Introduction to the Semantics of Law (Law and Visual Jurisprudence, 6) 3030956784, 9783030956783

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Law and Visual Jurisprudence 6 Series Editors: Sarah Marusek · Anne Wagner

Andrzej Malec

Introduction to the Semantics of Law

Law and Visual Jurisprudence Volume 6 Series Editors Sarah Marusek, University of Hawai'i Hilo, Hilo, HI, USA Anne Wagner, Lille University, Lille, France Advisory Editors Shulamit Almog, University of Haifa, Haifa, Israel Mark Antaki, McGill University, Montréal, Canada José Manuel Aroso Linhares, University of Coimbra, Coimbra, Portugal Larry Catá Backer, Pennsylvania State University, University Park, USA Kristian Bankov, New Bulgarian University, Sofia, Bulgaria, Sichuan University, Sichuan, China Vijay Bhatia, Chinese University of Hong Kong, Hong Kong, Hong Kong Katherine Biber, University of Technology Sydney, Sydney, Australia Nicholas Blomley, Simon Fraser University, Burnaby, Canada Patrícia Branco, University of Coimbra, Coimbra, Portugal John Brigham, University of Massachusetts, Amherst, USA Jan Broekman, KU Leuven, Leuven, Belgium, Pennsylvania State University, University Park, USA Michelle Brown, University of Tennessee, Knoxville, USA Sandrine Chassagnard-Pinet, Lille University, Lille, France Le Cheng, Zhejiang University, Hangzhou, China Paul Cobley, Middlesex University, London, UK Angela Condello, University of Turin, Turin, Italy Renee Ann Cramer, Drake University, Des Moines, USA Karen Crawley, Southport, Australia Marcel Danesi, University of Toronto, Toronto, Canada David Delaney, Amherst College, Amherst, USA Nicolas Dissaux, Lille University, Lille, France Michał Dudek, Jagiellonian University, Cracow, Poland Mark Featherstone, Keele University, Keele, UK Magalie Flores-Lonjou, La Rochelle University, La Rochelle, France Marcilio Toscano Franca-Filho, Federal University of Paraíba, Paraíba, Brazil Thomas Giddens, University of Dundee, Dundee, UK Nathalie Hauksson-Tresch, Linnaeus University, Växjö, Sweden Carlos Miguel Herrera, Cergy-Pontoise University, Cergy, France Daniel Hourigan, University of Southern Queensland, Toowoomba, Australia Lung-Lung Hu, Dalarna University, Falun, Sweden Stefan Huygebaert, Ghent University, Ghent, Belgium Miklós Könczöl, Pázmány Péter Catholic University, Budapest, Hungary Magdalena Łągiewska, University of Gdańsk, Gdańsk, Poland Anita Lam, York University, Toronto, Canada Massimo Leone, University of Turin, Turin, Italy David Machin, Zhejiang University, Hangzhou, China Samantha Majic, City University of New York, New York, USA Danilo Mandic, University of Westminster, London, UK Francesco Mangiapane, University of Palermo, Palermo, Italy Aleksandra Matulewska, Adam Mickiewicz University, Poznan, Poland Renisa Mawani, University of British Columbia, Vancouver, Canada Rostam J. Neuwirth, University of Macao, Taipa, Macao Arnaud Paturet, Paris Nanterre University, Naterre, France Andrea Pavoni, Lisbon University Institute, Lisbon, Portugal Timothy D. Peters, University of Sunshine Coast, Sippy Downs, Australia Andreas Philippopoulos-Mihalopoulos, University of Westminster, London, UK Richard Powell, Nihon University, Tokyo, Japan Kimala Price, San Diego State University, San Diego, USA Alison Renteln, University of Southern California, Los Angeles, USA Marco Ricca, University of Parma, Parma, Italy Peter W. G. Robson, University of Strathclyde, Glasgow, UK Austin Sarat, Amherst College, Amherst, USA Cassandra Sharp, University of Wollongong, Wollongong, Australia Julia J. A. Shaw, De Montfort University, Leicester, UK Richard K. Sherwin, New York Law School, New York, USA Lawrence M. Solan, Brooklyn Law School, New York, USA Mateusz Stępień, Jagiellonian University, Cracow, Poland Kieran Mark Tranter, Queensland University of Technology, Brisbane, Australia Farid Samir Benavides Vanegas, Ramon Llull University, Barcelona, Spain Denis Voinot, Lille University, Lille, France Honni von Rijswijk, University of Technology Sydney, Ultimo, Australia Marco Wan, University of Hong Kong, Hong Kong, Hong Kong Oliver Watts, Australian Government Art Collection, Canberra, Australia Xu Youping, Guangdong University of Foreign Studies, Guangzhou, China

The Series Law and Visual Jurisprudence seeks to harness the diverse and innovative work within and across the boundaries of law, jurisprudence, and the visual in various contexts and manifestations. It seeks to bring together a range of diverse and at the same time cumulative research traditions related to these fields to identify fertile avenues for interdisciplinary research. In our everyday lives, we experience law as a system of signs. Representations of legality are visually manifested in the materiality of things we see and spatially experience. Methodologically, aesthetic texts of legality semiotically emerge as examples of visual jurisprudence and illustrate the constitutive waltz between social governance, formal law, and materiality. In its tangled relationship to regulation, the visual complexity of law is semiotically articulated as an ongoing process of meaning imbued with symbolism, memory, and cultural markers. Through a legal semiotics framework of symbolic articulation and analysis, the examination of law that happens in conjunction with the visual expands understandings of how law is crafted and takes root. Additionally, such an inquiry challenges the positivist view of law based within the courtroom as disciplinary spatial practices, the observation of everyday phenomenon, and the visible tethering of regulation to cultural understandings of legality generate a framework of visual jurisprudence. The Series seeks to enliven such frameworks as those in which law happens precisely without formal institutions of law and through which a visual-based methodology of law is crafted through everyday instances of ordinariness that contextualize the relationship between law, culture, and banality. The Series welcomes proposals – be they edited collections or single-authored monographs – emphasizing the contingency and fluidity of legal concepts, stressing the existence of overlapping, competing and coexisting legal discourses, proposing critical approaches to law and the visual, identifying and discussing issues, proposing solutions to problems, offering analyses in areas such as legal semiotics, jurisprudence, and visual approaches to law. Keywords: Legal Visual Studies, Popular Culture, Everyday Law, Spatiality, Legal Semiotics, Legal Geography, Legal Materiality, Legal Transplant, Bioethics, Cyber Law, Communication, Heritage and Territory, Design, Marketing, Packaging, Digitalization, Arts.

More information about this series at https://link.springer.com/bookseries/16413

Andrzej Malec

Introduction to the Semantics of Law

Andrzej Malec Kochanski & Partners sp.kom. Warsaw, Poland

ISSN 2662-4532 ISSN 2662-4540 (electronic) Law and Visual Jurisprudence ISBN 978-3-030-95678-3 ISBN 978-3-030-95679-0 (eBook) https://doi.org/10.1007/978-3-030-95679-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 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 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

The world is the totality of facts, not of things1

1

Ludwig Wittgenstein, Tractatus logico-philosophicus (thesis 1.1).

Acknowledgments

The current version of this book is based on the first edition, originally published in Polish language. The discussion on the first edition allowed for a better writing of this version. I owe special thanks to Professor Witold Marciszewski, for his comments on problems related to the reistic approach and on the role of Wittgenstein in contemporary logic, to Professor Kazimierz Trzęsicki, for his encouragement to develop the discussed theory in an axiomatic form, to Doctor Roman Matuszewski, for his in-depth review of the Polish edition, and also to Doctor Paweł Stacewicz, for organizing an Internet discussion on it. I also thank my son Jakub Malec, who was the first reader of the present English version, for his help in improving the text. I would also like to thank the anonymous reviewer of the book for the present edition, who encouraged me to present the discussed issues in a more popular way, so that they could be of interest to a wider audience.

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Contents

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 What Makes a Book on the Semantics of Law Worth Reading? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 What Is the Law? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 What Are the Legal Events? What Are the Acts? . . . . . . . . . . . 1.4 Can the Legal Norms Be True? . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Can a Norm Result from Another Norm? . . . . . . . . . . . . . . . . . 1.6 Can an Order Result from Another Order? . . . . . . . . . . . . . . . . 1.7 Can Semantics Distinguish Between Natural Law and Statutory Law? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 How Shall the Semantic Issues Be Examined? . . . . . . . . . . . . . 1.9 How to Interpret the Language of Law? . . . . . . . . . . . . . . . . . . 1.10 How to Apply the Law? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11 Why Are the Legal Reasoning Rules Valid? . . . . . . . . . . . . . . . 1.12 How to Formalize Legal Logic? . . . . . . . . . . . . . . . . . . . . . . . . 1.13 How Can Legal Logic Help in the Computer-Assisted Application of Law? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.14 How Can Legal Logic Help in the Computer-Assisted Creation of Law? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.15 In Tribute to Ludwig Wittgenstein . . . . . . . . . . . . . . . . . . . . . . 1.16 Not So Easy Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic Semantic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Domains, Subject Languages, Meta-languages . . . . . . . . . . . . . 2.2 Kinds of Legal Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Kinds of Subject Language Expressions . . . . . . . . . . . . . . . . . . 2.4 Interpretation, Veracity and Semantic Relations . . . . . . . . . . . . 2.5 Semantic Models and Other Methods . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

A Model of the Domain of Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 A Concrete Basis of the Law Domain . . . . . . . . . . . . . . . . . . . 3.2 Wolniewicz’s Ontology of Situations . . . . . . . . . . . . . . . . . . . . 3.3 Legal Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Event as a Sequence of Situations . . . . . . . . . . . . . . . . . . . . . . 3.5 Accessibility Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 The R Relation of Direct Accessibility Between Possible Worlds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 The R+ Relation of Indirect Accessibility Between Possible Worlds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 The SER Relation of Direct Accessibility Between Proper Situations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.4 The SIR Relation of Direct Accessibility Between Alternative Situations . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.5 The SER+ Relation of Indirect Accessibility Between Proper Situations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.6 The SIR+ Relation of Indirect Accessibility Between Alternative Situations . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.7 The Accessibility Relations and the Set of Natural Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Legal Events in Terms of the Ontology of Situations . . . . . . . . . 3.7 Legal Rules as Specific Sets of Events . . . . . . . . . . . . . . . . . . . 3.7.1 Towards the Concept of Legal Rule . . . . . . . . . . . . . . . 3.7.2 Basic Conditions for Legal Rules . . . . . . . . . . . . . . . . . 3.7.3 Conditions Based on Properties of Acts . . . . . . . . . . . . . 3.7.4 Conditions Based on Properties of Situations . . . . . . . . . 3.7.5 Towards the Mathematization of the Law Domain . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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A Model of the Language of Law . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Vocabulary and Grammar . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Interpretation of the Subject Language Expressions . . . . . . . . . . 4.3 The First Order Logic as a Basis of Legal Theories . . . . . . . . . . 4.4 Theories of Legal Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Theory 1: All Legal Events Are Permitted (AEP) . . . . . . 4.4.2 Theory 2: All Legal Events Are Either Permitted or Forbidden (AEPF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Theory 3: All Legal Events Are Either Permitted or Ordered or Forbidden (AEPOF) . . . . . . . . . . . . . . . . 4.4.4 Theory 4: All Legal Events Are Either Permitted or Ordered or Forbidden or Irrelevant (AEPOFI) . . . . . . 4.4.5 Existence of Legal Events . . . . . . . . . . . . . . . . . . . . . . 4.4.6 Selected Theorems of Legal Events Theories . . . . . . . . .

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4.5

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Theories of Simple Acts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Specific Axioms of AAPOF . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Selected Theorems of AAPOF That Are Equivalent to Theorems of AEPOF . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Selected AAPOF Theorems Specific to Acts . . . . . . . . . . 4.6 Theories of Compound Acts . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Specific Axioms of AAPOF for Compound Acts . . . . . . . 4.6.2 Selected AAPOF Theorems Specific to Compound Acts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 First Order Deontic Theories as Descriptions of Law Domain Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85 86 87 88 89 91 92

Semantics of Norms and Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Norms and Normative Expressions . . . . . . . . . . . . . . . . . . . . . 5.2 Veracity and Falsity of Norms . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 True Norms, False Norms, and Popper’s Third World . . . . . . . . 5.4 Veracity of Norms And the Is: Ought Problem . . . . . . . . . . . . . 5.5 Entailment Between Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Contradiction, Opposition and Sub-Opposition Between Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Semantic Relations Between Orders . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 93 . 93 . 98 . 100 . 103 . 105

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Transformations of Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 The Continental Model of Applying the Law . . . . . . . . . . . . . . 6.2 General Norms and Individual Norms . . . . . . . . . . . . . . . . . . . 6.3 Concretization and Specification of Norms . . . . . . . . . . . . . . . . 6.4 Subsumption by Concretizing and Specifying Norms . . . . . . . . 6.5 Subsumption by Redescribing Facts of the Case . . . . . . . . . . . . 6.6 Schemas of the Legal Syllogism . . . . . . . . . . . . . . . . . . . . . . . 6.7 Creativeness of Norms Transformations . . . . . . . . . . . . . . . . . . 6.8 The Precedent Model of Applying the Law . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Epilogue: The Concept of Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

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Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 What Is the Law? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 What Are the Legal Events? What Are the Acts? . . . . . . . . . . . 8.3 Can the Legal Norms Be True? . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Can a Norm Result from Another Norm? . . . . . . . . . . . . . . . . . 8.5 Can an Order Result from Another Order? . . . . . . . . . . . . . . . . 8.6 Can Semantics Distinguish Between Natural Law and Statutory Law? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

8.7 8.8 8.9 8.10 8.11 8.12

How Shall the Semantic Issues Be Examined? . . . . . . . . . . . . . How to Interpret the Language of Law? . . . . . . . . . . . . . . . . . . How to Apply the Law? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Why Are Legal Reasoning Rules Valid? . . . . . . . . . . . . . . . . . . How to Formalize Legal Logic? . . . . . . . . . . . . . . . . . . . . . . . . How Can Legal Logic Help in the Computer-Assisted Application and Creation of Law? . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1

Introduction

In this chapter, the reader is introduced to the main issues that will be examined in the following parts of the book. These issues have been divided into philosophical, methodological, and logical ones. The adopted examination methods as well as the most important results that have been achieved, are also discussed.

1.1

What Makes a Book on the Semantics of Law Worth Reading?

In words of Confucius: if you want to repair the world, you shall begin with restoring the proper meanings to the words. To rephrase it, in order to change the world efficiently, you need its real picture. And semantics, being the science of how words relate to the world, teaches how to create such a picture. At first glance, it seems that the way language is related to the world is simple and intuitive. After all, everyone understands what the sentence ‘It’s snowing’ says. And yet, when one reads in a statute that ‘financial instruments shall be valued at the adjusted purchase price or at market value or at fair value’, they usually are confused about the meaning of such sentence. And more and more frequently laws are being written using this kind of sentences. Why is it that some words are understood clearly, and others are not at all or vaguely? It can be answered this way: the former have been already learned by us, and the latter—not yet. And while it is true, it is not the whole truth. The fact is that the meanings of some words are much easier to comprehend than the meanings of others. While walking, you meet people, cats and dogs, you see trees, cars, houses and streets. You can finally see that it is snowing. During such a walk, however, you do not encounter financial instruments, neither market value, nor fair value, let alone the adjusted purchase price. And no wonder, since these words do not directly refer to concrete objects. But then—to what objects do these words apply? What reality are we talking about in the language of law? It is © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Malec, Introduction to the Semantics of Law, Law and Visual Jurisprudence 6, https://doi.org/10.1007/978-3-030-95679-0_1

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1 Introduction

semantics that allows us to understand how words, including those we do not use while taking a stroll, relate to the world—and it is by no means a trivial matter! This issue should be of great interest to all lawyers, but it may also be quite interesting to any reader who wants to better understand the world around them. That is because the law plays an increasingly significant role in the modern world. It surrounds, more and more closely, everyone. And it is precisely the fact that this book shows what the laws and court judgments really say, that makes it worth reading. The semantic issues discussed in this book relate to several areas of legal theory, namely: a) the philosophy of law—helping to answer questions about the nature of law and legal duties; about the nature of legal events, including acts; about the veracity of legal norms, and semantic relations between legal norms, and semantic relations between orders, as well; about a relation between natural law and positive law, b) the methodology of law—helping to answer questions about methods of studying legal language semantics; about semantic grounds of legal language; about models of law application; about justifying the rules of legal reasoning, c) legal logic—helping to answer questions about an intuitive form of legal logic; about its application to the computerization of law. In this introduction, a few words will be devoted to each of these issues so that a reader may decide whether or not to read the relevant chapters of this book.

1.2

What Is the Law?

Have you ever thought about what the law is? It can be said that the law is a collection of inscriptions, specifying how people are supposed to behave in various situations, adopted by the authority. And, in some sense, it is true. In each legal system, it is possible to indicate such sets of inscriptions. Sometimes, it is a set of laws adopted by a parliament. In other instances, it is a set of court decisions. These collections allow us to classify behaviors as ordered, prohibited and permitted, or as recommended and not recommended, or in some other way. Generally speaking, these collections, which may be called ‘collections of legal texts’ or simply ‘legal texts’, allow us to assign some deontic modalities to behaviors. Thus, it can be assumed that: THE LAW ¼ THE SET OF LEGAL TEXTS: But it is also possible to think about the law in a different manner. The Polish sentence ‘Zakazane jest deptanie trawników’ and the English sentence ‘It is forbidden to trample the lawns’ seem to say the same, albeit in different words. So, is it not the case that there is the same meaning hidden behind these inscriptions? In fact, this is the case when the sentences describing the reality are considered. For example, it

1.2 What Is the Law?

3

is recognized that the Polish sentence ‘Pada śnieg’ has the same meaning as the English sentence ‘It is snowing’. What does it mean ‘to have the same meaning’ in this case? Provisionally, it can be answered as follows: two sentences describing the reality have the same meaning if and only if they describe the same situation. And what is the answer in the case of legal provisions? What does it mean that the Polish sentence ‘Zakazane jest deptanie trawników’ and the English sentence ‘It is forbidden to trample the lawns’ have the same meaning? Perhaps, it means that both sentences describe the same object, exactly as it was in the case of the sentences ‘Pada śnieg’ and ‘It is snowing’. However, this time, this object is not a situation. It is something else, a rule of behavior, a legal rule. Thus, the law can be treated not only as a collection of inscriptions, but also as something that is hidden behind this collection. Namely, it can be treated as the set of legal rules. A legal rule may be described in various legal texts, in many ways, in various languages, but it will remain one and the same. It is related to inscriptions in the sense that these inscriptions are used to code the rule, but it is not identical with them. A legal rule is something beyond the language. Respectively, it can be assumed that: THE LAW ¼ THE SET OF LEGAL RULES: If legal rules are admitted, the question arises about their ontological status. Are they only universal meanings of legal texts? Or, maybe, they are like mathematical ideas, which may be discovered or not, but are valid all the same? If the latter holds, the legal rules may be discovered by the lawmaker, or not, but their validity is independent from the lawmaker and legal texts created by them. Therefore, the question about the ontological status of legal rules is also a question about the existence of natural law and its relation to statutory law: STATUTORY LAW 6¼ NATURAL LAW? To consider these questions, it is not enough to understand that legal rules differ from legal texts. A clear understanding of the nature of these rules is also required. This is where semantics can help. In this book, the nature of legal rules will be explained. In Chap. 3, a law domain model will be proposed, and legal rules will be its parts. They will be defined as sets of legal events, so they will be described in terms of the set theory: LEGAL RULES ¼ SETS OF LEGAL EVENTS: This understanding of legal rules is a consequence of looking at the domain of law from the perspective of logical atomism. Namely, the law domain model will be

4

1 Introduction

defined in terms of Ludwig Wittgenstein’s1 ontology. In his Tractatus logicophilosophicus, Wittgenstein presented an atomic picture of the world. According to his theory the world is a totality of facts. The world itself is a ‘maximum fact’ consisting of ‘smaller’ facts, and those in turn consist of even ‘smaller’ ones, up to a level of atomic facts—the smallest facts that no longer contain constituent facts. The worlds that are also possible but do not exist are counterparts to the real world. These possible, but not existing, worlds are the totalities of non-facts forming hierarchies similar to the hierarchy of facts in the real world. A formal description of this ontology has been provided by Bogusław Wolniewicz2 in his Ontology of situations. His formalism will be used as a starting point of our model of the law domain. As a result, several abstract objects from the domain of law, such as legal events, including legal acts, and, finally, legal rules, are going to be defined in Wittgenstein’s terms. Thanks to this approach, the intuitive concept of law will be supplemented with a formal concept of legal rules. Thus, it will be possible to verify our intuition in respect to the law, and develop it, as well.

1.3

What Are the Legal Events? What Are the Acts?

The concept of legal events is among the most important ones in legal theory. A legal event is an event that brings legal effects. Usually, legal events are under human control. As a result, legal responsibility may be attributed to a person. Among legal events, the acts are distinguished. The acts are simple actions or omissions, fully depending on their perpetrators. Whereas these concepts are important, both in legal theory and in legal practice, the ontological status of legal events and acts is ambiguous. For example, it is not clear what kind of objects they belong to, and what their relation to legal rules is. In Chap. 3, an answer to the abovementioned questions is proposed. Namely, legal events and acts may be understood as sequences of situations:3 THE ACTS ⊂ THE LEGAL EVENTS ⊂ THE STRINGS OF SITUATIONS: Such an understanding is not only intuitive but also convenient, since there is a good philosophical theory of situations created by Wittgenstein and formalized by Wolniewicz. By developing this theory, a formal theory of legal events and acts is 1

Ludwig Wittgenstein (1889–1951), an Austrian-British philosopher of logic and language, the ‘godfather’ of neopositivism. 2 Bogusław Wolniewicz (1927–2017), a Polish philosopher and logician, translator and commentator of Ludwig Wittgenstein’s Tractatus, the creator of formal ontology (metaphysics) of situations. 3 The symbol ‘ ⊂ ’ that occurs below is the sign of inclusion. The expression ‘A ⊂ B’ is read ‘the set A is included in the set B’.

1.4 Can the Legal Norms Be True?

5

obtained, and as a result—legal rules are defined as sets of legal events. These results may be useful when one attempts to computerize the law. They also bring a comforting thought that behind intuitive concepts there are mathematical concepts, too.

1.4

Can the Legal Norms Be True?

One of the most important concepts of legal theory is the concept of legal norm. Legal norms express legal duties. A legal norm is neither a legal provision nor a legal rule. The legal norms are distinguished from the legal rules by the fact that they are language expressions, and the legal rules are not. In turn, the legal provisions, like the legal norms, belong to the language of law. However, every legal provision is a fragment of a legal text. And legal norms are first and foremost the expressions of a specific shape. While some legal norms are fragments of legal texts, others are not. Some of them are constructed on the basis of legal provisions, by transformations of these provisions. And some others are not. Moreover, some legal norms can contradict legal provisions. This will be the case, for example, when a legal norm expresses a rule of natural law that is contrary to statutory law. There is a common belief in legal theory that the legal norms are neither true nor false. The reason for this belief is that legal norms do not describe the world, but recommend how to change it. Indeed, the norm ‘It is forbidden to trample the lawns’ does not describe the condition of the lawns. It does not indicate whether lawns are being trampled on or not. It only recommends not to trample the lawns. Therefore, it is said that rather than being descriptions, legal norms are prescriptions. Thus, the norm ‘It is forbidden to trample the lawns’ says something similar to the order ‘Don’t trample the lawns!’. When, however, legal norms are assumed to be true or false, this truthfulness is usually associated with their validity. A norm is presumed to be true when it is valid, and false when it is not. Thus, the norm ‘It is forbidden to trample the lawns’ is true when a relevant authority has imposed such a prohibition. On the other hand, this norm is false when such prohibition has not been imposed by this authority. Consequently, a false norm becomes true as soon as it is established, and may later become false again as soon as it is repealed. Per contra, semantic inquiries prove that in respect of veracity, the legal norms do not differ from the theorems of mathematics. In fact, every declarative sentence, if interpreted, is either true or false. And there is no doubt that the norms, unlike the orders, are declarative sentences. Instead of the physical world, they describe a

6

1 Introduction

different world called the ‘Popper’s third world’.4 And that exact third world of Popper is described by the mathematical theorems. These inquiries are carried out in Chap. 5 of the book. First, legal norms are defined there, and then the truth conditions for such norms are examined. This examination reveals how the veracity of norms is rooted in the third world of Popper, and also questions the truth of Hume’s famous thesis—the claim that statements about the physical world are logically separated from deontic statements (i.e. statements about duties).

1.5

Can a Norm Result from Another Norm?

An issue on the border of the philosophy of law and the methodology of law is whether one norm may result from another norm. It is said that a sentence results from another sentence when the truth of the latter guarantees the truth of the former. Thus, if someone claims that norms are neither true nor false, they shall also claim that the resulting cannot exist between legal norms. On the other hand, everyone can see that the norm ‘It is forbidden to trample the lawns in the morning’ results from the norm ‘It is forbidden to trample the lawns’. The theory of law tries to somehow reconcile the objections in respect to the veracity of norms with the obviousness of the fact that norms result from norms. One way is to assume that the resulting between norms is a specific one, namely that it is not based on a guarantee of truthfulness. Semantic inquiries, however, lead to the conclusion that the resulting between norms is a standard one. Such a conclusion is not surprising, having in mind that the legal norms are supposed to be sentences in a logical sense, that is, they are supposed to be either true or false. These inquiries are presented in Chap. 5 of the book.

1.6

Can an Order Result from Another Order?

Of course, the orders, i.e. expressions such as ‘Do not trample the lawns!’ or ‘Drink milk!’, are not sentences in a logical sense, that is, they are neither true nor false. As a consequence, they cannot result from each other in a standard manner, i.e. when the truth of one sentence guarantees the truth of another. However, there is a relation between the orders ‘Eat an apple and a pear!’ and ‘Eat an apple!’ that fairly resembles the resulting. Namely, whoever obeys the order ‘Eat

4

Karl Popper (1902–1994), an Austrian philosopher of science and social philosopher, the creator of critical rationalism and the concept of three worlds—the physical world, the psychic world and the world of cultural creations (where abstract objects exist).

1.8 How Shall the Semantic Issues Be Examined?

7

an apple and a pear!’, also obeys the order ‘Eat an apple!’, but not vice versa. It is a semantic relation referring to the meanings of orders, although it is not based on a guarantee of truthfulness. In Chap. 5, this relation is defined formally with reference to the concept of event. In that chapter, other semantic relations between orders, similar to semantic relations between norms, are defined with reference to this concept, as well. They are: contradiction, opposition, and sub-opposition of orders. This way, the concept of event turns out to be a common ground for defining both the semantic relations between norms and the semantic relations between orders. This allows us to see the common roots of norms and orders, even though the former are sentences in a logical sense, and the latter are not. It may be said that whereas the norm ‘It is forbidden to trample the lawns’ is a description of a legal rule, i.e. a description of a set of legal events, the order ‘Do not trample the lawns!’ prescribes this legal rule, i.e. prescribes these events.

1.7

Can Semantics Distinguish Between Natural Law and Statutory Law?

After the Second World War, it has become apparent that not all statutes deserve to be respected. Hence, in the theory of law, the issue of determining the criteria that positive law should meet in order to deserve approval has arisen. There are many approaches to defining such criteria, including various concepts of natural law. This book deals with other issues, but the distinction between legal rules and legal norms allows for a formal semantic distinction between natural law and statutory law. Namely, statutory law can be identified with the set of norms derived from legal texts, and natural law—with the set of norms describing legal rules, or simply with the set of legal rules. Thus, the relation between statutory law and natural law can be presented as a relation between two sets of norms. This issue is discussed in Chap. 7, the epilogue of this book. In the same place, one more issue is mentioned. Namely, the possibility of inferring legal rules from evaluations of events.

1.8

How Shall the Semantic Issues Be Examined?

The first methodological issue of this book is choosing the method of examination. Studies of relations between the domain and the language of law are often carried out without going beyond the natural language. In this book, however, such studies will be supported by the tools of logical semantics and other formal tools. The method used in this book can be summarized in four main points as follows:

8

1 Introduction

1) formal models of the domain and the language of law are created—this way, the benefits of studying simpler objects instead of complex ones are achieved (the so-called ‘toy-model approach’), 2) in the law domain model, concrete objects (i.e. objects perceived by the senses) and abstract objects (i.e. objects not accessible to the senses) are distinguished; the constructing of the domain starts from defining concrete objects, and the abstract ones are introduced with reference to the concrete ones—this way, the entire domain is ultimately ‘anchored’ in a sensually perceivable part of the reality (the so-called ‘ultimate concretism’), 3) in the law domain model, the situations are the only concrete objects—this way, the basic objects are simple, intuitive, and equipped with a good philosophical theory and a good mathematical theory (the so-called ‘situational approach’), 4) in the legal language model, which is the classic language of first order predicate logic, with variables running through a set of situations and events, the meanings of many predicates are determined at first by direct reference to the law domain model—this way, these predicates obtain intuitive understanding prior to the meanings given to them by the axioms of the constructed deontic theories (the so-called ‘intuitive formalism’5). This method combines the intuitive reflection with formal accuracy. It can also be useful for other inquiries in the legal theory. This method is outlined at the end of Chap. 2, and the entire book is an illustration of it.

1.9

How to Interpret the Language of Law?

The second methodological issue of this book is the semantic perspective of interpreting the language of law. The word ‘interpretation’ (from Latin: interpretatio ¼ an explanation) has two meanings. In the first, ‘interpretation’ means clarifying unclear expressions, i.e. explaining their meanings in terms of other expressions.6 In the second, ‘interpretation’ means something more, namely—linking expressions with a domain the law speaks about. For interpretation in the latter sense, the law domain categorization is important, as it determines the objects of the domain. If it is assumed that the law speaks about persons, the legal norms shall be interpreted as applying to these persons. In this approach, when interpreting a legal norm, one should determine who this norm concerns and what duty it assigns

5

This term comes from Stanisław Leśniewski (1886–1939), a Polish logician and philosopher, the creator of three original systems of logic which are called ‘mereology’, ‘ontology’ and ‘protothetics’. In accordance with Stanisław Leśniewski’s intuitive formalism, formal theories, including logical ones, should have an intuitive interpretation. 6 In this sense, Romans had used to say Clara non sunt interpretanda—what is clear, does not need interpretation. It is needless to say that the lawyers are usually masters of such interpretation.

1.9 How to Interpret the Language of Law?

9

to them. For example, when interpreting the legal norm ‘It is forbidden to trample the lawns’, one will establish that this norm applies to every person, and the assigned duty is the obligation of avoiding lawns. However, if it is assumed that the law speaks about events, then the legal norms shall be interpreted as applying to these events, i.e. to some sequences of situations. In this approach, when interpreting a legal norm, one should determine what situations are being replaced and what situations are replacing them. For example, when interpreting the legal norm ‘It is forbidden to trample the lawns’, one will find that it applies to any situation in which a person is standing in front of a lawn. Further, one will establish that this situation shall be replaced by one of the following situations: the person has turned left; the person has turned right; the person has turned back; or by some other situation, but not by the one in which the person has stepped onto the lawn. In fact, in this first approach, the law speaks not only about persons, i.e. concrete objects, but also about legal duties, i.e. abstract objects. Therefore, the schema of interpretation looks as follows:

In the latter approach, however, the law speaks only about replacing some situations with others, that is about replacing something that can be seen, or at least imagined, with something else that can also be seen, or at least imagined. Thus, in the latter approach, the law speaks only about replacing concrete objects with other concrete objects. Therefore, the schema of interpretation looks here as follows:

Considering the categories of objects to which the norms refer, the first approach could be called a ‘person-duty interpretation’, and the latter—a ‘situational interpretation’ or a ‘concrete interpretation’. In this book, the latter semantic perspective is advised for interpreting the language of law. In Chap. 3, the domain of law is divided into categories in accordance with Wittgenstein’s ideas, with situations as a basic concept, and in Chaps. 5 and 6 this approach is used when issues of legal language interpretation are considered.

10

1.10

1 Introduction

How to Apply the Law?

Have you ever wondered which model of law application, continental or precedent, is more convenient? In particular, which of these models brings more predictable results of applying the law? And also which is more prone to computerization of the law? In the Anglo-Saxon (precedent) model, if two cases are sufficiently similar, then an individual norm once applied to the first case, will also be applied to the second. In this model, the only problem is to assess accurately whether the old and new facts are sufficiently similar. In the continental model, first, general and usually abstract norms are derived from legal texts. Then, they are compared with facts of the case. If the facts fit to the hypothesis of a legal norm, then so-called ‘legal syllogism’ is applied to the legal norm and the facts, and an individual norm relevant to these facts is derived as a result. Matching the facts to the hypothesis of the norm is called ‘subsumption’. And, in fact, the main problem in the continental model is to subsume accurately. This problem is not trivial at all, as the facts are usually described in individual and concrete terms, and the hypothesis of the norm is expressed in general and abstract terms. In this book, the basic objects of the law domain are situations. That leads to a new schema of general norms. This new schema, along with the distinction between concrete and abstract predicates, makes it possible to define subsumption in terms of two other processes that can be formally represented, namely the concretization and the specification of legal norms. Thanks to this approach, the subsuming is slightly more predictable and closer to computerization. Chapter 6 is devoted to these issues. The transformations of norms in the course of applying the law are examined in detail. The concepts of concretization and specification are explained. And, finally, the continental and the Anglo-Saxon models are compared.

1.11

Why Are the Legal Reasoning Rules Valid?

In legal practice, many rules of legal reasoning are applied. Some of them are dating back to the times of Roman lawyers. For example, let us look at the rules called a minori ad maius and a maiori ad minus. According to the first of these rules, if it is forbidden to do less, then, all the more, it is forbidden to do more. If it is forbidden to break into someone else’s house, all the more so, it is forbidden to break into someone else’s house and burn it down. If it is forbidden to cross the lawns, even more it is forbidden to trample the lawns in a systematic manner. According to the second rule, if it is permitted to do more, then, all the more, it is permitted to do less. If it is permitted to take all the fruits from the garden, all the more so, it is permitted

1.12

How to Formalize Legal Logic?

11

to take a part of them. If it is permitted to detain a criminal by force, even more it is permitted to detain them without the use of force. The methodology of law examines legitimacy of such rules. Are these rules binding only by the force of tradition, or is there a logic behind them, independent of this tradition? This book shows that it is the structure of the law domain, including the properties of legal rules, that determines which rules of legal reasoning are valid. In Chap. 4, theories of legal duty are examined. They describe properties of legal rules belonging to different hypothetical law domains. This examination shows how the properties of legal rules are ‘translated’ into axioms and theorems of particular theories. Thus, one can see how the properties of legal rules entail rules of legal reasoning which are expressed by these axioms and theorems. Some of these rules of legal reasoning are quite simple, e.g. ‘what is permitted, is not forbidden’, ‘what is ordered, is also permitted’, ‘there is always something permitted’ etc. However, others are not, e.g. some of these rules describe deontic relations between parts and the whole (‘if an act is permitted, every part of it is permitted’, ‘if all parts of an act are permitted, the act itself is permitted’ and so on). Such rules are a good point of reference when the reasonings a minori ad maius and a maiori ad minus are examined. Moreover, the axioms and theorems of the abovementioned theories also express rules of legal reasoning which are intuitive but nevertheless have not been discussed in the legal theory yet. The deontic logics presented in Chap. 4 are probably the first logics of this type. They are an outcome of the situational approach to the law domain. The relations between the structure of the law domain and the rules of legal reasoning are also examined in Chap. 5. Namely, it is examined how the properties of legal rules and the structure of legal events determine the validity of particular rules of legal reasoning, expressed by deontic sentences, including norms.

1.12

How to Formalize Legal Logic?

The term ‘legal logic’ is understood in many ways. Sometimes, it denotes all theories of legal reasoning, whether they are formal or not. In such a broad sense, legal logic intersects with the methodology of law. Here, however, this term is understood as denoting only formal systems of legal logic, also called ‘deontic logics’ (from Greek: deon ¼ duty). The deontic logics formalize the concepts of duty. However, this is not an easy task. The deontic logics date back to the 1950s. To a wide audience, the propositional deontic logics are the best known. In such logics, the deontic operators apply to sentences, including compound sentences. For example, the formula: O ð p ˅ qÞ is a sentence of this kind. It is read as follows: ‘It is ordered that p or q’ where the letters ‘p’ and ‘q’ stand for simple sentences of the natural language. Usually, the

12

1 Introduction

deontic operators of these logics are defined similarly to modal (aletic) operators.7 And, in fact, this is a source of the odd properties of these logics and many open paradoxes. Simply, the deontic modalities fundamentally differ from the aletic ones. For example, the very idea to apply deontic operators to any sentence is odd. Indeed: what is the meaning of the sentence ‘It is ordered that Mount Everest is the highest mountain in the world’, or ‘It is forbidden that 2 + 2 ¼ 4’?8 The paradoxes of propositional deontic logics are well known. But let us mention two of them. First, the following sentence is a theorem of propositional deontic logics: O p ! O ðp ˅ qÞ: This sentence is read as follows: ‘if p is ordered, then p or q is also ordered’. Thus, if it is ordered to save a drowning person, then it is also ordered to save a drowning person or drink coffee. The open paradox is obvious. Second, the sentence below also is a theorem of propositional deontic logics: F p ! O ðp ! qÞ: This sentence is read as follows: ‘if p is forbidden then it is ordered that if p then q’. Thus, if it is forbidden to kill a man, then it is ordered to rob this man after killing him’. For these reasons, intuitive and paradox-free legal logic is sought outside of the propositional logics. For example, there are approaches where obligation, prohibition and permission are defined in terms of action systems. It is the actions that are ordered, forbidden or permitted.9 To give another example, some logicians claim that simple approaches are not suitable for the description of the law domain and propose many-sorted framework for normative reasonings.10 This book also promotes the view that intuitive and paradox-free legal logic should be sought outside of the propositional logics. However, no special languages for deontic logics are created here. In Chap. 4, it is demonstrated that the language of first order predicate logic is sufficient for deontic theories. Thus, deontic logics can be formed as first order theories. The universe described in these theories is the universe of situations and events. As a result, prescriptions, prohibitions and permissions apply to events, including acts. These theories are intuitive and paradoxfree since they were constructed, from the very beginning, to describe well-defined models of the law domain.

The aletic operators are, first and foremost, the necessity operator and the possibility operator. It is also odd to apply deontic operators to compound sentences as well as to apply several deontic operators to one argument. See: Malec (2019), p. 109. 9 See: Czelakowski (2015). 10 This is the approach of Prof. Valentin Goranko from Stockholm University. 7 8

1.13

1.13

How Can Legal Logic Help in the Computer-Assisted Application of Law?

13

How Can Legal Logic Help in the Computer-Assisted Application of Law?

This book also fits in the inquiries of the computer-assisted application of law. Whether we like it or not, the computers are allowed to make decisions about people’s affairs more and more often. For example, the press reports that in China artificial intelligence systems, supported by the omnipresent monitoring systems, already discipline the citizens.11 Various devices, including cameras, record events, and computers recognize people, assess their behavior and apply sanctions, for the time being, only of the administrative nature. For example, they may reject a loan application, block the right to purchase airline or train tickets, depending on the citizen’s score. It seems that China is the leader in this respect, nonetheless in all other countries computers also make decisions with legal effects on people. In Poland, for example, during the COVID-19 pandemic, it was computers that qualified, or not, the companies affected by the pandemic for the government assistance. Artificial intelligence is also becoming important in the court application of law. Computers help to find relevant laws, regulations, rulings, court decisions or commentaries. They help to match a legal norm to facts of the case. Perhaps, soon, as it is already in China, they will be deciding on sanctions themselves. Perhaps, sooner or later, they will start to resolve civil disputes independently, as well.12 However, before the computers replace the judges they will be their assistants. Most probably, just as they validate mathematical proofs now, the computers will soon validate court decisions. The judge will decide the case at their discretion. But the computer will find and underline ‘in red’ all parts of the decision that violate legal norms, or rules of reasoning, or that are inconsistent with common sense. It will be easier for the judge to control the correctness of their own reasoning, for the parties to complain of errors in this reasoning, and for a higher court to resolve these complaints. Then, at last, the proverb: charta non erubescit will lose its raison d'être. To support this process, the law should be translated into a language understandable for the computers, that is into the language of computer programs, and earlier—algorithms.13 On the other hand, when computers will apply the law by themselves, they will also have to interpret this law. This process of moving from laws and regulations to individual legal norms, carried out by artificial intelligence, does not have to be

This is called ‘the system of social trust’. Another interesting example of computers deciding about humans is the use of artificial intelligence in combat robots. Some armies are already equipped with combat robots, e.g. drones, which independently, without human control, make decisions about attacking enemy troops. When it is left to the robot to decide whether to kill a human being, it is worth to understand and well-program the rules of the decision-making process. 13 An algorithm is a sequence of unambiguous instructions, the execution of which allows you to achieve the goal. The algorithm is an older relative of the computer program. 11 12

14

1 Introduction

understandable for humans.14 If we want to retain the ability to control the computers in this regard, we must train them how to justify their decisions in a way that is consistent with our perspective.15 In a word, we have to teach these computers to justify their decisions in a way the judges do. To achieve this, a formal description and systematization of legal reasonings are necessary. The proposed model of the law domain from Chap. 3, deontic logics from Chap. 4, and formal schemas of legal norms from Chap. 5, as well as a formal describing of subsumption from Chap. 6 of the book, are our contribution to the inquiries on the automatization of law. The deontic theories from Chap. 4, the semantics of norms and orders from Chap. 5, as well as a formal picture of law application from Chap. 6, are our further contribution to the inquiries on formalizing and systematization of the rules of legal reasoning.

1.14

How Can Legal Logic Help in the Computer-Assisted Creation of Law?

This book also fits in the inquiries of the computer-assisted creation of law. Everything said above about the role of legal logic in the computer-assisted application of law, remains relevant when the computer-assisted creation of law is considered. If the computers are able to control the validity of judges’ reasoning, they will also be able to control the process of lawmaking. Once the existing statutory law is formalized, the computers will ‘keep an eye’ on the legislator. They will indicate whether the newly created law complies with the existing law, and if it is not the case, how to deal with this inconsistency. Furthermore, when the lawmaker accepts some basic norms in a system of formalized law, then, at the same second, a multitude of derived norms are potentially created. Same as in mathematics, when accepted axioms and rules of inference give rise to an infinite number of theorems of a mathematical theory. The computers will be able to write down all these derived norms. They will automatically supply the law created by the lawmaker with all necessary auxiliary regulations. At the same time, they will maintain consistency of the new law with the old one. Moreover, by showing the consequences of basic norms created by the lawmaker, the computers will help us to better understand and assess whether these basic norms are reasonable. If some norms have ridiculous consequences, they should not be created. Therefore, the computers will also help us to create the reasonable law. 14 See: Jarek Gryz, Marcin Rojszczak, Black-Box Algorithms and the Rights of Individuals: No Easy Solution to the ‘Explainability’ Problem. 15 Of course, one may ask if human control on the computer decisions is needed at all. After all, in computations, the computers are more trusted than humans. Perhaps such trust in artificial intelligence is also justified in the law domain. On the other hand, the need for the possibility of appeal against a legal decision made by a computer seems to be deeply grounded in European thinking because of the tradition of Aristotelian intellectualism.

1.16

Not So Easy Introduction

15

In time, this is probably how Europe will cope with the present terrible quality of the Community law. Our inquiries on the automatization of law and the formalization and systematization of the rules of legal reasoning, mentioned in the previous section, are also a small contribution to the researches on the computer-assisted creation of law.

1.15

In Tribute to Ludwig Wittgenstein

There is a view that in the face of new developments in logic, especially in the field of deductive theories (Hilbert, Gödel, Turing, etc.), Wittgenstein’s Tractatus logicophilosophicus has become obsolete. This book shows that Wittgenstein’s idea of ‘depicturing’ facts in true sentences offers new possibilities for formalizing legal concepts and the legal language, and for the mechanization of legal reasonings. Thanks to the Wittgenstein perspective, legal rules and legal events can be described in terms of facts. The Wittgenstein-style deontic logics allow us to penetrate deeply into the structure of sentences, e.g. they reflect relations between simple and complex acts. Thanks to this, the structure of abstract norms and concrete ones can be described more precisely, as well as the process of transition from the former to the latter. This perspective opens up new possibilities for the computerassisted application of law. Finally, Wittgenstein’s approach brings valid arguments for the view that legal norms are sentences in a logical sense (i.e. they are either true or false). I believe that the Wittgenstein perspective presented in this book provides lawyers with arguments useful in studies devoted to legal norms, and legal interpretation. It may also be helpful for the law students in their attempt to understand the basic concepts of law as well as in the researches focused on formalizing the legal language and the automatization of legal reasonings.

1.16

Not So Easy Introduction

I still owe the readers an explanation of this book’s title. Namely, this book is an introduction to the semantics of law in the sense that it deals with the basic issues of legal semantics: how to categorize the law domain and the language of law, how to relate the language of law to the law domain, what is the nature of legal norms, etc. However, unfortunately, this kind of introduction does not mean avoiding mathematics. This is for the following three reasons. First, the very idea of mathematization of the law corresponds to modern trends in science. Mathematics is making its way into other sciences. It is impossible to imagine natural sciences, e.g. modern physics, without mathematics. No one is also surprised any more by mathematical theories in economics or in linguistics. The reason for this expansion of mathematics is practical: the mathematical

16

1 Introduction

description of any regularities that occur in any domain is the most accurate (though not always the most detailed). When someone makes the statement ‘The law is the highest form of moral knowledge’, and someone else makes a different statement ‘The law is a form of class oppression’, you may like these statements, or not. However, it is difficult to decide which of the debaters is wrong, and which is right. On the other hand, when someone says that ‘2 + 2 ¼ 4’, and someone else says that ‘2 + 2 ¼ 5’, most people will probably decide without hesitation that the first debater is right and the second is wrong. The same holds for applying mathematics to legal inquires. It leads to a more precise discourse. When the law is defined as a set of legal rules that meets some enumerated and strictly expressed conditions, and, in turn, legal rules are defined as sets of legal events that meet some enumerated and strictly expressed conditions, and legal events are defined as sequences of situations in the sense of Wittgenstein-Wolniewicz, it is much easier to discuss the drawbacks and benefits of this understanding of the law than in the case of statements such as ‘The law is the highest form of moral knowledge’ or ‘The law is a form of class oppression’. Secondly, it is impossible to avoid mathematics when you want to use the tools of logical semantics. In the twentieth century, logical semantics got many formal results that are worth using. It is enough to mention Tarski’s semantic definition of truth. According to Tarski’s approach, the truth is a property of sentences of one language that is expressed in another language (metalanguage). In order to determine the truth and falsehood of the sentences of a given language, another language is necessary, rich enough for a ‘comparison’ of these sentences with the reality they describe. This approach is adopted here, in particular, it will be used to resolve the issue of the veracity of legal norms, and this approach requires some math. Third, the law itself is asking for a mathematization. It has its own internal logic (of course, I am not thinking of modern administrative regulations, especially tax regulations). When a law student explores, for example, Roman law, they see how its particular elements are arranged and matched to each other. Entire fragments of Roman law could be presented axiomatically. Due to this internal logic, the law is closer to mathematics and physics than to the humanities. It is worth noting that Max Weber himself already preached the idea of a judging machine that would settle legal disputes mathematically. Perhaps, because of these formal tools, this book would be better called ‘Not so easy introduction to the semantics of law’. On the other hand, these formal tools are quite simple and a moment of concentration is enough to fully enjoy the benefits they bring. However, if the reader hates math, they can simply ignore them. The text of this book itself allows to understand quite well the essence of the arguments, without delving into the formulas. During my high school years, I stared at the sentence written on the wall of the physics classroom: ‘Not everything in physics is logic and mathematics - there is also beauty and poetry in it’. I believe that the same holds for the law. Thus, my aim here was to combine the formal and intuitive approaches.

References

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References Czelakowski J (2015) Freedom and enforcement in action. Trends in logic (Studia Logica Library), vol 42. Springer, Dordrecht Malec A (2019) Deontic logics as axiomatic extensions of first-order predicate logic: an approach inspired by Wolniewicz’s formal ontology of situations. Axioms 8(4):109

Chapter 2

Basic Semantic Concepts

In this chapter, the basic kinds of objects in the law domain and the basic kinds of expressions in the language of law will be distinguished. First, however, it will be explained what domains, subject languages and metalanguages are. These concepts will be helpful in explaining that the language of legal semantics is a metalanguage in which both the law domain and the language of law are described, and relations between this domain and this language are defined. Next, the following categories of the law domain objects will be discussed: the concrete individuals (persons and things), the concrete collectives (situations), the properties and the relations. The discussion will be especially focused on the concept of situation, as situations are a starting point for defining abstract objects of the law domain, including events and acts. In turn, the following categories of the legal language expressions will be discussed: the sentences, the names, and the functors. The concrete sentences and predicates, as well as the abstract ones, will be discussed in detail, as they are important for our further research, e.g. for defining subsumption in Chap. 6. Next, the concepts of interpretation, truth and falsehood, entailment, contradiction, opposition, and sub-opposition will be reminded to the reader. This chapter will be concluded with a discussion of the concept of model. It will be explained how models of the law domain and the legal language help in learning semantic relations between the real domain of law and language of law.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Malec, Introduction to the Semantics of Law, Law and Visual Jurisprudence 6, https://doi.org/10.1007/978-3-030-95679-0_2

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2.1

2 Basic Semantic Concepts

Domains, Subject Languages, Meta-languages

The word ‘semantics’ (from Greek: semeion ¼ a sign) means either a part of language or a field of knowledge. The rules of linking the expressions of a language with a domain constitute the semantics as a part of this language.1 And these rules are of interest to the semantics as a field of knowledge. This book examines the language of law from the perspective of logical semantics, i.e. some formal tools are applied.2 Before applying these formal tools, however, let us explain some basic semantic concepts first. To speak about any domain (e.g. about the world outside the window, or about natural numbers), a language (a system of signs) adapted to this domain is required. Such a language should include signs referring to the domain (e.g. the word ‘mountain’ refers to some terrain, in natural language, and the sign ‘1’ refers to a natural number, in the language of mathematics). A language that is used to communicate about a domain is called a ‘subject language’ in respect to this domain. For example, the sentences: ‘It is raining outside the window’, ‘John and Kate went to the forest’, ‘The highest mountain of the world is Mount Everest’ are sentences of the subject language in respect to the domain of physical objects (i.e. in respect to the reality3), whereas the sentences: ‘2 + 2 ¼ 4’, ‘5 > 3’, ‘1 ¼ 1’ belong to the subject language in respect to the domain of natural numbers. A language itself may constitute a domain that we are talking about. To talk efficiently about a language, another language is required.4 This latter language includes signs that refer to the first language and is called a ‘meta-language’ in respect to this first language. The sentences: ‘The word ‘rain’ consists of six letters’, ‘The word ‘forest’ is a noun’, ‘The name ‘Mount Everest’ is a proper name’, ‘The sentence ‘2 + 2 ¼ 4’ is true’ are examples of meta-language expressions. 1

According to logic, three sets should be defined to create a language: (i) a dictionary—a set of signs that will be used as words of this language (the strings of such words are called ‘expressions’), (ii) a grammar—a set of rules for distinguishing correct expressions from others (‘Kate has a cat’ is a correct expression in the English language—unlike the expression ‘Kate cat a has’; the first expression follows the rules of English grammar while the second does not), and (iii) a semantics—a set of rules for linking correct expressions with a domain outside the language. 2 Semantic studies date from antiquity. Currently, this word covers both linguistic and logical approaches. Logical semantics is a part of logical semiotics, i.e. logical theory of signs. Logical semiotics covers three fields of studies: (i) studies of syntax—examining relations between signs, (ii) logical semantics—examining relations between signs and domains to which the signs relate, (iii) logical pragmatics—examining relations between signs and sign users. In turn, logical semiotics is one of three parts of logic in a broad sense (next to formal logic and general methodology of sciences). 3 The word ‘reality’, in the broadest sense, refers to both concrete and abstract objects. In this paper, however, the reality is identified with the domain of physical objects. 4 Of course, it is possible to talk about natural language in natural language itself. However, this property of natural language leads to paradoxes. Therefore, logicians avoid such situations. See: Marciszewski (1987), pp. 174–181 (A close counterpart in English: Marciszewski 1981).

2.2 Kinds of Legal Objects

21

This way, the subject language in respect to a domain is used to communicate about this domain, and the meta-language of this subject language is used to speak about the subject language in question. Going further, if one wants to communicate about the aforementioned meta-language, it shall be done in the meta-language of the aforementioned meta-language, i.e. in a meta-meta-language, etc. The language of semantics is a meta-language, because it is used for talking about relations between another language (e.g. the subject language in respect to the reality or to the domain of natural numbers) and a domain (e.g. the reality or the domain of natural numbers). Thus, the language of semantics, as a meta-language, is equipped with both kinds of signs: the signs referring to a domain and the signs referring to a subject language in which we speak about this domain. In the next two subchapters, these two kinds of signs will be discussed: first the signs referring to the law domain, and then the signs referring to the language of law itself.

2.2

Kinds of Legal Objects

The word ‘object’ means something to which our thoughts are directed, something we are talking about. The objects to which the existence is attributed, are called ‘beings’. The basic kinds of beings are called ‘ontological categories’ (from Greek: onto ¼ something that exists). The question: ‘what exists?’ is an age-old philosophical question. However, we are not looking for an answer to this question in this book. It is enough for us to answer the question about the types of objects that are distinguished in the domain of law, regardless of whether these objects are beings or not. Kant5 claimed that the way the objects are distinguished in the real world depends on us—that is, it is not determined by the real world. This idea can be illustrated by the following example: when we enter the electronics store, we distinguish stands with TVs from stands with computers, or stands with radios, even when none of these devices are connected to electricity. Our ancestor, that moved from e.g. the Bronze Age, will only see more or less colorful lumps in the same store—because they have neither the concept of the television, computer, nor the radio. For them, those TVs, computers, or radios will be just lumps distinguishable by size or color, but not by the modern way of use. In Europe, we are used to distinguishing people and things in the real world, but a Buddhist (and maybe a modern physicist as well) will tell us that the world is a field or energy, and what we perceive as people or things, are just lumps of this field or energy.

5

Immanuel Kant (1724–1804), a German philosopher, the creator of critical philosophy. Kant made a ‘Copernican revolution in philosophy’, saying that it is not the thought that depends on the objects, but that the objects depend on the thought.

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2 Basic Semantic Concepts

Instead of people and things, the facts may be distinguished as the basic objects of the real world, as well. This is Wittgenstein’s approach. According to Wittgenstein, the real world is the totality of facts. The facts are real configurations of things. The situations constitute a wider category than facts. Namely, the situations are such configurations of things that do not have to be real (e.g. a configuration in which Mount Everest is lower than K2 is a situation, but it is not a fact). In other words, facts are real situations. The totality of situations creates a logical space, of which the real world is an element.6 What is the domain we are talking about in the language of law? In other words, what are the objects we are talking about in this language? Let us consider a few examples. First, lawyers are speaking about parents and children, people of age, minors, perpetrators, etc. These words apply only to natural persons, i.e. to objects that are perceived sensually. Of course, lawyers are also speaking about things and animals (e.g. as about objects of property right), i.e. other objects that are perceived sensually, too. Second, lawyers are speaking about owners, landlords and tenants, lenders and borrowers, plaintiffs and defendants, etc. These words may refer to natural persons, although they may also refer to institutions (e.g. companies or foundations), i.e. may refer to abstract objects. Third, lawyers are speaking about obligations, rights, claims, limitation periods, etc. And the words ‘obligation’, ‘right’, ‘claim’, ‘limitation period’ do not relate to objects that are perceived sensually. The meanings of these words are purely abstract. Fourth, lawyers are speaking about crimes and offenses, accidents, meetings and demonstrations, states of higher necessity, etc. The expressions such as ‘crime’, ‘accident’, or ‘demonstration’ do not apply to natural persons, but they do nonetheless apply to something that can be perceived sensually: we will say that Mr. Smith was a witness to the crime when he saw that John shot Bill. Having in mind the abovementioned examples, it may be concluded that the both kinds of objects are spoken of in the language of law, namely: a) concrete objects, i.e. objects that can be perceived sensually,7 and b) abstract objects, i.e. objects that are not available to the senses. We do not prejudge whether there are abstract objects, or even whether there are concrete ones. These two kinds of objects are simply implied by the language of law, that is, lawyers are speaking about both concrete and abstract objects. 6

A logical space is implied by language. It means that it is the conceptual power of a language that determines both the perception of the real world and the universe of alternative worlds. 7 ‘Sensually perceptible objects’ means ‘those objects that would be perceived by the senses if they existed’. So, dragons are sensually perceptible objects, although they are not real. Therefore, the expression ‘concrete object’ has a broader meaning than the expression ‘physical object’: dragons are concrete objects (because if they existed, they could be perceived sensually), but they are not physical objects (because they do not exist).

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23

Following the examples mentioned, two kinds of concrete objects may be distinguished, namely: a) individuals, and b) collectives. The concrete individuals are people and things. Tadeusz Kotarbiński8 claimed that only persons and things exist, and that abstract objects do not.9 For example, when we say that ‘Love pushed Kate into John’s arms’, we attribute the real existence to abstract objects that by definition do not exist. There are only (probably) people loving each other—but there is no such being as love. And since love does not exist, it could not push anyone into anyone’s arms. Similarly, there is no such being as the law—there are only lawyers and legal books. Others say that there are, in a sense, not just people and things. For example, Karl Popper distinguished three worlds filled with objects that can be considered—the physical world, the psychic world, and the world of cultural creations (where abstract objects exist). The only concrete collectives are situations, i.e. spatial configurations of concrete individuals.10 The situations may be considered to be concrete objects because they are sensually perceptible, just like people and things. Our example of the electronics store may be used: just as someone distinguishes TVs, computers and radios in this store, and someone else just colorful lumps, in the real world someone sees people and things, and someone else—their configurations, that is, situations. The situations that occur in the real world are facts.11 The crimes and demonstrations mentioned above are examples of concrete collectives.12 When someone has seen that John shot Bill, they have seen a situation that is called ‘crime’, that is, they have seen a crime. When someone has seen thousands of people in front of the seat of the government, they have seen a situation that is called ‘demonstration’, that is, they have seen a demonstration. Concrete collectives are sets in a mereological sense.13 This means that situations are treated here as a kind of solids, three-dimensional fragments of the world. Such a three-dimensional slice of the world is both a situation in which John

8

Tadeusz Kotarbiński (1886–1981), a Polish philosopher, logician and ethicist, the creator of reism. Kotarbiński’s thesis that only people and things exist is equivalent to the thesis that only concrete individuals are beings. Therefore, sometimes, reism is called ‘concretism’. 10 Wittgenstein wrote that a fact is the existence of states of affairs (thesis 2), and a state of affairs is a combination of objects (things) (thesis 2.01). Wittgenstein’s logical atomism, limited to the domain of concrete objects, is a variation of concretism, another of which is Kotarbiński’s reism. 11 Alternatively, it could be assumed that any situation is either concrete or abstract. This way, abstract facts will be admitted. For example, 2 + 2 ¼ 4 will be an abstract fact in the domain of natural numbers. Here, however, we stick to the convention that the situations are solely concrete collectives. 12 If they are understood as situations, i.e. spatial configurations of concrete individuals. Both, the crimes and the demonstrations, can also be understood differently, namely as sequences of situations. In such a case, these objects will prove to be abstract (see: Chap. 3). 13 The sets in a mereological sense are sets perceived as solids. Mereology, or the theory of collective sets, is a creation of Stanisław Leśniewski. 9

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shot Bill and a situation in which thousands of people are standing in front of the seat of the government. Deciding on whether an object is an individual or a collective, is, to some extent, a conventional matter. For example: is the swarm of bees an individual or a collective? In words of Kant: it is not the world that determines kinds of objects. These kinds depend on us. So, if anyone wants to consider a swarm of bees as an individual— please. However, the opposite view, that individual bees are individuals, while the swarm is a collective, or a mereological set, is at least equally legitimate. The examples considered earlier in this subchapter show that there are many abstract objects in the law domain. It is difficult to propose any classification of these objects without prior detailed analysis. To group abstract objects such as obligations, rights, claims, limitation periods, etc., their nature should be examined first. Without such a study, it is not even possible to declare whether they are individuals or collectives. Of course, such a study goes beyond the outlined framework of this book. However, in Chap. 3, several kinds of abstract objects which are important from the legal perspective, will be introduced in our model of the law domain, namely legal events, including acts, and legal rules. At this point, only two kinds of abstract objects, which are important in every domain, and therefore also in the domain of law, will be distinguished, namely: a) distributive sets of objects, b) distributive sets of strings of objects.14 Thanks to the concepts of these sets, it is possible to explain in mathematical terms, namely in terms of the set theory, what does it mean that objects have properties, and that objects are in relations to each other. The link between sets and properties and relations is as follows. It is said that objects have properties. And distributive sets of objects correspond to these properties. For example, the set of lenders corresponds to the property of being a lender. In a given set of objects, each property designates a subset of objects that have that property. For example, in the set of people, the property of being a lender designates the subset of lenders. Therefore, to have a property is to belong to a subset of objects. It is also said that objects remain in relations with each other. And distributive sets of strings of objects correspond to relations between objects. For example, the set of pairs corresponds to the lending relation. In a given set of strings, each relation designates a subset of strings of objects that remain in that relation. For example, in the set of pairs, the lending relation designates the subset of

14

The sets in a distributive sense (or distributive sets) are sets we learned about in primary school. It is said about these sets that elements belong to them, and that some of them are contained in the others, etc. To the contrary, the sets in a collective sense (or mereological sets) are sets perceived as solids. It is not said about such sets that one is contained in the other, but that one is a whole and the other is a part of this whole. In the following text, the word ‘set’ without further definition will mean a set in a distributive sense.

2.3 Kinds of Subject Language Expressions

25

pairs . Therefore, to be in a relation is to belong to a subset of strings of objects. Properties are not perceptible sensually, even when concrete objects are involved: I may perceive a red apple or a red car sensually, but not the red itself. Similarly, relations are not perceptible sensually, even when they occur between concrete objects: I may sensually perceive pairs of people such that the first person is a brother of the other, but not the brotherhood itself. Properties and relations are themselves (abstract) objects. As objects, these properties and relations may have properties (e.g. the relation of brotherhood is transitive, i.e. it has the property of transitivity). They may also be in relations (e.g. the set of natural numbers is contained in the set of integers). This way, the universe of abstract objects is expanding. Properties and relations are useful when other abstract objects are constructed. So, they shall be used when one constructs institutions, obligations, rights, claims, limitation periods, and other abstract objects of the law domain. As a conclusion of this subchapter, it shall be noted that the subdomain of concrete objects of the law domain is extremely important: legal rules, in the end, tell us how people are supposed to behave in various situations. This way, ultimately, the law speaks about people and situations, i.e. it speaks about concrete objects.

2.3

Kinds of Subject Language Expressions

In the language of semantics, next to expressions referring to a domain, there are expressions referring to the subject language in which we speak about this domain. This way, in the language of semantics, it is possible to talk about the subject language, e.g., it is possible to distinguish kinds of expressions of this language, to consider properties of these expressions (e.g. whether they follow grammar), relations between them (e.g. whether they have the same meaning), or relations between them and a domain (e.g. what do they refer to in this domain). The kinds of subject language expressions distinguished according to the way they refer to a domain are called ‘semantic categories’. In logical semantics, two basic semantic categories are distinguished: 1) names—referring to objects in a domain, and 2) sentences—stating properties of objects or relations between objects in this domain. Due to the type of objects being referred, the names are: a) concrete—that is, referring to concrete objects, or b) abstract—that is, referring to abstract objects. The words ‘parent’, ‘child’, ‘perpetrator’, ‘house’, ‘animal’ are examples of concrete names. They refer to concrete individuals. The words ‘crime’, ‘accident’,

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2 Basic Semantic Concepts

‘demonstration’, ‘state of higher necessity’ are other examples of concrete names. They refer to concrete collectives (situations). The words ‘obligation’, ‘right’, ‘claim’, ‘limitation period’ are examples of abstract names. They refer to some abstract objects. However, in the subject language of law there are also expressions like ‘owner’, ‘landlord’, ‘tenant’, ‘lender’, ‘borrower’, ‘plaintiff’, ‘defendant’, etc. These words may refer to natural persons, i.e. concrete objects, although they may also refer to institutions (e.g. companies or foundations), i.e. abstract objects. Therefore, in the language of law the same name may be either concrete or abstract—depending on a situation. The division of names into concrete and abstract is discussed in most textbooks on logic.15 However, it is not obvious whether it is possible to talk about sentences in a similar way. If sentences were to be divided, like names, into concrete and abstract ones, this division could be made at least in two ways. First, a distinction between distributive relations (to be an element of a set, to be a subset of a set) and mereological relations (to be a part of a whole, and other spatial relations) may be used. This way, the sentences may be divided into concrete and abstract ones, as follows: a) concrete—i.e. stating both mereological and distributive relations, b) abstract—i.e. stating distributive relations only. This way, the sentence ‘Every attorney is a lawyer’ is an abstract one since it states just that the set of attorneys is included in the set of lawyers. In other words, this sentence concerns a relation between two distributive sets, i.e. two abstract objects. No spatial relation between concrete objects is stated. In turn, the sentence ‘John is sitting in the chair’ is a concrete sentence. This sentence states that John is an element of the set of people sitting in chairs. So, this sentence states a distributive relation (between an element and a distributive set). However, this sentence also states a spatial relation between two concrete individuals: John, and the chair. Unfortunately, in this approach the sentence ‘This flower is red’ also is an abstract one since it states just that a flower is an element of the set of red objects. In other words, this sentence concerns a relation between a set and its element, i.e. an abstract 15

Traditionally, in Polish textbooks on logic, the names are also divided into the universal, unitary, and empty names, as well as into the general and individual names. Due to the number of referred objects, names can be: a) universal—i.e. referring to more than one object, or b) unitary—i.e. referring to exactly one object, or c) empty—i.e. referring to none object.

Due to the way an object is referred, names can be: a) general—i.e. referring to objects by indicating their properties, or b) individual—i.e. referring to objects by an arbitrary decision.

2.3 Kinds of Subject Language Expressions

27

object and a concrete object. No spatial relation between concrete objects is stated. The same is with ‘This liquid stinks’, ‘This apple is sour’, ‘This towel is soft’, ‘This car is noisy’, etc. To be in line with the strict meaning of the word ‘concrete’ (i.e. ‘sensually perceptible’), and to treat the abovementioned sentences as concrete, the division of sentences into concrete and abstract shall be done in a different way. In particular, it shall be taken into account that the sentences like ‘This flower is red’, ‘This liquid stinks’, ‘This apple is sour’, ‘This towel is soft’, ‘This car is noisy’ are depicturing situations, just as concrete sentences in the former division. However, in this case, this depicturing of situations is not limited to the spatial aspects of depicted situations. This way, the sentences may be divided into concrete and abstract ones, as follows: a) concrete—i.e. depicturing situations and stating distributive relations, b) abstract—i.e. stating distributive relations only. Therefore, every sentence states a distributive relation, but only concrete sentences depicture situations. We can paraphrase the famous Wittgenstein’s thesis that a logical picture of facts is a thought, by saying that a picture of facts (or other situations) is a concrete sentence. However, it should be noted that this approach is not as clear as the first one. Probably, we all agree that the sentence ‘Kate’s face is red’ is depicturing a situation. But what about the sentence: ‘Kate’s face is pretty’? Obviously, not every describing is depicturing. We will come back to this issue at the end of this subchapter. In addition to the basic semantic categories, there is also an auxiliary category, namely: functors. There are functors allowing to create: a) sentences from sentences (e.g. the phrase ‘It is not true that’ added to the sentence ‘The bag is heavy’ creates a new sentence ‘It is not true that the bag is heavy’), b) sentences from names (e.g. the phrase ‘is heavy’ added to the name ‘the bag’ creates the sentence ‘The bag is heavy’), c) names from other expressions (e.g. the expression ‘heavy’ added to the name ‘bag’ creates the name ‘heavy bag’), d) functors from other expressions (e.g. the phrase ‘very’ added to the expression ‘heavy’ creates a new ‘very heavy’ functor). The functors that allow to create sentences from names are called ‘predicates’. Just like names and sentences, they are directly linked to domains: 1) unary predicates indicate properties of objects, i.e. they indicate sets of objects in these domains (e.g. the unary predicate ‘. . . is a lawyer’ indicates the set of lawyers), 2) other predicates indicate relations between objects, i.e. they indicate sets of strings of objects in these domains (e.g. the binary predicate ‘. . . is a brother of . . .’ indicates a set of people pairs).

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If predicates were to be divided, like names and sentences, into concrete and abstract ones, this division could be made at least in two ways (just like in the case of sentences). First, the predicates could be divided into concrete and abstract ones, as follows: a) concrete—i.e. referring to both mereological and distributive relations, b) abstract—i.e. referring to distributive relations only. And, second, they could be divided into concrete and abstract ones, as follows: a) concrete—i.e. referring to ‘observable’ properties or relations, b) abstract—i.e. referring to other relations. However, properties and relations are abstract objects. So, strictly speaking, they cannot be observable. Therefore, referring to ‘observable’ properties or relations means only that the veracity of a sentence formed with ‘observable’ predicates is decided exclusively by observations (by sense perceptions). For example, when I look at a car, I can tell whether this car is blue or not, but I cannot tell whether this car is pledged or not. The ‘being blue’ predicate is therefore a concrete predicate and the ‘being pledged’ predicate—is an abstract one.16 Relating the abovementioned considerations to our discussion on the division of sentences based on the depicturing criterion, it may be concluded that depicturing is based solely on sense perceptions whereas describing involves using some theory. This way, in the last two subchapters, the law domain and the language of law have been categorized. Namely, selected kinds of legal objects have been discussed, i.e. two kinds of concrete objects (concrete individuals and collectives), and two kinds of abstract objects (properties and relations). Further, several kinds of legal language expressions have also been discussed, i.e. names, sentences, and functors. In particular, there have been discussed the concrete sentences and predicates, as well as the abstract ones. This categorization of the law domain will be referred to in Chap. 3, when a formal model of this domain is constructed, and in the following chapters, as well. The abovementioned categorization of the language of law will be referred to in the fourth chapter, when the language of first order predicate logic is selected as the model of the language of law, as well as in the following chapters. Before we get into these considerations, however, let us remind the reader a few other semantic concepts that will be used in this further discussion.

16

This second division is similar to the division of predicates into observable and theoretical ones, proposed by Marian Przełęcki (1923–2013), a Polish logician and philosopher. According to Professor Przełęcki, observable predicates are defined solely by pointing to objects (i.e. purely ostensibly). All other predicates are theoretical. However, in some cases there are procedures allowing to ‘reduce’ theoretical predicates to observable ones. See: Przełęcki (1988), p. 37, p. 68 and following (English edition: Przełęcki 1969).

2.4 Interpretation, Veracity and Semantic Relations

2.4

29

Interpretation, Veracity and Semantic Relations

A language is interpreted by linking expressions of this language with a domain. Respectively, sentences are interpreted when the expressions of which they are formed, are interpreted. A concrete sentence may be linked with a relevant situation instead. Further, names are interpreted by indicating objects to which these names will refer to. These objects are called ‘designates’. The set of designates of a name is called its ‘denotate’. Names can be linked with objects in two ways. The first is linking by pointing. This way is called ‘ostensive defining’ (Latin: ostensio ¼ I point). Of course, only concrete names can be defined in this way, because only concrete objects can be pointed at with a finger. The second way is purely verbal: pointing is replaced by describing. In turn, predicates are interpreted by indicating: a) for a unary predicate—a set of objects to which that predicate refers, or b) for a multi-argument predicate—a set of strings of objects to which that predicate refers. And finally, other functors are interpreted by indicating how these functors change the meanings of their arguments. An example of the interpretation of a language made by linking it with a domain can be found in Chap. 4. Namely, first order predicate language, which is our model of the language of law, will be linked with the model of the law domain, constructed in Chap. 3. When a language has been interpreted, its sentences become true or false. Truth and falsity are semantic properties of sentences. According to the classic definition of truth, a sentence is true if and only if it corresponds to the reality. Otherwise, that sentence is false. Taking our previous considerations into account, this definition may be developed as follows: a sentence that states that an object has a property is true if and only if that object has that property. Otherwise, that sentence is false. Similarly, a sentence that states that some objects remain in a relation is true if and only if those objects remain in that relation. Otherwise, that sentence is false. A concrete sentence that states a fact is true. Otherwise, that sentence is false. To interpret sentences stating distributive properties, a domain that contains abstract objects is required. For example: the sentence ‘This car is blue’ refers not only to a concrete object (a car), but to an abstract object (a set of blue objects), as well. As a consequence, this sentence has no interpretation in the real world understood solely as the totality of facts. Thus, the sentence ‘This car is blue’ is neither true nor false in the real world. The abovementioned sentence can be interpreted solely in domains broader than the real world, e.g. in the domain {the real world; the set of blue objects}. And solely in such domains this sentence is true or false. Similarly, the sentences ‘Every attorney is a lawyer’, ‘A debtor is obliged to pay their debts’, ‘A person that caused a damage is obliged to compensate it’ are neither true nor false solely in the real world. As most of people can distinguish blue objects from others, it is easy to speak about the domain {the real world; the set of blue objects}. However, it is not so easy to speak about debts, damages or

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compensations, if relevant abstract objects are not properly constructed and ‘added’ to the real world. In the theory of law, it is questioned whether legal norms can be true or false. It is argued that legal norms are not describing the real world. Of course, it is true if the real world is understood solely as the totality of facts. But the same could be said about the sentence ‘This car is blue’. Nevertheless, most of us agree that the sentence ‘This car is blue’ can be true or false, because most of us easily distinguish blue objects from others. So, maybe, legal norms cannot be assessed as true or false, just because, so far, the language of law is not properly interpreted. The question of the veracity of legal norms will be discussed in Chap. 5. The truth or falsity of one sentence entails the truth or falsity of others. This dependency is reflected in semantic relations between sentences. The most important of the forementioned relations is entailment: from a sentence A results a sentence B if and only if it cannot be that A is true and B is false, i.e. when the truth of A entails the truth of B. For example, from the sentence ‘All people are mortal’ results the sentence ‘Socrates is mortal’ because it cannot be that the first sentence is true in a given domain, and the second is false in the same domain (assuming that Socrates is a human in that domain). Contradiction, opposition and sub-opposition are other important semantic relations between sentences: 1) A and B are contradictory if and only if they cannot be true at the same time and they cannot be false at the same time, 2) A and B are opposite if and only if they cannot be true at the same time, 3) A and B are sub-opposite if and only if they cannot be false at the same time. The sentences ‘All people are mortal’ and ‘A man is not mortal’ are contradictory. The sentences ‘All people are mortal’ and ‘All people are not mortal’ are opposite. And the sentences ‘A man is mortal’ and ‘A man is not mortal’ are an example of sub-opposite sentences. In the theory of law, there is a disagreement about the semantic relations between legal norms. On the one hand, some researchers claim that the semantic relations mentioned above or similar relations occur between legal norms. On the other hand, other researchers point out that semantic relations between legal norms cannot occur because legal norms are not sentences in a logical sense, and therefore they are neither true nor false. The question of possibility of these semantic relations between legal norms, as well as the issue of semantic relations between orders, will be discussed in Chap. 5.

2.5

Semantic Models and Other Methods

In this final part of Chap. 2, let us take an ‘aerial view’ to see how a model-based method of studying the relations between the language of law and the law domain works, and mention other methods on which our approach is based.

2.5 Semantic Models and Other Methods

31

Thanks to semantics, it is possible to imagine how the language of law is interpreted in the domain of law. Of course, I will not interpret the whole language of law in this book. It would require interpreting all the names, predicates and other functors of that language. But I will show how this interpretation can be made. I will use a model of the law domain and a model of the language of law for this purpose. Models are systems of objects (concrete or abstract ones) that map, in a simplified way, systems of other objects. Thanks to simplification, it is easier to capture the properties of the modeled systems. Models are very popular in mathematics, physics and economics. In these sciences, models are created in order to quickly and easily learn the most important properties of the studied phenomenon. Sometimes these models are extremely simplified. This approach is called ‘toy-model approach’. Logical semantics also uses this method. Symbolic languages of modern logic are examples of models. Let us look at the language of classical sentential calculus. In its dictionary, there are solely symbols representing simple sentences and conjunctions of natural language. There are no names, predicates, nor other functors than sentence conjunctions. Consequently, according to its grammar, the only correct expressions of this language are some of the combinations of simple sentences and conjunctions. Its axioms define solely the meanings of the conjunctions, while the sentence symbols are not interpreted. However, this way, the language of classical sentential calculus maps (in a very simplified way) some properties of natural language. We learn these properties by studying the model. Thanks to this model, it is easy to understand how some conjunctions of natural language work. On the other hand, various mathematical theories, including theories of possible worlds, are used in logic as models of the reality. Having an artificial language—a model of natural language, and an artificial domain—a model of the reality, one can interpret this artificial language in this artificial domain. This way, semantic properties and relations of natural language can be examined by studying the models.17 In the following chapters, a similar approach will be taken. In Chap. 3, a model of the domain of law will be constructed. This model will include some concrete objects, as well as some abstract ones. Facts and other situations will be concrete objects of our model. Legal events, including acts, and legal rules will be the most important among the abstract objects of that model. First, the subdomain of concrete objects will be described formally and, after that, abstract objects will be constructed on the basis of the well-defined concrete ones. Thanks to this approach, a better understanding of legal events, legal acts, and legal rules, will be achieved. Then, in Chap. 4, the language of first order logic will be adapted as a model of the language of law. It will be shown that this model is rich enough to express 17

Studying relations between theories expressed in artificial languages and mathematical domains yielded many valuable results, including, for example, the famous Gödel incompleteness theorem, which may be understood as a theorem about the limits of formal methods. Simplified, this theorem states that if a formal system is rich enough, there are true sentences in it, which cannot be proven in this system.

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2 Basic Semantic Concepts

normative theories in it. Thanks to this approach, a better understanding of the structure of the language of law as well as better understanding of links between this language and the law domain, will be achieved. Then, in Chap. 5, semantic properties of legal norms and semantic relations between them will be discussed, in reference to the both forementioned models. In particular, the questions about the veracity of legal norms and the possibility of legal entailment, will be answered. Besides the toy model approach, other methods derived from, or inspired by, logic will be used here. First, all abstract objects constructed here will be based, directly or indirectly, on concrete ones. This way, the law domain will be ‘anchored’ in the reality. Such ‘anchoring’ reflects the empirical nature of law. In the methodology of sciences, theories are divided into empirical and non-empirical. To verify the theorems of non-empirical theories, it is not necessary to explore the physical world. The conceptual studies alone will suffice. On the other hand, in the case of empirical theories, at least some concepts are empirical, and therefore related to the physical world. From this point of view, any system of legal norms is an empirical theory, since every legal norm relates to the physical world. Indeed, prescriptions, prohibitions, and permissions apply to people. Thus, it is easier to discuss these legal duties when the law domain is ‘anchored’ in concrete objects (we call this approach an ‘ultimate concretism’). Further, having in mind that there are two kinds of concrete objects in the law domain, namely concrete individuals (persons and things) and concrete collectives (situations), solely the latter are chosen as the fundament of our law domain model. This is because describing the world in terms of situations is more universal than in terms of persons and things, and therefore it is easier to make the former description than the latter. Besides, there is a good philosophical theory of situations (i.e. Wittgenstein’s Tractatus), and a good mathematical theory of them (i.e. Wolniewicz’s Ontology). Thus, the concept of situation is not only simple, but intuitive and formal, as well (we call this approach a ‘situational approach’). The abovementioned methods will be used in the following chapters, starting with Chap. 3 where a formal model of the law domain will be proposed. In turn, in Chap. 4, a method called an ‘intuitive formalism’ will be applied. This term comes from Stanisław Leśniewski, a Polish logician and philosopher, who claimed that all formal theories, including logical ones, should have an intuitive interpretation. Usually, the meanings of formal theories terms are determined by the choice of axioms. That is, selecting axioms, accepting them to be true by definition, determines a way of understanding the terms these axioms are constructed of. In Chap. 4, indeed, axioms will be used for this purpose. However, independently, before the axioms selection, our model of the language of law will be interpreted by linking its variables to a set of situations and events, and some of its predicates to other objects from the law domain model. This way, these expressions will acquire intuitive meanings, independent of those given to them by deontic theories. So, let us get to work!

References

33

References Marciszewski W (ed) (1981) Dictionary of logic as applied in the study of language. Concepts, methods, theories. Martinus Nijhoff Publishers, The Hague/Boston/London Marciszewski W (ed) (1987) Logika formalna. Zarys encyklopedyczny z zastosowaniami do informatyki i lingwistyki. Państwowe Wydawnictwo Naukowe, Warszawa (in Polish: Formal logic. Encyclopedic outline with applications to computer science and linguistics) Przełęcki M (1969) The logic of empirical theories. Routledge & Kegan Paul Limited, London Przełęcki M (1988) Logika teorii empirycznych. Państwowe Wydawnictwo Naukowe, Warszawa (in Polish: Logic of empirical theories)

Chapter 3

A Model of the Domain of Law

In Chap. 2, it was explained that the concrete part of the law domain is composed of concrete individuals (i.e. persons and things) and concrete collectives (i.e. situations, which are configurations of persons and things). In this chapter, a mathematical theory of situations will be presented. This theory has been created by a Polish logician Bogusław Wolniewicz, and reflects Ludwig Wittgenstein’s philosophy of logical atomism. Then, Wolniewicz’s theory will be used as a starting point for extending the law domain. Namely, abstract objects, such as legal events and legal rules, will be constructed in this domain. Legal events will be constructed as specific sequences of situations. For this purpose, Wittgenstein’s theory will be developed into a theory of events. Several kinds of legal events will be defined, including acts. The concept of legal event will be needed to define the legal rules. Namely, the legal rules will be constructed as specific sets of legal events. This way, the law domain will be expanded by including some abstract objects which are required for interpreting the language of law. In particular, it is the legal rules that will become the meanings of legal norms.

3.1

A Concrete Basis of the Law Domain

Ultimately, the law is about changing the physical world, a world composed of concrete objects. In the end, it is them, that the law applies to. Therefore, without prejudging what exists and what does not, it may be said that the most important thing, for the lawyers, is the domain of concrete objects. Thus, the lawyers are bound by a sort of ‘ultimate concretism’: to understand the law is to understand what effects it will bring to the domain of concrete objects. In Chap. 2, however, it was noted that lawyers speak not only of concrete objects, but also of abstract objects. This is not some unique property of the language of law. Abstract objects are spoken of in many areas of practice, culture, or science. The abstract objects may be examined for their better understanding alone, or for a better © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Malec, Introduction to the Semantics of Law, Law and Visual Jurisprudence 6, https://doi.org/10.1007/978-3-030-95679-0_3

35

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3 A Model of the Domain of Law

understanding of concrete objects. And again, without prejudging what exists and what does not, it can be said that the lawyers are partial to abstract objects. This is because it is easier to express how to change the world of concrete objects when abstract objects are admitted. Of course, it is possible to describe the physical world solely in concrete terms. Instead of saying that ‘It is forbidden to hunt’, one can say that ‘The king forbade his subjects to press the trigger of a shotgun aimed at the animal’. Most probably, the prohibition ‘Tax deductible costs do not include expenses incurred for the personal needs of members of the company’s bodies and its shareholders’ may also be paraphrased in concrete terms only. The same holds for other prescriptions, prohibitions, and permissions. But, in fact, it is not reasonable to avoid abstract terms when the law is created or discussed. After all, it is easier to write down such prescriptions, prohibitions, and permissions in abstract terms. Only when one applies the law, a transition from talking about the prohibition of hunting, to saying that John should pay a fine for shooting the hare, is inevitable. Thus, the lawyers adhere to ‘ultimate concretism’ when the law is applied, while they take the ‘anything goes’ approach when the law is created or discussed. The ‘anything goes’ approach, although it works in practice, has a drawback, which effects ought to be limited. Namely, it is easy to define concrete objects well, because for this purpose it is enough to carefully look at them. On the other hand, the abstract objects cannot, by definition, be viewed, so to define them well, one has to think of them properly, first. And as the words are quicker than thoughts, quite often there is nothing hidden behind abstract expressions, or the hidden abstract objects are not clearly conceived. This disadvantage of the ‘anything goes’ approach can be avoided if the abstract expressions are introduced only when the relevant abstract objects are properly constructed. Properly constructed abstract objects shall ‘interact’ with the concrete ones. To explain such an interaction, one may refer to works devoted to theoretical terms of empirical sciences.1 According to these works, there is no need for reducing the abstract objects to the concrete ones. However, a link between these two kinds of objects, is required. Thus, there is no need for reducing the companies or financial institutions to configurations of people or things. However, one shall understand what the results of referring to these abstract objects in legal texts, are. They shall understand, e.g., whether a person referred to as a ‘shareholder of the company’ will be paid money referred to as ‘money of the company’ in a place referred to as a ‘financial institution’. When the abstract objects interact well with the concrete ones, and only then, it may be said that they are properly constructed. In this chapter, it will be shown how the legal events and legal rules ‘interact’ with the concrete objects of the law domain.

1

See, e.g.: Przełęcki (1969).

3.2 Wolniewicz’s Ontology of Situations

3.2

37

Wolniewicz’s Ontology of Situations

There are two kinds of concrete objects in the law domain, namely the concrete individuals, i.e. persons and things, and the concrete collectives, i.e. situations. When applying the law, lawyers sometimes talk about persons (e.g. ‘John should not have stepped on the lawn’), and at other times about situations (e.g. ‘It is forbidden to trample the lawns’). When a model of the law domain is constructed, however, the latter approach is more convenient. It is more general than the former. Thus, it is easier to formalize it. Therefore, we adopt the latter approach. Consequently, legal events, including acts, and legal rules will be defined in reference to the concept of situation. In philosophy, the concept of situation was profoundly established by Ludwig Wittgenstein. According to him, the real world is the totality of facts. The facts are real configurations of things. As counterparts to the real world, other worlds can be considered. These other worlds are possible, but they do not exist. These possible, but not existing, worlds are the totalities of non-facts. The situations constitute a wider category than facts. Namely, the situations include both the facts and the non-facts. Therefore, the facts are real situations, and the non-facts are imaginary situations. The facts form a hierarchy. The real world is a ‘maximum fact’ consisting of ‘smaller’ facts, and those in turn consist of even ‘smaller’ ones, up to a level of atomic facts—the smallest facts that no longer contain constituent facts. Similarly, the non-facts form alternative hierarchies, with alternative possible worlds on the tops of these hierarchies. The totality of situations creates a logical space, of which the real world is an element.2 It is easy to imagine such a hierarchy of situations. For example, the situation where John is sitting on the bench and the situation where Kate is sitting on the bench are two parts of a larger situation where John and Kate are sitting on the bench. In turn, the latter situation is a part of an even larger one, involving all the benches and alleys of the park, and this is a part of another larger situation, and so on, up to a maximum situation, i.e. to a possible world. A jigsaw puzzle analogy can also be used to describe these hierarchies of situations. Every possible world is like an assembled jigsaw puzzle (‘AJP’). Each situation is like a piece of an AJP. The smallest (‘atomic’) situations are like the smallest pieces of an AJP, i.e. they are like single puzzle pieces. The real world is like the chosen AJP. And the logical space is like all AJPs together. Probably, many readers will find that this John and Kate example, or the jigsaw puzzle analogy, suffices to explain the concept of the hierarchies of situations. However, these hierarchies may also be defined precisely in terms of mathematics. This way, the situations may be considered from two points of view: intuitive and formal. Same as in physics, where intuitive and formal

2

According to Wittgenstein’s original approach, situations do not have to be concrete, however, understanding them as solely concrete objects is convenient when the law domain is considered. See footnote 11 in Chap. 2.

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approaches are usually combined. The formal approach allows to verify the intuitive one, and develop it, as well. This formal theory of situations has been created by Bogusław Wolniewicz. He intended to offer us a mathematical model of Wittgenstein’s logical space.3 He imagined, and then formally described, a system of objects and relations, being a model of the Wittgenstein universe of situations. In fact, the abovementioned John and Kate example, and the jigsaw puzzle analogy, also are models of this kind. However, Wolniewicz’s model is a symbolic one, more accurate and detailed, and allowing for precise inferences.4 This model is based on the lattice theory and has been constructed as follows. Let us take a non-empty set SE. The elements of this set are elementary situations.5 In the SE set, a subset of elementary proper situations (SE”) and two inappropriate situations, namely an empty situation o (‘zero’) and an impossible situation λ, are distinguished, i.e. symbolically: SE ¼ SE” U f o, λ g: We read this expression as follows: ‘The set SE is the sum of the set SE” and the set {o, λ}’. Thanks to the empty situation and the impossible one, the set of proper situations can be limited both from the ‘bottom’ and from the ‘top’. Namely, the empty situation is smaller than any proper situation, and the impossible situation is greater than any proper situation.6 This, in turn, allows for defining atomic situations and maximum situations (possible worlds). In our jigsaw puzzle analogy, there were no counterparts for the empty situation and the impossible one. If we wanted to transfer these concepts to our analogy, the empty situation would be something smaller than a single puzzle piece, and the impossible one would be something greater than an AJP. In the world of puzzles, however, there are no such objects. Therefore, Wolniewicz calls the empty situation and the impossible one the ‘inappropriate situations’. Let us now define a partial order relation  in the set SE:

3

See: Wolniewicz (1985), pp. 23–29 (A counterpart in English: Wolniewicz 1982, pp. 381–413). Wolniewicz’s theory will be presented here with minor modifications. 4 To compare our jigsaw puzzle analogy with Wolniewicz’s theory, the axioms of the latter will also be interpreted in the domain of jigsaw puzzles. Perhaps it will make it easier for the readers to understand Wolniewicz’s ontology. Although, it may turn out that the readers will become just puzzled. 5 The elementary situations are depicted in sentences like ‘John is eating an apple’ or ‘John is eating an apple and Bob is eating a pear’. 6 To be precise, according to the axioms below: the empty situation is no greater than any proper situation, and the impossible situation is no smaller than any proper situation.

3.2 Wolniewicz’s Ontology of Situations

39

x  x, x  y and y  z ➔ x  z, x  y and y  x ➔ x ¼ y, for any x, y, z from SE. The formula ‘x  y’ shall be read: ‘x occurs in y’ or ‘x is a part of y’. According to these axioms: (i) each situation is a part of itself, (ii) if the first situation is a part of the second situation, and the second situation is a part of the third situation, then the first one is a part of the third one, (iii) if two situations are parts of each other, then they are identical. In other words, the partial order relation is reflexive, transitive, and antisymmetric. Other examples of the partial order relation, are: ‘being less than or equal to’ in the set of natural numbers, ‘being not older’ in the set of people, ‘being a subset’ in the set theory. This relation also takes place in our jigsaw puzzle analogy: (i) each puzzle piece is a part of itself, (ii) if the first puzzle piece is a part of the second puzzle piece, and the second puzzle piece is a part of the third puzzle piece, then the first one is a part of the third one, (iii) if one puzzle piece is a part of the other, and the latter one is a part of the former one, then these two pieces are identical. Now, let us assume that the structure < SE,  > is a complete lattice, so each subset of SE has its upper and lower limits,7 i.e. 8A ⊂ SE Ǝ x E SE ð x ¼ sup A Þ, 8A ⊂ SE Ǝ y E SE ð y ¼ inf A Þ: These expressions shall be read as follows: ‘For any A that is a subset of SE, there is an x belonging to SE such that x is the upper limit of A’ and ‘For any A that is a subset of SE, there is an y belonging to SE such that y is the lower limit of A’. The first of the abovementioned axioms states that for any subset A from the set of situations SE, there is, in the set SE, a situation x such that any situation from the subset A is a part of x. The second of the abovementioned axioms states that for any subset A from the set SE, there is, in the set SE, a situation y such that it is a part of any situation from the subset A. In fact, the empty situation is a situation of the latter kind, whereas the impossible one is a situation of the former kind. Since in our jigsaw puzzle analogy there are no counterparts of the empty situation and the impossible situation, there are also no jigsaw puzzle properties that would correspond to these axioms. Further, let us assume that o is the smallest element and λ is the largest, i.e. each elementary situation is contained between the empty situation and the impossible one:

By definition, x is the upper limit of the set A when for any z E A: z  x. Accordingly, y is the lower limit of the set A when for any z E A: y  z.

7

40

3 A Model of the Domain of Law

o  x  λ: This expression shall be read as follows: ‘Any situation x is not smaller than the empty situation and not greater than the impossible one’. Since in our jigsaw puzzle analogy there are no counterparts of the empty situation and the impossible one, there is no counterpart of this axiom, as well. Unlike in Wolniewicz’s model, in our jigsaw puzzle analogy there is nothing smaller than single puzzle pieces, nor anything greater than AJPs. Thus, our jigsaw puzzle analogy slightly differs from Wolniewicz’s model. Then, Wolniewicz defines joints and splices of elementary situations. The joint of x and y is the largest situation, which is a fragment of both x and y: x!y ¼ inf f x, y g: This expression shall be read as follows: ‘The joint of situations x and y is the lower limit of the set {x, y}’. And the splice of x and y is the smallest situation, which contains both x and y: x; y ¼ sup f x, y g: This expression shall be read as follows: ‘The splice of situations x and y is the upper limit of the set {x, y}’. In terms of our jigsaw puzzle analogy, the first of the abovementioned axioms states that the joint of two puzzle pieces is the largest puzzle piece that occurs in each of these two. In other words, the joint is their largest common fragment. On the other hand, the second of the abovementioned axioms states that the splice of two puzzle pieces is the smallest puzzle piece in which these two appear. The concepts of joints and splices allow for a formal description of creating larger puzzle pieces from smaller ones. When x occurs in y, then x is their lower limit (i.e. their joint), and y is their upper limit (i.e. their splice): x!y ¼ x ⟺ x  y ⟺ x; y ¼ y: This expression shall be read as follows: ‘An x is the joint of situations x and y if and only if x is a part of y if and only if y is the splice of situations x and y’. Thus, in terms of our jigsaw puzzle analogy, if one puzzle piece is a part of another, the former is their joint and the latter is their splice. Two elementary situations are compatible if there is a proper situation in which both occur: x is compatible with y ⟺ Ǝ z E SE” ð x  z and y  z Þ:

3.2 Wolniewicz’s Ontology of Situations

41

This expression shall be read as follows: ‘An x is compatible with an y if and only if there is a proper situation z that x is a part of z and y is a part of z’. Otherwise, elementary situations are incompatible, i.e. their splice is the impossible situation: x; y ¼ λ: This expression shall be read as follows: ‘The splice of x and y is the impossible situation’. Thus, in terms of our jigsaw puzzle analogy, two puzzle pieces are compatible if together they can become parts of some puzzle piece. On the other hand, they are incompatible if you cannot make any puzzle piece out of them. Wolniewicz assumes that the lattice of elementary situations is a lattice with relative complements, i.e. for any a, x, y E SE occurs: a  x  y ➔ Ǝ x’ E SE ð a ¼ x!x’ and y ¼ x; x’ Þ, and consequently, for each x E SE there is also such x’ E SE that: x!x’ ¼ o and x; x’ ¼ λ: These expressions shall be read as follows: ‘For any situations a, x, y, if a is a part of x, and x is a part of y, then there is a situation x’ such that a is the joint of x and x’ and y is the splice of x and x’, and ‘For any situation x there is a situation x’ such that their joint is the empty situation and their splice is the impossible one’.8 Thus, in terms of our jigsaw puzzle analogy, if a puzzle piece a is a part of a puzzle piece x, and this x is, in turn, a part of a puzzle piece y, then there is also a puzzle piece x’ such that a will be the largest common fragment of x and x’, and y will be the smallest fragment including both x and x’. Further, for each puzzle piece x, there is a puzzle piece x’ such that they do not have a common fragment, and they are not together parts of any puzzle piece. In Wolniewicz’s model, every proper elementary situation is an atom or is made of atoms, i.e. situations that cover only the empty situation: 8x E SE” Ǝ A ⊂ SA ð x ¼ sup A Þ, where:

8

Wolniewicz also assumes that the lattice of elementary situations is conditionally distributive, and thus there is only one relative complement for each proper compartment of the lattice. See: Wolniewicz (1985), pp. 24–25.

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3 A Model of the Domain of Law

SA ¼




 x E SE” : x covers only o :

These expressions shall be read as follows: ‘For any proper situation x there is a subset of the atomic situations A such that x is the upper limit of A’, and ‘The set of atomic situations SA is the set of proper situations such that the empty situation is the only situation smaller than any of these situations’. Thus, in terms of our jigsaw puzzle analogy, the atoms are puzzle pieces that do not have parts (i.e. they are single puzzle pieces). And each puzzle piece is a configuration of single puzzle pieces. If two proper elementary situations are incompatible, they contain incompatible atoms, i.e. for any x, y E SE”: x; y ¼ λ ➔ Ǝ s, t E SA ð s  x and t  y and s; t ¼ λ Þ: This expression shall be read as follows: ‘If the splice of situations x and y is the impossible situation, then there are atoms s and t such that s is a part of x, and t is a part of y, and the splice of s and t also is the impossible situation’. Thus, in terms of our jigsaw puzzle analogy, if two puzzle pieces are incompatible, then at least some of their atoms (single puzzle pieces) are incompatible as well. In the set of atoms, the relation of being a complement is transitive, and therefore for any x, y, z E SA we have: ð x; z ¼ λ and y; z ¼ λ Þ ! ð x ¼ y or x; y ¼ λ Þ: This expression shall be read as follows: ‘For any atoms x, y, z, if the splice of x and z is the impossible situation, and the splice of y and z is the impossible situation, then either x and y are identical or the splice of x and y is the impossible situation’. Thus, in terms of our jigsaw puzzle analogy, if two single puzzle pieces are incompatible to the third one, they are either identical or they are incompatible to each other. The last two statements about jigsaw puzzles will become clearer in a moment, when a convertibility relation is defined. The opposite of atoms are possible worlds, i.e. the elementary situations that are covered only by the impossible situation: SP ¼




 x E SE” : x is covered only by λ :

This expression shall be read as follows: ‘The set of possible worlds SP is the set of proper situations such that the impossible situation is the only situation greater than any of these situations’. The possible worlds are counterparts of our AJPs. Thus, in terms of our jigsaw puzzle analogy, any AJP is a puzzle piece that is not a fragment of another puzzle piece. In Wolniewicz’s model, every proper elementary situation is a fragment of a possible world:

3.2 Wolniewicz’s Ontology of Situations

43

8x E SE” Ǝ w E SP ð x  w Þ: This expression shall be read as follows: ‘For any proper situation x there is a possible world w such that x is a part of w’. Thus, in terms of our jigsaw puzzle analogy, each puzzle piece is a part of an AJP. Wolniewicz assumes that every two proper situations are separated by a possible world in such a way that one of these situations occurs in this world and the other does not, i.e. for any x, y E SE”: x 6¼ y ➔ Ǝ w E SP ð ð x  w and  y  wÞ or ð x  w and y  wÞ Þ: This expression shall be read as follows: ‘For any proper situations x and y, if x and y are not identical, then there is a possible world w, such that either x is a part of w and y is not a part of w, or x is not a part of w and y is a part of w’. Thus, in terms of our jigsaw puzzle analogy, there number of AJPs is so large that for any two different puzzle pieces there is an AJP such that only one of these two puzzle pieces is a part of this AJP. One of the possible worlds is singled out as the real world (w0). The elementary situations of the real world, namely: f x E SE : x  w0 g, are the real situations, or the facts.9 All other elementary situations are imaginary. Thus, in terms of our jigsaw puzzle analogy, only one AJP is real. All others are just possibilities. In the set of atoms, Wolniewicz defines a convertibility relation d, namely: x, y E SA ➔ ð x d y ⟺ ð x ¼ y v x; y ¼ λ Þ Þ: This expression shall be read as follows: ‘If an x and an y are atoms, then x is convertible to y if and only if x and y are identical, or the splice of x and y is the impossible situation’. Thus, in terms of our jigsaw puzzle analogy, two single puzzle pieces are convertible if and only if they are either identical or their splice is the impossible situation. For example, imagine ten AJPs produced on one cutting matrix, but each of a different color. This way, each of these AJPs consists of the same number of single puzzle pieces, and each single puzzle piece from any AJP has one counterpart that is identical in shape but different in color, in each of the other AJPs. All singe puzzle pieces that are identical in shape, are called ‘convertible puzzle pieces’. When one of these single puzzle pieces is replaced with the other in an AJP, then another, two colored, AJP is obtained.

9

This expression shall be read as follows: ‘The set of situations x such that x is a part of w0’.

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3 A Model of the Domain of Law

If the set SE” is non-empty, then a breakdown of the set of atoms due to the convertibility relation, i.e. D ¼ SA=d is the set of logical dimensions of the SP set.10 Two atoms belong to the same logical dimension of SP when they are convertible (i.e. either identical or incompatible): x, y E Di ➔ ð x ¼ y or x; y ¼ λ Þ: This expression shall be read as follows: ‘if atoms x and y belong to the same logical dimension Di, then either x and y are identical or the splice of x and y is the impossible situation’. The number of logical dimensions of SP indicates how many atoms does each of the possible worlds of SP consist of. Every possible world contains exactly one atom from each dimension. The abovementioned two axioms state that the set of single puzzle pieces may be divided into subsets of single puzzle pieces of the same shape. These subsets are called ‘logical dimensions’. If two single puzzle pieces belong to the same logical dimension, then either they are identical (they are not only of the same shape but also of the same color) or their splice is the impossible situation, i.e. they cannot appear together in one AJP. The concept of logical dimension is important: any AJP has exactly one single puzzle piece from each logical dimension. Thus, if someone is going to assemble their own AJP, then they have to take exactly one single puzzle piece from each logical dimension. For simplicity, Wolniewicz assumes that the number of logical dimensions is finite: D ¼ f D1, D2, D3, . . ., Dn g: This expression shall be read as follows: ‘The set D consists of dimensions D1, D2, D3, . . . , Dn’. Thus, in terms of our jigsaw puzzle analogy, every AJP is made of a finite number of single puzzle pieces. Of course, such a restriction is very natural for AJPs. Moreover, it is also in accordance with lawyers’ practical perception of the world. Based on the Wolniewicz concept of convertible atoms, the convertible situations can also be defined:

10 This expression shall be read as follows: ‘The set D is the breakdown of the set of atoms based on the convertibility relation’.

3.2 Wolniewicz’s Ontology of Situations

45

h x d y ⟺ 8s E Di ðs  x ➔ Ǝ t E Di t  yÞ and i 8t E  Di ðt  y ➔ Ǝ s E Di s  xÞ , for any x, y E SE” and Di E D.11 Two proper situations are convertible when they are identical as to logical dimensions, i.e. if one contains an atom from a given dimension then the other contains an atom from that dimension. This way, for every x E SE”, there is a set of situations convertible to x: SI ðxÞ ¼




 y E SE” : x d y :

They will be called ‘the alternative situations of x’ or ‘the alternatives of x’. Therefore, this expression shall be read as follows: ‘The set of alternatives of x is the set of proper situations y, such that x and y are convertible to each other’. This set will be labeled ‘SI’. All possible worlds are alternative situations of the real world. Thus, in terms of our jigsaw puzzle analogy, the alternatives of x are all puzzle pieces identical to x as to their logical dimensions. So, if x has a single puzzle piece of some shape, every alternative of x has a single puzzle piece of that shape. And, if x does not have a single puzzle piece of some shape, then none of the alternatives of x have a single puzzle piece of that shape. Each proper situation is an alternative situation of itself, i.e. x E SI (x), for any x E SE”. However, for every x E SE”, there is a set of strict alternative situations: .

This expression shall be read as follows: ‘The set of strict alternatives of x is the set of proper situations y, such that y is an alternative of x, and the splice of x and y is the impossible situation’. This set will be labeled ‘SI’. Thus, in terms of our jigsaw puzzle analogy, the strict alternatives of x are all fragments identical to x in respect of logical dimensions, but differing from x in color. So, if x has a single puzzle piece of some shape, every strict alternative of x has a single puzzle piece of that shape. And, if x does not have a single puzzle piece of some shape, then none of the strict alternatives of x have a single puzzle piece of that shape. However, no strict alternative of x is identical to x in terms of colors. Both concepts of alternative situations will be very useful when the properties of legal events are defined. This way, the core of Wolniewicz’s model of the Wittgenstein logical space has been presented. I only hope that this presentation has not been too puzzling to the reader.

This expression shall be read as follows: ‘Two proper situations x and y are convertible to each other if and only if for any dimension Di: if x contains an atom s from Di, then y contains an atom t from Di, and if y contains an atom t from Di, then x contains an atom s from Di’.

11

46

3 A Model of the Domain of Law

The model presented above may serve as a suitable mathematical theory of situations understood as concrete collectives introduced in Chap. 2. Namely, Wolniewicz’s elementary situations correspond to such concrete collectives, and the partial order relation () corresponds to the ‘being part of’ relation between them. In this model, situations may be compared. In particular, it may be stated that one situation is a part of another, that two situations have a common part, or both are contained in a third one. Also, it may be declared whether two situations are compatible or incompatible, whether they are convertible or inconvertible. Further, it may be declared whether they are alternatives of a third situation or not. And finally, in Wolniewicz’s theory, the alternative worlds are only a kind of the alternative situations. Therefore, when an act is ordered, prohibited or allowed, it can be formally expressed, if needed, that this act only changes a part of the world and the rest remains the same. These properties will be helpful when the particular types of legal events and legal rules are defined. However, if the reader did not have sufficient mathematical enthusiasm to look at Wolniewicz’s formal model, it is enough for them to remember that behind the terms such as: • situation—that is, a three-dimensional fragment from the physical world or from some alternative world, • fact—that is, a three-dimensional fragment from the physical world, • being a part—that is, the relation of partial order between situations, • splice of situations—that is, the smallest situation containing both of the spliced situations, • atom—that is, a situation that does not contain other situations, • possible world—that is, a situation that is not contained in other situations, • logical dimension—that is, a set of atoms that are convertible to each other, • alternatives—that is, situations of the same logical dimensions, stand not only Wittgenstein’s philosophical intuitions, but also a piece of solid mathematics. Therefore, the abovementioned terms may be trusted.

3.3

Legal Events

Let us start with some intuitions in respect of legal events. When the law prohibits killing, we can read it in two ways: (i) no one should kill, or (ii) events where someone kills someone else are prohibited. In the first case, the law applies to persons, and in the second—to events or, in other words, to the sequences of situations. Indeed, when John kills Bill, a situation where Bill is alive, is replaced by a situation where Bill is dead. Therefore, our basic intuition is that events are the sequences of situations. Whereas persons and situations are concrete objects, the sequences of situations are not. Any sequence of situations is an ordered set of situations. And every set is an

3.3 Legal Events

47

abstract object. So, the first abstract objects that shall be defined in our model are events. Some events are ordered by the law, some others are permitted, and some others are forbidden. To be ordered, permitted or forbidden, an event shall be under human control. Events that are under human control are legal events.12 It is possible to distinguish at least four kinds of legal events in accordance with the range of the perpetrator’s control over the event, namely: 1) 2) 3) 4)

acts, multi-acts, controlled causal events, induced causal events.

Acts are ‘quanta’ of law. Pulling gun’s trigger and throwing a punch are examples. The perpetrator has a full control over the act. Therefore, the attribution of responsibility for the act is simple. It is easy to observe who pulls the trigger or throws a punch. Multi-acts are sequences of acts. Gun firing and fist fighting are examples. The perpetrator has a full control over each and every act constituting the multi-act. Therefore, the attribution of responsibility for a multi-act requires the attribution of responsibility for all acts constituting the multi-act. In principle, the issue who participates in gun firing or fist fighting, can be determined by an observation. Controlled causal events are sequences of acts and events of a deterministic character. When a bullet from a gun moves through a thin air and kills a man, the whole such sequence of situations is a controlled causal event. Here, the attribution of responsibility needs not only an observation but also a theory of physical causality. It may seem in the beginning that a bullet caused a man’s death. But an in-depth examination may lead us to a conclusion that the bullet was irrelevant because the man died as a result of poisoning a second earlier. And finally, induced causal events are sequences of situations beginning with an act and linked in a way (not necessarily deterministic) that enables the attribution of responsibility for the final effect to the person responsible for the first act. Burning down a house as a result of careless handling of fireworks is an induced causal event. Here, the attribution of responsibility needs not only an observation and a theory of physical causality but also a legal theory of attribution. Usually, an event, to be recognized as an induced causal event, has to pass two tests: a) the but for test (the conditio sine qua non test), and b) the legal test.

12

Strictly speaking, in legal theory legal events are usually understood as events that bring legal effects. However, the fact that most of legal events are under human control is essential for the possibility of ordering, prohibiting or allowing events. Therefore, here the term ‘legal event’ is used in a narrower sense, that is, it is used exclusively for events under human control.

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3 A Model of the Domain of Law

This first test consists in checking whether the first event (the cause) was a necessary condition for the last one (the effect). So, the first test is a matter of logic and physics. To pass the second test, the cause and the effect have to meet the criterium of proximity, not clearly defined by lawyers. It is said that the effect should be: (i) ‘an adequate consequence of the cause’, or (ii) ‘a normal consequence of the cause’, or (iii) ‘a direct consequence of the cause’, or (iv) ‘an immediate consequence of the cause’, or (v) ‘a not-too-distant consequence of the cause’, or (vi) ‘a predictable consequence of the cause’, etc. The second test is purely legal: the law itself determines which physical relations are sufficient for attributing (legal) responsibility.13 In the following subchapters, the abovementioned objects will be formally constructed.

3.4

Event as a Sequence of Situations

In order to describe events as sequences of situations, Wolniewicz’s model shall be expanded, namely, a Wolniewicz’s structure shall be assigned to each moment of time. This way, instead of one logical space, this time, a sequence of logical spaces will be constructed. It will be achieved in the following way.14 Let S be a set of Wolniewicz’s structures described in the previous section, that is: S ¼ f Si : Si ¼ < SEi ,  >g and let T be a strictly linearly ordered set of moments: T ¼< T, < > : A set is strictly linearly ordered if the relation < is a strict linear order relation, i.e. if it fulfils the following conditions:  x < x, x < y and y < z ➔ x < z, x < y ➔  y < x, x < y or y < x,

13

See: Hart and Honore (2002). As the reader could learn the way of reading formulas in the subchapter devoted to the Wolniewicz ontology, from this moment on, the way of reading formulas, as a rule, will not be given. However, the way of reading will be provided should the formula so require.

14

3.4 Event as a Sequence of Situations

49

for any x, y, z from the set in question. The formula ‘x < y’ shall be read: ‘x occurs earlier then y’. According to these axioms, the considered time structure ought to have the following properties: (i) no moment is earlier than itself, (ii) if the first moment is earlier than the second one, and the second one is earlier than the third one, then the first one is earlier than the third one, as well, (iii) if the first moment is earlier than the second one, then the latter is not earlier than the former, (iv) for any two moments, first one is earlier than the second one, or vice versa. In other words, the strict linear order relation is antireflexive, transitive, strictly antisymmetric, and strongly connected. When such a structure of time is adopted, the linear and discrete nature of time is assumed. It is in accordance with common sense. And, it is consistent with legal practice. Now, let ψ assign exactly one structure Si E S to each moment t E T. This way, the function ψ orders strictly linearly sets of elementary situations, sets of proper elementary situations, sets of possible worlds, and sets of atoms. As a result, strings of sets are obtained: the string of sets of elementary situations (SE1, SE2, SE3, . . .), the string of sets of proper elementary situations (SE”1, SE”2, SE”3, . . .), the string of sets of possible worlds (SP1, SP2, SP3, . . .), and the string of sets of atoms (SA1, SA2, SA3, . . .). The structure M ¼ < S, T, ψ >, constructed as described, is a model of a sequence of logical spaces or, in other words, it is a model of a dynamic logical space. In this model, the reality (w0) and alternative worlds (w1, w2, w3, . . .) are represented not by individual Wolniewicz’s possible worlds, but by sequences of such worlds (the second index next to the letters reflects time order): w0 ¼< . . . , w01 , w02 , w03 , w04 , w05 , w06 , w07 , w08 , . . . > , w1 ¼< . . . , w11 , w12 , w13 , w14 , w15 , w16 , w17 , w18 , . . . > , w2 ¼< . . . , w21 , w22 , w23 , w24 , w25 , w26 , w27 , w28 , . . . > , w3 ¼< . . . , w31 , w32 , w33 , w34 , w35 , w36 , w37 , w38 , . . . > , ......... While the possible worlds of Wolniewicz’s original structure were like photos of the reality and the alternative worlds, in the current structure the possible worlds resemble film tapes. In this structure, events may be defined in the following way. First, a set being the sum of the abovementioned sets of proper elementary situations is introduced: SE” ¼




 SE” 1 U SE” 2 U SE” 3 U . . . :

Respectively, the other sets being sums of the abovementioned sets of elementary situations, sets of possible worlds, and sets of atoms can be considered: SE ¼

50

3 A Model of the Domain of Law

{SE1 U SE2 U SE3 U . . .), SP ¼ {SP1 U SP2 U SP3 U . . .), SA ¼ {SA1 U SA2 U SA3 U . . .). Then, the events are defined as finite strings formed with elements of SE”:
 EVENT ¼ < x1 , . . . , xn >: x1 , . . . , xn E SE” : As a consequence, it is assumed that any event is made up solely of proper situations. This assumption is in accordance with an intuition that events are sequences of concrete objects only. The empty situation and the impossible one shall not be elements of any event, because they are abstract. Further, it is assumed that any event is a finite sequence. This assumption is in accordance with an intuition that every event begins and ends. And finally, it is assumed that any event is a sequence of at least two situations. This final assumption is in accordance with an intuition that any event is a transition between situations. Therefore, one situation is required as the starting point and the second one as the ending point. This definition of event is very broad. According to it, e.g., a sequence where after the morning rain in Warsaw in 2018, Voldemort kills Socrates in 2050, and then Copernicus discovers America in 753, is an event. This sequence does not contradict logic, of course. However, it is not compatible (probably) with the laws of nature. Therefore, it would be great to have a narrower concept of event. According to this narrower concept, the situations constituting an event shall be linked in a natural way. In other words, a concept of events that are compatible with the laws of nature, is required. Such events will be called ‘natural events’, and will constitute a kind (a subset) of events: NAT ⊂ EVENT: However, to define them better, it is necessary to find what links between situations in the sequence are natural. Of course, this is very complicated issue and a proper definition of natural events will not be constructed in this book. Nevertheless, in the next subchapter, some accessibility relations will be examined, which will cover two aspects of such natural links, namely the time sequence and the identity of logical dimensions.

3.5

Accessibility Relations

The natural sequences of situations are described in natural sciences. Physics, for example, offers theories that closely describe how situations give rise to other situations, and how events produce other events. Thanks to the laws described by physics, spaceships that went to Mars and beyond were sent. The laws of chemistry allow us to transform minerals into medicines. Thanks to the laws of biology, people know how to get better crops, etc. In a book on the semantics of law, it is impossible

3.5 Accessibility Relations

51

to translate all this wealth of laws into formulas. There is no such need, as well. Our task is not to fully describe the real domain of law, but to create a model of this domain, i.e. its simplified description. So, it is enough for only some aspects of the natural link between situations to be described here. These aspects will be the sequence in time and the identity of logical dimensions. Undoubtedly, these aspects are important for the law domain. They will be presented by defining several accessibility relations between situations. In logical semantics, artificial structures for interpreting languages are being created. There are various structures of possible worlds among these artificial structures, where next to the real world, there are some alternative worlds. Most readers probably associate the concept of alternative worlds mainly with science fiction. However, in logical semantics, alternative worlds are being used for creating models, in particular, when some modalities are examined, such as aletic (necessary, possible), temporal (earlier, later), or deontic (obligatory, forbidden, allowed). These models are called the ‘semantics of possible worlds’. One of the first philosophers who used the concept of possible world was Saul Kripke.15 In Kripke-type semantics, the structures < W, R > are considered, where W is the set of objects that are called ‘possible worlds’ and R is a relation between possible worlds that is called ‘alternativity relation’ or ‘accessibility relation’. Various assumptions are made about accessibility relations, e.g. that they are reflexive, transitive, symmetric, etc. Depending on these assumptions, different models (models of different domains) are obtained. Here, some accessibility relations will be defined, as well. However, thanks to Wolniewicz’s structures, such relations will be defined not only for possible worlds, but also for any proper situations. As a result, our models will be more detailed than any Kripke-type models. In particular, in our structures, an event or act changes a part of the world, while the rest remains the same. Such relations cannot be reflected in any Kripke-type semantics, where only possible worlds are considered, but their parts are not. This property of Wolniewicz’s structures will turn out to be helpful when legal events and legal rules are defined. The accessibility relations constructed here, will reflect a succession in time between possible worlds, between any proper situations, or only between alternative situations. As the reader may remember, the properties of the T structure make time a strictly linearly ordered set of moments. Since each moment corresponds to a set of situations, the time sequences of situations may be considered. Depending on whether an accessibility relation takes place only between situations where the second situation is next in time to the first one, or whether this relation is allowed to take place between situations distant in time, the accessibility relations are divided into direct and indirect. Further, as the reader may remember, the alternative situations are situations of the same logical dimensions, i.e. situations that are identical in terms of their structures. Depending on whether an accessibility relation

15

Saul Aaron Kripke (1940), an American philosopher and logician, pioneer of research into the semantics of modal logic.

52

3 A Model of the Domain of Law

takes place only between alternative situations, or whether this relation is allowed to take place between any situations, the accessibility relations are divided into restricted to alternative situations, and those not restricted to them. As the reader may also remember, all possible worlds are identical in terms of logical dimensions, i.e. every possible world contains exactly one atomic situation from each logical dimension. So, all possible worlds are alternative to each other, and therefore there is no sense in dividing the accessibility relations between possible worlds into restricted to alternatives, and those not restricted to them. Therefore, exactly six accessibility relations will be considered here: 1) 2) 3) 4) 5) 6)

R—a relation of direct accessibility between possible worlds only, R+—a relation of indirect accessibility between possible worlds only, SER—a relation of direct accessibility between any proper situations, SIR—a relation of direct accessibility between alternative situations only, SER+—a relation of indirect accessibility between any proper situations, SIR+—a relation of indirect accessibility between alternative situations only.

Whereas we understand that the time succession and the identity of logical dimensions are only two of many aspects of the natural link between situations, we do not impose other conditions on accessibility relations here. Imposing such additional conditions requires further research, e.g. it may be considered whether any conditions imposed in modal or temporal logics could be used here.

3.5.1

The R Relation of Direct Accessibility Between Possible Worlds

In the set of possible worlds SP ¼ { SP1 U SP2 U SP3 U . . . ), the relation R of direct accessibility is defined as follows: x R y ➔ Ǝ wn E SPn , wnþ1 E SPnþ1 ð wn ¼ x and wnþ1 ¼ y Þ: This definition shall be read as follows: for any possible worlds x and y, if y is directly accessible from x, then there are possible worlds wn belonging to SPn and wn+1 belonging to SPn+1, such that x is identical to wn and y is identical to wn+1. In other words, if a world y is directly accessible from a world x, then x directly precedes y in time. The implication symbol ‘➔’ indicates that the relation R is defined by a partial definition. It means that a direct precedence is the necessary condition for the R accessibility, but it is not the sufficient condition: it may be the case, that the world x directly precedes the world y in time, and, nevertheless, the world y is not directly accessible from the world x. There is still a place for the laws of physics, chemistry, biology, etc. The R relation is a starting point for defining all other accessibility relations.

3.5 Accessibility Relations

3.5.2

53

The R+ Relation of Indirect Accessibility Between Possible Worlds

In the set of possible worlds SP ¼ { SP1 U SP2 U SP3 U . . . ), the relation R+ of indirect accessibility is defined as follows: x Rþ y ⟺ Ǝ wn , wnþ1 , wnþ2 , . . . , wnþm ðwn ¼ x and wnþm ¼ y and 8i E < n; n þ m  1 > wi R wiþ1 Þ: This definition shall be read as follows: for any possible worlds x and y, y is indirectly accessible from x if and only if there are possible worlds wn, wn+1, wn+2, . . ., wn+m, such that the first of them is identical to x, the last one is identical to y, and each successive one is directly accessible from the previous one.16 In other words, a world y is indirectly accessible from a world x if and only if x and y are respectively the first and the last element in a sequence of possible worlds, where each world is directly accessible from the previous one. As a consequence, if one world is directly accessible from some other world, then it is also indirectly accessible from that other world: x R y ➔ x Rþ y: Further, if from one world, a second world is directly accessible, and a third world is directly accessible from this second world, then from this first world this third world is accessible indirectly: x R y and y R z ➔ x Rþ z: Below, relations similar to the relation R and the relation R+ will be defined for any proper situations.

3.5.3

The SER Relation of Direct Accessibility Between Proper Situations

Thanks to Wolniewicz’s structures, the counterparts of the R and R+ relations may be defined for any proper situations. In particular, for any x and y E SE”:

The expression ‘8 i E ’ is read as follows: ‘for any i belonging to the range from n to n + m -1’.

16

54

3 A Model of the Domain of Law

x SER y ⟺ Ǝ wn , wm E SP ð x  wn and y  wm and wn R wm Þ: This definition shall be read as follows: for any proper situations x and y, y is directly accessible from x if and only if there are possible worlds wn and wm belonging to SP, such that x is a part of wn and y is a part of wm and the world wm is directly accessible from the world wn. In other words, a situation y is directly accessible from a situation x if and only if y is a part of a possible world R-accessible from a possible world that contains x. Since possible worlds are proper situations, and each possible world is a part of itself, if one world is R accessible from some other world, then it is also SER accessible from that other world: x R y ➔ x SER y: The SER relation is convenient for distinguishing natural events that are subject to legal assessments. For example, a gasoline tank explosion is SER accessible from throwing a cigarette butt.

3.5.4

The SIR Relation of Direct Accessibility Between Alternative Situations

For SER accessibility, the logical dimensions of situations are irrelevant. In other words, SER accessible situations may differ in logical dimensions. Thus, it may turn out that a rain in China is SER accessible from a flying butterfly in London. Sometimes, this lack of identity in respect to logical dimensions, is convenient. This was the case of the fuel tank explosion. Other times, however, it is more convenient if the initial situation corresponds in terms of logical dimensions to the ending situation. The concept of alternative situations may be used to define such an accessibility relation. Namely, for any x and y E SE”: x SIR y ⟺ x SER y and y E SI ðxÞ: This definition shall be read as follows: for any proper situations x and y, y is SIR accessible from x if and only if y is SER accessible from x and y is an alternative of x, i.e. x and y are identical with respect to logical dimensions. For example, standing on the first ladder step is SIR accessible from standing in front of the ladder. As a direct consequence, if one situation is SIR accessible from some other situation, then it is also SER accessible from that other situation: x SIR y ➔ x SER y:

3.5 Accessibility Relations

55

Since all possible worlds are alternatives of each other, if one world is R accessible from some other world, then it is also SIR accessible from that other world: x R y ➔ x SIR y: Similarly to the SER accessibility, the SIR relation is convenient for describing natural events that are subject to legal assessments. For example, pulling gun’s trigger is SIR accessible from taking a gun, and throwing a punch is SIR accessible from standing in front of a guy. Thus, the SIR accessibility will turn out to be helpful in defining acts.

3.5.5

The SER+ Relation of Indirect Accessibility Between Proper Situations

The relations of direct accessibility, namely R, SER, and SIR, are not suitable for describing events where the start and the end of an event, are distant in time, e.g. for expressing a link between a mistake in dispensing drugs to patients, and the next day death of a patient. In contrast, the R+ relation of indirect accessibility is not limited this way, but it is defined solely for possible worlds, and not for any proper situations. Therefore, a relation of indirect accessibility for any proper situations will be defined below. For any x and y E SE”: x SERþ y ⟺ Ǝ wn , wm E SP ð x  wn and y  wm and wn Rþ wm Þ: This definition shall be read as follows: for any proper situations x and y, y is indirectly accessible from x if and only if there are possible worlds wn and wm belonging to SP, such that x is a part of wn and y is a part of wm and the world wm is indirectly accessible from the world wn. In other words, a situation y is indirectly accessible from a situation x if and only if y is a part of a possible world indirectly accessible from a possible world that contains x. As it was for the R and SER relations, if one world is R+ accessible from some other world, then it is also SER+ accessible from that other world: x Rþ y ➔ x SERþ y: Similarly to the SER and SIR relations, the SER+ relation is convenient for describing natural events that are subject to legal assessments. For example, burning of a (distant) forest is SER+ accessible from throwing a cigarette butt.

56

3.5.6

3 A Model of the Domain of Law

The SIR+ Relation of Indirect Accessibility Between Alternative Situations

For SER+ accessibility, the logical dimensions of situations are irrelevant. In this respect, the SER and SER+ relations are similar. Thus, it may turn out that a rain in China in 2021 is SER+ accessible from a flying butterfly in London in 1967. If one does not want such situations to be called accessible, and at the same time admits that situations distant in time may be linked with an accessibility relation, they can use the SIR+ accessibility. Namely, for any x and y E SE”: x SIRþ y ⟺ x SERþ y and y E SI ðxÞ: This definition shall be read as follows: for any proper situations x and y, y is SIR+ accessible from x if and only if y is SER+ accessible from x and y is an alternative of x, i.e. x and y are identical with respect to logical dimensions. For example, standing on the last ladder step is SIR+ accessible from standing in front of the ladder. As it was for the SIR and SER relations, if one situation is SIR+ accessible from some other situation, then it is also SER+ accessible from that other situation: x SIRþ y ➔ x SERþ y: Further, as it was for the R and SIR relations, if one world is R+ accessible from some other world, then it is also SIR+ accessible from that other world: x Rþ y ➔ x SIRþ y: Similarly to the SER, SIR, and SER+ relations, the SIR+ relation is convenient for describing natural events that are subject to legal assessments. For example, pulling gun’s trigger is SIR+ accessible from loading a gun with bullets, and a gasoline tank explosion is SIR+ accessible from careless repairing this tank a few days earlier.

3.5.7

The Accessibility Relations and the Set of Natural Events

The abovementioned relations are linked as follows. Let ‘R’, ‘R+’, ‘SER’, ‘SER+’, ‘SIR’, ‘SIR+’ denote sets of situations (pairs) connected by, respectively, the R, R+, SER, SER+, SIR, SIR+ relations.17 The following will hold: 17

Strictly speaking, every binary relation is a set of pairs. However, when defining accessibility relations, the non-bold symbols ‘R’, ‘R+’, ‘SER’, ‘SER+’, ‘SIR’, ‘SIR+’ were used as predicate letters. Here, these symbols are used as set names. Thus, it is convenient to have them in bold, to distinguish this usage from the previous one.

3.6 Legal Events in Terms of the Ontology of Situations

57

R ⊂ Rþ , SER ⊂ SERþ , SIR ⊂ SIRþ , SIR ⊂ SER, SIRþ ⊂ SERþ : While each of the relations mentioned above brings a partial answer to the question of what links between situations in the sequence are natural, none of them fully express such natural links. In other words, the natural link is a proper subset of the sum of these accessibility relations, i.e.: NATLINK ⊂ R U Rþ U SER U SERþ U SIR U SIRþ and NATLINK 6¼ R U Rþ U SER U SERþ U SIR U SIRþ : Since each of the relations R, R+, SER, SIR, SIR+ is a subset of SER+, the abovementioned statement may be also expressed in a shorter way: NATLINK ⊂ SERþ and NATLINK 6¼ SERþ : Consequently, the natural events may be defined as events where subsequent situations are linked in a natural way: NAT

3.6

¼

n

< xn , xnþ1 , . . . , xnþm > E EVENT : 8i E < n; n þ m  1 > o ð< xi , xiþ1 > E NATLINKÞ :

Legal Events in Terms of the Ontology of Situations

Thus, among the events, the natural events may be distinguished. In turn, among the natural events, it is possible to distinguish legal events, i.e. the events under human control:

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LEV ⊂ NAT ⊂ EVENT: In the third subchapter of this chapter, four kinds of legal events have been distinguished: 1) 2) 3) 4)

acts, multi-acts, controlled causal events, induced causal events.

Probably, other types of legal events may also be distinguished. This guess may be expressed in symbols in the following way: ACT U MUL U CCE U ICE ⊂ LEV: Thanks to the formal concept of event, acts, multi-acts, controlled causal events, and induced causal events, may be defined in a formal way, too. As it was said before, acts are ‘quanta’ of law. Pulling gun’s trigger and throwing a punch are good examples of acts. A constitutive feature of acts is that the perpetrator has a full control over the act. The acts will be defined by providing conditions every act shall fulfill. First, since every legal event is a natural event, every act is a natural event, as well: ACT ⊂ NAT: In other words, people are acting within the limits set by the laws of nature. Second, during any act exactly one situation is replaced by another. Therefore, it is assumed that the acts are pairs of situations: x E ACT ➔ x ¼< a, b > , where a E SE” and b E SE”. Third, during any act solely a specific fragment of the reality is changed. Old and new situations are identical with respect to logical dimensions. Thus, it is assumed that the acts are pairs of alternatives:18 < a, b > E ACT ➔ b E SI ðaÞ: As a consequence, the situations constituting an act shall be bound either by the SIR accessibility or by the SIR+ accessibility, and thus the act is not a pair of any proper situations:

18 The first situation of the act is called ‘the situation of choice’ (or ‘the choice situation’) and the second one is called ‘the chosen situation’.

3.6 Legal Events in Terms of the Ontology of Situations

59

< a, b > E ACT ➔ ð< a, b > E SIR or < a, b > E SIRþ Þ, < a, b > E ACT ➔  ð< a, b > E SER∖SIR Þ, < a, b > E ACT ➔  ð< a, b > E SERþ ∖SIRþ Þ: The expression ‘SER\SIR’ shall be read as follows: ‘the difference of the sets SER and SIR’. The expression ‘SER+\SIR+ ’ shall be read in a similar way. An x belongs to the difference of sets A and B if and only if x belongs to the set A and does not belong to the set B. Thus, e.g., a pair of proper situations belongs to the difference of the sets SER and SIR if and only if the situations constituting this pair are of different logical dimensions. According to the SI definition, every situation is an alternative for itself: a E SI ðaÞ: Thus, it is possible that < a, a > E ACT, that is, keeping status quo is an act. From the perspective of legal theory, this feature of the acts, is an important one. In particular, a person may be legally responsible for an omission, i.e. for refraining from acting. In a sense, according to lawyers, non-acting is acting. And the same is in our theory of acts. Fourth, any act is a choice. Therefore, it is assumed that there are at least two alternatives in every choice situation: < a, b > E ACT ➔ Ǝ c E SE” ð< a, c > E ACT and b; c ¼ λ Þ: In other words, if there is no choice, there is no act, either. In law, pushing another person is not an act, if the pushing person had no control over their body, e.g., because they were pushed themselves by a third person. All acts relevant in a choice situation are called ‘alternative acts’. So, when two people, say John and Kate, are drowning, and only one of them can be saved, the following acts will be alternative acts: saving John, saving Kate, and not taking any rescue action. Fifth, acts are short in time. Therefore, it is assumed that the SIR accessibility fits more than SIR+ to define acts: ACT ⊂ SIR: Indeed, it is not intuitive to consider two situations as an act when one precedes the other by two years. Therefore, the SIR accessibility fits to acts better than the SIR+ accessibility. Sixth, any act is a manifest of the will of a person. We cannot formally express this condition. However, formally, it may be written down that:

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ACT 6¼ SIR: In other words, it may formally be written that not every pair of situations bound by the SIR relation is an act. Of course, this statement is not equal to saying that any act is a manifest of the will of a person. However, some intuitions are difficult to be formalized in full. Perhaps, the acts should also meet other conditions. However, the six abovementioned definitely are the basic ones. It may be noted that these conditions are not independent of each other. Namely, the fifth condition entails the second one and the third one. In other words, every event that belongs to the SIR set is an event composed of solely two situations, and these two situations are alternative to each other. Further, the sixth condition entails the fourth one. In other words, if there was no choice, the will of the person could not be manifested. As a result, it may be said that the acts are the SIR natural events that manifest human will. Based on the definition of act, the other three types of legal events: multi-acts, controlled causal events, and induced causal events, may be defined. As mentioned earlier, multi-acts are sequences of acts. Gun firing and fist fighting are good examples of multi-acts. A constitutive feature of multi-acts is that the perpetrator has a full control over each and every act constituting the multi-act. This feature leads to the following definition of multi-acts: MUL

¼

n

< xn , xnþ1 , . . . , xnþm > E NAT : 8i E < n; n þ m  1 > o ð< xi , xiþ1 > E ACTÞ :

Thus, the multi-acts are natural events where every two consecutive elements constitute an act. The next type of legal events are controlled causal events. When a bullet from a gun moves through a thin air and kills a man, this sequence of situations is a controlled causal event. In this case, one needs a theory of physical causality to attribute legal responsibility. Thus, the controlled causal events may be defined as sequences of acts and deterministic events: CCE

¼

n

< xn , xnþ1 , . . . , xnþm > E NAT : 8i E < n; n þ m  1 > o ð< xi , xiþ1 > E ACT or < xi , xiþ1 > E DETÞ ,

where ‘DET’ designates the set of deterministic events. Thus, the controlled causal events are natural events where every two consecutive elements constitute an act or a deterministic event. Intuitively, an event is deterministic when each of its elements inevitably brings the next one. The deterministic events may be formally defined in our theory of events, but I believe that the reader will find this informal definition as satisfactory.

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The last type of legal events distinguished in this chapter, are induced causal events. The induced causal events are sequences of situations beginning with an act, where the situations are linked in a way (not necessarily deterministic) that enables the attribution of responsibility for the final effect to the person responsible for the first act. Burning down a house as a result of careless handling of fireworks is an induced causal event. In this case, the attribution of responsibility needs not only an observation and a theory of physical causality, but also a legal theory of attribution. Thus, to define the induced causal events, a legal relation is required: ICE

¼

n

< xn , xnþ1 , . . . , xnþm > E NAT :< xn , xnþ1 > E ACT o and < xn , xnþ1 , xnþm > E LEG :

As it was said before, an event, to be recognized as an induced causal event, has to pass two tests: a) the but for test (the conditio sine qua non test), and b) the legal test. Therefore, the LEG relation should be able to pass these two tests. Passing the first test may be formally expressed in the following way. In accordance with common sense, an act is a necessary condition for an effect if and only if none of the alternative acts can produce this effect. Thus, an event with the beginning E ACT and the ending s E SE” satisfies the but for test (the conditio sine qua non test), when any event with the beginning E ACT, where y 6¼ v, does not contain s. Therefore:
NEC ¼ < x, y, s > E NAT : 8 < x, y, . . . , s > E NAT, 8 < x, v, . . . , s’ > E NAT ð< x, y > E ACT and < x, v > E ACT and y 6¼ v ➔ s 6¼ s’ Þ g: It is also possible to express what the but for test means for the real events, having in mind that attributing responsibility for the real events is a standard situation in legal practice. Let us assume that SF is the set of facts in a strict sense, i.e. the set of real situations. Of course, a natural event may be a sequence of facts. Let us call such events ‘facts in a broad sense’ and signify them with a symbol ‘FEV’: FEV ¼ f< x1 , x2 , . . . , xn > E NAT : x1 , x2 , . . . , xn E SF g: Respectively, the following set of facts in a broad sense may be constructed: FACT ¼ f< x1 , x2 > E ACT : x1 , x2 E SF g,

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In other words, the set FACT is the set of acts that are facts in a broad sense. An act is a fact in a broad sense if and only if all of its constituent situations are facts in a strict sense. Other legal events may be facts in a broad sense, too. Respectively: FMUL ¼ { E MUL: x1, x2, . . ., xn E SF}, FCCE ¼ { E CCE: x1, x2, . . ., xn E SF}, and FICE ¼ { E ICE: x1, x2, . . ., xn E SF}. An event with the beginning E FACT and the ending s E SF satisfies the but for test (the conditio sine qua non test), when any event with the beginning E ACT, where y 6¼ v, does not contain s. In other words, a real act is a necessary condition for a real effect if and only if none of the alternative acts can produce this effect. Therefore:
NEC ¼ < x, y, s > E FEV : 8 < x, y, . . . , s > E FEV, 8 < x, v, . . . , s’ > E NAT   < x, y > E FACT and < x, v > E ACT and y 6¼ v ➔ s 6¼ s’ g: Anyway, in either meaning of the but for test: LEG ⊂ NEC: To pass the legal test, the act and the presumed effect s should be linked by legal proximity. As it was said before, the criterium of proximity is not clearly defined by lawyers. The effect should be: (i) ‘an adequate consequence of the cause’, or (ii) ‘a normal consequence of the cause’, or (iii) ‘a direct consequence of the cause’, or (iv) ‘an immediate consequence of the cause’, or (v) ‘a not-too-distant consequence of the cause’, or (vi) ‘a predictable consequence of the cause’, etc. In fact, the second test is purely legal: the law itself determines which physical relations are sufficient for attributing (legal) responsibility. Respectively, the LEG relation may be a relation of indirect accessibility between any proper situations: LEG ⊂ SERþ , or it may be a relation of direct accessibility between any proper situations: LEG ⊂ SER, Although, probably, it is not a relation of direct accessibility between alternative situations:  ð LEG ⊂ SIR Þ:

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63

The last conclusion is a consequence of the fact that lawyers recognize a LEG relation between throwing a cigarette butt and burning of a (distant) forest, i.e. between events differing in logical dimensions.19 To conclude this section: ACT U MUL U CCE U ICE ⊂ LEV ⊂ NAT ⊂ EVENT: Due to the need for simplicity, the following considerations will usually be limited to events as such, or acts. However, it should be remembered that there are several types of legal events and all of them are important in legal theory.

3.7

Legal Rules as Specific Sets of Events

Now, the concept of legal rule will be introduced. According to an intuition expressed in the Introduction to this book, the legal rules are supposed to be semantic correlates of legal norms, i.e. they are supposed to be objects in the law domain to which legal norms may be referred. The legal rules will be defined in relation to the events formally described in the previous subchapters.

3.7.1

Towards the Concept of Legal Rule

Let us think of a set of all legal events, that is, events under human control. Let the following diagram represent this set:

LEGAL EVENTS

Many events under human control are of interest to the law, but not all. The legal events that are of interest to the law will be called ‘legal events in the strict sense’. On the other hand, the events under human control that are not of interest to the law will be called ‘irrelevant events’. Thus, in the set of legal events, it is possible to distinguish two subsets, namely the set of legal events in the strict sense and the

19

The accessibility relations defined in this book are binary. On the other hand, the LEG relation has been defined above as a three-argument relation. So, strictly speaking, in order to compare the LEG relation with accessibility relations, the LEG relation should be modified into a binary form. For this purpose, for example, a LEG’ relation may be introduced between the chosen situation and the supposed effect. However, for the sake of simplicity, we limit ourselves to this footnote.

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set of irrelevant events, whereby each legal event is either a legal event in the strict sense, or is irrelevant:

LEGAL EVENTS IN A STRICT SENSE

IRRELEVANT EVENTS

The legal events in the strict sense may be divided into two groups, namely those that are permitted by the law, and those that are forbidden by the law. In other words, in the set of legal events in the strict sense, a subset of permitted events and a subset of forbidden events, may be distinguished, where each legal event in the strict sense is either permitted or forbidden:

PERMITTED EVENTS

FORBIDDEN EVENTS

IRRELEVANT EVENTS

Some of the permitted events are simultaneously ordered. Others are permitted but are not ordered. Thus, in the set of permitted events, it is possible to distinguish a subset of ordered events:

PERMITTED EVENTS

ORDERED EVENTS

FORBIDDEN EVENTS

IRRELEVANT EVENTS

This way, the sets of ordered, forbidden, permitted and irrelevant events were distinguished in the set of legal events. In fact, these sets are perfect candidates for the semantic correlates of legal norms. Namely, each time a legal norm is stated, it is also stated that an event belongs to one of these sets, or that a set of events is a subset of one of these sets. The former is

3.7 Legal Rules as Specific Sets of Events

65

the case of norms that are usually called ‘individual norms’, and the latter—norms that are usually called ‘general norms’. First, let us examine the former case. When one looks at the legal norms: ‘It is ordered that John helps Kate’, ‘It is permitted that John takes advantage of the tax credit’, or ‘It is forbidden that John steps on the lawn’, they can see that these norms state properties of events. Indeed, the very sentences ‘John helps Kate’, ‘John takes advantage of the tax credit’, ‘John steps on the lawn’, are just descriptions of events, and the sentences ‘It is ordered that John helps Kate’, ‘It is permitted that John takes advantage of the tax credit’, or ‘It is forbidden that John steps on the lawn’, state that the events in question are either ordered, permitted, or forbidden, i.e. they attribute some deontic properties to these events. In Chap. 2, it was said that distributive sets of objects correspond to the properties of objects. For example, the set of lenders corresponds to the property of being a lender. Respectively, when one states that an object has a property, they also state that this object belongs to a set of objects. For example, when one claims that John is a lender, they also claim that John belongs to the set of lenders. The same is true for legal events. Namely, when one claims that an event is ordered, permitted, or forbidden, they also claim that this event belongs to, respectively, the set of ordered, permitted, or forbidden events. For example, when one says that ‘It is ordered that John helps Kate’, they state that the event that John helps Kate belongs to the set of ordered events. Similarly, when one says that ‘It is permitted that John takes advantage of the tax credit’, they state that the event that John takes advantage of the tax credit belongs to the set of permitted events. Likewise, when one says that ‘It is forbidden that John steps on the lawn’, they state that the event that John steps on the lawn belongs to the set of forbidden events. And so on. The norms: ‘It is ordered that John helps Kate’, ‘It is permitted that John takes advantage of the tax credit’, or ‘It is forbidden that John steps on the lawn’, are called ‘individual norms’, because each of them decides about the deontic modality of a single event. However, in the theory of law, the norms that determine the deontic modality of the classes of events are also considered. These norms are the general norms. The norms: ‘It is ordered to help others’, ‘It is permitted to take advantage of tax credits’, or ‘It is forbidden to step on the lawns’, are the examples. When someone states that ‘It is ordered to help others’, they state that every event consisting of helping people belongs to the set of ordered events, i.e. that the set of events consisting of helping others is a subset of the set of ordered events. Similarly, when someone states that ‘It is permitted to take advantage of tax credits’, they say that every event consisting of taking advantage of a tax credit belongs to the set of permitted events, and therefore that the set of events consisting of taking advantage of a tax credit is a subset of the set of permitted events. Similarly, when one states that ‘It is forbidden to step on the lawns’, they say that every event consisting of stepping on a lawn belongs to the set of forbidden events, i.e. that the set of events consisting of stepping on a lawn is a subset of the set of forbidden events. And so on. A glance at these examples leads to the conclusion that behind all ordering norms, there is a set of ordered events; behind all forbidding norms, there is a set of

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forbidden events; and behind all permitting norms, there is a set of permitted events. Thus, in the law domain, the set of ordered events is a correlate of all ordering norms, the set of forbidden events is a correlate of all forbidding norms, and the set of permitted events is a correlate of all permitting norms. These three extra-linguistic correlates of legal norms will be called the ‘general legal rules’. Thus, precisely three general legal rules, namely, the prescription, the prohibition, and the permission, are distinguished: GLR ¼ f OBL, FOR, PER g: In each of these three sets, it is possible to distinguish subsets that are correlates of particular norms, as well. For example, in the set of forbidden events, it is possible to distinguish a subset of lawn trampling events, a subset of crossing the street at a red light events, a subset of killing people events, etc. The first of these subsets is a specific correlate of the norm ‘It is forbidden to trample the lawns’, the second one is a specific correlate of the norm ‘It is forbidden to cross the street at a red light’, and the third one—a specific correlate of the norm ‘It is forbidden to kill’. Such specific correlates of legal norms will be called the ‘specific legal rules’. The number of specific legal rules is vast. Potentially, this number may be equal to the sum of the number of all subsets of the set of ordered events, the number of all subsets of the set of forbidden events, and the number of all subsets of the set of permitted events: SLR ¼ 2OBL U 2FOR U 2PER : This expression shall be read as follows: ‘The set of specific legal rules is the sum of three sets: the power set of the set of ordered events, the power set of the set of forbidden events, and the power set of the set of permitted events’.20 This way, three general legal rules and many specific legal rules, may be defined. Of course, the practical question arises how to divide the set of legal events into these subsets? Which events belong to the set of ordered events, which belong to the set of forbidden events, which belong to the set of permitted ones, and which belong to the set of irrelevant ones? The answer to the abovementioned question depends on legal culture. Some will say that this division is a matter of the statutory law, that these subsets are created by enacting laws and regulations, just as mathematical worlds are created by adopting axioms. Others will say that this division was made once in God’s mind. Some others will say that this division is the result of court decisions. And others still, that such a division may be a result of scientific analysis. Anyway, the answer to the abovementioned question is beyond the scope of this book. Therefore, it will not be indicated here which events constitute a given legal rule.

The power set of the set X is the set of all subsets of the set X. For example, if the set X ¼ {1, 2, 3}, then the power set of the set X ¼ { , {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}. The power set of X is usually marked as ‘2X’. The symbol ‘ ’ means an empty set. 20

3.7 Legal Rules as Specific Sets of Events

67

However, we are going to indicate here some conditions that all legal rules should meet to be in line with intuition.

3.7.2

Basic Conditions for Legal Rules

Before all else, in order to be in line with intuition, the legal rules should meet the conditions specified above with the help of diagrams. These conditions may be also presented in the form of axioms. First, the relevant events (i.e. legal events in a strict sense) shall be separated from the irrelevant events:

that is, no event is relevant and irrelevant at the same time. Second, the permitted events shall be separated from the forbidden events:

that is, no event is permitted and forbidden at the same time. Third, the set of ordered events shall be a proper subset of the set of permitted events: OBL ⊂ PER, OBL 6¼ PER, that is, everything that is ordered is also permitted, but not everything that is permitted is ordered.

3.7.3

Conditions Based on Properties of Acts

To reflect the specific features of acts, other conditions may also be provided. Thus, while the conditions listed above concern all legal events, the following relate to the acts only. To express these additional conditions, the formal concept of act, created in previous subchapters, will be used. First of all, according to intuition, the prescriptions should not contradict each other. It may be expressed by the condition that the alternative acts cannot be ordered at the same time. For example, when two people are drowning, say John and Kate, and only one of them can be saved, the alternative acts will be: (i) saving John, (ii) saving Kate, and (iii) not taking any rescue action. Therefore, in this situation

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exactly one of these acts can be ordered, i.e. either to save John, or to save Kate, or not to take any rescue action. Therefore, fourth, the set of ordered events shall not contain alternative acts: ð< a, b > E ACT and < a, c > E ACT and b 6¼ c Þ ➔  ð< a, b > E OBL and < a, c > E OBL Þ: This expression shall be read as follows: if the sequence of situations is an act, and the sequence of situations is an act, and b differs from c, then the acts and cannot be ordered at the same time. In other words, only one of alternative acts shall be ordered. Similar conditions do not apply to FOR and PER. Alternative acts may be forbidden at the same time. They may be also permitted at the same time. For example, it may be forbidden to turn left and turn right at a crossroads—then everyone shall go straight through this crossroads. Similarly, a left turn and a right turn at a crossroads may be permitted at the same time. Further, according to intuition, if one of the alternative acts is ordered, the others shall be prohibited. For example, if it is ordered to turn right at a crossroads, then it shall be forbidden to turn left and go straight ahead at this crossroads. Therefore, fifth, if one of alternatives is ordered, then all others shall be forbidden: ð< a, b > E ACT and < a, c > E ACT and b 6¼ c Þ ➔ ð< a, b > E OBL ➔ < a, c > E FOR Þ: This expression shall be read as follows: if the sequence of situations is an act, and the sequence of situations is an act, and b differs from c, then, if the act is ordered, then the act is forbidden. According to intuition, it is not the case that everything is forbidden. Indeed, there is no sense in setting road signs at a crossroads in such a way that at the same time they forbid going straight ahead, turning left, turning right, U-turning, and stopping the car. Therefore, sixth, if exactly one of alternatives is forbidden, all others shall be either permitted or irrelevant: ð< a, b > E ACT and < a, c > E ACT and b 6¼ c Þ ➔ ð< a, b > E FOR ➔ ð ð< a, d > E FOR ! d ¼ b Þ ➔ ð< a, c > E PER U IRR Þ Þ:

This expression shall be read as follows: if the sequence of situations is an act, and the sequence of situations is an act, and b differs from c, then, if the act is forbidden, and every forbidden act is identical to the act (i.e. the act is the only forbidden act), then the act is permitted or irrelevant. Additionally, seventh, if there are exactly two alternatives, it shall not be a case that both are forbidden:

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ð< a, b > E ACT and < a, c > E ACT and b 6¼ cÞ ➔ ðð< ad > E ACT ➔ ðd ¼ b or d ¼ cÞÞ ➔  ð< a, b > E FOR and < a, c > E FORÞÞ: This expression shall be read as follows: if the sequence of situations is an act, and the sequence of situations is an act, and b differs from c, then, if every act is identical to either the act or the act (i.e. the acts and are the only alternative acts), then the act and the act cannot be forbidden at the same time. According to intuition, since it is not the case that all alternative acts are forbidden, then there is an allowed act in every choice situation. In particular, if there are only two alternative acts in a given choice situation, then at least one of them shall be either strongly allowed (i.e. permitted) or softly allowed (i.e. irrelevant). Thus, eighth, if there are exactly two alternatives, at least one of them shall be permitted or irrelevant: ð< a, b > E ACT and < a, c > E ACT and b 6¼ cÞ ➔ ðð< a, d > E ACT ➔ ðd ¼ b or d ¼ cÞÞ ➔ ðð< a, b > E PER U IRRÞ or ð< a, c > E PER U IRRÞÞÞ: This expression shall be read in the following way: if the sequence of situations is an act, and the sequence of situations is an act, and b differs from c, then, if every act is identical to either the act or the act (i.e. the acts and are the only alternative acts), then the act is permitted or irrelevant, or the act is permitted or irrelevant. Clearly, at this point, it would be possible to give even more conditions by referring to the concept of act, or to the concept of alternative acts. However, these examples alone are sufficient to demonstrate that the forementioned concepts allow us to formally grasp intuitions that are not expressible with the Venn diagrams presented above. While these diagrams allow to state, for example, that if an event belongs to a set then this event does not belong to some other set, the reference to alternative acts allows to state, among others, that if an event belongs to a set then another event (which is an alternative to the former) does not belong to that or another set.

3.7.4

Conditions Based on Properties of Situations

Moreover, thanks to the fact that Wolniewicz’s ontology allows to compare situations (in particular, it allows stating that one situation is a part of another situation), it is also possible to define some specific conditions for the acts consisting of other acts.

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In particular, it may be expressed that if all parts of an act have the same deontic modality (that is, they are all either ordered, forbidden, or permitted), then this act has this modality, as well. Thus, for example, if two persons are drowning, and it is permitted to save the first person, and it is permitted to save the second person, then it is also permitted to save both persons. And symmetrically: if an act has a deontic modality, then all its parts have this modality, too. Thus, for example, if two persons are drowning, and it is ordered to save both, then, simultaneously, it is ordered to save the first person, and it is ordered to save the second one. These dependencies may be expressed in the form of the following conditions. Ninth, if all parts of an act are ordered, the whole act shall be ordered: < a; c , b; d > E ACT ➔ ðð< a, b > E OBL and < c, d > E OBLÞ ➔ < a; c , b; d > E OBLÞ: This expression shall be read in the following way: if the sequence whose first element is the splice of situations a and c, and the second element is the splice of situations b and d, is an act, then, if the act is ordered and the act is ordered, then this first sequence also is ordered. Tenth, if all parts of an act are forbidden, the whole act shall be forbidden: < a; c , b; d > E ACT ➔ ðð< a, b > E FOR and < c, d > E FORÞ ➔ < a; c , b; d > E FORÞ: This expression shall be read in the following way: if the sequence whose first element is the splice of situations a and c, and the second element is the splice of situations b and d, is an act, then, if the act is forbidden and the act is forbidden, then this first sequence also is forbidden. Eleventh, if all parts of an act are permitted, the whole act shall be permitted: < a; c , b; d > E ACT ➔ ðð< a, b > E PER and < c, d > E PERÞ ➔ < a; c , b; d > E PERÞ: This expression shall be read in the following way: if the sequence whose first element is the splice of situations a and c, and the second element is the splice of situations b and d, is an act, then, if the act is permitted and the act is permitted, then this first sequence also is permitted. Twelfth, if an act is ordered, all its parts shall be ordered: ð< a, b > E ACT and < c, d > E ACTÞ ➔ ð< a; c , b; d > E OBL ➔ ð< a, b > E OBL and < c, d > E OBLÞÞ: This expression shall be read in the following way: if the sequence is an act, and the sequence is an act, then, if the sequence whose first element is the

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splice of situations a and c, and the second element is the splice of situations b and d, is ordered, then the act is ordered and the act is ordered. Thirteenth, if an act is forbidden, all its parts shall be forbidden: ð< a, b > E ACT and < c, d > E ACTÞ ➔ ð< a; c , b; d > E FOR ➔ ð< a, b > E FOR and < c, d > E FORÞÞ: This expression shall be read in the following way: if the sequence is an act, and the sequence is an act, then, if the sequence whose first element is the splice of situations a and c, and the second element is the splice of situations b and d, is forbidden, then the act is forbidden and the act is forbidden. Fourteenth, if an act is permitted, all its parts shall be permitted: ð< a, b > E ACT and < c, d > E ACTÞ ➔ ð< a; c , b; d > E PER ➔ ð< a, b > E PER and < c, d > E PERÞÞ: This expression shall be read in the following way: if the sequence is an act, and the sequence is an act, then, if the sequence whose first element is the splice of situations a and c, and the second element is the splice of situations b and d, is permitted, then the act is permitted and the act is permitted.

3.7.5

Towards the Mathematization of the Law Domain

Clearly, it is possible to put forward more conditions, or even put forward other conditions for legal rules. It is also possible to introduce more than three general legal rules as well as less than three general legal rules. For example, in ethical rigorism every event would probably be just either ordered or forbidden. Thus, the Venn diagram for ethical rigorism would be as follows:

ORDERED EVENTS

FORBIDDEN EVENTS

So, the conditions presented here are only examples (although they are not arbitrary). The point is that legal rules are constructed here in terms of the set theory and their elements are well defined in terms of the theory of situations. Thus, a mathematical model of a fragment of the law domain is constructed. Whereas only a few of many abstract objects of the law domain have been defined here, they are the fundamental ones. In Chap. 4, this model will be used for interpreting a model of the language of law. In turn, thanks to the results of Chap. 4, the issue of the veracity of

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legal norms as well as the issue of entailment between legal norms, will be solved in Chap. 5.

References Hart HLA, Honore T (2002) Causation in the law, 2nd edn. Clarendon Press, Oxford Przełęcki M (1969) The logic of empirical theories. Routledge & Kegan Paul Limited, London Wolniewicz B (1982) A formal ontology of situations. Studia Logica 41:381–413 Wolniewicz B (1985) Ontologia sytuacji. Państwowe Wydawnictwo Naukowe, Warszawa (in Polish: Ontology of situations)

Chapter 4

A Model of the Language of Law

In the fourth chapter, a model of the language of law suitable for describing the law domain model will be presented. First order predicate logic language will be chosen as this model. To introduce it, first the types of its symbols and its grammar will be described. Then, in accordance with our ‘intuitive formalism’ method, some of this language’s expressions will be directly linked with the law domain model built in Chap. 3. This way, these expressions will obtain their meanings, and our model of the language of law will be ready for use, in the sense that legal norms might be expressed in it, and their semantic properties might be studied. However, besides this direct method of attributing meanings to expressions, an indirect method will be used as well. Namely, several deontic theories will be constructed in our language. The axioms of these theories will give meanings to the expressions that appear in them. This way, it will be demonstrated that first order predicate logic language is well suited to describe our law domain model, in particular, that the meanings of the prescription, prohibition, and permission predicates may be defined in it. In addition, the reader may learn the axioms and theorems that serve as a formal justification for various rules of legal reasoning.

4.1

The Vocabulary and Grammar

In this chapter, an artificial language that may serve as a model of the language of law shall be defined. A well-chosen model is a simplification of the studied object, that retains its most important properties. In our case, the simplification will not be excessive if the artificial language in question turns out to be sufficient for describing the general legal rules of our law domain model. The language of first order predicate logic will be chosen as our model of the language of law, since its categorization fairly corresponds to the categorization of the language of law made in Chap. 2. In particular, the sentences, names, predicates,

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Malec, Introduction to the Semantics of Law, Law and Visual Jurisprudence 6, https://doi.org/10.1007/978-3-030-95679-0_4

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sentence conjunctions, and quantifiers, are distinguished in this artificial language.1 Moreover, some logicians believe that everything we have to say can be said in the first order language. Among others, this is what Wittgenstein assumed in his Treaty. Therefore, it is possible that this language is not only sufficient to describe the law domain model, but is sufficient to describe the law domain itself, as well. As it was said in Chap. 2, according to logic, three sets should be defined to create a language: (i) a dictionary—a set of signs that will be used as words of this language, (ii) a grammar—a set of rules for distinguishing correct expressions from others, and (iii) a semantics—a set of rules for linking correct expressions with a domain outside the language. In this subchapter, the dictionary and grammar of first order predicate logic language will be described. We take a vocabulary of the language of first order logic as it is. We do not add any specific operators. Moreover, we do not use name-forming functors. Thus, the symbols in our model are as follows: (1) a1, a2, a3, a4, . . . .. (2) x1, x2, x3, x4, . . . . (3) P1, P2, P3, P4, . . . . (4) ¼ (5) 8, Ǝ (6) ˄, ˅, !, $, ⅂ (7) ( , ) , ,

constants for individuals, variables for individuals, predicates (unary, binary and multi-argument), a binary equality (identity) predicate, a universal quantifier, and an existential quantifier, symbols for logical conjunctions: a conjunction, an alternative, an implication, an equivalence, and a negation, technical signs: a left bracket, a right bracket, and a comma.2

The individual constants are counterparts to the proper names (individual names) of the language of law (e.g. the names of countries, or surnames of persons). On the other hand, the individual variables are counterparts to the common names (general names) of the legal language (e.g. the names of professions, or goods). The predicates of our model are counterparts to the predicates of the language of law. The logical conjunctions are counterparts to the sentence conjunctions of the legal language. And, finally, the quantifiers correspond to the expressions ‘every’ and ‘some’ of the language of law. In our model, there are neither name-forming, nor functor-forming functors, discussed in Chap. 2. We take a grammar of the language of first order logic as it is, as well. Thus, only names (terms) and sentence formulas are correct expressions of our language. The only names are constants and variables for individuals. The only sentence formulas are expressions of the following structure: (1) predicates combined with an appropriate number of terms enclosed in brackets and separated by commas (e.g. ‘P1(x1)’, ‘P2(a1,x2)’, ‘P3(x1,a2)’), A standard description of the language of first order logic can be found, for example, in Marciszewski (1987), pp. 24–26. 2 For the sake of transparency and simplicity, other letters will be also used as well as a strict separation of subject language and meta-language symbols will be abandoned. 1

4.2 Interpretation of the Subject Language Expressions

75

(2) sentence formulas preceded by a quantifier, or preceded by a negation, or connected by a binary conjunction, enclosed in brackets in a standard way (e.g. ‘8 x1 P1(x1)’, ‘8 x1 P3(x1,a2)’, ‘Ǝ x1 P3(x1,a2)’, ‘⅂ P3(x1)’, ‘8 x1 P3(x1,a2) ˅ Ǝ x1 P3(x1,a2)’, ‘8 x1 ⅂ P3(x1,a2) $ ⅂ Ǝ x1 P3(x1,a2)’). The concept of sentence formula is more general than the concept of sentence, namely, every sentence is a sentence formula, but not every sentence formula is a sentence. This is because the sentence formulas may contain the so-called ‘free variables’, i.e. individual variables that are not bound by any quantifier. On the other hand, the sentences do not contain free variables. Thus, ‘P1(a1)’ is a sentence— because it does not contain variables, ‘8x1P1(x1)’ is a sentence—because the variable ‘x1’ is bound by a quantifier, and ‘P1(x1)’ is not a sentence—since the variable ‘x1’ is not bound by a quantifier in this last expression. The expression ‘man went to the forest’ is an equivalent of a sentence formula that is not a sentence, in everyday language. While it may be decided whether the sentences ‘A man went to the forest’, or ‘Every man went to the forest’, are true or false, it is not the case when the expression ‘man went to the forest’ is considered. This is because it is not determined to whom that last expression refers to. As the reader may see, the dictionary and grammar of first order predicate logic language are not very complicated. To complete the description of this language, it is still required to interpret its expressions. It will be done in the next subchapter.

4.2

Interpretation of the Subject Language Expressions

Natural language expressions receive their meanings in two ways. The first is pointing a finger at what an expression refers to. The second is describing the meaning of an expression with other words of natural language. Providing axioms and rules of inference for an artificial language is similar to the latter method. The axioms and inference rules of an axiomatic theory determine the meanings of logical constants and non-logical expressions of this theory. We will use this approach later in this chapter. However, the expressions of our artificial language may also obtain their meanings by a direct reference to our model of the law domain. It slightly resembles pointing a finger at an object. However, in this case, it is the abstract objects that are pointed at. Now, this approach will be used to interpret some expressions of the first order logic language. Let us assume the function φ which assigns an object from our law domain model to each name and predicate in our law language model. First, the φ function assigns situations and events to names:

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φ ðan Þ, φ ðxn Þ E SE” U EVENT: In other words: (1) each constant for individuals corresponds to exactly one situation or event, (2) each variable for individuals runs the set of situations and the set of events, i.e. it can correspond to any situation or event. This way, the names refer to solely situations and events. There are no concrete individuals (i.e. persons and things). Nonetheless, this Wittgenstein-style simplification does not limit us in describing the law domain, since it is the configurations of persons and things that constitute situations and events. Thus, the domain in question is just described in a simpler manner. Second, the φ function links some predicates to objects defined in Chap. 3, as follows: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30)

φ (SE(. . .)) ¼ SE , φ (SE”(. . .)) ¼ SE” , φ (o(. . .)) ¼ { o } , φ (λ(. . .)) ¼ { λ } , φ (ɛ(. . .,. . .)) ¼ { < m , n >: m  n } , φ (γ(. . .,. . .,. . .)) ¼ { < k , m , n >: k = m ; n } , φ (SP(. . .)) ¼ SP , φ (SA(. . .)) ¼ SA , φ (SF(. . .)) ¼ SF , φ (SI(. . .,. . .)) ¼ SI , φ (SI(. . .,. . .)) ¼ SI , φ (EVENT(. . .)) ¼ EVENT , φ (FEV (. . .)) ¼ FEV , φ (NAT(. . .)) ¼ NAT , φ (R(. . .,. . .)) ¼ R , φ (R+(. . .,. . .)) ¼ R+ , φ (SER(. . .,. . .)) ¼ SER , φ (SIR(. . .,. . .)) ¼ SIR , φ (SER+(. . .,. . .)) ¼ SER+ , φ (SIR+(. . .,. . .)) ¼ SIR+ , φ (LEV(. . .)) ¼ LEV , φ (ACT(. . .)) ¼ ACT , φ (ACT(. . .,. . .)) ¼ ACT , φ (FACT(. . .)) ¼ FACT , φ (MUL(. . .)) ¼ MUL , φ (FMUL(. . .)) ¼ FMUL , φ (DET(. . .)) ¼ DET , φ (CCE(. . .)) ¼ CCE , φ (FCCE(. . .)) ¼ FCCE , φ (LEG(. . .)) ¼ LEG ,

4.3 The First Order Logic as a Basis of Legal Theories

(31) (32) (33) (34) (35) (36) (37) (38) (39) (40) (41) (42)

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φ (ICE(. . .)) ¼ ICE , φ (FICE(. . .)) ¼ FICE , φ (REL(. . .) ) ¼ REL , φ (IRR(. . .) ) ¼ IRR , φ (OBL(. . .)) ¼ OBL , φ (FOR(. . .)) ¼ FOR , φ (PER(. . .) ) ¼ PER , φ (REL(. . .,. . .)) ¼ { < m , n >: < m , n > E REL } , φ (IRR(. . .,. . .)) ¼ { < m , n >: < m , n > E IRR } , φ (OBL(. . .,. . .)) ¼ { < m , n >: < m , n > E OBL } , φ (FOR(. . .,. . .)) ¼ { < m , n >: < m , n > E FOR } , φ (PER(. . .,. . .)) ¼ { < m , n >: < m , n > E PER } .

In other words, sets of situations or events have been assigned to some of the predicates of our artificial language.3 For example, the predicate ‘SA (. . .)’ (which shall be read: ‘. . . is an atomic situation’) is linked to the set of atoms SA; the predicate ‘ACT (. . ., . . .)’ (which shall be read: ‘the pair of situations . . . and . . . is an act’)—to the set of acts ACT; the predicate ‘OBL (. . ., . . .)’ (which shall be read: ‘it is ordered to replace the situation . . . . with the situation . . .’)—to the set of ordered events OBL, etc. This way, the vocabulary, the grammar, and a kind of semantics of our artificial language, have been provided.4 Thus, if the reader is not interested in an alternative method of assigning meanings to this language expressions, namely by adopting axiomatic theories, they may skip the rest of this chapter and go directly to Chap. 5 where semantic properties of legal norms will be examined. However, if the reader would like to explore first order axiomatic theories describing various models of the law domain, and if they would like to learn about the nature of legal reasoning, then they should read this chapter to the end.

4.3

The First Order Logic as a Basis of Legal Theories

The meanings of language expressions can be determined by referring directly to a domain, as it was done in the previous subchapter. However, these meanings can also be determined by theories expressed in this language. Thus, here, several theories about the law domain model will be stated. The axiomatic method will be used, i.e. the theories in question will be constructed by adopting axioms (basic theorems accepted without a proof), and

Except the objects assigned to the predicates ‘ɛ(. . .,. . .)’ and ‘γ(. . .,. . .,. . .)’. These objects may or may not be events. This is because the events are sequences of solely proper situations, and ‘being a part’ relation and ‘being a splice’ relation are defined for any situations. 4 Solely the above-listed predicates have been linked to the law domain model. Thus, the meanings of other predicates are not specified yet. It can be done later, if needed. 3

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rules of inference (rules allowing to infer theorems from the axioms). When the axioms and inference rules are adopted, all theorems of a theory are determined (potentially). The axioms are expressed in a subject language, whereas the inference rules—in the meta-language of this subject language. The axioms and inference rules may be divided into logical and non-logical. Logical axioms and rules determine the meaning of logical constants (i.e., in our case, the logical conjunctions and quantifiers). Axioms and rules that determine the meaning of other expressions are called ‘non-logical’.5 Theories based solely on logical axioms and logical inference rules are called ‘logical theories’ or simply ‘logics’. An example of such a theory is the first order classical logic. Its axioms and inference rules reflect our understanding of quantifiers and logical conjunctions. Theories based on both logical and non-logical axioms and inference rules are called ‘theories built on logic’ or simply ‘theories’. Here, the first order logic will serve as a logical basis for theories of legal events and theories of acts.6 The first order logic with identity will be used. Thus, logical axioms and logical rules of our theories are as follows: (1) axioms of the first order logic, (2) axioms for the identity predicate: 8x ðx ¼ xÞ, 8x y ðx ¼ y $ y ¼ xÞ, 8x y z ðx ¼ y ˄ y ¼ z ! x ¼ zÞ, (3) rules of the first order logic.7 On this logical basis, several theories will be constructed. They can be divided into: (1) theories of legal events, (2) theories of simple acts, (3) theories of compound acts. The legal events theories will describe deontic relations that can be represented in Venn diagrams. The simple acts theories will describe deontic relations between alternative acts. And, finally, the compound acts theories will describe deontic relations between acts where one act is a part of the other.

The concept of logical constant is relative: the necessity symbol ‘□’ is a non-logical symbol of the classical propositional logic, while the same symbol is a logical constant of the modal propositional logic. See: Tarski (1983). Further, logical axioms will also be called the ‘non-specific axioms’, and non-logical axioms, the ‘specific axioms’. 6 These theories will also be called ‘deontic theories’ or ‘deontic logics’. They were first presented in: Malec (2019), p. 109. 7 Standard axioms and rules of the first order logic can be found, for example, in Marciszewski (1987), pp. 26–28. 5

4.4 Theories of Legal Events

4.4

79

Theories of Legal Events

Let us start with a few theories of legal events. Each of these theories will define the properties of legal rules in a different way. The simplest theory (AEP) will describe a hypothetical domain where all events are allowed. The second one (AEPF) will describe a hypothetical domain where every event is either permitted or forbidden, and the third one (AEOF)—a hypothetical domain where every event is either ordered or prohibited. Next, there will be presented a theory (AEPOF) that describes a hypothetical domain where every event is permitted, ordered, or forbidden. Finally, there will be presented a theory (AEPOFI) that corresponds to the legal rules described in Chap. 3, that is every event will be permitted, ordered, forbidden, or irrelevant. The AEP, AEPF, AEOF, and AEPOF theories describe some alternative models of the law domain. Thus, some of their theorems are false in our law domain model constructed in Chap. 3. In contrast to them, all AEPOFI theorems are true in that model. Looking at these theories one by one, it can be seen how the set of axioms defining the properties of legal rules is changing. The domain of these theories is the set of events described in Chap. 3. Thus, all their propositions are propositions about events. Only five unary predicates will be used here: LEV (x)—which shall be read ‘x is a legal event’, PER (x)—which shall be read ‘x is a permitted event’, FOR (x)—which shall be read ‘x is a forbidden event’, OBL (x)—which shall be read ‘x is an ordered event’, IRR (x)—which shall be read ‘x is an irrelevant event’. Thus, the specific axioms of these theories determine relations between ordered, forbidden, permitted, and irrelevant events.

4.4.1

Theory 1: All Legal Events Are Permitted (AEP)

The following Venn diagram shows a domain where everything is permitted:

PERMITTED EVENTS = LEGAL EVENTS

This domain can be described in our first order language by adding only one specific axiom to the logical basis:

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A1. 8 x ( LEV (x) $ PER (x) ).8 Likewise, theories describing domains where all events would be irrelevant, ordered, or forbidden, could be created. For this purpose, it is enough to replace the predicate ‘PER (x)’ with one of the predicates ‘IRR (x)’, ‘ORD (x)’, or ‘FOR (x)’ in the presented axiom. However, neither AEP nor the other such theories seem interesting from the point of view of legal logic.

4.4.2

Theory 2: All Legal Events Are Either Permitted or Forbidden (AEPF)

This is not the case for the next two theories which describe interesting domains of, respectively, a soft and hard legal rigorism. A domain where every event is either permitted or forbidden is presented in the following Venn diagram:

PERMITTED EVENTS

FORBIDDEN EVENTS

This domain can be described in our first order language by adding two specific axioms to the logical basis: A1. 8 x ( LEV (x) $ ( PER (x) ˅ FOR (x) ) ) , A2. ⅂ Ǝ x ( PER (x) ˄ FOR (x) ).9 This way, the AEPF theory has been defined. The domain described by AEPF is a domain of soft legal rigorism where no legal event is ordered, however, every legal event is either permitted or forbidden. If, in the adopted axioms, the predicate ‘PER’ is replaced with the predicate ‘OBL’, then we will get a hard rigorism instead of a soft one: A1. 8 x ( LEV (x) $ ( OBL (x) ˅ FOR (x) ) ) , A2. ⅂ Ǝ x ( OBL (x) ˄ FOR (x) ) .

8 This expression shall be read: ‘For any event x, x is a legal event if and only if x is a permitted event’. 9 The first axiom shall be read as follows: ‘For every event x, x is a legal event if and only if x is a permitted event or x is a forbidden event’. The second axiom shall be read as follows: ‘It is not true that there is an event x such that x is a permitted event and x is a forbidden event’. I assume that after reading these few axioms together, the reader will be able to easily read other axioms and theorems to be presented here.

4.4 Theories of Legal Events

81

From this perspective, every legal event in either ordered or forbidden. This theory is called ‘AEOF’ (all legal events are either ordered or forbidden) and corresponds to the following Venn diagram:

ORDERED EVENTS

FORBIDDEN EVENTS

These two domains are the simplest non-trivial structures of deontic modalities. A legal system that is consistent with the hard rigorism leaves no place to the freedom. Every behavior is determined by the law. On the other hand, a legal system that is in line with the soft rigorism embodies the liberal principle: ‘Everything which is not forbidden is allowed’.

4.4.3

Theory 3: All Legal Events Are Either Permitted or Ordered or Forbidden (AEPOF)

By adding three specific axioms: A1. 8 x ( LEV (x) $ ( OBL (x) ˅ PER (x) ˅ FOR (x) ) ) , A2. ⅂ Ǝ x ( PER (x) ˄ FOR (x) ) , A3. 8 x ( OBL (x) ! PER (x) ) , to the logical basis, we will get a deontic theory AEPOF. This corresponds to the following Venn diagram:

PERMITTED EVENTS

ORDERED EVENTS

FORBIDDEN EVENTS

This domain is very similar to our model. However, it is simpler since it does not contain irrelevant events. Therefore, if someone wants a theory that distinguishes ordered, forbidden, and permitted events, but does not distinguish strong permission (i.e. permitted events) and weak permission (i.e. irrelevant events), then the AEPOF theory is a perfect theory for them.

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4.4.4

Theory 4: All Legal Events Are Either Permitted or Ordered or Forbidden or Irrelevant (AEPOFI)

In Chap. 3, four kinds of legal events were distinguished: ordered events, permitted, forbidden, and irrelevant. Relations between the sets of ordered, permitted, forbidden, and irrelevant events described in Chap. 3, are reflected in the following Venn diagram:

PERMITTED EVENTS

ORDERED EVENTS

FORBIDDEN EVENTS

IRRELEVANT EVENTS

To describe this domain in our first order language, five specific axioms shall be added to the logical basis: A1. A2. A3. A4. A5.

8 x ( LEV (x) $ ( OBL (x) ˅ PER (x) ˅ FOR (x) ˅ IRR (x) ) ) , ⅂ Ǝ x ( PER (x) ˄ FOR (x) ) , ⅂ Ǝ x ( IRR (x) ˄ FOR (x) ) , ⅂ Ǝ x ( PER (x) ˄ IRR (x) ) , 8 x ( OBL (x) ! PER (x) ) .

This theory describes the legal rules as defined in Chap. 3. This is an approach to defining deontic modalities, where strong and weak permissions are distinguished.

4.4.5

Existence of Legal Events

It is worth noting that in the aforementioned deontic theories it is not prejudged whether there are legal events. If one wants to prejudge this fact, a specific axiom should be added to each of these systems: A0. Ǝ x LEV (x). Such an assumption is not necessary to consider deontic theories, but it is consistent with legal intuition and legal practice.

4.4 Theories of Legal Events

4.4.6

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Selected Theorems of Legal Events Theories

Selected theorems of the theories of legal events are presented below. The proofs are omitted, as simple and intuitive. AEPF, AEPOF, AEPOFI include, in particular, the following theorems: T1. T2. T3. T4.

8 x ⅂ ( PER (x) ˄ FOR (x) ) , 8 x ( ⅂ PER (x) ˅ ⅂ FOR (x) ) , 8 x ( PER (x) ! ⅂ FOR (x) ) , 8 x ( FOR (x) ! ⅂ PER (x) ) .

Thus, the relations between permission and prohibition are described in a standard way, in every of these three theories. Of course, in AEPOF and AEPOFI, the following theorems hold: T5. 8 x ( OBL (x) ! ⅂ FOR (x) ) , T6. 8 x ( FOR (x) ! ⅂ OBL (x) ) , T7. 8 x ( ⅂ PER (x) ! ⅂ OBL (x) ) . These relations between prescription, prohibition and permission are also described in a standard way. In fact, the theorems T1–T7 have close equivalents in deontic propositional logics. On the other hand, in AEPF and AEPOF, we have T8. 8 x ( LEV (x) ! ( PER (x) ˅ FOR (x) ) ) , T9. 8 x ( LEV (x) ! ( ⅂ PER (x) ! FOR (x) ) ) , T10. 8 x ( LEV (x) ! ( ⅂ FOR (x) ! PER (x) ) ) , and consequently, we also have T11. 8 x ( LEV (x) ! ( PER (x) $ ⅂ FOR (x) ) ) which follows from T3, T10, T12. 8 x ( LEV (x) ! ( FOR (x) $ ⅂ PER (x) ) ) which follows from T4, T9. The theorems T8–T12 have equivalents in deontic propositional logics. In particular, in these logics, permission is usually defined as the negation of prohibition (similar to T11), and prohibition is usually defined as the negation of permission (similar to T12). However, the theorems T8–T12 differ from their counterparts in deontic propositional logics. Namely, they concern solely legal events as it is indicated by the predecessor of these theorems. Our first order language is richer than the languages of deontic propositional logics, so our theorems are more ‘subtle’ than these logics theorems. All the same, the theorems of our theories of legal events, as a rule, have their counterparts in other systems of deontic logic. However, the reverse relation does not take place. Namely, there are constraints resulting from the very construction of

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our language that prevent: (i) applying deontic modalities to sentences regardless of their form, (ii) iterating deontic modalities, (iii) expressing deontic paradoxes presented in the Introduction and similar ones. Thus, the very construction of our language ‘cuts off’ the sources of non-intuitiveness and paradoxicality of many other deontic logics. This is due to the fact that our theories arose of a carefully constructed model of the law domain. Some might say that since there are theorems of other systems that have no counterparts in ours, the systems presented here are poor. However, it is not the case. Firstly, our first order language eliminates non-intuitive and paradoxical theorems of other deontic logics—of course, getting rid of such theorems is only an advantage, not a loss. Secondly, our first order language allows us to express intuitive and non-paradoxical theorems that have no counterparts in other deontic systems. The theorems of this kind are expressed in theories of simple and compound acts presented below.

4.5

Theories of Simple Acts

Each of the above-presented theories of legal events has its counterpart in a theory of simple acts. For the sake of simplicity, here, it will be considered only one such counterpart, namely the counterpart of AEPOF. This counterpart will be called ‘AAPOF’ (that is ‘all acts are either permitted or ordered or forbidden’). So, it will be a theory where strong and weak permissions are not distinguished. As the counterpart of AEPOF, AAPOF will have equivalents of all AEPOF axioms and theorems. However, it will also have axioms and theorems that cannot be expressed in AEPOF since they refer to the concepts of act and alternative acts. The domain of all theories of simple acts, including AAPOF, is the set of situations described in Chap. 3. Thus, all propositions of these theories are propositions about situations. The specific axioms of AAPOF will determine relations between ordered, forbidden, and permitted acts. Thus, only four binary predicates are used here: ACT (x, y)—which shall be read ‘replacement x by y is an act’, PER (x, y)—which shall be read ‘replacement x by y is permitted’, FOR (x, y)—which shall be read ‘replacement x by y is forbidden’, OBL (x, y)—which shall be read ‘replacement x by y is ordered’.

4.5.1

Specific Axioms of AAPOF

Every act is a legal event. Thus, the first three AAPOF-specific axioms are the exact counterparts of the AEPOF-specific axioms:

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85

A1. 8 x y ( ACT (x, y) $ ( OBL (x, y) ˅ PER (x, y) ˅ FOR (x, y) ) ) , A2. ⅂ Ǝ x y ( PER (x, y) ˄ FOR (x, y) ) , A3. 8 x y ( OBL (x, y) ! PER (x, y) ) . These three axioms determine the relations between any situations x and y that form one legal event (i.e. a sequence of situations < x, y >). The next three AAPOF-specific axioms define relations involving three situations: x, y, z that form two legal events (i.e. two sequences of situations: < x, y > and < x, z >). The axiom A4 states that every act is a choice: A4. 8 x y ( ACT (x, y) ! Ǝ z ( ACT (x, z) ˄ y 6¼ z ) ) (In each choice situation, there are at least two alternatives). The axiom A5 confirms that the prescriptions are consistent: A5. 8 x y z ( OBL (x, y) ! ( y 6¼ z ! FOR (x, z) ) ) (If in a choice situation x, an alternative y is ordered, then all other alternatives are prohibited in x). On the other hand, the axiom A6 states that not everything is forbidden: A6. 8 x y ( FOR (x, y) ! Ǝ z ( ACT (x, z) ˄ y 6¼ z ˄ ⅂ FOR (x, z) ) ) (If in a choice situation x, an alternative y is forbidden, then some other alternative is not forbidden in x). As in the case of the theories of legal events, it is not prejudged whether acts exist. If one wants to prejudge this fact, a specific axiom should be added to AAPOF: A0. Ǝ x y ACT (x, y) (There are choice situations / There are acts).

4.5.2

Selected Theorems of AAPOF That Are Equivalent to Theorems of AEPOF

In AAPOF, there are exact equivalents of the theorems T1–T12 of AEPOF: T1. T2. T3. T4. T5. T6. T7. T8. T9.

8 x y ⅂ ( PER (x, y) ˄ FOR (x, y) ) , 8 x y ( ⅂ PER (x, y) ˅ ⅂ FOR (x, y) ) , 8 x y ( PER (x, y) ! ⅂ FOR (x, y) ) , 8 x y ( FOR (x, y) ! ⅂ PER (x, y) ) , 8 x y ( OBL (x, y) ! ⅂ FOR (x, y) ) , 8 x y ( FOR (x, y) ! ⅂ OBL (x, y) ) , 8 x y ( ⅂ PER (x, y) ! ⅂ OBL (x, y) ) , 8 x y ( ACT (x, y) ! ( PER (x, y) ˅ FOR (x, y) ) ) , 8 x y ( ACT (x, y) ! ( ⅂ PER (x, y) ! FOR (x, y) ) ) ,

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T10. 8 x y ( ACT (x, y) ! ( ⅂ FOR (x, y) ! PER (x, y) ) ) , T11. 8 x y ( ACT (x, y) ! ( PER (x, y) $ ⅂ FOR (x, y) ) ) , T12. 8 x y ( ACT (x, y) ! ( FOR (x, y) $ ⅂ PER (x, y) ) ) . This is not surprising as every act is an event.

4.5.3

Selected AAPOF Theorems Specific to Acts

In AAPOF, however, there are also theorems that do not have their exact counterparts in AEPOF, e.g.: T13. 8 x y z ( OBL (x, y) ! ( y 6¼ z ! ⅂ PER (x, z) ) ) (If an alternative y is ordered in a choice situation x, then no other alternative is permitted in x), T14. 8 x y z ( OBL (x, y) ! ( y 6¼ z ! ⅂ OBL (x, z) ) ) (If an alternative y is ordered in a choice situation x, then no other alternative is ordered in x), T15. 8 x y z ( OBL (x, y) ˄ OBL (x, z) ! y ¼ z ) (If, in a choice situation, two alternatives are ordered, they are identical), T16. 8 x y z ( y 6¼ z ! ⅂ (OBL (x, y) ˄ OBL (x, z) ) ) (In any choice situation, different alternatives cannot be ordered together), T17. 8 x y ( FOR (x, y) ! Ǝ z ( y 6¼ z ˄ PER (x, z) ) ) (If an alternative y is forbidden in a choice situation x, then some other alternative z is permitted in x), T18. 8 x y ( OBL (x, y) ! Ǝ z ( y 6¼ z ˄ FOR (x, z) ) ) (If an alternative y is ordered in a choice situation x, then some other alternative z is forbidden in x), T19. 8 x y z ( y 6¼ z ! ( OBL (x, y) ! ⅂ PER (x, z) ) ) (If an alternative y is ordered in a choice situation x, then no other alternative is permitted in x), T20. 8 x y z ( ACT (x, y) ˄ ACT (x, z) ˄ y 6¼ z ˄ 8 w ( ACT (x, w) ! ( w ¼ y ˅ w ¼ z ) ) ! ⅂ ( FOR (x, y) ˄ FOR (x, z) ) ) (If there are exactly two alternatives in a choice situation, both cannot be forbidden), T21. 8 x y z ( ACT (x, y) ˄ ACT (x, z) ˄ y 6¼ z ˄ 8 w ( ACT (x, w) ! ( w ¼ y ˅ w ¼ z) ) ! ( FOR (x, y) ! PER (x, z) ) )

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(If, in a choice situation, there are exactly two alternatives, then if one of them is forbidden, the other is permitted), T22. 8 x y z ( ACT (x, y) ˄ ACT (x, z) ˄ y 6¼ z ˄ 8 w ( ACT (x, w) ! ( w ¼ y ˅ w ¼ z ) ) ! ( PER (x, y) ˅ PER (x, z) ) ) (If, in a choice situation, there are exactly two alternatives, then at least one of them is permitted), T23. 8 x y z ( ACT (x, y) ˄ ACT (x, z) ˄ y 6¼ z ˄ 8 w ( ACT (x, w) ! ( w ¼ y ˅ w ¼ z ) ) ! ( ⅂ PER (x, y) ! PER (x, z) ) ) (If, in a choice situation, there are exactly two alternatives, then if one of them is not permitted, the other is permitted), T24. 8 x y z w ( FOR (x, y) ˄ ( FOR (x, z) ! y ¼ z ) ! ( ACT (x, w) ˄ w 6¼ y ! PER (x, w) ) ) (If, in a choice situation, exactly one alternative is forbidden, then any other alternative is permitted). These theorems are consequences of adding the specific axioms A4–A6 to the system. These theorems prove that the properties of legal rules which are based on the concepts of act and alternative acts may be described in our first order language. The abovementioned theorems show, inter alia, how the deontic modalities of alternative acts depend on each other. This dependency may help, for example, when the so called ‘conflict rules’ are to be examined and formalized.10 Moreover, these theorems have no counterparts in deontic propositional logics, and, most probably, in other deontic logics.

4.6

Theories of Compound Acts

Each theory of simple acts can be developed into a theory of compound acts. For the sake of simplicity, here, only one such theory will be considered, namely, an extension of the AAPOF theory. It will be constructed by adding to the AAPOF axioms new ones for acts, some of which are parts of the others. This extended theory will express an intuition that the deontic modality of an act may depend on the deontic modalities of its parts. The domain of this theory is the set of situations described in Chap. 3. Thus, all propositions of this theory are propositions about situations. One unary predicate

10

The conflict rules are the rules of legal reasoning that allow for the elimination of contradictions in the set of legal norms. These rules can be formalized in first order language where the domain is the set of norms. See: Malec (2001), pp. 97–101. Undoubtedly, it would be interesting to formalize these rules in first order language where the domain is the set of acts.

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‘AT (x)’, one binary predicate ‘ɛ (x, y)’, and one ternary predicate ‘γ (x, y, z)’ are added to the previous ones. They shall be read in the following way: AT (x)—‘x is an atomic situation’, ɛ (x, y)—‘x is a part of y’, γ (x, y, z)—‘x is the sum (composition) of y and z’. Below, it will be used ‘x ɛ y’ instead of ‘ɛ (x, y)’ and ‘x ¼ y + z’ instead of ‘γ (x, y, z)’.

4.6.1

Specific Axioms of AAPOF for Compound Acts

First, let us list axioms that will determine when a situation is a part of another situation, when a situation is the sum (composition) of other situations, and when a situation is an atomic situation. Let us use Wolniewicz’s approach to define the relation of ‘being a part of’: A7. 8 x x ɛ x (Every situation is a part of itself) , A8. 8 x y z ( x ɛ y ˄ y ɛ z ! x ɛ z ) (If a situation is a part of another situation, and this second situation is a part of the third one, then the first one is a part of the third one, as well), A9. 8 x y ( x ɛ y ˄ y ɛ x ! x ¼ y ) (If a situation is a part of another situation, and the latter is a part of the former, then these situations are identical).11 Then, let us add the A10 axiom for atomic situations: A10. 8 x ( AT (x) $ 8 y ( y ɛ x ! y ¼ x ) ) (Every atom is a situation that has no proper parts). Further, let us introduce the sum (composition) of situations: A11. 8 x y z ( x ¼ y + z $ y ɛ x ˄ z ɛ x ˄ 8 w ( AT (w) ! ( w ɛ x ! ( w ɛ y ˅ w ɛ z ) ) ) ) (A situation x is the sum (composition) of situations y and z, when they are parts of it, and each atom of x is a part of y or a part of z).

11

This is not the only possible approach. Stanisław Leśniewski used two axioms for this purpose: 8 x ⅂ x ɛ x, 8 x y z ð x ɛ y ˄ y ɛ z ! x ɛ z Þ:

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Using the concept of a part of situation, it is possible to express an intuition that a part of an act has the same deontic modality as this act: A12. 8 x x1 y y1 ( x1 ɛ x ˄ y1 ɛ y ! ( OBL (x, y) ! ( ACT (x1, y1) ! OBL (x1, y1) ) ) ) (If it is ordered to replace x by y, and x1 is a part of x, and y1 is a part of y, and the sequence < x1, y1 > constitutes an act, then it is ordered to replace x1 by y1) , A13. 8 x x1 y y1 ( x1 ɛ x ˄ y1 ɛ y ! ( PER (x, y) ! ( ACT (x1, y1) ! PER (x1, y1) ) ) ) (If it is permitted to replace x by y, and x1 is a part of x, and y1 is a part of y, and the sequence < x1, y1 > constitutes an act, then it is permitted to replace x1 by y1) , A14. 8 x x1 y y1 ( x1 ɛ x ˄ y1 ɛ y ! ( FOR (x, y) ! ( ACT (x1, y1) ! FOR (x1, y1) ) ) ) (If it is forbidden to replace x by y, and x1 is a part of x, and y1 is a part of y, and the sequence < x1, y1 > constitutes an act, then it is forbidden to replace x1 by y1) . In turn, using the concept of the sum (composition) of situations, it is possible to express an intuition that any act has the same deontic modality as its parts: A15. 8 x x1 x2 y y1 y2 ( x ¼ x1 + x2 ˄ y ¼ y1 + y2 ! ( OBL (x1, y1) ˄ OBL (x2, y2) ! OBL (x, y) ) ) (If it is ordered to replace x1 by y1, and it is ordered to replace x2 by y2, and x is the sum of x1 and x2, and y is the sum of y1 and y2, then it is ordered to replace x by y) , A16. 8 x x1 x2 y y1 y2 ( x ¼ x1 + x2 ˄ y ¼ y1 + y2 ! ( PER (x1, y1) ˄ PER (x2, y2) ! PER (x, y) ) ) (If it is permitted to replace x1 by y1, and it is permitted to replace x2 by y2, and x is the sum of x1 and x2, and y is the sum of y1 and y2, then it is permitted to replace x by y) , A17. 8 x x1 x2 y y1 y2 ( x ¼ x1 + x2 ˄ y ¼ y1 + y2 ! ( FOR (x1, y1) ˄ FOR (x2, y2) ! FOR (x, y) ) ) (If it is forbidden to replace x1 by y1, and it is forbidden to replace x2 by y2, and x is the sum of x1 and x2, and y is the sum of y1 and y2, then it is forbidden to replace x by y) .

4.6.2

Selected AAPOF Theorems Specific to Compound Acts

The following theorems are among the consequences of adding the specific axioms A7–A17: T25. 8 x x1 x2 y y1 y2 ( x ¼ x1 + x2 ˄ y ¼ y1 + y2 ! ( OBL (x, y) ! ( ACT (x1, y1) ˄ ACT (x2, y2) ! ⅂ ( OBL (x1, y1) ˄ FOR (x2, y2) ) ) ) ) (If an act is ordered, it is not that one part of it is ordered and the other is forbidden),

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T26. 8 x x1 x2 y y1 y2 ( x ¼ x1 + x2 ˄ y ¼ y1 + y2 ! ( OBL (x, y) ! ( ACT (x1, y1) ˄ ACT (x2, y2) ! ⅂ ( PER (x1, y1) ˄ FOR (x2, y2) ) ) ) ) (If an act is ordered, it is not that one part of it is permitted and the other is forbidden), T27. 8 x x1 x2 y y1 y2 ( x ¼ x1 + x2 ˄ y ¼ y1 + y2 ! ( OBL (x, y) ! ( ACT (x1, y1) ˄ ACT (x2, y2) ! ⅂ ( FOR (x1, y1) ˅ FOR (x2, y2) ) ) ) ) (If an act is ordered, it is not that any part of it is forbidden), T28. 8 x x1 x2 y y1 y2 ( x ¼ x1 + x2 ˄ y ¼ y1 + y2 ! ( OBL (x1, y1) ˄ OBL (x2, y2) ! PER (x, y) ) ) (If acts are ordered, their composition is permitted), T29. 8 x x1 x2 y y1 y2 ( x ¼ x1 + x2 ˄ y ¼ y1 + y2 ! ( PER (x1, y1) ˄ PER (x2, y2) ! ⅂ FOR (x, y) ) ) (If acts are permitted, their composition is not forbidden), T30. 8 x x1 x2 y y1 y2 ( x ¼ x1 + x2 ˄ y ¼ y1 + y2 ! ( FOR (x1, y1) ˄ FOR (x2, y2) ! ⅂ PER (x, y) ) ) (If acts are forbidden, their composition is not permitted). The abovementioned theorems are useful when one wants to examine legal reasonings a maiori ad minus and a minori ad maius, as well as other similar reasonings. According to the a maiori ad minus rule, if it is permitted to do more, then, all the more, it is permitted to do less. If it is permitted to save two drowning people, say John and Kate, then it is also permitted to save Kate. It is easy to notice that the a maiori ad minus rule understood in this way is expressed by the axiom A13. Interestingly, in the theory of compound acts, the a maiori ad minus rule also applies to prescriptions and prohibitions. For prescriptions, it is expressed by the axiom A12, and for prohibitions, it is expressed by the axiom A14. If it is ordered to save John and Kate, it is also ordered to save Kate. If it is forbidden to save John and Kate, it is also forbidden to save Kate. In the theory of compound acts, the axioms A12 and A14 are no less intuitive than the axiom A13. Meanwhile, in the theory of law, the a maiori ad minus rule is usually presented as applying solely to the permissions. Probably, it results from the assumption that this rule also expresses some other dependencies between acts. According to the a minori ad maius rule, if it is forbidden to do less, all the more, it is forbidden to do more. If it is forbidden to save Kate, it is also forbidden to save John and Kate. If it is forbidden to step on the lawns, it is also forbidden to systematically trample the lawns. In our theory, there is no strict equivalent of such a rule. A somewhat similar dependence is expressed by the axiom A17: if two acts are forbidden then their composition is also forbidden. If it is forbidden to save John and it is forbidden to save Kate, it is also forbidden to save John and Kate.

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Thus, if one wants to have a strict equivalent of the a minori ad maius rule, an additional axiom should be adopted: A14’. 8 x x1 y y1 ( x1 ɛ x ˄ y1 ɛ y ! ( FOR (x1, y1) ! ( ACT (x, y) ! FOR (x, y) ) ) ) (If it is forbidden to replace x1 by y1, and x1 is a part of x, and y1 is a part of y, and the sequence < x, y > constitutes an act, then it is forbidden to replace x by y) . In general terms, the A12–A17 axioms and the T25–T30 theorems express dependencies of the same nature as the a maiori ad minus and a minori ad maius rules. Therefore, they may be useful when the a fortiori reasonings are examined by lawyers. As a conclusion, it shall be noted that the discussed theorems of AAPOF prove that the properties of legal rules which are based on the concepts of a part and the composition of situations may be described in our first order language. Moreover, these theorems have no counterparts in deontic propositional logics, and, most probably, in other deontic logics.

4.7

First Order Deontic Theories as Descriptions of Law Domain Models

In this chapter, an artificial language being a model of the language of law has been discussed. Its dictionary and grammar have been presented. Some of its expressions were interpreted directly by relating them to our model of the law domain. Then, several theories of events and acts were proposed where the meanings of deontic predicates have been defined in several ways. Undoubtedly, none of these theories fully describe our model of the law domain. For example, none of them describe specific properties of multi-acts, controlled causal events, or induced causal events. Also, without a doubt, some theorems of these theories may not be true in our model. For example, from the perspective of AEP, AEPF, AEPOF and AAPOF, irrelevant events do not exist, and this issue has not been settled in Chap. 3. Therefore, none of these theories shall be presumed an adequate theory of our law domain model.12 Nevertheless, these theories prove that first order language is suitable for describing law domain models of the kind constructed in Chap. 3. Thanks to the models of the law domain and the language of law, it will be finally possible to examine the veracity of legal norms and the semantic relations between them. This will be done in Chap. 5.

A theory is called an ‘adequate theory’ of a domain if and only if all true propositions about that domain are theorems of this theory, and all theorems of this theory are true propositions about that domain. 12

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References Malec A (2001) Legal reasoning and logic. In: Language, mind and mathematics. University of Bialystok, Bialystok, pp 97–101 Malec A (2019) Deontic logics as axiomatic extensions of first-order predicate logic: an approach inspired by Wolniewicz’s formal ontology of situations. Axioms 8(4):109 Marciszewski W (ed) (1987) Logika formalna. Zarys encyklopedyczny z zastosowaniami do informatyki i lingwistyki. Państwowe Wydawnictwo Naukowe, Warszawa (in Polish: Formal logic. Encyclopedic outline with applications to computer science and linguistics) Tarski A (1983) On the concept of logical consequence. In: Tarski A (ed) Logic, semantics, metamathematics, 2nd edn. Hackett Publishing, Indianapolis

Chapter 5

Semantics of Norms and Orders

The fifth chapter is devoted to the semantic properties of norms, the semantic relations between norms, as well as the semantic relations between orders. This is where it will be shown that legal norms are sentences in a logical sense, i.e. that every legal norm which has been interpreted, is either true or false. It will be shown that the legal norms do not differ in this respect from mathematical theorems. In addition, the famous Hume’s thesis about the logical separation of deontic sentences from sentences about the physical world, will be questioned. Then, it will be demonstrated that there are classic semantic relations between norms, i.e. the same as between other sentences in a logical sense. First and foremost, it will be demonstrated that a norm may result from another norm. The nature of this resulting will be examined. As an aside, an important issue of the semantic relations between orders will be discussed. It will be shown that there are semantic relations between the orders, even though the orders are neither true nor false. Because of this set of topics, it might be said that the fifth chapter is the main chapter of this book.

5.1

Norms and Normative Expressions

In this chapter, our task is to answer questions about the veracity of legal norms and the possibility of semantic relations between them, such as entailment, contradiction, or opposition. We are also going to answer the question about the possibility of semantic relations between orders, i.e. between normative expressions that are neither true nor false. In order to address the question of whether norms can be true or false, and to consider semantic relations between norms, it is first necessary to define what a norm is. In natural language, some sentences are considered to be norms, and others are not. It should be similar in our model of the language of law. So, let us first look at some sentences we consider to be norms of natural language, and then at some © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Malec, Introduction to the Semantics of Law, Law and Visual Jurisprudence 6, https://doi.org/10.1007/978-3-030-95679-0_5

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sentences which we do not consider to be such norms. Let us try to map the sentences of the both kinds in our model, at the same time. Example 1 The sentence ‘Killing is prohibited’ is considered to be a norm of natural language, and can be mapped as follows: 8x1 ð P1 ðx1 Þ ! FOR ðx1 Þ Þ, where x1 is an event, P1 is read ‘... involves killing a person’. The whole reads as follows: any event that involves killing a person is prohibited. The same sentence can also be mapped as follows: 8x1 8x2 ð P1 ð x1, x2 Þ ! FOR ð x1 , x2 Þ Þ, where x1 and x2 are situations, P1 is read ‘the transition from ... to . . . involves killing a person’. The whole reads as follows: the sequence of any two situations such that the transition from the first to the second involves killing a person, is prohibited. Example 2 The sentence ‘It is forbidden to cause damage’ is considered to be a norm of natural language, and can be mapped as follows: 8x1 ð P1 ðx1 Þ ! FOR ðx1 Þ Þ, where x1 is an event, P1 is read ‘... includes causing damage’. The whole reads as follows: any event that includes causing damage is prohibited. The same sentence can also be mapped as follows: 8x1 8x2 ð P1 ð x1, x2 Þ ! FOR ð x1 , x2 Þ Þ, where x1 and x2 are situations, P1 is read ‘the transition from ... to . . . includes causing damage’. The whole reads as follows: the sequence of any two situations such that the transition from the first to the second includes causing damage, is prohibited. The same sentence can be mapped in the following way, as well: 8x1 8x2 8x3 ð ACT ð x1 , x2 Þ ˄ P1 ð x2 , x3 Þ ˄ LEG ð x2 , x3 Þ ! FOR ð x1 , x2 Þ Þ, where x1 , x2 and x3 are situations and P1 is read ‘the transition from ... to . . . includes causing damage’. We read the whole: for any three situations where the first two constitute an act, and the transition from the second situation to the third situation includes causing damage, if there is a legal causation between the second situation and the third situation, then the act itself is prohibited.

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Example 3 The sentence ‘John Smith is obliged to appear at the Court of Appeal on June 3, 2018’ is considered to be a norm of natural language and can be mapped like this: OBL ð a5 Þ, where ‘a5’ is an individual constant for the event described above, as well as by the sentence: 8x1 OBL ð x1, a7 Þ, where ‘x1’ is an individual variable for situations and ‘a7’ is an individual constant for the outcome situation of the event described above. The first mapping is more convenient where the past is discussed (e.g. when one states that John Smith was obliged to appear at the Court of Appeal on June 3, 2018, but he failed). The second mapping is more convenient where the future is discussed (e.g. when one states that wherever John Smith is staying at the moment, he is obliged to appear at the Court of Appeal on June 3, 2018). Example 4 On the other hand, the sentence ‘If killing is prohibited then killing by shooting is prohibited’ speaks about a relation between norms, but it is not considered to be a norm of natural language itself. It can be mapped, e.g., as follows: 8x1 ð P1 ðx1 Þ ! FOR ðx1 Þ Þ ! 8x2 ð P2 ðx2 Þ ! FOR ðx2 Þ Þ, where x1 and x2 are events, P1 is read ‘... involves killing a person’ and P2 is read ‘... involves killing by shooting a person’. The same sentence can also be mapped as follows: 8x1 8x2 ð P1 ð x1, x2 Þ ! FOR ð x1 , x2 Þ Þ ! 8x3 8x4 ð P2 ð x3, x4 Þ ! FOR ð x3 ,x4 Þ Þ, where x1, x2, x3 and x4 are situations, P1 is read ‘the transition from ... to . . . involves killing a person’ and P2 is read ‘the transition from ... to . . . involves killing by shooting a person’. Example 5 Similarly, the sentence ‘If killing is prohibited then St. Valentine's Day Massacre is prohibited’ speaks about applying a norm, but it is not considered to be a norm of natural language itself. It can be mapped, e.g., as follows: 8x1 ð P1 ðx1 Þ ! FOR ðx1 Þ Þ ! FOR ða1 Þ, where x1 is an event, P1 is read ‘... involves killing a person’ and ‘a1’ is the individual constant for St. Valentine’s Day Massacre.

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These examples show that the norms prescribe, prohibit, or permit events, and that such prescriptions, prohibitions, or permissions, are either unconditional or conditional. If such a prescription, prohibition, or permission, is unconditional then it concerns exactly one, fully determined event (e.g. ‘OBL ( a5 )’ ), or all events with exactly one, fully determined outcome situation (e.g. ‘8 x1 OBL ( x1 , a7 )’ ). On the other hand, if such a prescription, prohibition, or permission, is conditional, its antecedent describes events that are prescribed, prohibited or permitted. Thus, the following schemas of norms can be proposed in our legal language model: (1) unconditional norms for events: OBL ðaÞ, FOR ðaÞ, PER ðaÞ (where ‘a’ is an individual constant for any but determined event), (2) conditional norms for events: 8x ð DESCRIPTION ðxÞ ! OBL ðxÞ Þ, 8x ð DESCRIPTION ðxÞ ! FOR ðxÞ Þ, 8x ð DESCRIPTION ðxÞ ! PER ðxÞ Þ (where ‘x’ is an individual variable for events, and ‘DESCRIPTION (x)’ stands for a non-deontic predicate or a conjunction of non-deontic predicates), (3) unconditional norms for situations: OBL ð am , an Þ, FOR ð am , an Þ, PER ð am , an Þ, 8xm OBL ð xm , an Þ, 8xm FOR ð xm , an Þ, 8xm PER ð xm , an Þ (where ‘am’ and ‘an’ are individual constants for any but determined situation, and ‘xm’ is an individual variable for situations),

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(4) conditional norms for situations: 8xm 8xn . . . ð DESCRIPTION ð xm , xn Þ ! OBL ð xm , xn Þ Þ, 8xm 8xn . . . ð DESCRIPTION ð xm , xn Þ ! FOR ð xm , xn Þ Þ, 8xm 8xn . . . ð DESCRIPTION ð xm , xn Þ ! PER ð xm , xn Þ Þ (where ‘xm’ and ‘xn’ are individual variables for situations, and ‘DESCRIPTION ( xm , xn )’ stands for a non-deontic predicate or a conjunction of non-deontic predicates). The schemas described above are the basic ones, but they are not the only possible ones. Clearly, norms based on other schemas are also possible. For example, it may be considered that the following sentence is a norm: P ðaÞ ! OBL ðaÞ because it prescribes conditionally an event, although it does not fall under any of the abovementioned schemas. On the other hand, any expression of our language that contains OBL, FOR, or PER predicates will be called a ‘normative expression’. Consequently, every norm is a normative expression, but there are normative expressions that are not norms, e.g.: P ðxÞ ! OBL ðaÞ, OBL ðaÞ ! FOR ðxÞ: These expressions are not norms, because they contain a free variable x, and therefore they are not sentences. And solely sentences may be qualified as norms. In turn, any normative expression without free variables will be called a ‘deontic sentence’. Thus, every norm is a deontic sentence, but there are deontic sentences that are not norms, e.g.: ð Pm ðaÞ ! OBL ðaÞ Þ ˄ Pm ðaÞ ! OBL ðaÞ, 8x ð Pm ðxÞ ! OBL ðxÞ Þ ˄ Pm ðaÞ ! OBL ðaÞ: The forementioned deontic sentences are not norms, because they contain ‘!’ as well as ‘OBL’ signs in the antecedent, and this is against our understanding of conditional norms. The following expressions are other examples of deontic sentences that are not norms: Pm ðaÞ ˄ Pm ðbÞ ! OBL ðaÞ ˅ OBL ðbÞ,

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8x8y ð Pm ðxÞ ˄ Pn ðyÞ ! OBL ðxÞ ˅ OBL ðyÞ Þ: The deontic sentences mentioned above are not norms, because they contain ‘˅’ sign in the consequent, and therefore they do not prescribe, prohibit, or permit a ‘determined event’. These terminology findings may be summarized as follows: NORM ⊂ DEONTIC SENTENCE ⊂ NORMATIVE EXPRESSION:

5.2

Veracity and Falsity of Norms

In the first order logic, the following inductive definition of veracity, based on Tarski’s truth conception, is usually accepted: (1) (2) (3) (4) (5) (6) (7) (8)

‘P (t1,t2,. . .,tn)’ is true ⟺ < φ (t1), φ (t2),. . ., φ (tn) > E φ (P) , ‘8 x α’ is true ⟺ for any φ (x) ‘α’ is true, ‘Ǝ x α’ is true ⟺ for some φ (x) ‘α’ is true, ‘⅂ α’ is true ⟺ it is not true that ‘α’ is true, ‘α ˄ β’ is true ⟺ ‘α’ is true and ‘β’ is true, ‘α ˅ β’ is true ⟺ ‘α’ is true or ‘β’ is true, ‘α ! β’ is true ⟺ it is not true that at the same time ‘α’ is true and ‘β’ is not true, ‘α $ β’ is true ⟺ ‘α’ is true if and only if ‘β’ is true.

The first point of this definition determines the truth condition for simple (atomic) sentences, i.e. the sentences that do not contain logical constants. According to the first point, the simple sentence is true if and only if the sequence of objects being this sentence terms’ interpretations belongs to the set being this sentence predicate’s interpretation. For example, if ‘t1’ means John, ‘t2’ means Kate, and the predicate ‘P(. . .,. . .)’ reads ‘... and ... love each other’, then the sentence ‘P(t1, t2)’ is true if and only if the sequence < John, Kate > belongs to the set of couples who love each other. In other words, the abovementioned sentence is true if and only if John and Kate love each other.1 The next points of this definition determine how the truth of a compound sentence depends on the truth of the sentences this compound sentence is made of. For example, the compound sentence ‘⅂ α’ is true if and only if the sentence ‘α’ is false (this is point 4 of the definition). Similarly, the compound sentence ‘α ˄ β’ is

1

The reader might notice that a distinction between the subject language and meta-language is used to define truthfulness in the forementioned way. For example, the expression ‘P(t1, t2)’ belongs to the subject language. On the other hand, the phrase ‘The sentence ‘P(t1, t2)’ is true if and only if John and Kate love each other’ belongs to the meta-language. The reader might notice, as well, that this comment is made in a meta-meta-language.

5.2 Veracity and Falsity of Norms

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true if and only if the sentence ‘α’ is true and the sentence ‘β’ is true (this is point 5 of the definition). Thus, this definition allows us to say about each sentence of our language (which has been interpreted) whether it is true or false. When one applies this definition to unconditional norms, and considers objects constructed in Chap. 3, the following equations for events are obtained: (1) ‘OBL (a)’ is true ⟺ φ (a) E φ (OBL) ⟺ φ (a) E OBL , (2) ‘FOR (a)’ is true ⟺ φ (a) E φ (FOR) ⟺ φ (a) E FOR , (3) ‘PER (a)’ is true ⟺ φ (a) E φ (PER) ⟺ φ (a) E PER . And the same for situations: (4) ‘OBL (a1, a2)’ is true ⟺ < φ (a1), φ (a2) > E { < m, n > : < m, n > E OBL } , (5) ‘FOR (a1, a2)’ is true ⟺ < φ (a1), φ (a2) > E { < m, n > : < m, n > E FOR } , (6) ‘PER (a1, a2)’ is true ⟺ < φ (a1), φ (a2) > E { < m, n > : < m, n > E PER } , and (7) ‘8 x OBL (x, a)’ is true ⟺ for any φ (x) < φ (x), φ (a) > E { < m, n > : < m, n > E OBL } , (8) ‘8 x FOR (x, a)’ is true ⟺ for any φ (x) < φ (x), φ (a) > E { < m, n > : < m, n > E FOR } , (9) ‘8 x PER (x, a)’ is true ⟺ for any φ (x) < φ (x), φ (a) > E { < m, n > : < m, n > E PER } . The Equations (1)–(9) are telling us when unconditional norms are true. Further, when one applies the forementioned veracity definition to conditional norms, the following equations for events are obtained:2 (10) ‘8 x (DESCRIPTION (x) ! OBL (x))’ is true ⟺ for any φ (x) it is not true that at the same time ‘DESCRIPTION (x)’ is true and ‘OBL (x)’ is not true, (11) ‘8 x (DESCRIPTION (x) ! FOR (x))’ is true ⟺ for any φ (x) it is not true that at the same time ‘DESCRIPTION (x)’ is true and ‘FOR (x)’ is not true, (12) ‘8 x (DESCRIPTION (x) ! PER (x))’ is true ⟺ for any φ (x) it is not true that at the same time ‘DESCRIPTION (x)’ is true and ‘PER (x)’ is not true. And the same for situations: (13) ‘8 xm 8 xn . . . ( DESCRIPTION ( xm , xn ) ! OBL ( xm , xn ) )’ is true ⟺ for any φ (xm), φ (xn), . . . it is not true that at the same time ‘DESCRIPTION ( xm , xn )’ is true and ‘OBL ( xm , xn )’ is not true, (14) ‘8 xm 8 xn . . . ( DESCRIPTION ( xm , xn ) ! FOR ( xm , xn ) )’ is true ⟺ for any φ (xm), φ (xn), . . . it is not true that at the same time ‘DESCRIPTION ( xm , xn )’ is true and ‘FOR ( xm , xn )’ is not true,

2

Here, the definitions of truth for ‘8 x α’ and ‘α ! β’ are applied.

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(15) ‘8 xm 8 xn . . . ( DESCRIPTION ( xm , xn ) ! PER ( xm , xn ) )’ is true ⟺ for any φ (xm), φ (xn), . . . it is not true that at the same time ‘DESCRIPTION ( xm , xn )’ is true and ‘PER ( xm , xn )’ is not true. The Equations (10)–(15) are telling us when conditional norms are true. So, in our model of the language of law, the norms are sentences in a logical sense, i.e. every norm is either true or false. If, however, everything about the issue of veracity and falsity of norms is so clear and obvious, why do some researchers claim that norms are neither true nor false?

5.3

True Norms, False Norms, and Popper’s Third World

The answer to this question is quite simple: norms are not related to facts. And, according to common sense, veracity shall be based on compliance with facts. Let us examine this issue a bit. Tarski’s truth conception is based on the classic, Aristotelian, veracity definition. This classic definition says that a sentence is true if and only if the reality is as this sentence says. Having this definition in mind, let us apply the veracity definition provided in the previous subchapter to some elementary sentences constructed of the predicates linked to the objects defined in Chap. 3: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22)

‘SE (a)’ is true ⟺ φ (a) E SE , ‘SE” (a)’ is true ⟺ φ (a) E SE” , ‘o (a)’ is true ⟺ φ (a) E { o } , ‘λ (a)’ is true ⟺ φ (a) E { λ } , ‘ɛ (a1, a2)’ is true ⟺ < φ (a1) , φ (a2) > E { < m, n > : m  n } , ‘γ (a1, a2, a3)’ is true ⟺ < φ (a1), φ (a2), φ (a3) > E { < k, m, n > : k ¼ m ; n } , ‘SP (a)’ is true ⟺ φ (a) E SP , ‘SA (a)’ is true ⟺ φ (a) E SA , ‘SF (a)’ is true ⟺ φ (a) E SF , ‘SI (a1, a2)’ is true ⟺ < φ (a1) , φ (a2) > E SI , ‘SI (a1, a2)’ is true ⟺ < φ (a1) , φ (a2) > E SI , ‘EVENT (a)’ is true ⟺ φ (a) E EVENT , ‘FEV (a)’ is true ⟺ φ (a) E FEV , ‘NAT (a)’ is true ⟺ φ (a) E NAT , ‘R (a1, a2)’ is true ⟺ < φ (a1) , φ (a2) > E R , ‘R+ (a1, a2)’ is true ⟺ < φ (a1) , φ (a2) > E R+ , ‘SER (a1, a2)’ is true ⟺ < φ (a1) , φ (a2) > E SER , ‘SIR (a1, a2)’ is true ⟺ < φ (a1) , φ (a2) > E SIR , ‘SER+ (a1, a2)’ is true ⟺ < φ (a1) , φ (a2) > E SER+ , ‘SIR+ (a1, a2)’ is true ⟺ < φ (a1) , φ (a2) > E SIR+ , ‘LEV (a)’ is true ⟺ φ (a) E LEV , ‘ACT (a)’ is true ⟺ φ (a) E ACT ,

5.3 True Norms, False Norms, and Popper’s Third World

(23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36) (37)

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‘ACT (a1, a2)’ is true ⟺ < φ (a1), φ (a2) > E ACT , ‘FACT (a)’ is true ⟺ φ (a) E FACT , ‘MUL (a)’ is true ⟺ φ (a) E MUL , ‘FMUL (a)’ is true ⟺ φ (a) E FMUL , ‘DET (a)’ is true ⟺ φ (a) E DET , ‘CCE (a)’ is true ⟺ φ (a) E CCE , ‘FCCE (a)’ is true ⟺ φ (a) E FCCE , ‘LEG (a)’ is true ⟺ φ (a) E LEG , ‘ICE (a)’ is true ⟺ φ (a) E ICE , ‘FICE (a)’ is true ⟺ φ (a) E FICE , ‘REL (a)’ is true ⟺ φ (a) E REL , ‘IRR (a)’ is true ⟺ φ (a) E IRR , ‘OBL (a)’ is true ⟺ φ (a) E OBL , ‘FOR (a)’ is true ⟺ φ (a) E FOR , ‘PER (a)’ is true ⟺ φ (a) E PER .

And the same for (33)–(37) in binary versions: (38) (39) (40) (41) (42)

‘REL (a1, a2)’ is true ⟺ < φ (a1), φ (a2) > E { < m, n > : < m, n > E REL } , ‘IRR (a1, a2)’ is true ⟺ < φ (a1), φ (a2) > E { < m, n > : < m, n > E IRR } , ‘OBL (a1, a2)’ is true ⟺ < φ (a1), φ (a2) > E { < m, n > : < m, n > E OBL } , ‘FOR (a1, a2)’ is true ⟺ < φ (a1), φ (a2 ) > E { < m, n > : < m, n > E FOR } , ‘PER (a1, a2)’ is true ⟺ < φ (a1), φ (a2) > E { < m, n > : < m, n > E PER } .

When one compares the forementioned equations with the classic, Aristotelian, veracity definition, they note that the reality, understood as the totality of facts, is the criterion of truth only in a few of these equations. Namely, the totality of facts is the criterion of truth only for elementary sentences constructed of the predicates: SF, FEV, FACT, FMUL, FCCE and FICE.3 For the remaining elementary sentences, including norms, the reality, understood as the totality of facts, is not the criterion of truth. So, do these equations violate the concept of truth which requires that a true sentence shall correspond to the reality? Of course not. All predicates refer to sets, and therefore to abstract objects. Thus, no predicate relates directly to the reality, understood as the totality of facts. They all relate to objects from the third world of Popper. Hence, a compliance with the reality is the criterion of truth only in the case of sentences concerning situations and events, that state that these situations and events are real (are facts). Example 1 Let us consider the sentence ‘Kate kissed John’. According to the veracity definition provided in the previous subchapter, this sentence is true if and only if Kate kissed John, i.e. when the fact is that Kate kissed John:

3

The SF, FEV, FACT, FMUL, FCCE and FICE predicates denote sets of facts. Other of the abovementioned predicates denote sets of objects other than facts.

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FACT ðam Þ’ is true ⟺ φ ðam Þ E FACT:

Here, the set FACT is a set of facts, so one needs a contact with the reality to determine whether the sentence ‘FACT (am)’ is true or false. Let us now consider the sentence ‘It is ordered that Kate kissed John’. According to the veracity definition provided in the previous subchapter, this sentence is true if and only if the event in question belongs to the set of ordered events: ‘

OBL ðam Þ’ is true ⟺ φ ðam Þ E OBL:

Here, the set OBL is not a set of facts, so one does not need a contact with the reality (understood as the totality of facts) to determine whether the sentence ‘OBL (am)’ is true or false. In both cases, the veracity criterion is that an object belongs to a set, i.e. it is a relation in the third world of Popper. The difference between these cases is as follows: in the first case, this set was defined in some reference to the reality, understood as the totality of facts, and in the second—not. Many sentences share the fate of norms when it comes to the criterion of their veracity. Example 2 Let us consider the sentence ‘Kate had to kiss John’. According to the veracity definition provided in the previous subchapter, this sentence is true if and only if the event in question belongs to the set of necessary events: ‘

DET ðam Þ’ is true ⟺ φ ðam Þ E DET:

Also in this case, the veracity criterion is that an object belongs to a set, and this set is defined without reference to the reality, understood as the totality of facts. Let us now consider the common sentence ‘Every attorney at law is a lawyer’. According to the veracity definition provided in the previous subchapter, this sentence is true if and only if the set of attorneys is a subset of the set of lawyers: ‘Every attorney at law is a lawyer’ is true ⟺ for any man x it is not true that at the same time ‘x is an attorney at law’ is true and ‘x is a lawyer’ is not true ⟺ φðattorney at lawÞ ⊂ φðlawyerÞ: It turns out that this common sentence tells us nothing about Popper’s first world, i.e. about the reality, understood as the totality of facts. This sentence just states a relation between two sets. So, it states something about Popper’s third world, and compliance with this third world makes this sentence true.

5.4 Veracity of Norms And the Is-Ought Problem

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Thus, if Aristotle-Tarski’s conception of truth is combined with Wittgenstein’s conception of fact, then it turns out that the criterion of truth is not the compliance with the reality of Popper’s first world (the totality of facts). This criterion is the compliance with the ‘reality’ of Popper’s third world (the world of cultural creations). When this dependence is learnt, all doubts as to whether norms can be true or false disappear. While norms are incapable of being in compliance with the facts, they can be in compliance with the creations of Popper’s third world.

5.4

Veracity of Norms And the Is-Ought Problem

The issue of veracity (falsity) of norms shall not be confused with the is-ought problem. According to this famous Hume’s4 thesis, moral distinctions cannot be derived from reason: Reason is the discovery of truth or falsehood. Truth or falsehood consists in an agreement or disagreement either to the real relations of ideas, or to real existence and matter of fact. Whatever, therefore, is not susceptible of this agreement or disagreement, is incapable of being true or false, and can never be an object of our reason.5 Laudable or blameable, therefore, are not the same with reasonable or unreasonable. The merit and demerit of actions frequently contradict, and sometimes control our natural propensities. But reason has no such influence. Moral distinctions, therefore, are not the offspring of reason.6

Thus, according to Hume, moral statements are neither true nor false. As a consequence, it is said that deontic statements, including norms, are neither true nor false. However, in contemporary philosophy, the is-ought problem is restated in a narrower way: Deontic statements are logically separated from non-deontic statements, i.e. neither deontic statements can be derived from non-deontic statements (simple Hume’s thesis), nor non-deontic statements can be derived from deontic statements (reverse Hume’s thesis).7

If this thesis holds, norms are logically separated from statements about facts. Thus, the is-ought problem may be understood in two ways. First, it may be understood in accordance with the contemporary approach: as a logical separation of the deontic sentences from the sentences about facts. Second, it may be understood

4

David Hume (1711–1776), a Scottish empiricist and moral philosopher. He drew our attention to the subjectivity of experience, the structural nature of causality, and the logical separation of sentences about the world from sentences about obligations. 5 Hume (1985), p. 510. 6 Ibidem. 7 Woleński (2003), pp. 293–303.

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in accordance with Hume’s original statement: it is not only about logical separation, but also about the lack of logical values in normative reasonings. Of course, these two understandings are different. While the first understanding of the is-ought problem does not exclude that norms can be either true or false, the second one does exclude this possibility. Thus, the conclusions obtained in the previous subchapter are compatible with the first understanding of this thesis, but are incompatible with its second understanding. To illustrate this compatibility with the contemporary understanding of the is-ought problem, let us return to the ‘Kate kissed John’ example: ‘

FACT ðam Þ’ is true ⟺ φ ðam Þ E FACT, ‘

OBL ðam Þ’ is true ⟺ φ ðam Þ E OBL:

Clearly, truth of the sentence ‘Kate kissed John’ depends on the set FACT and truth of the sentence ‘It is ordered that Kate kissed John’ depends on the set OBL, and these two sets are different. Therefore, it may be the case that the sentences in question are logically separated. Nevertheless, the sentence ‘It is ordered that Kate kissed John’ has the same right to be true or false as the sentence ‘Kate kissed John’, when Popper’s third world is considered. Moreover, the research of the previous subchapter seems to call into question the truth of Hume’s thesis in its original understanding. Namely, this research shows that from the point of view of semantics there is no difference between the sentences ‘It is ordered that Kate kissed John’ and ‘Kate had to kiss John’ or ‘Every attorney at law is a lawyer’. Indeed, the veracity of any of these sentences depends solely on the set relations described above, where the reality, understood as the totality of facts, is not directly involved. Even so, nobody doubts that each of the sentences ‘Kate had to kiss John’ or ‘Every attorney at law is a lawyer’ is either true or false. However, if these two sentences are either true of false, the same should hold for the sentence ‘It is ordered that Kate kissed John’. Moreover, if the veracity of a sentence depends on a set of events, and the veracity of another sentence depends on another set of events, it does not mean that they must be logically separated. In this respect it is sufficient to notice that truth of the sentence ‘Kate had to kiss John’ also depends on a set of events which may differ from FACT: ‘

DET ðam Þ’ is true ⟺ φ ðam Þ E DET:

Nonetheless, nobody claims that modal (alethical) statements are logically separated from statements about facts. On the contrary, it is assumed in modal logic that: □p ! p (if it is necessary that p then p).

5.5 Entailment Between Norms

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It is worth noting that the truth of this theorem is probably a consequence of some relations between the DET and FACT sets. It cannot be excluded that the OBL, FOR, and PER sets are also related to the FACT set in a similar way, although the nature of this relation has not yet been explored. Thus, while the conclusions of the previous subchapter do not contradict Hume’s thesis in its modern understanding, they call it into question. All the more, they call into question Hume’s thesis in its original understanding.

5.5

Entailment Between Norms

In logic it is assumed that from a sentence α results a sentence β if and only if the truth of the sentence α entails the truth of the sentence β, i.e. when it is impossible that the sentence α is true and at the same time the sentence β is false: from α results β ⟺ if α is true, then β is true: Entailment is based on expressions’ meanings. For example, from ‘Everyone is mortal’ results ‘Socrates is mortal’ since according to the meaning of the word ‘everyone’, it cannot be that ‘Everyone is mortal’ is true and ‘Socrates is mortal’ is false. Respectively, from ‘Mr. Smith is an attorney at law’ results ‘Mr. Smith is a lawyer’ since due to the meanings of the expressions ‘lawyer’ and ‘attorney at law’, it cannot be that the sentence ‘Mr. Smith is an attorney at law’ is true and the sentence ‘Mr. Smith is a lawyer’ is false (namely: because every attorney at law is a lawyer). Since norms are sentences in a logical sense, some of them should result from others. Indeed, there are examples when deontic sentences result from other deontic sentences, and in particular when norms result from other norms. In our model, entailment between norms or deontic sentences is based on the meanings of: (1) logical constants (logical conjunctions and quantifiers), or (2) predicates linked to the objects defined in Chap. 3. In the first order logic, the following meta-theorem is valid: ╞ ðα ! βÞ if and only if α╞ β, i.e. the sentence ‘α ! β’ is a tautology if and only if the sentence α entails the sentence β.8 Thus, entailment occurs between the predecessor and successor of any tautology, including tautologies being norms or deontic statements.

Following the conventions accepted in logic, the sign ‘╞’ when used in the expression ‘╞ (α ! β)’ indicates that the sentence ‘(α ! β)’ is a tautology, that is, it is always true. The same sign ‘╞’ when used in the expression ‘α╞ β’ indicates that the sentence β follows from the sentence α. In the latter case, the sign ‘╞’ is therefore a symbol of entailment.

8

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Hence, the following sentences are examples of entailment based on logical constants’ meanings: OBL ðaÞ ╞ ⅂⅂OBL ðaÞ, OBL ðaÞ ! OBL ðbÞ ╞ ⅂OBL ðbÞ ! ⅂OBL ðaÞ, ð OBL ðaÞ ! OBL ðbÞ Þ ˄ OBL ðaÞ ╞ OBL ðbÞ, ð OBL ðaÞ ˅ OBL ðbÞ Þ ˄ ⅂OBL ðaÞ ╞ OBL ðbÞ, 8x ð P ðxÞ ! OBL ðxÞ Þ ╞ ð P ðaÞ ! OBL ðaÞ Þ, 8x ð P ðxÞ ! OBL ðxÞ Þ ˄ P ðaÞ ╞ OBL ðaÞ, ⅂8x ð P ðxÞ ! OBL ðxÞ Þ ╞ Ǝ x ð P ðxÞ ˄ ⅂OBL ðxÞ Þ: The first of the forementioned expressions shall be read in the following way: ‘The sentence OBL (a) entails the sentence ⅂ ⅂ OBL (a)’. This entailment between the sentences OBL (a) and ⅂ ⅂ OBL (a) takes place because the sentence ‘OBL (a) ! ⅂ ⅂ OBL (a)’ is a tautology of our language. The same applies for the other forementioned examples. Similarly, the following sentences are examples of entailment based on the OBL, FOR, PER predicates’ meanings: OBL ðaÞ╞ ⅂FOR ðaÞ, FOR ðaÞ╞ ⅂OBL ðaÞ, OBL ðaÞ╞ PER ðaÞ, ⅂PER ðaÞ╞ ⅂OBL ðaÞ: Similarly, the following sentences are examples of entailment based on the OBL, FOR, PER, ɛ and γ predicates’ meanings: OBL ð a, b Þ ˄ b 6¼ c╞ ⅂OBL ð a, c Þ, OBL ð a, b Þ ˄ ɛ ð a1 , a Þ ˄ ɛ ð b1 , b Þ ˄ ACT ð a1 , b1 Þ╞ OBL ð a1 , b1 Þ, OBL ða1 , b1 Þ ˄ OBL ða2 , b2 Þ ˄ γ ða, a1 , a2 Þ ˄ γ ðb, b1 , b2 Þ ˄ ACT ða, bÞ╞ OBL ða, bÞ: And so on. From these examples, several are examples of entailment between norms and other deontic sentences, and one is an example of entailment strictly between norms:

5.6 Contradiction, Opposition and Sub-Opposition Between Norms

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OBL ðaÞ╞ PER ðaÞ: It also is worth noting that the schema: 8x ð P ðxÞ ! OBL ðxÞ Þ ˄ P ðaÞ╞ OBL ðaÞ is commonly called ‘legal syllogism’.9 Thus, ‘being true’ or ‘being false’ are not the only semantic terms applicable to legal norms. There are entailment relations between legal norms, as well. In this respect, the legal norms do not differ from other sentences (in a logical sense), including other deontic statements. Moreover, the semantic grounds of such entailment may be easily pointed out, as it was done in this subchapter. Thus, when lawyers claim that a norm entails other norm, they are supported by semantics. At the same time, no separate concept of legal entailment is needed since the entailment relations between legal norms are the classic ones.

5.6

Contradiction, Opposition and Sub-Opposition Between Norms

It is easy to observe that norms and other deontic statements may contradict each other, oppose each other, or be in the relation of sub-opposition, exactly as all other sentences in a logical sense. The sentences α and β are contradictory if and only if they cannot be true at the same time and they cannot be false at the same time. Thus, e.g. (1) norms: OBL (a) and FOR (a) are contradictory in a domain of hard rigorism, (2) deontic sentences: OBL (a) and ⅂ OBL (a) are contradictory, (3) deontic sentences: 8 x (P (x) ! OBL (x)) and ⅂ 8 x (P (x) ! OBL (x)) also are contradictory. Let us check, for example, that the first of these statements holds. It can be found out that the OBL (a) and FOR (a) norms are contradictory in a domain of hard rigorism, e.g. by referring to the relevant Venn diagram from Chap. 3:

ORDERED EVENTS

9

FORBIDDEN EVENTS

The legal syllogism will be discussed in detail, in Chap. 6.

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where it can be seen that the set of ordered events and the set of forbidden events are separate (no event belongs to both sets) and complementary (each event belongs either to one of these sets or to the other). Thus, the norms OBL (a) and FOR (a) cannot be true at the same time—because the event a cannot belong to both sets at the same time. Further, these norms cannot be false at the same time—because the event a cannot not belong to both sets at the same time. Since the norms OBL (a) and FOR (a) cannot be true at the same time, and they cannot be false at the same time, they are contradictory. The sentences α and β are opposite if and only if they cannot be true at the same time. Therefore, e.g. (1) norms: OBL (a, b) and OBL (a, c) are opposite, provided that b is not identical to c, (2) deontic sentences: 8 x (P (x) ! OBL (x)) and ⅂ Ǝ x (P (x) ˄ OBL (x)) also are opposite. Also in this case, let us check that the first of these statements holds. It can be proven that the norms OBL (a, b) and OBL (a, c) are opposite, e.g. by referring to the appropriate AAPOF theorem: T16.

8 x y z ( y 6¼ z ! ⅂ (OBL (x, y) ˄ OBL (x, z) ) ) .

This theorem states that for any x, y, z the norms OBL (x, y) and OBL (x, z) cannot be true at the same time provided that y 6¼ z (that is, if these norms are not identical). As a direct consequence, the norms OBL (a, b) and OBL (a, c) cannot be true at the same time provided that b 6¼ c. Thus, the norms OBL (a, b) and OBL (a, c) are opposite, provided that b is not identical to c. The sentences α and β are sub-opposite if and only if they cannot be false at the same time. Thus, e.g. (1) norms: PER (a, b) and PER (a, c) are sub-opposite, provided that b and c are the only alternatives in the choice situation a, (2) deontic sentences: Ǝ x (P (x) ˄ OBL (x)) and ⅂ 8 x (P (x) ! OBL (x)) also are sub-opposite. And again, let us check that the first of these statements holds. It can be proven that the norms PER (a, b) and PER (a, c) are sub-opposite, e.g. by referring to the relevant theorem of the AAPOF theory: T22.

8 x y z ( ACT (x, y) ˄ ACT (x, z) ˄ y 6¼ z ˄ 8 w ( ACT (x, w) ! ( w ¼ y ˅ w ¼ z ) ) ! ( PER (x, y) ˅ PER (x, z) ) ) .

This theorem states that for any x, y, z the norms PER (x, y) and PER (x, z) cannot be false at the same time provided that y and z are the only alternatives in a choice situation x, and that y 6¼ z. As a direct consequence, the norms PER (a, b) and PER (a, c) cannot be false at the same time provided that b and c are the only alternatives in a choice situation a, and that b 6¼ c. Thus, the norms PER (a, b) and

5.7 Semantic Relations Between Orders

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PER (a, c) are sub-opposite, provided that b and c are the only alternatives in the choice situation a. Some of the abovementioned relations can be presented in the form of a diagram:

There are presented: contradictory deontic sentences—on the diagonals of the diagram, opposite ones—on the upper edge, sub-opposite ones—on the lower edge, and entailed ones—along the side edges. Thus, the forementioned semantic relations are applicable to norms and deontic sentences, exactly as to other sentences. Therefore, when lawyers claim that a norm contradicts or opposes another norm, they are supported by semantics. At the same time, no separate concepts of legal contradiction, opposition, or sub-opposition are needed. The classic ones will do. This way, the main goal of this book has been achieved: it has been demonstrated that norms are sentences in a logical sense. They can be true or false, they can entail other norms, they can contradict, oppose, or sub-oppose other norms. Yet, before the reader moves on to Chap. 6 devoted to norms transformations, let us take a short look at the semantics of orders, i.e. expressions which - unlike norms—are not sentences in a logical sense.

5.7

Semantic Relations Between Orders

Orders are not sentences in a logical sense: they are not statements about a domain, but they are demands to change it. The expressions ‘John, drink this juice!’, and ‘Kate, kiss John!’ are examples of orders. Because orders are not sentences in a logical sense, they are neither true nor false. Therefore, norms differ significantly from orders since norms are sentences in a logical sense. They are either true or false. However, Kate, who kisses John, complies at the same time with the order saying ‘Kate, kiss John!’ and with the order saying ‘Kate, kiss a boy!’. One act may comply with several orders. Therefore, there is a semantic relation between the order ‘Kate, kiss a boy!’ and the order ‘Kate, kiss John!’. This is certainly a specific relation, because it is not based on the succession of truth (i.e. it is not based on entailment). Nevertheless, it is based on the meanings of the expressions used in both orders, i.e. it is a semantic relation. Let us examine this issue in more detail. When John, Boris, Donald, and Kate are the only persons in a room, Kate fulfills the order ‘Kate, kiss a boy!’ when she performs one of the acts:

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, , , , , , , where < - j , j > means kissing John, < - b , b > means kissing Boris, and < - d , d > means kissing Donald. The expression ‘< - j ; - b ; - d , j ; - b ; - d >’ shall be read in accordance with Chap. 3: a choice situation where John, Boris, and Donald are un-kissed, is changed for a chosen situation where John is kissed, but Boris and Donald remain un-kissed. Similarly, the other of the forementioned formulas shall be read. In the same circumstances, Kate fulfills the order ‘Kate, kiss John!’ when she performs one of the following acts: , , , : From this discussion follows that the set of events where ‘Kate, kiss John!’ is fulfilled, is the subset of the set of events where ‘Kate, kiss a boy!’ is fulfilled. It may be a good basis for defining here a relation between orders similar to entailment between sentences in a logical sense. As it was said before, in logic it is assumed that from a sentence α results a sentence β if and only if the truth of the sentence α entails the truth of the sentence β, i.e. when it is impossible that the sentence α is true and at the same time the sentence β is false: from α results β ⟺ if α is true, then β is true: Wolniewicz has expressed the abovementioned relation in the following way: from α results β ⟺ V ðαÞ ⊂ V ðβÞ

5.7 Semantic Relations Between Orders

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(from α results β if and only if the set of α verifiers is included in the set of β verifiers).10 Let us apply a similar approach to orders, just changing ‘verifiers’ for ‘fulfillments’. It may be said that from ‘Kate, kiss John!’ results ‘Kate, kiss a boy!’ in the sense that the set of ‘Kate, kiss John!’ fulfillments is included in the set of ‘Kate, kiss a boy!’ fulfillments. Indeed, in a given situation it is impossible that the order ‘Kate, kiss John!’ is fulfilled, and the order ‘Kate, kiss a boy!’ is not. Let now α mean ‘Kate, kiss John!’, and β mean ‘Kate, kiss a boy!’. Our findings may be written down in the following way: from α results β ⟺ fx E ACT : x is a fulfillment of αg ⊂ fy E ACT : y is a fulfillment of βg: And this is our definition for a specific entailment between orders. Moreover, the converse of this specific entailment turns out to be equally interesting. Kate fulfills the order ‘Kate, kiss a boy!’ when she kisses John. But she does not fulfill the order ‘Kate, kiss John!’ when she kisses Boris or Donald. It may be said that the order ‘Kate, kiss a boy!’ justifies kissing John. And the order ‘Kate, kiss John!’ does not justify kissing a boy other that John. Therefore, it may be concluded that the order ‘Kate, kiss John!’ is ‘justified’, in some sense, by the order ‘Kate, kiss a boy!’. Namely: the former is a kind of specification (concretization) of the latter. And in a general way: β justifies α ⟺ from α results β, or: β justifies α ⟺ fx E ACT : x is the fulfillment of αg ⊂ fy E ACT : y is the fulfillment of βg: Similarly, just by changing ‘verifiers’ for ‘fulfillments’, it is possible to introduce the contradiction, opposition, and sub-opposition relations for orders. The sentences α and β are contradictory if and only if they cannot be true at the same time, and they cannot be false at the same time. Respectively, the orders α and β are contradictory if and only if they cannot be fulfilled at the same time, and they

10

For all α, β E L and for all x, y E SE: from α results β ⟺ V ðαÞ ⊂ V ðβÞ,

that is, α results from β if and only if the set of α verifiers is included in the set of β verifiers. So, any situation that makes α true also makes β true. See: Wolniewicz (1985), p. 59.

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cannot be unfulfilled at the same time. So, ‘Kate, kiss John!’ is contradictory to ‘Kate, don’t kiss John!’. Further, the sentences α and β are opposite if and only if they cannot be true at the same time. Respectively, the orders α and β are opposite if and only if they cannot be fulfilled at the same time. Therefore, ‘Kate, kiss John and nobody else!’ is opposite to ‘Kate, kiss Boris and nobody else!’. Further, the sentences α and β are sub-opposite if and only if they cannot be false at the same time. Respectively, the orders α and β are sub-opposite if and only if they cannot be unfulfilled at the same time. So, ‘Kate, kiss one of the boys!’ and ‘Kate, don’t kiss any of the boys!’ are an example of sub-opposite orders. When the semantics of norms and the semantics of orders are compared, it may be noticed that both refer to sets of events, i.e. to legal rules. It is the legal rules that determine the semantic relations between norms as well as the semantic relations between orders. Thus, the norms and the orders have the common semantic roots, even though the former are sentences in a logical sense, and the latter are not. While the norm ‘It is forbidden to step on the lawns’ describes a legal rule, the order ‘Do not step on the lawns!’ prescribes this rule.

References Hume D (1985) A Treatise of human nature. Penguin Classics, p 510 Woleński J (2003) Uogólniona teza Hume’a (in Polish: A Generalized Hume’s Thesis). In: Bogucka I, Tobor Z (eds) Prawo a wartości. Księga jubileuszowa Profesora Józefa Nowackiego. Zakamycze Kraków, pp 293–303 Wolniewicz B (1985) Ontologia sytuacji. Państwowe Wydawnictwo Naukowe, Warszawa (in Polish: Ontology of situations)

Chapter 6

Transformations of Norms

In this chapter, the semantic limitations of transforming norms in the algorithmic way will be examined. This is a small contribution to a more general issue of computer-assisted application of the law. First, an outline of the stages of applying the law in the continental model will be provided. This will allow us to distinguish two stages at which norms are transformed. These stages are the subsumption and the application of legal syllogisms. Next, it will be explained why the concepts of conditional norm and unconditional norm, introduced in Chap. 5, shall be useful for describing the abovementioned stages. For this purpose, two more concepts will be introduced, as well, namely the concepts of norm concretizing and norm specifying. When the abovementioned two stages are described, it will be examined whether they are creative or algorithmic in nature. Finally, the continental model of applying the law will be compared with the Anglo-Saxon one to find that the latter is based on logically simpler mechanisms.

6.1

The Continental Model of Applying the Law

The two best known systems of applying the law are the continental system and the Anglo-Saxon (precedent) one. The first assumes a passive role of the courts in respect to the law, while the second recognizes that the courts are creative in this respect. To put it simply, in the continental model, the court resembles a classic computer into which a program (legal texts) and data to be processed (facts of the case) are introduced in order to obtain a result predetermined by this program and the introduced data. On the other hand, the Anglo-Saxon court resembles a self-learning computer, i.e. one based on artificial intelligence, where the rules for data processing may be created by the computer itself, so the calculation result is not predetermined by introduced programs and data. This picture of the application of law in the continental model reflects the traditional concept of the tripartite separation of power, according to which the government rules, the parliament makes the law, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Malec, Introduction to the Semantics of Law, Law and Visual Jurisprudence 6, https://doi.org/10.1007/978-3-030-95679-0_6

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and the courts apply it. The court should be passive with respect to the law, as law-making belongs to the parliament, not to the court. In this context, the abovementioned comparison of a court to a classic computer that only calculates and adds nothing from itself, is very accurate. It is worth noting that Max Weber himself already preached the idea of a judging machine that would settle legal disputes mathematically. This idea of the continental courts passiveness is not true. Every lawyer knows that the continental courts make the law. In this chapter, however, it will be examined whether this idea has a semantic basis. Namely, it will be examined whether the continental model allows in principle, in terms of semantics, the application of the law in a passive way, that is, in a way that avoids the law creation. Our study will be limited to norms transformations in the continental model, where a ‘norm transformation’ means a situation in which one norm is replaced by or gives rise to another norm. It will be examined whether such transformations are algorithmic, and therefore whether they could be performed by a classical computer. This limitation of the scope of the study is strictly technical: we can say more about the transformations of norms than about the other stages of applying the law in the continental model. However, first, these stages will be outlined in order to select those at which norms transformations take place. When the continental model is considered, it makes sense to distinguish the following stages of applying the law: (1) (2) (3) (4) (5)

finding that a case requires applying the law, outlining a legal norm to be applied, finding facts of the case, subsuming the facts under the norm in question, applying a legal syllogism.

The abovementioned distinction is, of course, a matter of convention, but it corresponds to logical and methodological properties of these stages. The first stage is to decide whether the law should be applied to a situation in question. Sometimes, it is obvious, like when someone crosses the street at a red light. Sometimes, however, it is not, as when someone pitches a tent on the beach— perhaps the law is not concerned with pitching tents on the beach, but the reverse is just as likely. Of course, obviousness is subjective. For many sunbathers, the need of applying the law in the latter situation may be as obvious as in the former. On the other hand, for our ancestor, who moved to the present time from the Bronze Age, the former situation may be as unobvious as the latter. Thus, it can be said that deciding whether the law should be applied to a situation in question depends on a person’s legal awareness. This process is not a formal one, and is more of interest of psychologists and physiologists than of logicians and legal methodologists. Thus, it will not be the subject of our further consideration. The second stage is outlining a legal norm. It means transforming the inscriptions constituting legal texts in order to obtain an inscription of a special form, namely, the form of general and abstract norms. According to a traditional approach, this form could be presented in the following way:

6.1 The Continental Model of Applying the Law

115

8x ðPðxÞ ! Q ðxÞÞ: This expression states that any subject who finds themselves in a situation of the kind P is obliged to behave in the manner specified by Q. The predecessor of this implication (i.e. ‘P (x)’) is called the ‘hypothesis’ of this legal norm, while its successor (i.e. ‘Q (x)’) is called the ‘disposition’ of this legal norm. However, this form of legal norms is not universal enough: only selected norms fall under the above-presented schema. In particular, if the deontic modality of a norm depends on relations between objects, the norm does not fall under the schema. Let us consider such a norm, e.g. ‘If an animal left unattended caused damage, the owner of this animal is obliged to compensate for the damage’. This norm shall be formalized in the following way: 8x8y8z ðAðxÞ ˄ DðyÞ ˄ PðzÞ ˄ Cðx, yÞ ˄ Oðz, xÞ ! Rðz, yÞÞ, where x, y, z are any objects (concrete or abstract ones), and the predicates shall be read as follows: ‘A(. . .)’ – ‘. . . is an animal’, ‘D(. . .)’ – ‘. . . is a damage’, ‘P(. . .)’ – ‘. . . is a person’, ‘C(. . .,. . .)’ – ‘. . . is the cause of ...’, ‘O(. . .,. . .)’ – ‘. . . is the owner of . . .’, ‘R(. . .,. . .)’ – ‘. . . shall repay for . . .’. This example shows that even quite simple norms need more complex schemas to be expressed. One may only imagine the complexity of schemas which would be needed to write down the norms of the booming branches of administrative law in this way. However, the situational approach advised in this book leads to alternative schemas of general and abstract norms. Namely, in Chap. 5, three basic schemas of conditional norms were proposed:1 8x ð Pm ðxÞ ˄ . . . ˄ Pn ðxÞ ! OBL ðxÞ Þ, 8x ð Pm ðxÞ ˄ . . . ˄ Pn ðxÞ ! FOR ðxÞ Þ, 8x ð Pm ðxÞ ˄ . . . ˄ Pn ðxÞ ! PER ðxÞ Þ: In these schemas, ‘x’ is an individual variable for events (i.e. ‘x’ refers to an event), the predicates ‘Pm’, . . ., ‘Pn’ indicate properties of this event, and the predicates ‘OBL’, ‘FOR’, and ‘PER’ state that this event is, respectively, ordered, prohibited, or permitted. Therefore, the hypothesis of a legal norm is a characteristic of an event, and the disposition of this legal norm determines a deontic modality of that event (‘ordered’, ‘forbidden’, ‘permitted’). Thus, this approach differs from the traditional one. If this approach is accepted, outlining a legal norm means transforming inscriptions of legal texts into statements about the properties of some ordered, forbidden,

1

These formulas are about events. In Chap. 5, similar formulas for situations were also presented. However, for the sake of simplicity, these two kinds of formulas (one for events and the other for situations) will not be considered in this chapter, and the examples will be limited to events.

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or permitted events. On the one hand, this process may be perceived as a kind of translation. Namely, legal texts are translated into the language of norms. Nowadays, the computers can translate texts from one language to another. Therefore, it seems that automatic translation of legal texts into the language of norms should not be a problem. On the other hand, such a translation should discard expressions of legal texts that are not needed to characterize the events in question. Thus, such a translation should divide all terms of legal texts into those that are directly needed for applying the law and the others. This division may be called an ‘empirical reduction’, because the language of legal texts is reduced to the language of describing events, and these events are sequences of situations, i.e. empirical objects. Certainly, teaching the computers to make such an empirical reduction is not trivial. Although the process of transforming legal texts into legal norms, is undoubtedly important for assessing the possibility of algorithmic application of the law, it will not be discussed in more detail in this book, as we do not have more to say about it yet.2 The third stage is describing facts of the case. These facts are already present at the first stage of applying the law. At this stage, however, their description should be adjusted, as far as possible, to the hypothesis of the legal norm outlined. This is not a trivial matter. On the one hand, such a hypothesis is expressed, solely or predominantly, in abstract terms. On the other hand, facts of the case are usually described, solely or predominantly, in concrete terms.3 In the extreme, but not uncommon, cases, the legal norm is outlined solely in abstract terms: 8x ð Am ðxÞ ˄ . . . ˄ An ðxÞ ! OBL ðxÞ Þ while the facts of the case are expressed solely in concrete terms: Km ðaÞ ˄ . . . ˄ Kn ðaÞ: (The letter ‘A’ is used for abstract predicates, and the letter ‘K’ for concrete ones.) How may such a description of facts fit the hypothesis of a legal norm? Certainly, in practice, there is a significant role for intuition in this respect. When facts of the case are being described in concrete terms, one already has a feeling of some relevance between this description and the abstract hypothesis of a legal norm.

2

It is worth noting that it is usually at this stage that the conflict of laws rules and the rules of legal inference are applied. The conflict of laws rules require us to reject given provisions of legal texts if their use would introduce a contradiction to the set of constructed norms. On the other hand, the rules of legal inference allow us to add new norms based on already constructed norms. The conflict of law rules and the rules of legal inference can be examined in respect of formalization and mechanization. For example, the conflict of law rules may be defined by an axiomatic theory built on the first order logic. See: Malec (2001), pp. 97–101. 3 By the way, it is worth noting that there are no fundamental obstacles to describing facts of the case in abstract terms. Describing facts in concrete terms is common simply because it is easier to verify whether such a description is true.

6.1 The Continental Model of Applying the Law

117

This feeling may be called a ‘pre-subsumption’. There is no doubt that the understanding of describing the facts is very important for the assessment of the possibility of algorithmic application of the law. However, like in the case of stage two, stage three will not be discussed in more detail in this book, as we do not have more to say about it yet.4 The fourth stage of applying the law is subsuming, i.e. ‘official’ linking of facts of the case to a norm. This linking may be formally described in two ways which may be called respectively ‘subsuming by concretizing and specifying norms’ and ‘subsuming by redescribing facts of the case’. The first way means interpreting the legal norm outlined. Such an interpretation aims at answering the question: ‘do the facts of the case fall under the hypothesis of the norm?’ and may be described as two processes: concretizing the norm in question and its specifying. This concretizing is a kind of replacing of abstract predicates that make up this norm’s hypothesis with concrete ones. On the other hand, this specifying is a kind of adding new predicates to this hypothesis in order to cover facts of the case, whenever possible. The second way is nothing more than redescribing facts of the case in terms in which the hypothesis in question is expressed. While both forms of subsuming will be examined in this chapter in detail, we will especially focus on the former, because it is this form of subsuming that involves transforming norms. The fifth and final stage is applying the legal syllogism. The schema of legal syllogism is being presented in various ways, e.g. in the following way:

It reads as follows: ‘From the premises: (i) it is ordered that every x in the circumstances described by P shall behave in the manner described by Q, and (ii) an a is in the circumstances described by P, results that (iii) it is ordered that a shall behave in the manner described by Q’. Thus, by application of the legal syllogism an individual norm is derived from a general norm and facts of the case. So, the norms are transformed at this stage, as well. In this chapter, it will be inquired in detail how this process shall be formalized correctly. To sum up, the legal norms are transformed during the fourth and the fifth of the aforementioned stages, and these stages will be examined below in detail. In particular, it will be examined whether these transformations can be performed in an algorithmic manner.

4

It is worth noting that among the logical issues related to describing facts there are also issues of legal defining. See: Malec (2000).

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6.2

6 Transformations of Norms

General Norms and Individual Norms

In Chap. 5, two kinds of norms have been introduced in our model of the language of law:5 (1) unconditional norms: OBL ðaÞ, FOR ðaÞ, PER ðaÞ (where ‘a’ is an individual constant for any but determined event), and (2) conditional norms: 8x ð DESCRIPTION ðxÞ ! OBL ðxÞ Þ, 8x ð DESCRIPTION ðxÞ ! FOR ðxÞ Þ, 8x ð DESCRIPTION ðxÞ ! PER ðxÞ Þ (where ‘x’ is an individual variable for events, and ‘DESCRIPTION (x)’ stands for a non-deontic predicate or a conjunction of non-deontic predicates). While every unconditional norm orders, prohibits, or allows only one single event, every conditional norm orders, prohibits, or allows all events of a relevant kind. These conditional and unconditional norms can be perceived as counterparts to the norms of real legal systems. Namely, the conditional norms resemble general norms coded in laws and regulations, whereas the unconditional norms resemble individual norms which constitute court verdicts or administrative decisions. Indeed, every general norm applies to all events of a specific kind. And the same is for our conditional norms. On the other hand, every individual norm concerns exactly one event. And the same is for our unconditional norms. This likeness encourages us to explicate the process of transformation of the norms coded in laws and regulations into the norms constituting court verdicts or administrative decisions, in terms of conditional and unconditional norms. The first step of such an explication is to define the concepts of concretizing and specifying.

5 These schemas are for events. In Chap. 5, similar schemas were introduced for situations as well. However, for simplicity reasons, in this chapter the examples will be limited to schemas for events.

6.3 Concretization and Specification of Norms

6.3

119

Concretization and Specification of Norms

In Chap. 2, the predicates have been divided into concrete and abstract ones. This division has been done in two ways. In accordance with the first approach, concrete predicates refer to both mereological and distributive relations, whereas abstract ones refer to distributive relations only. According to the second approach, concrete predicates refer to ‘observable’ properties or relations,6 whereas abstract ones refer to other properties or relations. If the predecessor of a conditional norm is constructed solely with concrete predicates, in the second meaning described above, i.e. when: 8x ð DESCRIPTION ðxÞ $ Km ðxÞ ˄ Kmþ1 ðxÞ ˄ . . . ˄ Kn ðxÞ Þ, we will call this predecessor a ‘concrete description’, and will call this norm a ‘concrete norm’. On the other hand, if the predecessor of a conditional norm is constructed solely with abstract predicates, in the second meaning described above, or with concrete ones and abstract ones, i.e. when: 8x ð DESCRIPTION ðxÞ $ Am ðxÞ ˄ Amþ1 ðxÞ ˄ . . . ˄ An ðxÞ Þ, or 8x ð DESCRIPTION ðxÞ $ Km ðxÞ ˄ Kmþ1 ðxÞ ˄ . . . ˄ Kn ðxÞ ˄ Anþ1 ðxÞ ˄ Anþ2 ðxÞ ˄ . . . ˄ Ar ðxÞ Þ, we will call this predecessor an ‘abstract description’, and will call this norm an ‘abstract norm’. According to the abovementioned definitions, the predecessor of an abstract norm may contain concrete predicates and abstract predicates together. Thus, abstract norms can be ‘less abstract’ or ‘more abstract’ depending on what predicates their predecessors are made of. Consequently, the process of replacing abstract predicates in the predecessor of an abstract norm with concrete ones, will be called a ‘concretization’ of this norm. In the process of concretization, an abstract norm becomes less and less abstract, and eventually becomes a concrete norm. Further, as it was said in the previous subchapter, while every unconditional norm orders, prohibits, or allows only one single event, every conditional norm orders, prohibits, or allows all events of a relevant kind. This kind is determined by the predecessor of this norm.

When we talk about ‘observable’ properties or relations, we only mean that the truth or falsity of any sentence formed with ‘observable’ predicates is determined exclusively on the basis of observations (sense perceptions). See: Przełęcki (1988), p. 37, p. 68 and following. 6

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If this predecessor indicates exactly one event, i.e. when: 8x8y ð DESCRIPTION ðxÞ ˄ DESCRIPTION ðyÞ ! x ¼ y Þ, it will be called a ‘definite description’. If this predecessor indicates more than one event, i.e. when: Ǝ x Ǝ y ð DESCRIPTION ðxÞ ˄ DESCRIPTION ðyÞ ˄ x 6¼ y Þ, it will be called an ‘indefinite description’. And, finally, if this predecessor does not indicate any event, i.e. when: ⅂ Ǝ x DESCRIPTION ðxÞ, it will be called an ‘empty description’. Adding new predicates to the predecessor of a norm will be called a ‘specification’ of this norm. In the process of specification, a conditional norm becomes more and more determined, and eventually becomes a norm with a definite description, i.e. a norm that applies to exactly one event.

6.4

Subsumption by Concretizing and Specifying Norms

Let us now reconstruct formally the fourth stage of applying the law, i.e. subsuming. As it was said before, subsuming means ‘official’ linking of facts of the case to a norm. This linking may be formally described in two ways which may be called respectively ‘subsuming by concretizing and specifying norms’ and ‘subsuming by redescribing facts of the case’. The first way means interpreting a given general and abstract norm, in order to determine whether facts of the case fall under the hypothesis of this norm. This process starts with a conditional norm, e.g.: 8x ð Pm ðxÞ ˄ . . . ˄ Pn ðxÞ ! OBL ðxÞ Þ, or 8x ð Pm ðxÞ ˄ . . . ˄ Pn ðxÞ ! FOR ðxÞ Þ, or 8x ð Pm ðxÞ ˄ . . . ˄ Pn ðxÞ ! PER ðxÞ Þ: Here, any ‘Pm’ stands either for ‘Am’ or for ‘Km’, i.e. either for an abstract predicate or for a concrete one. Such a norm shall be interpreted, in order to determine whether given facts of the case, described by the sentence:

6.4 Subsumption by Concretizing and Specifying Norms

121

ð Km ðaÞ ˄ . . . ˄ Kr ðaÞ Þ, fall under the hypothesis of this norm. First, such a norm is subject to concretization, i.e. all abstract predicates from its predecessor shall be replaced by concrete ones, i.e. a sentence formula: ð Pm ðxÞ ˄ . . . ˄ Pn ðxÞ Þ, is transformed into a sentence formula: ð Km ðxÞ ˄ . . . ˄ Kn ðxÞ Þ: Here, for the sake of simplicity, each letter P with an index has been replaced by only one letter K with the same index. In practice, any abstract predicate would be replaced by some combination of concrete predicates. In the real life, such a concretization takes place, e.g., when a judge interprets a word ‘burglary’ from a penal code as ‘taking property belonging to a third party from a locked room, preceded by breaking the locks of this locked room’. Consequently, an abstract term used by the legislator is replaced by a concrete term developed by the doctrine or case law. Replacing abstract predicates with concrete ones, is not trivial. In his Logic of Empirical Theories, Marian Przełęcki gives us examples of such an ‘empirical reduction’.7 Sometimes, it takes the form of an equivalence definition: 8x ð Am ðxÞ $ α ðxÞ Þ, where ‘α (x)’ is a formula with one free variable ‘x’ which does not contain ‘Am (x)’ predicate.8 In other instances, it takes the form of a conditional definition: 8x ð β ðxÞ ! ð Am ðxÞ $ α ðxÞ Þ Þ, where ‘α (x)’ and ‘β (x)’ are formulas with one free variable ‘x’ which do not contain ‘Am (x)’ predicate.9 Other times, it takes the form of a partial definition: 8x ð ð α ðxÞ ! Am ðxÞ Þ ˄ ð β ðxÞ ! ⅂ Am ðxÞ Þ Þ, where ‘α (x)’ and ‘β (x)’ are understood as before.10 Other forms of such an ‘empirical reduction’ also are possible. However, only equivalence definitions are

7

See: Przełęcki (1988), pp. 68–93. See: Przełęcki (1988), p. 68 and following. 9 See: Przełęcki (1988), p. 73 and following. 10 See: Przełęcki (1988), p. 77 and following. 8

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suitable for concretizing, because the expression ‘DESCRIPTION (x)’ stands only for a non-deontic predicate or a conjunction of non-deontic predicates.11 As a result of concretizing, concrete norms are obtained: 8x ð ð Km ðxÞ ˄ . . . ˄ Kn ðxÞ Þ ! OBL ðxÞ Þ, 8x ð ð Km ðxÞ ˄ . . . ˄ Kn ðxÞ Þ ! FOR ðxÞ Þ, 8x ð ð Km ðxÞ ˄ . . . ˄ Kn ðxÞ Þ ! PER ðxÞ Þ: The predecessors of these norms are indefinite descriptions. Thus, every of such norms applies to more than one event. These norms are adapted to facts of the case in the process of their specification, i.e.: ð Km ðxÞ ˄ . . . ˄ Kn ðxÞ Þ, is replaced by: ð Km ðxÞ ˄ . . . ˄ Kn ðxÞ ˄ Knþ1 ðxÞ ˄ . . . ˄ Kr ðxÞ Þ, where the Kn+1 (x), . . ., Kr (x) predicates complement the predecessor according to the facts (who, whom, when, in what place, etc.). In the real life, specifying takes place when a judge expands the result of concretization into a complete description of an event, e.g., the description: ‘taking property belonging to a third party from a locked room, preceded by breaking the locks of this locked room’, is expanded into the description: ‘taking by Mr. John Smith, on December 15th, 2018, in the evening, a TV set that had belonged to Mr. John Bean, and had been locked in Mr. Bean’s apartment at Elwood Street, Highbury, North London, preceded by breaking by Mr. John Smith two locks to the apartment with a crowbar’. The result of this step is a concrete definite description, i.e. a description that exactly one event fulfills: 8x8y ð ð ð Km ðxÞ ˄ . . . ˄ Kr ðxÞ Þ ˄ ð Km ðyÞ ˄ . . . ˄ Kr ðyÞ Þ Þ ! x ¼ y Þ: Thus, at this stage, we obtain norms in the form of: 8x ð ð Km ðxÞ ˄ . . . ˄ Kr ðxÞ Þ ! OBL ðxÞ Þ, 8x ð ð Km ðxÞ ˄ . . . ˄ Kr ðxÞ Þ ! FOR ðxÞ Þ,

11

The rules of concretization are classic rules of interpreting legal expressions. These are linguistic, systemic and functional rules.

6.5 Subsumption by Redescribing Facts of the Case

123

8x ð ð Km ðxÞ ˄ . . . ˄ Kr ðxÞ Þ ! PER ðxÞ Þ, whose predecessors are solely concrete and definite descriptions. This way, general and abstract norms coded in laws and regulations, are finally transformed into norms that are concrete and, despite of their general form, refer exactly to one event. As a result, it may be declared that they fit to the facts of the case in question, described by the sentence: ð Km ðaÞ ˄ . . . ˄ Kr ðaÞ Þ: Having this subsumption, the legal syllogism may be applied.

6.5

Subsumption by Redescribing Facts of the Case

The second way of ‘matching’ the hypothesis of a legal norm and facts of the case is redescribing these facts in terms corresponding to the hypothesis in question. We can reconstruct this process as follows. Let us suppose that a legal norm is expressed by the formula:12 8x ð α ðxÞ ! OBL ðxÞ Þ, and facts of the case are expressed by the formula: α’ ðaÞ: Since ‘α (x)’ and ‘α’ (a)’ are different expressions, there is a mismatch between the description of the facts and the hypothesis of the legal norm. This will be the case, for example, of the legal norm: ‘

Stealing is forbidden’

and the description of facts: ‘Mr. John Smith, on December 15th, 2018, in the evening, took a TV set that had belonged to Mr. John Bean’. If we were subsuming by concretizing and specifying, we would strive to replace the norm: 8x ð α ðxÞ ! OBL ðxÞ Þ by the norm: 12

Likewise, we could choose 8 x ( α (x) ! FOR (x) ), or 8 x ( α (x) ! PER (x) ).

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8x α’ ðxÞ ! OBL ðxÞ : In terms of our example, we would strive to replace the abstract and general norm: ‘

Stealing is forbidden’

by a concrete and determined norm: ‘It is forbidden to take, on December 15th, 2018, in the evening, a TVset belonging to Mr. John Bean’. However, when we are subsuming by redescribing facts of the case, we do not interpret the legal norm, but instead we add to the description of the facts the following sentence: ‘Taking, on December 15th, 2018, in the evening, a TV set that had belonged to Mr. John Bean, is a stealing’ or we simply redescribe these facts by stating that: ‘

Mr:John Smith committed an act of stealing’ :

This first approach is expressed formally as accepting the thesis:
8x α’ ðxÞ ! α ðxÞ , whereas the second approach is expressed formally as accepting the thesis: α ðaÞ: As a side note, it
is worth noting that the implication sign in the thesis ‘8x α’ ðxÞ ! α ðxÞ ,’ indicates that the description in concrete terms and the description in abstract terms may be not equivalent. Thus, in this case of subsumption by redescribing facts of the case, we are not looking for the full meaning of the hypothesis of a legal norm. It is enough for us to make sure that the considered facts are within the meaning of this hypothesis. In both approaches, however, the description of the facts and the hypothesis of the legal norm become matched to each other. Indeed, in both cases the respective description of the facts includes the same ‘α’ which constitutes the predecessor of the norm in question. In fact, both kinds of subsuming, the one by concretizing and specifying a legal norm, and the one by redescribing facts of the case, are based on defining. However, while the former is based on defining expressions constituting the hypothesis of a legal norm in less abstract terms, the latter is based on defining expressions describing facts in more abstract terms. In practice, quite often both types of subsuming occur simultaneously: the hypothesis of a legal norm undergoes a certain concretization and specification,

6.6 Schemas of the Legal Syllogism

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and the description of facts of the case changes, adequately to this already transformed hypothesis.

6.6

Schemas of the Legal Syllogism

When the hypothesis of a legal norm and the description of facts of the case have been matched by subsuming, the legal syllogism can be applied. The simplest schema of the legal syllogism, one that is taught to law students in their first year of studies, is the following one:

The first premise of this syllogism (i.e. ‘8 x (P(x) ! Q (x))’ ; the so-called ‘greater premise’) is an abstract and general norm obtained through the interpretation of legal texts. For example, the norm ‘Every adult has the right to vote in elections’ is a norm of this kind.13 The second premise of this syllogism (i.e. ‘P(a)’; the so-called ‘minor premise’) is the description of facts of the case, for example, ‘John is an adult’. Taking into account both of these premises, a conclusion (i.e. ‘Q(a)’) can be derived, for example, ‘John has the right to vote in elections’. This schema of the legal syllogism is a deductive one, i.e. it guarantees that if the premises are true then the conclusion is true, as well. Indeed, according to the rule of omitting the large quantifier, from the sentence ‘8 x (P(x) ! Q (x))’ results the sentence ‘P(a) ! Q (a)’. This way, the predecessor of the transformed major premise becomes identical to the minor premise, which makes it possible to apply the modus ponens rule, and as a result to conclude that ‘Q(a)’.14 However, this is not the only possible schema of the legal syllogism. If it is taken into account that the greater premise is usually an abstract norm, but can also be a concrete norm, and the minor premise is usually described in concrete terms, but can also be described in abstract terms, and if the schemas of conditional norms adopted in this work are applied, then many alternative schemas of the legal syllogism can be considered. Namely, when facts of the case are described in concrete terms, and: 13 Consequently: ‘P(. . .)’ shall be read as ‘. . . is an adult’, ‘Q(. . .)’ shall be read as ‘. . . has the right to vote in elections’, and ‘x’ is a variable for persons. 14 According to the rule of omitting the large quantifier, the large quantifier at the beginning of the formula can be omitted, and the variables previously bound by this quantifier are replaced with any constants. According to the modus ponens rule, if the formula in the form of an implication is accepted, and its predecessor is accepted, the successor of this formula should also be accepted. The large quantifier omitting rule and the modus ponens rule are deductive ones.

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(1) the predecessor of a norm is a concrete and definite description (in the real life, it is rather unique, but still possible), or (2) the predecessor of a norm is an abstract and indefinite description, but this norm has been concretized and specified, the following forms of the legal syllogism are applicable:

and

and

Further, when facts of the case are described in abstract terms, e.g. as a result of redescribing these facts in terms of the hypothesis of a norm, and: (1) the predecessor of a norm is an abstract and definite description (in the real life, it is rather unique, but still possible), or (2) the predecessor of a norm is an abstract and indefinite description, but this norm has been specified, the following forms of the legal syllogism are applicable:

6.6 Schemas of the Legal Syllogism

127

and

and

Further, when facts of the case are described in concrete terms, but as a result of redescribing these facts, their description becomes linked with the hypothesis of an abstract norm, the following forms of the legal syllogism are applicable:

and

and

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6 Transformations of Norms

Of course, mixed situations may arise, e.g. where a norm or facts of the case have been expressed partly in abstract terms, and partly in concrete terms. In such cases, the abovementioned schemas shall be appropriately modified. All these examples can be generalized into two meta-schemas showing formal relations between the premises and the conclusion, namely:

and

In the first meta-schema, the expression ‘α’ (x)’ is either a concrete description or an abstract one. Usually, when ‘α’ (x)’ is a concrete description, it is the result of subsumption by concretizing and specifying a legal norm. On the other hand, when ‘α’ (x)’ is an abstract description, it is usually the result of subsumption by redescribing the facts in terms of the legal norm hypothesis. In the second meta-schema, the expression ‘α (x)’ is an abstract description, and the expression ‘α’ (x)’ is a concrete description. These descriptions become linked to each other thanks to the middle premise of the syllogism, being a result of subsumption by redescribing facts of the case in terms of the legal norm hypothesis. This middle premise is a kind of translation of the minor premise terms into the greater premise terms. In both meta-schemas, the expression ‘RULE (x)’ denotes the deontic modality of the event in question: prescription, prohibition, or permission. It may be regarded as the disposition of the legal norm. Importantly, in all these schemas, the expressions ‘a’ and ‘x’ refer to events, not people. As a result of applying the legal syllogism rules to conditional norms and the descriptions of facts of the case, unconditional norms are obtained, in the forms: OBL ðaÞ, and

6.7 Creativeness of Norms Transformations

129

FOR ðaÞ, and PER ðaÞ: Such norms correspond to individual norms that constitute court verdicts, or administrative decisions. As a side note, it is worth noting that in legal theory one can also encounter other schemas of the legal syllogism, e.g.

which shall be read: ‘from the premises: (i) it is ordered (‘o’) that any person (‘x’) who finds themselves in a situation of the kind P, shall behave in the manner specified by Q, and (ii) a person a is in a situation of the kind P, may be derived that: (iii) the person a is obliged to behave in the manner specified by Q’, or:

which shall be read: ‘from the premises: (i) any person who finds themselves in a situation of the kind P is obliged to behave in the manner specified by Q, and (ii) a person a is in a situation of the kind P, may be derived that: (iii) the person a is obliged to behave in the manner specified by Q’. It shall be noted that both of these schemas are not expressed in the language of first-order logic (since the ‘o’ operator is not an expression of this language). As a result, they are not the laws of first-order logic. On the other hand, all the legal syllogism schemas advised by us are infallible rules of inference (laws) of classical first-order logic.

6.7

Creativeness of Norms Transformations

According to a popular view, in the Anglo-Saxon system of law, the courts create the law, while in the continental system, the law is created solely by parliaments. This view is not true. In the Anglo-Saxon system, parliaments also create laws, and in the continental system, the courts do the same. In the Anglo-Saxon model, however, the courts are empowered to create the law (as the Anglo-Saxon model is a precedent

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6 Transformations of Norms

model). Contrary to this, in the continental model, the courts are supposed to only apply the law. However, there are many reasons for which in fact the continental courts also create the law. It is sufficient to mention that legal provisions turn out, from time to time, to be contradictory, and there are contradictory rules of legal interpretation, as well. And if the axioms or rules of a theory contradict each other, then any conclusion may be achieved in it. This is what Duns Scotus famous law proclaims: ð α ˄ ⅂α Þ ! β, where ‘α’ and ‘β’ represent any sentences (read: ‘From contradictory sentences any sentence results’). However, is it even possible to transform norms in a non-creative manner? Is it possible to entrust this process to a computer? Let us look from this point of view at the fourth and fifth stages of applying the law described in the previous subchapters. First, it may be concluded that specifying concrete norms is a non-creative process. Any specification of a legal norm of this kind consists in adding new concrete predicates to the predecessor of this norm to enable subsumption. Thus, if a norm and the description of facts of the case are provided: 8x ð ð Km ðxÞ ˄ . . . ˄ Kn ðxÞ Þ ! OBL ðxÞ Þ, ð Km ðaÞ ˄ . . . ˄ Kn ðaÞ ˄ Knþ1 ðaÞ ˄ . . . ˄ Kr ðaÞ Þ, one needs only to add the Kn+1 (x), . . ., Kr (x) predicates to the predecessor of the norm to transform it in a large premise of the legal syllogism rule:

The Kn+1 (x), . . ., Kr (x) predicates that ought to be added, can be identified by a simple comparison of the norm and the description of facts in question. Second, it may be concluded that applying the legal syllogism rules is a non-creative process, too. This is because all schemas of the legal syllogism advised in this book are the rules for replacing some symbols with other, strictly defined, symbols. If someone knows the premises of the syllogism, they also know its conclusion. In other words, applying the legal syllogism is mechanical. It follows an algorithm that can be executed by any computer. Thus, if the laws and regulations contained solely conditional norms of the forms: 8x ð ð Km ðxÞ ˄ . . . ˄ Kn ðxÞ Þ ! OBL ðxÞ Þ,

6.7 Creativeness of Norms Transformations

131

8x ð ð Km ðxÞ ˄ . . . ˄ Kn ðxÞ Þ ! FOR ðxÞ Þ, 8x ð ð Km ðxÞ ˄ . . . ˄ Kn ðxÞ Þ ! PER ðxÞ Þ, i.e. concrete norms with indefinite descriptions, transformations of norms in the process of applying the law would be non-creative. However, the point is that some of our norms are abstract. Thus, they need to be concretized. If there were provided the relevant definitions of abstract predicates in the form of an equivalence definition: 8x ð Ak ðxÞ $ ð Km ðxÞ ˄ . . . ˄ Kn ðxÞ Þ Þ, the transformations of norms would still be algorithmic. Indeed, having an abstract conditional norm: 8x ðAk ðxÞ ! OBL ðxÞ Þ, and an equivalence definition: 8x ð Ak ðxÞ $ ð Km ðxÞ ˄ . . . ˄ Kn ðxÞ Þ Þ, one will use an appropriate tautology of logic, and receive as a result a concrete conditional norm: 8x ð ð Km ðxÞ ˄ . . . ˄ Kn ðxÞ Þ ! OBL ðxÞ Þ, which can be further transformed in a non-creative manner. However, in the real systems of law, as a rule, the equivalence definitions of the abovementioned kind are not provided. For this reason, in practice, transforming conditional norms into unconditional ones, is a creative process, although theoretically the opposite might be possible. Similarly, subsuming by redescribing facts of the case also turns out to be creative. When a conditional norm takes the form: 8x ð α ðxÞ ! OBL ðxÞ Þ and facts of the case are described by the formula: α’ ðaÞ, then this description shall be adjusted to the hypothesis of the norm in question by adopting an additional premise:

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6 Transformations of Norms


8x α’ ðxÞ ! α ðxÞ : When such a premise is found, the unconditional norm can be obtained algorithmically using the legal syllogism in the form:

However, such an additional premise is usually not included in legal texts. Thus, in practice, it is created by judges, which means that applying the law is no longer algorithmic. The same will happen if, instead of searching for this additional premise, the authority applying the law describes the facts of the case directly in abstract terms: α ðaÞ: Indeed, without clear rules for describing facts in abstract terms, such a description is obviously creative. Of course, the more definitions legal texts contain, the less creative subsumption is. However, in modern legal texts such definitions are scarce compared to the number of abstract terms used by the legislator. Moreover, usually abstract terms are defined by other abstract terms. Such a way of defining does not facilitate subsuming, because subsuming is about linking abstract terms to concrete ones. Of course, if there were algorithms for creating adequate definitions of abstract expressions in concrete terms, subsuming would become a non-creative process. However, creating such algorithms does not seem to be an easy task. Thus, in view of the small number of definitions in legal texts, their poor quality, and the lack of algorithms to create such definitions, in practice, subsuming remains a creative process. Finally, it is worth noting that computers will be able to apply the law on their own, only if each of the five stages discussed in the first part of this chapter turns out to be algorithmic. However, this research has been focused solely on the fourth and fifth of these stages. Nevertheless, the mechanization of these two stages alone would allow the computers to act as ‘quality overseers’ of applying the law by people. Nowadays, this is how the computers work in many areas of science and practice. For example, there are programs that validate mathematical proofs. The computer itself is not able to prove a given theorem, yet. However, it can verify whether the proof of this theorem, proposed by a human, is faultless. In the same role, the computers may soon appear, for example, in court proceedings. The computer may not be able to decide the case on its own. But it will be able to show the judge where their

6.8 The Precedent Model of Applying the Law

133

reasoning is incorrect. Undoubtedly, it will fundamentally change the process of applying the law.

6.8

The Precedent Model of Applying the Law

Let us compare the continental model of applying the law with the precedent one, where the law-making role of courts is expressly admitted. First, when a precedent is created, the reasoning schema is as follows: ╞ OBL ðbÞ what shall be read: ‘it is true that b is ordered’. Second, when a court’s decision is based on an existing precedent, the reasoning schema is as follows:

what shall be read: ‘from the premises: (i) the event b is ordered, and (ii) the event a is similar to the event b, it may be derived that (iii) the event a also is ordered’. This second reasoning is called a ‘reasoning by analogy’ (or simply ‘analogy’). Undoubtedly, analogy belongs to rational reasonings, i.e. it frequently brings true conclusions provided that premises are true. However, analogy is not a deductive reasoning, i.e. it may be the case that the premises are true and, nevertheless, the conclusion is false. Does the fact that in the continental model unconditional norms are derived from conditional ones by the deductive rules of the legal syllogism, while in the precedent model unconditional norms are derived only by non-deductive analogy, give the continental model an advantage over the precedent model? Probably not. Indeed, in the continental model, the final stage of applying the law is based on deductive rules, i.e. it is based on logic. But the very process of applying the law is much more complicated than in the precedent model. As a result, the court has a lot of opportunities to shape the law in any way, and the outcome of applying the law is often unpredictable. Only if all the stages of applying the law in the continental model turned out to be mechanizable, could we speak of the advantage of the continental system over the precedent system.

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References Malec A (2000) Zarys teorii definicji prawniczej (in Polish: An Outline of the Theory of Legal Definition). PHILOMAT, Warsaw Malec A (2001) Legal reasoning and logic. In: Language, mind and mathematics. University of Bialystok, Bialystok, pp 97–101 Przełęcki M (1988) Logika teorii empirycznych. Państwowe Wydawnictwo Naukowe, Warszawa

Chapter 7

Epilogue: The Concept of Law

Let us conclude this book with a few remarks on the nature of the law from the perspective of semantics. From this perspective, the nature of the law appears to be similar to the nature of mathematics. Some people will say that mathematical theories discover mathematical objects that exist in Popper’s third world (or in Plato’s world of ideas). Others will say that there are no mathematical objects without mathematical theories. It is probably the same with the law. Some will say that lawyers only discover the law that exists in Popper’s third world (or in Plato’s world of ideas). Others might say that there is no law without a relevant legislation or precedent. The latter cannot question the validity of properly introduced norms, while the former may say that some norms are true (because they correctly describe the law), and others are false (because they distort the law). In terms of semantics, this controversy may be expressed in the following way. Let NORM_ST will be the set of norms properly introduced by the lawmaker. It may be also assumed that some norms are not properly introduced: NORM ST ⊂ NORM: The set of logical consequences of the properly introduced norms will be called ‘CN (NORM_ST)’. From the perspective of semantics, every norm of our language is either true or false. Further, from this perspective, neither norm of our language is true and false. Thus: NORM ¼ NORM T U NORM F, NORM T \ NORM F ¼ ϕ: The relation between CN (NORM_ST) and NORM_T is one of the following ones:

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Malec, Introduction to the Semantics of Law, Law and Visual Jurisprudence 6, https://doi.org/10.1007/978-3-030-95679-0_7

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7 Epilogue: The Concept of Law

(1) (2) (3) (4) (5)

CN (NORM_ST) ¼ NORM_T , or CN (NORM_ST) ⊂ NORM_T , or NORM_T ⊂ CN (NORM_ST) , or CN (NORM_ST) \ NORM_T ¼ ᴓ , or CN (NORM_ST) \ NORM_T 6¼ ᴓ and Ǝx ( x E CN (NORM_ST) and ~ x E NORM_T ) and Ǝy ( ~ y E CN (NORM_ST) and y E NORM_T ).

In other words, the sets CN (NORM_ST) and NORM_T may be identical (1), the first of them may be a proper subset of the second (2), the second may be a proper subset of the first (3), they may be disjoint (4), they may be intersecting (5). If CN (NORM_ST) ¼ NORM_T, the distinction between the positive (statutory) and natural law is not important. For in such a case, both it is true that only properly introduced norms and their consequences are the law, and it is true that the legislators and judges can only discover the law. It may be said as well that in such a case the law has been discovered by the legislators and judges in its totality. However, if CN (NORM_ST) 6¼ NORM_T, then the distinction between the positive (statutory) and natural law is important. For in such a case, the positive law, which is the set CN (NORM_ST), and the natural law, which is the set NORM_T or the set of legal rules itself, differ in their contents. In our model, it is not prejudged whether CN (NORM_ST) ¼ NORM_T or not. Thus, it is admitted that positive law may differ from natural law. Moreover, in our model, the set of legal rules is defined earlier than the NORM_T set and the NORM_ST and CN (NORM_ST) sets. Thus, in a sense, in this work the natural law precedes the positive law. Further, our model of the language of law is convenient for expressing some theories of natural law, e.g., for expressing Leibniz’s1 formula that the best alternative is obligatory: 8x8y ð BEST ð x, y Þ $ OBL ð x, y Þ Þ, that is: replacing x with y is ordered if and only if y is the best alternative in the choice situation x. This kind of linking deontic modalities with the evaluation of events seems to be a very perspective direction of research. But this is a different story.2

Reference Malec A (2000) Zarys teorii definicji prawniczej (in Polish: An Outline of the Theory of Legal Definition). PHILOMAT, Warsaw

1 Gottfried Wilhelm Leibniz (1646–1716), a German rationalist philosopher, the creator of monadology. 2 See: Malec (2000), pp. 134–143.

Chapter 8

Conclusions

Was this book worth reading? Each reader will answer this question for themselves. The aim of this book was to show a semantic view of law, an approach using both the results of logical semantics and the ‘situational perspective’ flowing from Wittgenstein’s philosophy. The book was intended to give a good theory of legal phenomena, and their description, thus also serving practice, in accordance with the motto that there is nothing more practical than a good theory. Let us summarize the most important conclusions of this book in the form of short answers to the questions posed in the Introduction.

8.1

What Is the Law?

The legal rules can be thought of as sets of events. As a consequence, the law itself can be thought of as a set of sets of events. Does this approach change the perception of law? It seems to be a fresh look at the nature of law: through the prism of lattice theory and set theory.

8.2

What Are the Legal Events? What Are the Acts?

The legal events can be thought of as sequences of situations in the sense of Wittgenstein-Wolniewicz. The acts can be thought of as a kind of legal events. Namely, any act can be thought of as a two-element sequence of situations where the first element is a choice situation, and the second one is an alternative situation directly accessible from the choice situation. Does this conclusion change the perception of legal events and acts? It certainly shows that the intuitive concepts of legal events and acts, can also be considered in a strict manner. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Malec, Introduction to the Semantics of Law, Law and Visual Jurisprudence 6, https://doi.org/10.1007/978-3-030-95679-0_8

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8.3

8

Conclusions

Can the Legal Norms Be True?

The main thesis of this book is that legal norms, if have been interpreted, are sentences in a logical sense. In other words, every legal norm, if has been interpreted, is either true or false. This thesis has been proved by presenting the truth conditions for legal norms. The compliance of legal norms with the reality consists in their compliance with Popper’s third world, just as it is in the case of mathematical theorems. Does this thesis change the perception of legal norms? It certainly gives strong arguments against the popular view today that legal norms have nothing to do with truth and falsehood. I also put forward and justify the hypothesis that Hume’s thesis on the logical separation of sentences about the world from sentences about obligations may be false. This hypothesis is a proposal to reject the approach that has been determining the way of justifying legal norms for several hundred years.

8.4

Can a Norm Result from Another Norm?

If legal norms can be true and false, they can also be in the entailment relation to each other, as well as in other semantic relations, exactly as it is the case with other sentences in a logical sense. Thus, legal norms can contradict each other, they can be opposite, or sub-opposite. Does this thesis change the perception of legal norms? It is certainly a voice in the debate on the legitimacy of using terms such as ‘entailment’ or ‘contradiction’ to describe relations between norms. A voice showing how much time was wasted.

8.5

Can an Order Result from Another Order?

Although the orders are not sentences in a logical sense, and therefore are neither true nor false, they can be in some specific semantic relations to each other. I define several relations of this kind, namely: 1) 2) 3) 4) 5)

entailment for orders, justifying for orders, contradiction for orders, opposition for orders, and sub-opposition for orders.

It turns out that norms and orders have common roots: while the norms describe legal rules, the orders prescribe these rules. Does this thesis change the perception of legal norms or orders? Certainly, it allows to better understand why do the norms differ from the orders, even though both are based on legal rules.

8.8 How to Interpret the Language of Law?

8.6

139

Can Semantics Distinguish Between Natural Law and Statutory Law?

When the natural law is defined as a set of sets of events, and the positive (statutory) law is defined as a set of legal provisions and their consequences, then the natural law and the positive law are being distinguished in a semantic and completely formal way. Does this proposal change the perception of law? Probably not, but it shows that the distinction between the natural law and the positive law is not only a matter of philosophy, but also semantics.

8.7

How Shall the Semantic Issues Be Examined?

There are four methods useful in studying relations between the legal language and the law itself: 1) using formal models of the law domain and the language of law (‘toy-model approach’), 2) defining abstract objects of the law domain model in terms of concrete ones, whenever possible (‘ultimate concretism’), 3) basing the law domain model on situations (‘situational approach’), 4) linking expressions of the language model directly to the domain model (‘intuitive formalism’). Does this proposal change the perspective of studying the language of law? Well, the toy-model approach, which has been used in logical research for over a hundred years, is now boldly entering many sciences, so it is not new. On the other hand, the remaining three methods seem to be new tools that legal semantics can use.

8.8

How to Interpret the Language of Law?

When interpreting the language of law, it is advised to adopt a situational semantic perspective. Thus, instead of the traditional schema of a general legal norm: 8x ðPðxÞ ! Q ðxÞÞ, where ‘x’ is a variable denoting a person, ‘P’ is a predicate denoting this person’s properties, ‘Q’ is a predicate denoting this person’s obligations, and ‘8’ is a general quantifier (corresponding to the word ‘everyone’ in everyday language), it is worth to adopt the following schemas:

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Conclusions

8x ð Pm ðxÞ ˄ . . . ˄ Pn ðxÞ ! OBL ðxÞ Þ, 8x ð Pm ðxÞ ˄ . . . ˄ Pn ðxÞ ! FOR ðxÞ Þ, 8x ð Pm ðxÞ ˄ . . . ˄ Pn ðxÞ ! PER ðxÞ Þ, where ‘x’ is a variable denoting an event, the predicates from ‘Pm’ to ‘Pn’ indicate properties of this event, and the predicates ‘OBL’, ‘FOR’, and ‘PER’ state that this event is ordered, prohibited, or allowed. Consequently, interpreting legal norms is not thought of as assigning obligations to persons, but as finding legal events which are ordered, prohibited, or allowed. Does this proposal change the perception of interpreting the language of law? It certainly offers lawyers an additional perspective of such interpreting.

8.9

How to Apply the Law?

The continental model of applying the law is more complicated than the precedent model. If applying the law in the continental model is to be predictable, inter alia, the nature of subsuming shall be explained. Subsuming can be expressed formally as the sum of two processes, namely the concretization and specification of legal norms. It can also be expressed formally as a process of redescribing facts of the case in the legal norm terms. Does this conclusion change the perception of subsuming? It certainly allows for a better understanding of this concept and, consequently, helps in applying the continental law in an algorithmic way.

8.10

Why Are Legal Reasoning Rules Valid?

The validity of the rules of legal reasoning is rooted in the structure of the law domain. This rooting can be investigated either directly—by comparing the rules of legal reasoning with properties of the law domain, or indirectly—by studying deontic logics. The deontic logics presented in Chap. 4 explain, inter alia, the validity of the rules of legal reasoning rooted in deontic relations between the parts of the act and the act itself. Probably, they are the first deontic logics that allow such an explanation. Does this result change the perception of the rules of legal reasoning? Certainly, it helps to better understand relations between the rules of legal reasoning and the structure of the law domain.

8.12

8.11

How Can Legal Logic Help in the Computer-Assisted Application and Creation. . .

141

How to Formalize Legal Logic?

The deontic logics do not require a special, non-classical language. They can be constructed as theories built on the first-order logic, with variables running through a set of situations or events. Does this conclusion change the perception of legal logic? It is definitely a bold voice in the debate concerning the best special language for legal logic—explaining that there is no need for any such special language.

8.12

How Can Legal Logic Help in the Computer-Assisted Application and Creation of Law?

Legal logic helps in these processes by formalizing legal concepts and explaining legal reasonings. In this book the efforts were undertaken to formalize, inter alia, the concepts of legal event, act, legal rule, legal norm, entailment and other semantic relations for orders. There also were efforts to formalize and explain subsuming and applying the legal syllogism. Several deontic logics were also constructed. Do these results change the perception of the computer-assisted application or creation of law? Of course not. But this work is a contribution in this respect, somewhat technical, and somewhat methodological.

*** The formal apparatus inspired by the philosophy of Wittgenstein was an invaluable tool for me to organize my own views on the nature of law, on the nature of legal norms and rules, on the interpretation of law, and on other issues in the field of legal theory, which I have presented above. I hope that this apparatus will also be appreciated by the readers. If, however, the reader finds at least some of the conclusions contained in this book worthy of attention, and rejects the formal apparatus itself, like no longer needed ladder is being rejected, the purpose of this book will be achieved, because the ‘semantic sting’ will prove to be useful.