s → t) → e → ♦(s → t) λp, x, i, s.∀s  [s ≤ s  ∧ π(i)(s)(s  )] → ∀s  [P R O Vx (s)(s  ) → p(κ(x)(s  ))(s  )] : ♦(s → t) → e → ♦(s → t) λi, s.ιx : prime.number (i)(s)(x(i)) : ♦(se)

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ABBREVIATION spider man inductive

love

λp, x, i, s. bel(i, x, s, p(κ(x)(s  ))) λp, x, i, s. pr ove(i, s, x, p(κ(x)(s  ))) the. primes

4.1 Lexical Entries and Examples There are four comments to make about these lexical entries in Table 2.30 Firstly, h denotes the agent Harold, E denotes the enlightened perspective and u denotes the perspective index of the utterer of a sentence. The enlightened perspective index is the perspective index which, if supplied to an expression whose denotation takes a perspective index as an argument, returns the intension of that expression. Secondly, these lexical entries are simplified. For example, we are assuming (for expository simplicity) that prove is a guarded universal quantifier over worlds–but we haven’t specified what sort of universal quantifier it is–and that Prove is factive, and so presupposes the truth of its complement. Thirdly, a crucial aspect of the lexical entries for verbs, is our formalisation of the intuition of AG that such complements are always interpreted relative to a perspective which the utterer thinks is the perspective of another person. Consider the denotation of believe, which combines with a proposition of type ♦t (i.e, a function from perspective indices to states to truth values), an individual and a perspective index. We assume that, in the case of propositional attitude verbs, this perspective index must always be the utterer’s perspective index. Where u is the utterer’s perspective index

30 Lambda

abstractions of the form λx1 .λx2 , ..., λxn .φ are abbreviated as as λx1 , x2 , ..., xn .φ.

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at state s,31 Believe then universally quantifies over both (i) all the states s  ≥ s, such that s  ∈ π(u, s), where s is the state in which the sentence is being evaluated and (ii) all the states s  which are doxastically accessible from s ◦ s  . x believes p is then true iff p is true in s  relative to the perspective index associated with x at s  .

4.2 ‘Prove’ and the Complements of Attitude Verbs Consider (17) and (18): (17)

Harold proves that the primes are not inductive.

(18)

Harold proves that the primes are infinite.

Using the lexical entries above, we can derive (19) for a sentence like (17) and (20) for a sentence like (18): (19)

λs.∀s  [ s ≤ s  ∧ s  ∈ π(u)(s)] → ∀s  [P R O Vh (s)(s  ) → ¬I (κ(h)(s  ))(s  )(ιx. prime.number (κ(h)(s  ))(s  )(x(κ(h)(s  ))))]

(20)

λs.∀s  [ s ≤ s  ∧ s  ∈ π(u)(s)] → ∀s  [P R O Vh (s)(s  ) → ¬ f inite(κ(h)(s  ))(s  )(ιx. prime.number (κ(h)(s  ))(s  )(x(κ(h)(s  ))))]

Crucially, (19) and (20) will differ in truth value, if Harold associates distinct denotations with inductive and finite. We can also derive the following readings for (17): (21)

λs.∀s  [ s ≤ s  ∧ π(E)(s)(s  )] → ∀s  [P R O Vh (s)(s  ) → ¬I (E)(s  )(ιx. prime.number (E)(s  )(x(E)))]

(22)

λs.∀s  [ s ≤ s  ∧ π(u)(s)(s  )] → ∀s  [P R O Vh (s)(s  ) → ¬I (u)(s  )(ιx. prime.number (u)(s  )(x(u)))]

(23)

λs.∀s  [ s ≤ s  ∧ π(u)(s)(s  )] → ∀s  [P R O Vh (s)(s  ) → ¬I (u)(s  )(ιx. prime.number (κ(h)(s  ))(s  )(x(κ(h)(s  ))))]

(24)

λs.∀s  [ s ≤ s  ∧ π(u)(s)(s  )] → ∀s  [P R O Vh (s)(s  ) → ¬I (κ(h)(s  ))(s  )(ιx. prime.number (u)(s  )(x(u)))]

In [5] we present the P-HYPE derivations of (19) and (21)–(24). Here we present (21) and (22), using the abbreviations in Table 2 and abbreviating types s → e and

a domain of contexts of utterance C, and a function utter er : C → S → e, u = κ(utter er (c)(s))(s)

31 Given

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s → t with se and st. Equation (21) is derived via the tree in (26) below. On this interpretation, (17) receives its usual intension, since when we supply the enlightened perspective index to an expression that expression has its ordinary intension as its semantic value. For this derivation we feed the enlightened perspective index to ‘the primes’ and ‘inductive’, using η to ensure that these functions can combine, and to ensure that ‘the primes are inductive’ is fixed to its ordinary intension before combining with ‘prove’. Then we feed in the enlightened perspective index as the top-level perspective index. The reason why supplying the enlightened perspective index to an expression results in its intensional interpretation can be illustrated with an example. Let P R O Vx be an accessibility relation of provability for an agent x. Above, we laid down the condition that, for all s, s  : π(E)(s)(s  ) iff s  ∈ {s}, whence s  = s. From this condition we can reduce (25) and via the sequence in (1)–(5) below: (25)

λs.∀s  (s ≤ s  ∧ π(E)(s)(s  ) → ∀s  (P R O Vx (s  )(s  ) → A p→st (κ(E)(s  ))(s  )

λs.∀s  (s ≤ s  ∧ π(E)(s)(s  ) → ∀s  (P R O Vx (s  )(s  ) → A p→st (κ(E)(s  ))(s  ) (1) = λs.∀s  , s  (s ≤ s  ∧ π(E)(s)(s  ) ∧ (P R O Vx (s  )(s  ) → A p→st (κ(E)(s  ))(s  ) (2) = λs.∀s  , s  (s ≤ s  ∧ s  = s ∧ (P R O Vx (s  )(s  ) → A p→st (κ(E)(s  ))(s  ) = λs.∀s, s  (s ≤ s ∧ (P R O Vx (s  )(s  ) → A p→st (κ(E)(s  ))(s  )

(3) (4)

= λs.∀s  (P R O Vx (s)(s  ) → A p→st (κ(E)(s  ))(s  ))

(5)

(1) reduces to (2) via the equivalence A → B → C ≡ A ∧ B → C, and the quantifier ∀s  is moved to the front by classical distributivity laws. Since π(E)(s)(s  ) implies s  = s, we can substitute s  with s, deriving (4) and then s ≤ s can be eliminated (being trivially true), whence we derive (5), the interpretation of prove on which it is a simple universal quantifier over accessible states. For readability, we represent the derivation of (22) by two trees: (27) (for the upper half of the tree) and (28) (for the lower half). (22) is the reading on which both ‘inductive’ and ‘the primes’ are interpreted from the utterer’s perspective. This reading requires we scope both ‘inductive’ and ‘the primes’ above , so that they escape κ:

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(26)

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(27)

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(28)

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Expressions scoping above  are evaluated relative to the utterer’s perspective, and those scoping below are evaluated according the perspective index associated with the subject. This allows us to generate ‘mixed interpretations’ ((23) and (24)) in which some expressions in a sentence are evaluated at the utterer’s perspective and others are interpreted relative to the subject’s perspective, depending on where they scope.32 In theory, this would allow some expressions in a sentence to be evaluated with respect to the enlightened perspective and others not, but we think this results in interpretations which do not correspond to possible readings of sentences. For this reason, whenever we use , the top-level perspective index (that is, the final perspective index at the top of the tree) that we feed to the sentence denotation must be the utterer’s perspective index. We only use the enlightened perspective in one case: when all expressions in a sentence are to express their usual intensional values. In cases where  is not used, the top-level perspective index can be either the enlightened perspective index or the utterer. We use  to derive the readings in (22)–(24).33 But, to derive (22), we could have produced a tree isomorphic to (26) without , but in which the utterer’s perspective index replaces every instance of the enlightened perspective index. Similarly, (23) and (24) could be derived without  with trees in which ‘inductive’ and ‘the primes’ remain in-situ under the denotation of prove, which we omit due to space constraints. So we might think  is unnecessary. If, however, we want the priviledged status of the utterer’s perspective to be syntactically represented in the trees that we give for various sentences, one option is to require that the utterer’s perspective on the interpretation of expressions in a sentence is only available if those expressions scope above , and that terms that scope above  are always interpreted relative to the utterer’s perspective. If we choose this option, then we preserve one feature of the account of [2], according to which perspectivally-sensitive expressions are somewhat like expressions which take wide-scope when interpreted de re. This might be a reason to suggest keeping , at least if we think this analogy between perspectivallysensitive expressions and de-re, scoping expressions is significant.

32 We hope in future work to explore the relations between such mixed interpretations and various de re interpretations of sentences in which some constituents are interpreted de re and others are not. Such mixed interpretations need to be constrained in some way, so as to avoid overgeneralising, and we hope to explore strategies of constraining such overgeneralisation. 33 Patrick Elliott (p.c) has asked whether we can make do with an applicative functor as opposed to a monad. In a monad we have ♦♦A ≡ ♦A. In an applicative we have η : α → ♦α, but we do not have the ‘join’ μ : ♦♦α → ♦α of the monad, nor . We think certain perspective relative phenomena require μ, but don’t have the space to discuss these here. However, the argument of this paragraph would also require , if correct.

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5 Hyperintensionality and Metalinguistic Focus One interesting group of cases which might be thought to pose a challenge to our account involves contrastive focus on co-intensional predicates. Consider an utterance of (28b) in the scenario (29a) ([ e ] F indicates that the expression ‘e’ has focus): (29)

a. Scenario: Harold has written a proof for Bill in which the conclusion is that the set of prime numbers do not belong to set of finite sets, but there is no explicit mention of the inductive sets. Harold and Bill are enlightened about the predicates ‘inductive’ and ‘finite’ (i.e, they both know that the prime numbers are finite if, and only, if they are inductive) and they interpret ’the prime numbers’ in the same (enlightened) way as denoting the set of prime numbers. It is mutual knowledge that they are enlightened about all these expressions and that each understands the predicates ‘inductive’ and ‘finite’ as they are defined in ZFC.34 No one else’s opinion (other than Harold and Bill’s opinions) about the predicates is salient to them in the context. Bill utters (28b) to Harold:35 b. You proved the prime numbers are not [finite] F , not that the prime numbers are not [inductive] F .

(28b) is felicitous in the scenario (29a). However, our semantics predicts that, if the utterer is aware of Harold’s being enlightened, (28b) would be infelicitous, given the assumption that a speaker who is enlightened about two co-intensional expressions assigns them the same intension. The felicitous reading of this sentence in the scenario described is arguably (Chris Barker, p.c) a case of metalinguistic focus [29], in which Bill is rejecting the appropriateness of using ‘not inductive’ as opposed to ‘not finite’, perhaps because the conclusion of the proof explicitly states that the set of prime numbers do not belong to the set of finite sets. In a typical case of metalinguistic focus negation, focus and negation are used to object to the words or expressions someone has used to express their thought, as opposed to its intuitive meaning. For example, consider the dialogue (30): (30)

a. A: Look! Some gooses are flying b. B: No. Some [geese] F are flying.

34 Here we used ZFC just to illustrate a certain type of scenario, but the scenario could be modified so that Harold and Bill both take the predicates to be necessarily co-intensional, without depending on their agreeing to the definition in some particular formal framework. 35 We have made this scenario rather complicated in order to rule out alternative readings of (28b). For example (28b) could be felicitously uttered by Bill, if Bill is considering some constructive form of set theory on which the predicates are not equivalent, or (28b) could be uttered by Bill if there is someone salient in the context– either by being physically present or by having been discussed– who distinguishes the predicates ‘finite’ and ‘inductive’.

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In this dialogue, B is not objecting to the truth of what A says, but to the linguistic appropriateness of A’s use of ‘gooses’, which is not the correct plural form of ‘goose’ in standard English. The contribution of B thus has an ‘expressive’ dimension [34]: it conveys the non-at-issue information about what expression the speaker uses in her utterance and about what alternative expressions could be used. Reference [29] shows that cases of metalinguistic focus can be captured via two monads. We shall now examine how this works and then, integrating P-HYPE with these two monads, show that a simple variant of (29b) can be treated via the same method. Reference [29] follows [33], who proposes a two-dimensional semantics of linguistic expressions. We add two further domains to P-HYPE: a domain Dc of utterance contexts, in the sense of [25], which fix information about the denotation of context-sensitive expressions (indexicals, etc), and (ii) a domain Du of possible phonological strings, which is such that, if a, b are phonological strings, their concatenation a ∩ b is a phonological string. Let   be a function from strings to utterance contexts to the content expressed by the utterance of u in c, which in the case of P-HYPE is just its semantic value. For example, if someone uses ‘gooses’ to mean ‘geese’ in a given context c in the actual world w0 , then gooses(w0 )(c) = geese. Let  apply to a string u, producing a pair of the meaning of u in the context c and a formula ex p(c, u, u(c)) which says that the string u was used to express u(c) in c:  u (c), ex p(c, u, u (c)) if u is a meaning bearing element in c c u = (31) otherwise, undefined Since the expression component of the two pairs is distinct, inductivec =  f initec even though, at least in the scenario described inductive(c) =  f inite(c). It turns out we can use two monads to capture both the contribution of focus and the contribution of . First, let Foc(a) denote the focus semantic value of a, which, following [38], is a pair consisting of a and a set of alternatives alt (a) of the same semantic type of a. Standardly, this set of alternatives captures the intuition that focus involves contrast with salient alternatives; for example, an utterance of ‘John likes [tea] F with milk’ suggests that there are alternative things that John likes to have with milk that are salient in the context. In Roothian focus semantics the semantic value of ‘John likes [tea] F with milk’ in a given context would be a pair consisting of the ordinary intension of the sentence (a set of worlds), and a set of alternative propositions which are salient in the context (for example, they might be the propositions expressed by the sentences ‘John likes coffee with milk’, ‘John likes cereal with milk’, etc). The focus value of a use of ‘tea’ in the described utterance might then be a pair of teag and a set of alternatives alt (tea) of the same type as teag where alt (tea) might be, for example {coffeeg , orange juiceg , teag }.

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In Roothian focus semantics, each constituent of the sentence denotes a pair of its ordinary semantic value and salient alternatives, and whenever there are no salient alternatives, the second member of the pair is the singleton containing the first member. We can then compositionally derive the meaning of a sentence like ‘John likes [tea] F with milk’ as follows. Suppose johng =  john, { john} and likes[tea] F with milkg = λx.with(like(x)(tea))(milk), {λx.with(like(x)(a))(milk) | a ∈ alt (tea)}.

We combine these by first applying the first coordinate of likes[tea] F with milkg to each member of the first coordinate of johng and then applying each member of the second coordinate of likes[tea] F with milkg to the first coordinate of johng and pairing the result. Thus we have (6): (6) John likes[tea] F with milkg =  with(like( john)(tea))(milk), {with(like( john)(a))(milk) | a ∈ alt (tea)} 

This operation is known as pointwise function application [38].36 Roothian focus semantics can be implemented via the ‘pointed power-set’ monad [8, 29, 41]. The monad is defined as follows (πi are projection functions returning the ith argument of the expression they apply to, and F : T Y P E → T Y P E is the monadic type constructor of the monad (F, η F ,  F )): (32)

η F (x) = x, {x}

(33)

a F f = π1 ( f (π1 (a))),

 x  ∈π2 (a)

π2 ( f (x  ))

η F forms a pair consisting of the argument x that η F is applied to, and the singleton containing it. Impure monadic types are pairs whose first members are ordinary semantic values and whose second members are sets of alternatives of the same type. For example, we might have a pair  john, alt ( john) whose first member is the constant john and whose second member are various other individuals salient in the context of utterance. a F f forms a pair consisting of the first member of the function f applied to the first member of a, and the set of things produced by applying f to the members of π2 (a).

Roothian focus semantics α : σ → τ and β : σ are shifted to sets of alternatives so that we have α g ⊆ Dσ →τ and β g ⊆ Dσ . According to pointwise function application: if α : σ → τ and β : σ , then the application of α to β, α(β) : τ , is a set of things of type τ and α(β)g = {g(a) | g ∈ α g and a ∈ β g }.

36 In

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The job done by , which was described above, can also be implemented monadically [29], as follows (U : T Y P E → T Y P E is the monadic type constructor of the monad (U, ηU , U )): (34)

ηU (x) = x, 

(35)

aU f = π1 ( f (π1 (a))), π2 (a) ∧ π2 ( f (π1 (a)))

ηU forms a pair consisting of the argument x that ηU is applied to, and . Impure monadic types are pairs of the form a(c), ex p(c, a, a(c)), as this was described above. For example, consider geese(c), ex p(c, gooses, gooses(c)) whose first member is the constant geese denoting a one place property and whose second member is a metalinguistic statement that the string ‘gooses’ was used to express this property in the context c. Note, that geese(c), ex p(c, gooses, gooses(c))c = geese(c), ex p(c, gooses, geese(c))c , since the expression to the right of = has a different meaning in its ‘expression’ component; the second member of it indicates that the string ‘geese’ was used in c to express the relevant property, whereas the expression to the left of = indicates that the string ‘gooses’ was used in c to express the relevant property. Using these two monads, [29] shows that we can capture the metalinguistic character of B’s response in the dialogue (30) above. By using the two monads above, we can generate the expressive meaning of B’s utterance of ‘Some [geese] F are flying’ as (36), and we can generate this compositionally from the meaning of the parts of the sentence (36)

c  Some[geese] F are flying  =  ∃x.geese(c)(x) ∧ f ly(x), ex p(c, geese, geese(c)) ,















 ∃x.u (c)(x) ∧ f ly(x), ex p(c, u , u (c))  | u ∈ alt (geese : u)

(36) consists of a pair whose first and second members are pairs. Its first member is a pair whose first member is the ordinary intension of the sentence ‘Some geese are flying’, and whose second member indicates that the string ‘geese’ was used in c to express the relevant property. The second member of (36) consists of a set of meaning expression pairs containing the metalinguistic alternatives u  that are relevant in the context and could be used in the context to express the ordinary semantic value of ‘geese’. These might, for example, include ‘gooses’. As is shown in detail in [29], we can derive (36) via the tree below:

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(37)

In the derivation, we can see that the focused phonological form ‘geese’ takes scope at the top of the tree. At the root of the tree is the formula derived after β-reduction. The reader can consult [29] for details. We now show that a similar analysis can be given in P-HYPE of the second sentence of dialogue (38c) (in the scenario (38a)), a simpler variant of (29b)): (38)

a. Scenario: Harold has written a proof for Bill in which the conclusion is that the set of prime numbers do not belong to set of finite sets, but there is no explicit mention of the inductive sets. Harold and Bill are enlightened about the predicates ‘inductive’ and ‘finite’ (i.e, they both know that the prime numbers are finite if, and only, if they are inductive) and they interpret ’the prime numbers’ in the same (enlightened) way as denoting the set of prime numbers. It is mutual knowledge that they are enlightened about all these expressions and that each understands the predicates ‘inductive’ and ‘finite’ as they are defined in ZFC.

b. Harold: I proved that the prime numbers are not inductive. c. Bill: No. You proved the prime numbers are not [finite] F .

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In scenario (38a) ‘inductive’ and ‘finite’ have the same denotation from Harold and Bill’s perspective, and we suppose that they are ‘enlightened’ with respect to these predicates and so both assign them their standard intensional interpretation. In this situation Bill is rejecting the use of a particular word and so interpreting his utterance as involving metalinguistic focus in the scenario described is quite natural. The truth conditions of ‘You proved the prime numbers are not finite’ in which ‘finite’ and ‘the prime numbers’ are interpreted from the enlightened perspective are as in (39) (we ignore tense, and treat addr essee(c) as denoting the individual picked out by ‘you’ in the context c). The target truth conditions of Bill’s utterance in (38c) in the scenario described in (38a) are (40) (we use  c as a function from surface strings to lambda terms which represent the logical form of these strings parameterised to the context c): (39)

∀s  [ s ≤ s  ∧ π(E)(s)(s  )] → ∀s  [P R O Vaddr essee(c) (s)(s  ) → ¬ f inite(c)(E)(s  )(ιx. prime.number s(E)(s  )(x(E)))]

(40)

c You proved that the prime numbers are not [ finite] F  =  ∀s [ s ≤ s  ∧ π(E)(s)(s  )] → ∀s  [P R O Vaddr essee(c) (s)(s  ) →

¬ f inite(c)(E)(s  )(ιx. prime.number s(E)(s  )(x(E)))],

ex f inite,  f inite(c)) , p(c,  ∀s  [ s ≤ s  ∧ π(E)(s)(s  )] → ∀s  [P R O Vaddr essee(c) (s)(s  ) → ¬x  (c)(E)(s  )(ιx. prime.number s(E)(s  )(x(E)))], 

ex p(c, x  , x  (c))  | x  ∈ alt ( f inite : u) : F(U (t)) (40) consists of a pair whose first and second members are pairs. Its first member is a pair whose first member is the semantic value of the sentence ‘You proved that the primes are finite’ relative to the enlightened perspective in P-HYPE, and whose second member indicates that the string ‘finite’ was used to express the property of being finite. Thus the truth conditions encode which words were used in the utterance, and make this semantically significant. The second member of (40) consists of a set of meaning expression pairs containing the metalinguistic alternatives u  that are relevant in the context and could be used in the context to express the semantic value of ‘finite’ relative to an enlightened speaker. These might, for example, include ‘inductive’.

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The truth conditions of the sentential complement ‘that the prime numbers are not [finite] F ’ in (38c) are in (41): (41)

 that the prime numbers are not [finite] F c = η(λs  .¬ f inite(c)(E)(s  )(ιx. prime.number s(E)(s  )(x(E)))), ex p(c, f inite,  f inite(c)),  { η(λs  .¬x  (c)(E)(s  )(ιx. prime.number  s(E)(s )(x(E)))), ex p(c, x  , x  (c))  | x  ∈ alt ( f inite : u)} : F(U (♦st))

The formula in (41) can be derived compositionally. For legibility, we have split its derivation into two connecting parts ((43) and (42)). The root of (42) is identical to the bottom right leaf node of (43) and (41) is at the root of (43) . Let us consider each tree in turn. (42)

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(43)

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(43) begins similarly to the derivation of (26), and derives the enlightened interpretation of ‘the primes are not finite’ by feeding the enlightened perspective to ‘the prime numbers’ and to ‘finite’, which is lifted via η. In (43), we then lift this denotation with ηU , and this tree is in fact isomorphic to (37) above, which derived the metalinguistic interpretation of ‘some [geese] F are flying’. In this tree u of type U (♦e → ♦t) is introduced, and then abstracted over by λu further in the tree. This ensures that we can create something of type (u → F(U (♦t)) which can combine with the metalinguistic focus alternatives in f inite F : F(u), which has been shifted via  F . Consider (41), repeated here as (44): (44)

 that the prime numbers are not [finite] F c = η(λs  .¬ f inite(c)(E)(s  )(ιx. prime.number s(E)(s  )(x(E)))), ex p(c, f inite,  f inite(c)), { η(λs  .¬x  (c)(E)(s  )    (ιx. prime.number s(E)(s  )(x(E)))), ex p(c, x , x (c))  | x  ∈ alt ( f inite : u)} : F(U (♦st))

We need to combine  that the prime numbers are not [finite] F c of type F(U (♦t)), which is at the root of (43), with the denotation of pr ove, given in (45): (45)

pr ove := λp, x, i, s.∀s  [s ≤ s  ∧ π(i)(s)(s  )] → ∀s  [P R O Vx (s)(s  ) → p(κ(x)(s  ))(s  )] : ♦st → e → ♦st

We need to ensure that the inner argument i and s of prove are saturated, prior to combining with prove with  that the prime numbers are not [finite] F c ; for, otherwise we will not be able to saturate these arguments. For this reason, we provide the operator shift, in (46) below, which saturates the arguments of prove, producing (47): (46)

λz, p, x.z( p)(x)(E)(s  ) : (♦st → e → ♦st) → ♦st → e → t

(47)

λp, x.∀s  [s ≤ s  ∧ π(E)(s)(s  )] → ∀s  [P R O Vx (s)(s  ) → p(κ(x)(s  ))(s  )] : ♦st → e → t)

We abbreviate the application of shi f t to pr ove as pr ove(E)(s  ) in (52) below. shift(prove) is not of the right type to take  that the prime numbers are not [finite] F c as an argument, so we need to shift the type of prove from ♦st → e → t to F(U (♦t)) → F(U (e)) → F(U (t)). This can be done by applying the functor of each monad U and F to pr ove. Every monad M, η M ,  M  has a functor G : M(α → β) → M(α) → Mβ, for arbitrary types α, β. The lambda term F1 in (48), implements the application of the functor U (α → β) → U (α) → U (β), where α := ♦st and β := e → t. The lambda term F2 in (49), implements the application of

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the functor F(α → β) → F(α) → F(β), where α := U ♦st and β := U (e) → U t. If we apply F1 to pr ove, abbreviating the result as pr ove1 , and then we apply F2 in (49) to pr ove1 , then we derive pr ove2 : F(U (♦(t))) → F(U (e)) → F(U (t)) (see (50)), which can take  that the prime numbers are not [finite] F c as an argument: (48)

F1 := λ f ♦st→e→t , xU (♦st) , yU e .xU (λx  .yU (λy  .ηU ( f x  y  )))

(49)

F2 := λ fU (♦st)→U e→U (t) , x F(U (t)) , y F(U e) .xU (λx  .yU (λy  .ηU ( f x  y  )))    pr ove2 := λx F(U (♦st)) , y F(U e)) .  pr ove(π1 (π1 (x  )))(π1 (π1 (y  ))),

(50)

π2 (π1 (x  )) ∧ π2 (π1 (y  )) ,



{ pr ove(π1 (xb ))(π1 (xa )), π2 (xb ) ∧ π2 (xa )  | xb ∈ π2 (x  ), xa ∈ π2 (y  )} If we then apply pr ove2 to a lambda term a : F(U (♦t)) such that π1 (π1 (a)) = t and a lambda term b : F(U e)) such that π1 (π1 (b)) = h to (50), this will result in a pair of the form (51):  (51)  pr ove (t) (h), π2 (π1 (a)) ∧ π2 (π1 (b)),  { pr ove π1 (xb ) π1 (xa ), π2 (xb ) ∧ π2 (xa ) | xb ∈ π2 (a), xa ∈ π2 (b)} The first member of the pair (51) will consist of a pair whose first member is the denotation of pr ove(t)(h), and the second member is a series of expressions of the form ex p(c, u, u(c)), where u is a string that was used to express pr ove(t)(h). The second member of the pair (51) is a set of pairs: the first members are alternative formulas of the form ‘ pr ove(y)(x)’, where x, y are focus alternatives and whose second member is a series of expressions of the form ex p(c, u, u(c)), where u is a string which could be used to convey a given instance of the first pair. Once we’ve shifted the type of pr ove, it can combine with  that the prime numbers are not [ finite] F c , and it can combine with the rest of the sentence when we lift addr essee(c) two times via ηU and η F . The rest of the tree is as follows, and the root of the tree β-reduces to (40) above:

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(52)

Let us end this section by briefly considering a remaining problem with our account of hyperintensionality, which, having discussed the monadic operator U , we can now present a solution to. Consider a state s in π(κ(d, s), s), for some agent d where (53) is true:

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(53)

237

is.white(κ(speaker (c))(s))(s)(snow(κ(speaker (c))(s))(s))

Abbreviate (53) with A, and let B be an arbitrary formula. All HYPE validities will also hold at s. So, in particular, A → (B → A), being a HYPE validity, will hold at all states in the perspective π(κ(d, s), s) even though d may intuitively reject A → (B → A). To avoid this consequence of HYPE, we can incorporate explicit mention of the expressions used in a particular speech act, via the monadic operator U .37 In the case of believe, this simply involves using the functions F1 discussed above, to shift the denotation of believe from something of type ♦st → e → ♦st) to something of type U (♦st) → U e → U (♦st)). Then ‘John believes that φ’ could have a different semantic value from ‘John believes that ψ’, even when φ and ψ are logically equivalent in HYPE, since the sentences in this case would have a semantic value which included the fact that a certain expression was used as opposed to another. This shows that, via monads, we can actually conservatively incorporate into HYPE finer grained semantic values when required. Reference [28] remarks that “if the goal were to determine a logic and semantics for an operator expressing the beliefs of realistic agents...then it is highly questionable whether any background theory short of syntax or some similarly fine-grained theory of structured propositions could do the job...The logic and semantics of the system HYPE...would not be able to contribute to investigations into hyperintensional operators of any such quasi-syntactic kind. Indeed, the appropriate semantics for such operators is likely to be proof-theoretic in nature”. However, our example shows that HYPE can easily be combined with monads which enable us to distinguish sentences which would appear to be logically equivalent in HYPE, so long as we allow metalinguistic resources into our semantics, resources which have been found useful elsewhere in semantic theory [29, 33].

6 Conclusion Monads are a powerful device for modelling various natural language phenomena. In previous work [5, 6], we have introduced P-HYPE as a compositional semantic theory for analysing hyperintensionality and anaphora which combines HYPE and the [2]’s perspective relative semantics, via the reader monad. P-HYPE is a hyperintensional semantic theory that provides the means to compositionally derive sentence semantic values based on their parts. Here we have shown that we can also use other monads to enrich P-HYPE with other kinds of fine-grainedness, where necessary. We hope in the future to explore the extent to which monads can be used to implement other hyperintensional semantic theories which have been discussed in the literature. In

37 Whether this consequence of HYPE is actually something which requires rectifying depends, of course, on whether one thinks HYPE itself provides a satisfactory level of fineness of grain in the first place.

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this way, monads could serve as a way of implementing hyperintensional semantic theories in a conservative, compositional fashion whilst preserving the advances that have been made in natural language semantics. Acknowledgements We thank Alessandro Giordani, Michael Mendler and Hannes Leitgeb for discussion, and audiences at Sinn und Bedeutung 23 in Barcelona and at Fine-Grained Semantics for Modal Logic: Formal and Foundational Issues in Milan. Particular thanks to Chris Barker, who brought my attention to [29]’s analysis of metalinguistic focus alternatives, which led to my analysis of certain kinds of hyperintensionality as cases of metalinguistic focus.

References 1. Asudeh, A. and Giorgolo, G. (2012). (m, η, ): Monads for conventional implicatures. In C. Guevara and Nouwen (Eds.), Proceedings of Sinn und Bedeutung 16 (pp. 265–278). http://mitwpl. mit.edu/catalog/sube01/. 2. Asudeh, A., & Giorgolo, G. (2016). Perspectives. Semantics & Pragmatics., 9(21), 1–53. 3. Barker, C., & Shan, C.-C. (2014). Continuations and natural language (Vol. 53)., Oxford studies in theoretical linguistics Oxford: OUP. 4. Bumford, D. (2015). Incremental quantification and the dynamics of pair-list phenomena. Semantics and Pragmatics, 8, 89–1. 5. Burke, L. (2019). P-HYPE: a monadic situation semantics for hyperintensional side effects. In Teresa Espinal, M., Castroviejo, E., Leonetti, M., McNally, L., & Real-Puigdollers, C. (Eds.) Proceedings of Sinn und Bedeutung 23 (Vol. 1, pp. 201–217). https://ojs.ub.uni-konstanz.de/ sub/index.php/sub/article/view/511/427 6. Burke, L. (2019). Monads for hyperintensionality? A situation semantics for hyperintensional side effects and intra-sentential anaphora. In Proceedings of the sixth workshop on natural language and computer science (pp. 34–43), Association for Computational Linguistics, https:// www.aclweb.org/anthology/W19-1104 7. Cameron, P. J. (2012). Sets, logic and categories. London: Springer. 8. Charlow, S. (2014). On the semantics of exceptional scope. Ph.D. thesis, New York University. 9. Charlow, S. (2017). The scope of alternatives: Indefiniteness and islands. Manuscript: Rutgers University. 10. Cresswell, M. J., & Von Stechow, A. (1982). De re belief generalized. Linguistics and Philosophy, 5(4), 503–535. 11. Cresswell, M. J. (1985). Structured meanings. Cambridge, MA: MIT. 12. Edelberg, W. (1995). A Perspectivalist Semantics for the Attitudes. Noûs, 316–342. 13. Égré, P. (2014). Hyperintensionality and de re beliefs. In F. Lihoreau & M. Rebuschi (Eds.), Epistemology, context, and formalism (vol. 369). Springer. 14. Fairtlough, M., & Mendler, M. (1997). Propositional Lax logic. Information and Computation, 137(1), 1–33. 15. Fine, K. (2017). Truthmaker semantics. In Hale, C. W. B., Miller, A. (Eds.) A Companion to the Philosophy of Language. Chichester: Wiley. 16. Forbes, Graeme. (1999). Enlightened semantics for simple sentences. Analysis, 59(2), 86–91. 17. Fox, C., & Lappin, S. (2008). Foundations of intensional semantics. Oxford: Blackwell. 18. Gallin, D. (1975). Intensional and higher-order modal logic: With applications to Montague semantics, (Vol. 19). Elsevier. 19. Giorgolo, G., & Unger, C. (2009). Coreference without discourse referents: a nonrepresentational DRT-like discourse semantics. LOT Occasional Series, 14, 69–81. 20. Heim, I., & Kratzer, A. (1998). Semantics in generative grammar (Vol. 1185). Oxford: Blackwell .

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239

21. Hyland, M., Plotkin, G., & Power, J. (2002). Combining computational effects: Commutativity and sum. In Foundations of information technology in the era of network and mobile computing. Springer. 22. Hyland, J. M. E., & P. G. P. A., (2006). Combining effects: Sum and tensor. Theoretical Computer Science, 357(1–3), 70–99. 23. Jago, M. (2014). The impossible: an essay on hyperintensionality.Oxford: OUP. 24. Karttunen, L. (1976). Discourse referents. In McCawley, J. D. (Ed.), Syntax and semantics, volume 7: Notes from the linguistic underground (pp. 363–385). New York: Academic. 25. Kaplan, D. (1989). Demonstratives. In Almog, J., Perry, J., & Wettstein, H. Themes from Kaplan (pp. 481–563). 26. Klein, Ewan, & Sag, Ivan A. (1985). Type-driven translation. Linguistics and Philosophy, 8(2), 163–201. 27. Lasersohn, P. (2017). Subjectivity and perspective in truth-theoretic semantics. Oxford: Oxford University Press. 28. Leitgeb, H. (2018). HYPE: A system of hyperintensional logic (with an application to semantic paradoxes). Journal of Philosophical Logic, 1–101. 29. Li, H. (2017). Semantics of metalinguistic focus. In Cremers, A., van Gessel, T., & Roelofsen, F. (Ed.) Proceedings of the 21st Amsterdam Colloquium (pp. 354–363). ILLC-University of Amsterdam. 30. MacLane, S. (1978). Categories for the Working Mathematician. Graduate Texts in Mathematics (Vol. 5). Springer. 31. (2019). Monad (in computer science). Retrived from https://ncatlab.org/nlab/show/monad+ 32. Pearson, Hazel. (2012). A Judge-free semantics for predicates of personal taste. Journal of Semantics, 30(1), 103–154. 33. Potts, C. (2007). The dimensions of quotation. In Barker, C., Jacobson, P. (Eds.) Direct compositionality. Oxford studies in theoretical linguistics. (Vol. 14) Oxford: Oxford University Press. 34. Potts, C. (2007b). The expressive dimension. Theoretical Linguistics, 33(2), 165–198. 35. Pym, D. J. (2013). The semantics and proof theory of the logic of bunched implications. (Vol. 26). Springer Science & Business Media. 36. Rabern, Brian. (2012a). Against the identification of assertoric content with compositional value. Synthese, 189(1), 75–96. 37. Rabern, Brian. (2013). Monsters in Kaplan’s logic of demonstratives. Philosophical Studies, 164(2), 393–404. 38. Rooth, Mats. (1992). A theory of focus interpretation. Natural language semantics, 1(1), 75– 116. 39. Saul, Jennifer M. (1997). Substitution and simple sentences. Analysis, 57(2), 102–108. 40. Sauerland, U., & Schenner, M. (2009). Content in embedded sentences. In Multimodal signals: Cognitive and algorithmic issues (pp. 197–207), Springer. 41. Shan, C.-C. (2002). Monads for natural language semantics. In Proceedings of the ESSLLI 2001 student session (pp.285–298). 42. Shan, C.-C. (2007). Linguistic side effects. In Barker, C., & Jacobson, P. (Eds.) Direct compositionality. Oxford studies in theoretical linguistics (vol. 14). Oxford: Oxford University Press. 43. Unger, C. (2011). Dynamic semantics as monadic computation. In JSAI international symposium on artificial intelligence (pp.68–81). Springer. 44. Quine, W. V. (1956). Quantifiers and propositional attitudes. The Journal of Philosophy, 53(5), 177–187. 45. Van Eijck, J., & Unger, C. (2010). Computational semantics with functional programming. Cambridge: Cambridge University Press. 46. Veltman, F. (1985). Logic for Conditionals. Ph.D. thesis, University of Amsterdam. 47. Wadler, P. (1995). Monads for functional programming. In Jeuring, J., & Meijer, E. (Eds.) Advanced functional programming, LNCS 925. Springer Verla. 48. Yablo, S. (2014). Aboutness. Princeton: Princeton University Press.

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49. Yalcin, S. (2014). Semantics and metasemantics in the context of generative grammar. In Burgess, A., & Sherman, B. (Eds.) Metasemantics: New essays on the foundations of meaning. Oxford: OUP. 50. Zimmermann, T. E. (1991). Kontextabhängigkeit. In Stechow, A. V., & Wunderlich, D. (Eds.) Semantik-Semantics. Ein internationales Handbuch der zeitgenössischForschung – An International Handbook of Contemporary Research (pp. 156–229). Berlin: de Gruyter. 51. Zimmermann, T. E. (2005). What’s in two names? Journal of Semantics, 22(1), 53–96.

Index

A Action inclusion, 119 Action specification, 119 Action token, 134 Action trope, 134 Action types, 118, 120, 132, 134, 151 Agent-relative action types, 151, 154 Alternatives, 227 Aristotle Thesis, 50, 53, 55, 56, 60 First, 50 Second, 50 Third, 55 Attitude verbs, 205, 213, 219, 220 B Barbershop paradox, 49, 50, 60–63 Believe, 219, 237 Bilateral exact equivalence, 111, 112 Boethian Thesis, 50, 54–56, 60, 62, 63 First, 50 Second, 50 Third, 55 Bunched logic, 214 But-for test, 153, 162 C Category theory, 203, 208 Causal responsibility, 150 Compositional, 205–207, 238 Compositional semantics, 203 Compositional semantic theory of natural language, 205, 210 Conduct, 118–120, 134 Conjunction, 54, 57, 58

Context-sensitive expressions, 227 Continuous actions, 157 D Deontic paradoxes, 133, 142, 145 E Enlightened perspective, 213, 219, 231, 234 Enlightened perspective index, 221 Expected result, 152, 160 Expressive meaning, 229 F Falsification, 99 classical, 101 exact, 100 inexact, 100 Fictional names, 210 Fineness of grain, 237 Focus, 226, 227 Focus semantics, 227 Focus semantic value, 227 Free choice permission, 142, 146 Functional programming, 203 Fusion, 213, 214, 217, 218 G Generic act, 118 Group responsibility, 167 H HYPE, 205, 213, 214, 218, 237

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Giordani and J. Malinowski (eds.), Logic in High Definition, Trends in Logic 56, https://doi.org/10.1007/978-3-030-53487-5

241

242 Hyperintensional deontic action logic, 133 Hyperintensionality, 78, 81, 84, 98, 178, 181, 182, 203, 205–208, 226, 236 Hyperintensional operators, 210, 213 Hyperintensional semantic theories, 203, 205, 207, 208, 210, 237 Hyperoffice, 181, 183, 186, 193, 194 Hyperrequisite, 180, 193, 194 I Illegal action, 119 Imagination ceteris paribus, 79, 99, 102 counterpossible, 90, 91 indeterminacy in, 83, 84, 93, 103 nature of, 98, 99 Implication, 49, 51–53, 55, 57, 58, 63 connexive, 50, 51, 54, 59, 60, 62, 64 emphatic, 51 material, 49–52, 54, 55, 58, 61, 62 relatedness, 57, 58 relating, 58, 59, 63 strict, 51, 52, 54, 55 Implicit content, 85 Impossibilia, 178, 183 Impossible individual, 178, 181, 186, 201 Impossible worlds, 78, 91, 92, 113 open, 92 Impure monadic types, 228 Incompatibility, 213 Independence of agency, 150, 156 Individual action, 119 Individual concept, 179, 181 Individual-in-intension, 180, 185 Intension, 221 Intensional deontic action logic, 125 Intensional equivalence, 204 Intensional operators, 206 Intensional semantics, 203 Intentional content, 98 Intentional object, 98 L Lambda calculus, 182 Legal action, 119 Liar paradox, 61 Logic Abelardian, 55 anti-paradox, 56 classical, 49, 55, 59, 62 conjunction-idempotent, 56 connexive, 49–51, 54–56, 59, 60, 62, 63

Index Boolean, 50 minimal, 55 subminimal, 55 constructive, 49 inconsistent, 49, 55 intuitionistic, 49 Johansson, 49 Kapsner-strong, 56 modal, 50 non-classical, 49 of action, 57 quasi-connexive, 55 relatedness, 57, 58 relating, 57–59 simplificative, 56 trivial, 49 Logical consequence classical, 101 exact, 101 inexact, 101 Logical equivalence classical, 101 exact, 101 inexact, 101 Logically equivalent, 205, 206, 237 Logical omniscience, 83–85, 92, 93 M Mental representations, 211 Mereology, 81, 83, 98, 99 Metalinguistic alternatives, 229, 231 Metalinguistic focus, 203, 227 Metalinguistic focus alternatives, 234 Metalinguistic interpretation, 234 Modalized situation space, 99 Monads, 203, 205–208, 227–229, 237, 238 N Natural language semantics, 203, 238 Negation, 50, 53–55 NESS test, 153, 162 Non-deterministic operators, 206 Non-monotonicity, 83, 84, 93 O Object-based approach, 179, 181 Obligation, 124, 126, 133 Opposing relation, 157 P Paralogism, 61

Index Permission, 120, 124, 126, 128, 133, 136, 142 Perspective, 205, 211, 219, 231 Perspective indices, 219 Perspective relativity, 207 Perspective-sensitive semantic theory, 203 P-HYPE, 205, 208, 210, 227, 237 Pointed power-set monad, 228 Possible world, 99, 113 Procedural semantics, 182 Prohibition, 120, 126, 128, 133, 136 Proposition bilateral, 99 structured, 78 Propositional attitudes, 207 Prove, 213, 219, 220 R Reader monad, 208 Reductio, 178 Referential opacity, 206 Relating semantics, 49, 50, 56, 58–60 Relation of opposing, 152 Requisite, 180, 184, 191 S Selection function, 82, 102 Semantic values, 203, 237 Side effects, 203, 205–207 Situations, 78, 98, 113, 213 impossible, 98, 99, 114

243 Situation semantics, 203 State-changing operators, 206 States, 213, 220, 237 STIT logic, 149 Strong permission, 120, 126, 133, 142 Structure-based approach, 181

T Topic-frame, 82 Topic-model, 82 Topics, 78, 81, 102, 113 Transition, 132 Transparent intensional logic, 180, 187 Truthmaker semantics, 98 Two-dimensional semantics, 227

U Utterer’s perspective, 225

V Verification classical, 100, 101 exact, 98, 100 inexact, 100

X XSTIT operators of causal responsibility, 160