Contextual Approaches to Truth and the Strengthened Liar Paradox 9783110324587, 9783110324365

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Christine Schurz Contextual Approaches to Truth and the Strengthened Liar Paradox

LOGOS Studien zur Logik, Sprachphilosophie und Metaphysik

Herausgegeben von / Edited by Volker Halbach • Alexander Hieke Hannes Leitgeb • Holger Sturm Band 20 / Volume 20

Christine Schurz

Contextual Approaches to Truth and the Strengthened Liar Paradox

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2012 ontos verlag P.O. Box 15 41, D-63133 Heusenstamm www.ontosverlag.com ISBN: 978-3-86838-172-6 No part of this book may be reproduced, stored in retrieval systems or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use of the purchaser of the work. Printed on acid-free paper ISO-Norm 970-6 FSC-certified (Forest Stewardship Council) This hardcover binding meets the International Library standard Printed in Germany by CPI buchbücher.de GmbH

Preface This book contains my phD-thesis, which I have submitted at the University of Salzburg in August 2011. I want to thank my supervisor Reihnhard Kleinknecht for his great and constant help during the years of my doctorate study. Furthermore, I am highly indebted to my co-supervisors Alexander Hieke and Hannes Leitgeb for their very helpful comments on my thesis and their improvement suggestions. I also want to thank my father Gerhard Schurz for proof-reading the pen-ultimate version of my thesis, which has helped me much to complete my work. Moreover, I wish to express my deepest gratitude to the jury of the Marie Andeßner Dissertationsstipendium for awarding me the Marie Andeßner Dissertationsstipendium, which supported me from June 2009 to May 2010. Last, but not least I want to thank Markus for all his patience and encouragement during the years of preparing my thesis.

Contents 1 Introduction 2 The Problem of Truth and the Liar Paradox Languages 2.1 The Formal Language . . . . . . . . . . . . . . 2.2 The Undefinability of Truth . . . . . . . . . . . 2.3 The Conditions of the Liar Paradox . . . . . . .

7 in Formal . . . . . . . . . . . . . . . . . .

15 15 17 29

3 Theories of Partial Truth 3.1 Classical Approaches to Truth . . . . . . . . . . . . . . . . 3.2 Iterative Approaches to Truth and Groundedness . . . . .

31 32 37

4 The lem 4.1 4.2 4.3

65 65 69 73

Contextual Approach to the Strengthened Liar ProbThe Strengthened Liar Problem . . . . . . . . . . . . . . . The Contextual Approach . . . . . . . . . . . . . . . . . . Three Instances of CR1 . . . . . . . . . . . . . . . . . . .

5 Contextual Approaches by Iterating Partial Truth Predicates 89 5.1 A Hierarchy of Models According to CR1 . . . . . . . . . 89 5.2 A Hierarchy of Models According to CR2 . . . . . . . . . 100 5.3 Extending the Hierarchies to Transfinite Ordinal Numbers 119 5.4 Are Contextual Accounts Unnatural? . . . . . . . . . . . . 123 6 Summary and Conclusions

133

Subject Index

137

Bibliography

143

1 Introduction One of the oldest and most extensive problems of philosophy is the problem of truth, which is that of explicating the concept of truth. The problem of truth is of course related to other fundamental questions of philosophy, beginning with the question of what the truth bearers are, or how we can find out whether a proposition is true. The task is then to specify the nature of truth and whether truth can be properly reduced to other concepts. But there is also another dimension of the problem of truth that is related to the liar paradox. The liar paradox seems to show that truth is an incoherent concept. It occurs by placing only a minimal demand on the concept of truth, namely convention T :1 X is true i↵ p for every sentence p, where ‘X’ is a name of p. The liar paradox is equally a↵ecting all positions concerning the nature of truth. Thus the liar problem is usually investigated independently of any position to the nature of truth. Accordingly S. Blackburn and K. Simmons ([5], p. 22) state: “Sooner or later any truth theorist deflationary or not must confront with the notorious liar paradox.” Contextual approaches to the liar paradox try to avoid and solve this problem by assuming that the concept of truth is context-dependent. Contextual approaches argue that there is a semantic shift in the course of the liar argument (which is presented below). This semantic shift induces a change from one initial concept of truth to another concept of truth. In 1 As

is well known, convention T owes its name to A. Tarski. He was the first who realized the great importance of this principle, for the concept of truth as well as for the liar paradox (see Tarski[36]).

8

Introduction

this book, I would like to present current contextual approaches to the liar paradox, extract their common structure and evaluate their prospects. To begin with, I shall briefly introduce the sentential liar paradox. A more detailed introduction can e.g. be found in Sainsbury[31]. Above I have assumed that propositions are the bearers of truth. Ever since philosophers began to apply and to study the concept of a proposition, there has been a great dispute on what propositions are. I shall not go into this debate but instead usually assume that sentences are the primary bearers of truth. This spares a lot of troubles, as there is no debate concerning the nature of sentences. Also many works on the liar problem that prefer propositions as truth bearers can be found in the literature. Most of these approaches consider their proposal as independent of any particular ontological view on the nature of propositions. Some of them will be subject of the fourth chapter. The liar paradox consists in a contradiction, which results from a socalled liar sentence. A common version of a liar sentence is, for instance, the following sentence: (L1 ) This very sentence is false. Another popular version of a liar sentence is the following: (L2 ) What I am now saying is false. Let us assume a liar sentence L, such that L says of itself that it is false and nothing else. The liar paradox arises from any attempt to answer the question whether L is true or not. Assume the initial answer is ‘yes’. Then what L says, namely that it is false, is true. So, by convention T , L is false, which contradicts the initial answer. The only alternative (in classical logic) is then that L is false. But if so, what L says, namely that L false, is true. Therefore L turns out to be true, if and only if it is false. The just given argument is called a ‘(sentential) liar argument’, or, since a contradiction is derived from exclusively harmless looking assumptions, a ‘(sentential) liar paradox’. It turns out that natural languages provide several ways to create liar sentences. If we e.g. want to avoid indexicals

9 as ‘this’, ‘I’ or ‘now’, we might prefer the following liar sentence version (cf. K. Simmons[33]). Assume that in room 320 of the Department of Philosophy at the University of Salzburg there is only one blackboard, on that only the following sentence is written on the left upper corner. (L3 ) The sentence written on the left upper corner of the blackboard in room 320 of the Department of Philosophy at the University of Salzburg is false. Also for L3 a liar argument, such as the one stated above, can be obtained. We learn from the liar paradox that at least one of its seemingly harmless premises must in fact be problematic. The first reaction to the liar paradox is mostly to wonder whether there is something wrong with the liar sentence. Literally interpreted it says of itself that it is false. But is this literal interpretation really adequate? Many philosophers have argued that the liar sentence in fact does not say anything at all. What might seem suspicious is that L1 and L2 both contain a context dependent expression. This could lead to the impression that a liar sentence must necessarily be context dependent.2 So there are situations, in which the utterance of the sentences would do no harm, and other situations, in which they give rise to a paradox. However, due to the outstanding works of K. G¨odel [13] and A. Tarski[36], liar sentences without any context dependent expression became known. Thus the contextual elements in the above stated liar sentences (such as ‘this’ etc.) are not essential to the problem of the liar paradox. G¨odel and Tarski have discovered their construction of a liar sentence independently in 1930 (although Tarski published his discovery later). Both logicians have carried out their investigations in context of formal languages of predicate logic. G¨odel’s and Tarski’s liar sentence can roughly be reproduced in informal English as follows: let the norm of any English expression A be the result of appending the citation ‘A’ of A to A. Then consider the following sentence: 2 Sentence

L3 is usually not regarded as containing a context-dependent expression. One could however argue that ‘is a corner’ is a vague expression.

10

Introduction (L4 ) Among the false sentences is the norm of ‘Among the false sentences is the norm of’.

Again an argument analogous to the liar argument stated above can be derived. The norm of ‘Among the false sentences is the norm of’ is the sentence ‘Among the false sentences is the norm of ‘Among the false sentences is the norm of’ ’. Assume that L4 is true. Then, by convention T , the norm of ‘Among the false sentences is the norm of’ is among the false sentences, in other words, the norm of ‘Among the false sentences is the norm of’ is a false sentence. Since the norm of ‘Among the false sentences is the norm of’ is equal to the sentence ‘Among the false sentences is the norm of ‘Among the false sentences is the norm of’ ’ it follows, by substitution of identities, that ‘Among the false sentences is the norm of ‘Among the false sentences is the norm of’ ’ is false. So it follows that L4 is false. Assume that L4 is false. Then by convention T , the norm of ‘Among the false sentences is the norm of’ is not among the false sentences, in other words, the norm of ‘Among the false sentences is the norm of’ is not a false sentence. Since the norm of ‘Among the false sentences is the norm of’ is equal to the sentence ‘Among the false sentences is the norm of ‘Among the false sentences is the norm of’ ’ it follows, by substitution of identities, that ‘Among the false sentences is the norm of ‘Among the false sentences is the norm of’ ’ is not a false sentence. Therefore L4 is true. Thus L4 is true, if and only if it is false, just like the above informal liar sentence L. So liar sentences do not necessarily contain a context dependent expression. One could still think of many other reasons for why the liar sentence does not express a proposition. There are many kinds of grammatically well formed sentences for which it is doubtful whether they express anything that is true or false. For example sentences that are neither true, nor false because of a category mistake (e.g. ‘Two is a green number.’).3 But whatever semantical diagnosis we propose to explain that liar sentences are neither true nor false, it will not be of much help to solve the liar paradox, because of another phenomenon which is called the ‘strengthened liar paradox’. This paradox arises from the so-called strengthened liar 3A

lot of other examples can be found in Sainsbury[31].

11 sentence S, such that S says of itself that it is not true and nothing else. S must be either true, or not true. But this leads straight back into a paradox: Assume S is true. Then what S says, namely that it is not true, is true. So, by convention T , L is not true, which contradicts the initial answer. The only alternative is that S is not true. But if so, what S says, namely that S not true, is true. Therefore S turns out to be true, if and only if it is not true. As stated above, contextual approaches argue that a semantic shift obtains in a certain step of the just given argument. This semantic shift induces a change from one initial concept of truth to another concept of truth. Accordingly, the strengthened liar sentence is true according to one of these di↵erent concepts, and it is not true according to the other concept. To doubt that S is either true or not true would mean to treat ‘is not true’ equal to ‘is false’, and to assume ‘not true’ to display a weak concept of negation. In classical logic the negation concept is displayed by the following truth table: A

not-True(‘A’)

true f alse

f alse true

If we extend our logic by an extra truth value, which represents an alternative semantic state, there are several ways of how to extend the above truth table. Here are the three-valued truth tables for the weak and, respectively, the strong negation concept: A

notW -True(‘A’)

A

notS -True(‘A’)

true f alse def ective

f alse true def ective

true f alse def ective

f alse true true

It is certainly a radical restriction of the expressive means of a language to assume that its concept of ‘not’ is generally understood in the sense

12

Introduction

of a weak negation. Also no such restriction can be found in the natural languages. Even if someone would insist on a strict weak reading of ‘not’ the problem is not solved, as we can construct a strengthened liar sentence from the following concept of ‘untrueness’: Assume a sentence is called ‘semantically defective’ i↵ it is neither true, nor notW true, which is represented by the following three-valued truth table: A

SemDef(‘A’)

true f alse def ective

f alse f alse true

Let a sentence be called ‘untrue’ i↵ it is either notW true or semantically defective, and let S be a sentence denoting ‘S is untrue’. Then (if we assume that no sentence can be true as well as false, and that no sentence can be true as well as semantically defective) we are back in a paradox, as S can neither be true, nor it can be false, nor it can be semantically defective. To avoid this unpleasant consequence, one has to assure that no concept of semantical defectiveness, such as the one just stated, is expressible in the given language. This is, again, a radical limitation of the expressive means of the language, which cannot be found in natural languages. We see that the strengthened liar problem in fact consists in the following dilemma. First, there is the problem that many solutions for the original liar problem give rise to other strengthened liar sentences which cannot be solved by the solution. This is often called the ‘revenge paradox’, or the ‘revenge problem’. To avoid revenge paradoxes resulting from a strengthened liar sentence these solutions must admit grave expressive limitations of the language proposed. But this leads to another problem, which has been called the ‘unable-to-say problem’: because of the expressive limitations of the language, many semantical statements about the strengthened liar sentence, such as declaring the liar sentence as not true in the sense of a strong negation, or declaring the liar sentence as semantically defective in the sense stated above, cannot be expressed in the very language itself, but only in a more comprehensive meta-language. If these statements were

13 expressible in the object language, we would immediately land in another revenge paradox. So, in order to avoid revenge paradoxes, we are forced to accept serious expressive limitations of our object language with respect to the means to express its own semantics, which implies that we cannot express certain semantic statements about strengthened liar sentences. To many, in particular to proponents of contextual accounts, this strengthened liar paradox is the core of the liar problem. Therefore, the terms ‘liar’ and ‘strengthened liar’ are often used synonymously (cf. Glanzberg[12], p. 3, footnote 1). Contextual approaches argue that there is a semantic shift in the course of the above strengthened liar argument, which induces a change from one initial concept of truth to another concept of truth. This means that the strengthened liar sentence is true according to one of these di↵erent concepts and not true according to the other one. As I have mentioned above, Tarski and G¨odel have carried out their investigations within a formal language of predicate logic. This formalization is a historical step in the philosophical research on the problem of truth, which has, ultimately, led to a new research field of philosophical logic. Also contextual approaches belong to this field. For this reason, I begin with introducing a formal language of first order predicate logic in the second chapter. Furthermore, I shall demonstrate how the liar problem can be formally reconstructed in this language. In chapter three, I shall present several formal truth theories, which I call ‘theories of partial truth’. They form the essential components of the contextual reconstructions of the strengthened liar argument, which are introduced in chapters four and five. In the fourth chapter, I shall formally reconstruct the strengthened liar argument in case of the formal language, which was introduced in chapter two, and its partial truth predicate, which was introduced in chapter three. I shall also clarify how contextual approaches to the strengthened liar problem are formally realized, and I shall be concerned with the question at which position of the strengthened liar argument a semantic shift is presumed by contextual approaches. I shall demonstrate that three prominent contextual approaches each agree with respect to this question and I shall put forward the corresponding contextual reconstruction scheme CR1 of the strengthened liar argument in the framework of the formal language from chapter two. In the fifth chapter, I shall show how another

14

Introduction

instance of CR1 can be obtained by an iteration of Kripke’s account. Furthermore, I shall propose an alternative contextual reconstruction scheme CR2 of the strengthened liar argument, in which a di↵erent position of the semantic shift is assumed. I will also show how an instance of CR2 can be obtained by a minor variation of the iteration of Kripke’s account, which was previously introduced in case of CR1. Finally, I shall analyze a very serious objection against contextual approaches, namely the objection that they are unnatural.

2 The Problem of Truth and the Liar Paradox in Formal Languages In this chapter, I shall present a reconstruction of the liar problem in formal languages. There are many techniques of formally presenting the liar problem and its conditions. Mostly this is done within a first order language of arithmetic. To o↵er a clear presentation of my account, I shall introduce a formal language in section 2.1. In section 2.2, I shall demonstrate how a liar problem can be reconstructed in this formal language. In its formally reconstructed version the liar problem is expressed by the so-called undefinability theorem of truth, which is the central result of section 2.2. In section 2.3, I shall give a list of conditions that lead to the undefinability theorem.

2.1 The Formal Language Let a signature S be a set, the elements of which are called constants, function symbols and predicate symbols. I shall use ‘c’, ‘c1 ’, ‘c2 ’, . . . to denote constants in S, ‘f ’, ‘f1 ’, ‘f2 ’, . . . to denote function symbols in S, and ‘P ’, ‘P1 ’, ‘P2 ’, . . . to denote predicate symbols in S. Furthermore, let ‘⌘’ be the identity symbol in S. If S is finite, S can be given by a list, i.e., S = (c1 , . . . , cl , f1 , . . . , fm , P1 , . . . , Pn ). Let AS be the alphabet over S, which apart from all symbols in S contains first order variables, logical symbols and auxiliary symbols. I shall use ‘x’, ‘y’, ‘z’, ‘u’, ‘x1 ’, ‘x2 ’, . . . as meta-variables denoting variables of S. The terms and formulas over AS shall be formed by the usual inductive rules. Let L be the set of formulas over AS and let T be the set of terms over AS . As usual, I shall call L a (first order) language (over AS )indexfirst order language L a over AS . Furthermore, I shall sometimes refer to S by ‘the signature of L’, to AS by ‘the alphabet of L’, and to T ‘the terms of L’. I shall use ‘'’, ‘ ’, ‘'1 ’, ‘'2 ’,

16 The Problem of Truth and the Liar Paradox in Formal Languages . . . as meta-variables denoting formulas, and ‘t1 ’, ‘t2 ’, . . . as meta-variables denoting terms. I shall denote the result of simultaneously substituting the terms t1 , t2 , . . . , tn as meta-variables denoting terms. I shall denote the result of simultaneously substituting the terms t1 , t2 , . . . , tn for the n variables x1 , x2 , . . . , xn in ' by ‘[']tx11,t,x2 ,...,t ’. Finally, for any formula ', 2 ,...,xn let F ree(') be the set of free variables occurring in '. The formulas ' of L with F ree(') = ? are called ‘the sentences of L’. To specify a semantics of L, I shall assume an S-structure M for L, i.e.: M = (D, I S ), where D is a non-empty set, called the domain of discourse of M, and I S is an interpretation function that assigns elements of D to the constants in S, n-ary functions on D to the n-place function symbols in S, and subsets of D n to the n-place predicates in S. If it is clear from the context to which S it is referred, I shall omit ‘S’ in ‘I S ’. Each Sstructure M uniquely determines a first order language L, which I shall simply call ‘the first order language of M’. A pair (L, M), such that M is a structure for L, is called an interpreted language. For any L and M, I shall sometimes refer to the corresponding interpreted language (L, M) by ‘LM ’. ‘V alM ’ shall denote the semantic valuation function, which assigns a semantic value to each term of L according to the well-known rules of predicate logic with identity. Let ‘✏’ denote the usual satisfaction relation between a structure M and a sentence ', i.e. M ✏ ' is equivalent to V alM (') = t. If M ✏ ', I shall say ‘M is a model for ', or ‘' is made true by M’, or‘' is a true sentence of LM ’.AnysignatureS+ , such that S ✓ S + , is called ‘an extension of S by the symbols in S + \S’. The formal language L+ determined by S + is then called ‘an (S + -)extension of L’. Any S + -structure M+ = (D+ , I + ), such that D + = D and I + |S = I is called ‘an (S + -)expansion of M = (D, I)’. Up to now, all we have assumed is that LM is a first order language of predicate logic with identity. Additionally to this I shall mostly also assume that L is a recursive language, in the sense that its alphabet is a recursive set.

2.2 The Undefinability of Truth

17

2.2 The Undefinability of Truth In order to contain a liar sentence, LM must have the ability to “describe its own syntax” (cf. McGee[26], p. 18). This means, first, that certain objects in D serve as names, in other words, meta-language codes for the expressions of L. Furthermore, for each such meta-language code of an expression of L, there is a term of L, which denotes this meta-language code. Finally, certain relations and functions that belong to the theory of the syntax of LM must be definable in LM . LM provides meta-language codes for its expressions, if to each expression of LM an unique object in D can be assigned. This can be accomplished by any injective function from L [ T into D. In English a natural option for the code of a sentence can simply be derived by quoting the sentence. The quotation of a sentence is also a structural descriptive name (cf. Tarski[36]), that is, it is functional with respect to the logical connectives, quantifiers and substitution. ‘Socrates is mortal and Plato is mortal’ is, e.g., the concatenation of ‘Socrates is mortal’, ‘and’ and ‘Plato is mortal’. An injective mapping from L [ T into D can be obtained by combining two of the following types of mappings: An injective mapping from AS S into D, and a bijective mapping from n 0 D n to D. First, each symbol is substituted by its corresponding object d 2 D according to the first injective mapping 1S . Secondly, the resulting finite sequence of objects in D is mapped to its corresponding d 2 D according to the second bijective mapping 2 . I shall from now on assume that there is an injective function S S from A into D, and a bijective mapping n 1 from 2 S 1 n 0 D to D. If it is clear from the context to which S it is referred, I shall sometimes omit ‘S’ in ‘ 1S ’. We then obtain an injective function S : L [ T ! D by combining 1S and 2 , i.e.: S (↵

1

↵2 . . . ↵n ) =

S S S 2 ( 1 (↵1 ), 1 (↵2 ), . . . , 1 (↵n )),

where ↵1 ↵2 . . . ↵n is the concatenation of the symbols ↵1 , ↵2 . . . ↵n 2 AS . I shall call S a coding function for M, and I shall call S (') ( S (t)) the 1 Instead

of an injective function that AS ✓ D.

S 1

from AS into D, we can, for simplicity, also assume

18 The Problem of Truth and the Liar Paradox in Formal Languages meta-language code of ' (of t). Let us introduce some further useful notions related to S . First, let p'q (ptq) denote a term the interpretation of which is the meta-language code of ' (of t), i.e., I(p'q) =

S (')

(I(ptq) =

S (t)).2

I shall call p'q (ptq) the object language code of ' (of t). Furthermore, let _. be a function symbol in S representing the primitive recursive function that takes the meta-language codes of any formulas ' and to the metalanguage code of ' _ . Analogously, let the primitive recursive functions corresponding to all other logical symbols be represented by the respective dot-notation in S. I shall frequently use brackets ‘(’ and ‘)’ to improve the legibility of nested applications of dot-notations. Let num be a function symbol in S representing the primitive recursive function that takes each object d such that I(t) = d to the object d0 which is the meta-language code of t. Thus M ✏ num(t) ⌘ ptq. L needs to contain enough terms to denote each d 2 D (or at least enough terms to denote all those objects that are meta-language codes of some expressions). Otherwise some expressions could have no object language code in the language L although they have a meta-language code in D. Therefore, I shall assume that for each d 2 D, there is a term t1 2 T , such that I(t1 ) = d. This means that I assume that I is a function from T onto D. For notational simplicity, I shall even assume that for each d 2 D, ¯ denote a there is a closed term t1 2 T denoting d. For each d 2 D, let ‘d’ ¯ = d. closed term in T such that I(d) Another condition LM has to meet to “describe its own syntax” is that certain predicates and functions, which belong to the theory of the syntax of LM , must be definable in LM . The specification of ‘LM describes its own syntax’ will ultimately lead us to the notion of an acceptable structure (definition 2.3), which is frequently utilized in the literature on formal theories of truth (see e.g. McGee[26]). 2 There

is at least one term t1 , such that I(t1 ) = S (') (I(t1 ) = S (t)), since for every d 2 D, there is a term t1 denoting d. The arguments below will not depend on the assumption that the set of terms denoting S (') ( S (t)) is recursively enumerable.

2.2 The Undefinability of Truth

19

By ‘definability in LM ’ I refer to the following usual notion. Definition 2.1 Definability of an n-place relation on D in LM and definability of an n-place function on D in LM . 1. An n-place relation R on D (i.e. a subset of Dn ) is definable in LM i↵ there is a formula ' with F ree(') = {x1 , x2 , . . . , xn } such that for ¯ ¯ d¯n any (d1 , d2 , . . . , dn ) 2 Dn , M ✏ [']dx11,,xd22,..., ,...,xn i↵ (d1 , d2 , . . . , dn ) 2 R. I shall say that ' represents R in LM . 2. An n-place function F on D is definable in LM i↵ there is a term t with F ree(t) = {x1 , x2 , . . . , xn } such that for any (d1 , d2 , . . . , dn+1 ) 2 ¯ ¯ d¯n ¯ Dn+1 , M ✏ [t]dx11,,xd22,..., ,...,xn ⌘ dn+1 i↵ F (d1 , d2 , . . . , dn ) = dn+1 . I shall say that t represents F in LM . I also want to introduce the following related notions: Definition 2.2 Implicit definition, explicit definition of P . Let LP be the language L extended by a new predicate symbol P , and let M = (D, I) be a structure for L. For any E ✓ D, let (M, E) the expansion of M to a model of LP with I(P ) = E. Then 1. A sentence of LP is an implicit definition of P i↵ for any two E1 , E2 with E1 ✓ D and E2 ✓ D: If (M, E1 ) ✏ and (M, E2 ) ✏ , then E1 = E2 . I shall say that implicitly defines P in LP . 2. A sentence is an explicit definition of P i↵ there is a one-place formula ' of L such that = 8x(P (x) $ '). I shall say that explicitly defines P in LP . By Beth’s definability theorem there is an explicit definition of P , if and only if there is an implicit definition of P . Thus, if is an explicit or implicit definition of P , then there is an uniquely determined E ✓ D, such that (M, E) ✏ . Let this uniquely determined E be called the extension of P determined by and M. If the relation R is definable in LM , then there is an implicit (explicit) definition of an n-place predicate P in LP such that R is equal to the extension of P determined by and M. If is

20 The Problem of Truth and the Liar Paradox in Formal Languages an implicit (explicit) definition of an n-place predicate P in LP , then the extension of P determined by and M is definable in LM. Some fundamental syntactic concepts, such as the length of an expression of LM , are based on a concept of natural numbers. Therefore it is useful to assume that there is a set N , N ✓ D, and there is a relation symbol ,  2 S, such that the -structure N := (N, I |N ()) is isomorphic to (N [ {0}, ), i.e. the structure of natural numbers (where ‘’ denotes the usual ordering on natural numbers). Definition 2.3 An acceptable structure M. I. Let M = (D, I) be any structure such that S 1. there is a bijective mapping 2 : n 0 D n ! D,3 and

2. there is a set N , N ✓ D, and there is a relation symbol  2 S, such that (N, I |N ()) is isomorphic to the structure of natural numbers (N [ {0}, ).

Then, the triple C = hN, I |N (),

2i

is called a coding scheme for M.

II. M is called an acceptable structure i↵ there is a coding scheme C for M, and the following subset of D and functions on D (D 2 ) are definable in LM : 1. d 2 Sequence i↵ for some d1 , d2 , . . . , dn 2 D, d = 2 ((d1 , d2 , . . . , dn )) 8 > / Sequence < 0 if d 2 2. Length(d) = > : n if d 2 Sequence and d = ((d , d , . . . d )) 2 1 2 n 8 > di if for some d1 , d2 , . . . , dn : > > < d = 2 ((d1 , d2 , . . . , dn )) 3. Element(d, i) = > > > : 0 else

3 In

fact condition I.1 of definition 2.3 is superfluous, as we have already assumed that S there is a bijective mapping 2 : n 0 D n ! D. However, in order not to confuse the reader, I retain this condition, as it is usually presupposed in the definition of ‘acceptable structure’.

2.2 The Undefinability of Truth

21

The set Sequence is the set of all elements in D that are sequences. The function Length(d) returns the length of the sequence associated with d, if existent. Element(d, i) returns the i-th element of the sequence associated with d, if existent. From now on, I shall assume that M is an acceptable structure. Furthermore, let us assume that a one-place predicate symbol SeqL , a one-place function symbol lhL and a two-place function symbol qL are available in S such that SeqL (x) represents the set Sequence in LM , lhL (x) represents the function Length in LM , and qL (x, y) represents the function Element in LM . If it is clear from the context which L is referred to, I shall omit the subscript ‘L’. For any signature S, let VS be the set of all objects d 2 D that are S -codes of variables of A , let C be the set of all objects d 2 D that are S S 1 S -codes of constants of A , let F be the set of all objects d 2 D that S S 1 S are 1 -codes of function symbols of AS , and let PS the set of all objects d 2 D that are 1S -codes of predicates of AS . It is necessary to assume that these four sets VS , CS , FS and PS are definable in LM . Let VS , CS , FS and PS be defined by the predicates V arL (x), CstL (x), F ctL (x) and P rdL (x), I assume to be available in S. Any recursive language, i.e. any language such that VS , CS , FS and PS are recursive sets, meets these conditions. Up to now, we have assumed that LM is a a first order language of predicate logic with identity that is interpreted by a structure M = (D, I) such that S1 There is a coding function

S

for M.

S2 I S is a function from T onto D. S3 M is an acceptable structure. S4 The sets V S , C S , F S and P S are definable in LM . The prime example for languages that meet these conditions are first order languages of arithmetic (cf. McGee[26], p. 39). Thus, e.g. all structures that meet the axioms of Peano arithmetic are acceptable structures. As a consequence of the four conditions above, the following function Sub is definable in LM . A proof of this can be found in many textbooks,

22 The Problem of Truth and the Liar Paradox in Formal Languages such as Shoenfield[32]. Definition 2.4 The substitution function Sub. Where d1 , d2 and d3 are any objects in D,

8 ([']tx ) if there is a formula ', > > > > > and there is a variable x, > > > > and there is a term t, < Sub : D3 7! D, Sub(d1 , d2 , d3 ) = such that d1 = ('), > > > d2 = (x) and d3 = (t) > > > > > > : (?) else

Let 'Sub (x, y, z) be a term expressing the function Sub in LM . In addition I shall assume that there is a function symbol sub in S such that M ✏ sub(x, y, z) ⌘ w $ 'Sub (x, y, z) ⌘ w, and in consequence I(sub) = Sub. Then sub(x, y, z) is defined as the object language code of the formula that results from the formula with object language code x by substituting all free occurrences of the variable with object language code y by the term with object language code z. The following diagonalization function Diag maps the meta-language p'(x)q code of each '(x) to the meta-language code of ['] x . Definition 2.5 The diagonalization function Diag. Diag(d1 ) = Sub(d1 , (x), d1 ), i.e.:

8 > ([']px'q ) if there is a formula ' > > > > such that d1 = (') < Diag : D 7! D, Diag(d1 ) = and x 2 F ree(') > > > > > : (?) else

Since Sub is definable in LM , also Diag is definable in LM . Let 'Diag (x) be any term expressing Diag in LM . In addition I shall assume that there is a function symbol diag in S such that M ✏ diag(x) ⌘ y $ 'Diag (x) ⌘ y. Therefore, I(diag) = Diag. In the rest of this section I shall abbreviate

2.2 The Undefinability of Truth

23

[']tx by '(t), for any formula ' with F ree(') = {x}. In consequence, (⇤) M ✏ diag(p'q) ⌘ p'(p'q)q, where ' is a formula such that x 2 F ree('). Definition 2.6 Fixed point sentences. Let be any one-place formula of LM , such that F ree( ) = {x}. Then a sentence ' is called a fixed point sentence of i↵ M ✏ ' $ (p'q). Lemma 2.1 Diagonalization lemma. For every one-place formula of LM , such that F ree( ) = {x}, there is a fixed point sentence ' of . Proof Consider the formula ' = (diag(p (diag(x))q)). By (⇤), we obtain M ✏ diag(p (diag(x))q) ⌘ p (diag(p (diag(x))q))q. By substitution of identities, it follows that M ✏ (diag(p (diag(x))q)) $ (p (diag(p (diag(x))q))q) or, resp., M ✏ ' $ (p'q)

A formalized liar paradox arises from the question whether the set of true sentences of LM is definable in LM . The set of true sentences of LM is the set of objects in D that are meta-language codes of sentences that are made true by M. Let me denote this set by ‘T rue(L,M) ’, and let L0 be the set of all sentences of L. Then, for each d 2 D, it holds that d 2 T rue(L,M) i↵ there is a ' 2 L0 , such that d = I(p'q) and V alM (') = t. Assume there is a one-place predicate T r in S, such that T r(x) represents T rue(L,M) in LM . Then

24 The Problem of Truth and the Liar Paradox in Formal Languages

I(T r) = {there is a ' 2 L0 , such that d = I(p'q) and V alM (') = t}. Note that this expression is circular, as V alM is defined in terms of I and I is defined in terms of V alM . I shall now show that T rue(L,M) is not definable in LM . Lemma 2.2 If T rue(L,M) is definable in LM , then there is a one-place formula such that M ✏ (p'q) $ ' for all formulas '. Proof T rue(L,M) is definable in LM i↵ represents T rue(L,M) in LM i↵

there is a

2 L, such that

there is a

¯ i↵ d 2 T rue(L,M) . 2 L, such that for all d 2 D, M ✏ (d)

Therefore, for any d 2 D, such that there is a ' 2 L with d = I(p'q): M ✏ (p'q) i↵ V al(') = t, which is equivalent to M ✏ (p'q) $ ' for every formula '. From now on, I shall assume the following terminology: Definition 2.7 A complete truth predicate. represents complete truth in LM i↵ M ✏ (p'q) $ ' for all formulas '. Let any one-place predicate T r in S be called a complete truth predicate with respect to LM i↵ T r(x) represents complete truth in LM . Theorem 2.1 Undefinability theorem of (complete) truth. 1. T rue(L,M) is not definable in LM . 2. There is no formula in LM .

of LM such that

represents complete truth

2.2 The Undefinability of Truth

25

3. There is no complete truth predicate in LM . Proof The first claim follows from the second by lemma 2.2. The third claim follows immediately from the second claim. The second claim can be proven by considering the fixed point sentence ' of ¬ (where is M assumed to define complete truth w.r.t. L ). The existence of ' leads to a contradiction because of M ✏ ' $ ¬ (p'q) (since ' is the fixed point sentence of ¬ ), M ✏ ' $ (p'q) (since

defines complete truth).

Thus M ✏ ¬ (p'q) $ (p'q), which proves that there cannot be any formula in LM that completely represents truth in LM . The liar paradox plays an indirect role in the proof of the undefinability theorem of truth. Assuming one of the following three properties for LM , CT1 T rue(L,M) is definable in LM ; CT2 There is a formula

that represents complete truth in LM ;

CT3 There is a complete truth predicate in LM ; implies the existence of a formal liar sentence, which is the fixed point sentence of the negated one-place formula that defines complete truth. I.e., in case that T r(x) is a complete truth predicate in LM , the formal liar sentence is a sentence such that M ✏ $ ¬T r(p q). In the corresponding informal liar argument, the existence of a complete truth predicate is assumed for reductio, but it is taken as a feature of natural language semantics. This leads to a genuine paradox. The undefinability theorem gives a very general and negative answer to the problem of defining truth. I shall now show another related aspect of this result, which exhibits the problem of characterizing truth by a theory

26 The Problem of Truth and the Liar Paradox in Formal Languages in LM . Up to now I have only considered the predicate T r and one-place formulas that define this predicate. But we can also “partially define” T r by a theory. A theory in L is a deductively closed subset of sentences in L, i.e. it is closed under the first order derivability relation `.4 A consistent theory is any theory that does not entail a contradiction, or, equivalently stated, that is di↵erent from the set of all sentences. Definition 2.8 E is characterizable in (L+ , M+ ), is a theory of P and characterizes P in L+ . Let LM be an interpreted language, and let S be the signature of L. Let S + be any extension of S by a one-place predicate P , and let L+ be the S + -extension of L. For each E ✓ D, let (M, E) denote the S + -expansion of M according to which P is interpreted by the set E. Let E be a collection of subsets of D. Then I shall say that E is characterizable in (L+ , M+ ) i↵ there is a theory in L+ , such that E := {E ✓ D : (M, E) ✏ }5 In this case, I shall say that is a theory of P . Furthermore I shall + say that characterizes P in L . A theory is often stated by a set of sentences of L+, such that the closure of under first order consequence is . If is recursive, then is called ‘an axiom system’, and is called ‘axiomatizable’. In this case I shall call ‘an axiomatic theory of truth’. Sometimes is also stated within a suitable and usually more “comprehensive” meta-language. For example, Kripke gives a semi-formal description of his theory within English enriched by mathematical presuppositions such as parts of informal set-theory (cf. section 5.1). Another option for more “comprehensive” meta-languages of L are languages of second order predicate logic. The advantage of an 4`

is defined as usual (cf. e.g. Rautenberg[29]. A sentence ' is derivable from ( ` ') i↵ there is a finite sequence of sentences ending with ', each member of which is either an axiom of first order predicate logic, or it is a sentence in , or it follows from preceding members of the sequence by the modus ponens rule. 5 ✏, now, is the simultaneous satisfaction relation.

2.2 The Undefinability of Truth

27

axiomatizable theory is that there is no need for a more “comprehensive” meta-language (compare Halbach[16]). On the other hand, some interesting extensions of T r do not allow for theories that are axiomatizable, and therefore are not amenable to the axiomatic approach. As a consequence of the above undefinability theorem, not all T r-biconditionals will be derivable from any theory in L+ , unless is inconsistent. Definition 2.9 A theory of complete truth in (L+ , M+ ). Let LM be an interpreted language, and let S be the signature of L. Let S + be an extension of S by the one-place predicate symbol ‘T r’, and let be a theory in L+ . If for all ' of L+ , ` T r(p'q) $ ', then let me say that characterizes complete truth in (L+, M+), and that is a theory of complete truth in (L+ , M+ ). Thus the following corollary expresses that there is no consistent theory of complete truth in (L+ , M+ ). Corollary 2.1 Let S + be an extension of S by the one-place predicate symbol ‘T r’. Let be a theory in L+ , such that CT4 Then

is a theory of complete truth in (L+ , M+ ). is inconsistent.

Corollary 2.1 is an substantial strengthening of theorem 2.1. Not only is there no definition of complete truth, but also there is no theory of complete truth. At this point, let me note that before we step from our initial model M to a new expanded model M+ , we should be aware of the following problem: it is desirable for M+ to meet conditions S1 S4. If M+ does not meet conditions S1 S4, M+ would not help to explain the liar problem as it occurs in natural language. Indeed it can hardly be doubted that, in natural language, a rich theory of syntax that allows for claims corresponding to S1 S4, and in consequence, functions such as Sub and Diag, is available. However, supposing M+ to meet S1 S4 is a rather modest assumption. Of course S1 will hold in M+, as the injective func-

28 The Problem of Truth and the Liar Paradox in Formal Languages tion 1S can be adapted to the extended signature S + . S2 will also be met by M+ as I S + |S = I S . Any coding scheme C for M will also be a coding scheme for M+ , and Sequence, Length(d) and Element(d, i) will also be definable in (L+ , M+ ). So S3 will be met in M+ . Only regarding S4 we have to be cautious. Since M+ is an expansion of M, the interpretations of V ar(x), Cst(x), F ct(x) and P rd(x) according to M+ are V S , C S , F S and + P S . But note that 1S will in general be di↵erent from 1S . Thus in general + + + + V S 6= V S , C S 6= C S , F S 6= F S and P S 6= P S . For M+ to meet S4 + + + + however, we need V S , C S , F S and P S to be definable in (L+ , M+ ). This requirement does not follow from what we have assumed for M+ so far. I propose to accommodate this by introducing certain kinds of ex+ tensions of 1S to a coding function 1S of S + ; I shall call them standard extensions. Assume that there is a recursively enumerable subset D0 of D, such that for no element d 2 D0 , there is a symbol s 2 AS , such that d is the 1S -code of s.6 Let d0 , d1 , d2 , . . . be an enumeration of D0 . In the following definition 2.10, only one d 2 D, such there is no symbol s 2 AS with d is the 1S (s) = d, is required. In the fifth chapter, we will however need an infinite set of such d 2 D. Definition 2.10 Standard extensions of of M. S+ S 1 is a standard extension of 1 i↵ •

S+ 1 (T r)

•

S+ 1 (s)

S 1

and standard S + -expansion

:= d0 ;

:=

S 1 (s)

for each symbol s 2 AS . +

Let any S + -expansion M+ , such that the injective function 1S from AS + to D is a standard extension of 1S , be called a standard S + -expansion of M. If we assume that

S+ 1

is a standard expansion of

+

• V S = V S; +

• CS = CS;

6 Otherwise,

S 1

can be rearranged accordingly.

S 1,

then

2.3 The Conditions of the Liar Paradox

29

+

• F S = F S. +

• P S = P S [ {d0 }.

2.3 The Conditions of the Liar Paradox To summarize the results of the previous sections, let me state a list of all conditions for our formal reconstruction of the liar problem. We have assumed that our first order language of predicate logic LM has the following features: S1 There is a coding function

S

for M.

S2 I S is a function from T onto D. S3 M is an acceptable structure. S4 The sets V S , C S , F S and P S are definable in LM . CT1 T rue(L,M) is definable in LM . CT2 There is a formula

which represents complete truth in LM .

CT3 There is a complete truth predicate in LM . CT4

is a theory of complete truth in (L+ , M+ ).

As demonstrated in this chapter, assuming S1 S4 together with either CT1, CT2, CT3 or CT4 for (L, M) ((L+ , M+ )) leads to a contradiction, or, in case of CT4, to the inconsistency of . It seems that each of these conditions display very strong intuitions concerning natural language and its concept of truth, and it is not easy, or even not possible to rationally decide which of these intuitions is the strongest and most fundamental one. To solve the liar problem we need to reconsider these conditions, and find alternative conditions by which we can replace them. Any such alternative approaches will thus be to some extent counter-intuitive. Nearly all accounts to solve the liar problem are focusing only on the conditions CT1, CT2, CT3 or CT4 concerning the formal representation of truth in

30 The Problem of Truth and the Liar Paradox in Formal Languages LM .7 Restricting conditions S1 S4 would mean to take away from LM the ability of expressing a basic theory of syntax, or resp. of arithmetic. One would have to put forward an alternative restricted theory of syntax, resp., arithmetic, such that not all recursive functions (such as Diag) are definable. Although this could be an interesting challenge, I personally doubt that such a system would really help to gain any new insights for the liar problem in natural language, as in natural language a rich theory of arithmetic and syntax, that allows for claims corresponding to S1 S4, and in consequence allows for the expressibility of functions corresponding to Sub and Diag, is available.

7 Let

me note that besides conditions CT1 CT4, there is indeed another feature we have assumed of our formal representation of truth: Truth is represented by a predicate. It might also be represented by an operator, which is not interpreted by a subset of D and which in general does not meet the rule of substitution of identities. There are only very few and not much elaborated approaches that have seriously investigated this option. Such approaches shall not be discussed here.

3 Theories of Partial Truth In this chapter, I shall introduce two examples for theories of partial truth. Theories of partial truth are formal theories of truth that give up condition CT4 from above. They are the essential components of the contextual reconstructions of the strengthened liar argument that are introduced in chapters four and five. Two di↵erent sorts of theories of partial truth, which I shall call the theories of classical truth and the theories of iterative truth, are dominant in the literature. The theories of classical truth are inspired by Tarski’s influential work (Tarski[36], 1935 and Tarski[37], 1969). In section 3.1, I shall state a simple version of a classical theory. Most current theories of iterative truth originate from Kripke’s concept of truth, which is presented in Kripke[21], 1975. Many of these theories introduce their concept of truth together with a concept of groundedness. In section 3.2, I shall demonstrate how an axiom system for truth and groundedness can be developed. We will need this axiom system in chapters four and five. For the rest of this chapter, we assume that LM is a first order language of predicate logic, that meets the conditions S1 S4, which were explicated in section 2.2. Let S be the signature of L and let S + always be the extension of S by a one-place predicate T r, let L+ be the S + -extension of L, and let M+ be any S + -expansion of M. Furthermore, let L0 be the set + of all sentences of L, and let L+ 0 be the set of all sentences of L . Then I shall assume theories of partial truth to be defined as follows: Definition 3.1 Partial truth predicate and strictly partial truth predicate. Let be any set such that L0 ✓ ⇢ L+ 0. 1. Then T r is called a partial truth predicate in (L+, M+) i↵ M+ ✏ ' $ T r(p'q) for all ' of . Let be any theory in L+. Then I shall say that

32

Theories of Partial Truth

T r is a partial truth predicate with respect to i↵ ` ' $ T r(p'q) for all ' of . Furthermore, any theory in L+ , such that T r is a partial truth predicate with respect to , shall be called a theory of partial truth. 2. T r is called a strictly partial truth predicate in (L+ , M+ ) i↵ M+ ✏ ' $ T r(p'q) for all ' 2 , and M+ ✏ ¬(' $ T r(p'q)) for all ' 2 / . So any theory of partial truth will give rise to a division of the sentences of L+ into two groups, one group of sentences that meet the T r-scheme and the other group of sentences, for which we do not know whether they meet the T r-scheme. In consequence of the undefinability theorem of truth, 6= L+ . If T r is a strictly partial truth predicate, then the sentences outside of are often called ‘pathological’ or ‘semantically defective’ to indicate that they cannot be evaluated by a conventional, classical semantics. The rest of this chapter is devoted to the question of how a suitable subset of L+ can be found.

3.1 Classical Approaches to Truth The basic suggestion of classical approaches is to set = L0 (i.e. all sentences in L0 meet the T r-scheme), and to stipulate that all other sentences outside of L0 are not true. Let me define: Definition 3.2 Classical truth predicate in (L+ , M+ ). T r is called a classical truth predicate in (L+, M+) i↵ M+ ✏ ' $ T r(p'q) for all ' in L0 , and M+ ✏ ¬T r(p'q) for all ' in L+ be any 0 \L0 . Let + theory in L . Then I shall say that T r is a classical truth predicate with respect to i↵ ` ' $ T r(p'q) for all ' in L0 , and 0 T r(p'q) for all ' in L+ in L+ , such that T r is a classical 0 \L0 . Furthermore, any theory truth predicate with respect to , shall be called a theory of classical truth. Let M be the standard model of arithmetic, and T r be interpreted by the set of all meta-language codes of all sentences of L that are true in LM , i.e. M+ ✏ T r(p'q) i↵ V alM (') = t or, equivalently stated,

3.1 Classical Approaches to Truth

33

I + (T r) = {d 2 D : d 2 L, I(p'q) = d, V alM (') = t}, where L is the set of objects d in D, such that d is the meta-language code of a sentence of L. Then M+ ✏ ' $ T r(p'q), for all ' of L0 , and M+ ✏ ¬T r(p'q) for all ' in L+ 0 \L0 . Thus, T r is a classical truth predicate in (L+, M+). Note that there are sentences ' 2 / L0 , which do + meet the T r-scheme. E.g., for all ' 2 / L0 with M ✏ ¬', it holds that M+ ✏ ' $ T r(p'q). Thus classical truth predicates are no strictly partial truth predicates. Halbach[16] gives a detailed survey on theories of classical truth. The most simple theory of classical truth is T B (cf. Halbach[16]), which is the set consisting of the axioms P A of the theory of Peano arithmeticindexPeano arithmetic, of the set of all T r-biconditionals T r(p'q) $ ' where ' 2 L0 , and of all instances of the induction scheme '(¯0) ^ 8x('(x) ! '(S(x))) ! 8x'(x) where ' 2 L+ 0 . Usually a classical truth predicate is introduced by the axiom system T (P A), which characterizes T r by induction over formulas (cf. Halbach[16]). Before stating T (P A), some notations are in order. Let valL be a one-place function symbol, such that valL (x) ⌘ y represents the function that assigns to each meta-language code of a closed term of L the meta-language code of its semantic value in D according to M. Consider, e.g., the term p'q of L, where ' is a sentence of L. Then the function represented by valL assigns I(p q) = (') to (p'q). Furthermore, it follows M ✏ valL (ptq) ⌘ t for each t 2 T . Let ClT rmL be a one-place predicate expressing the set of all meta-language codes of closed terms of L. Furthermore I assume for each n-place predicate symbol P , there is a function symbol P., such that P.(x1 , . . . , xn ) ⌘ y represents the function that assigns to each n-tuple of meta-language codes of n terms t1 , . . . , tn the meta-language code of P (t1 , . . . , tn ). Let T rAtL be a one-place predicate expressing the set of all meta-language codes of true atomic sentences

34

Theories of Partial Truth

of L, which can be implicitly defined as follows (cf. Halbach[16], p. 31):

h

8x T rAtL (x) $

h

9x1 9x2

⇣

ClT rmL (x1 ) ^ ClT rmL (x2 ) ^ x ⌘ (x1 ⌘ . x2 ) !

T rAtL (x) $ valL (x1 ) ⌘ valL (x2 )

h

9x1 . . . 9xn

⇣

⌘i

_

ClT rmL (x1 ) ^ . . . ^ ClT rmL (xn ) ^ x ⌘ P.(x1 , . . . , xn ) !

T rAtL (x) $ P (val(x1 ), . . . , val(xn ))

⌘ii

It is obvious how this implicit definition of T rAtL can be extended to arbitrary recursive signatures of L. Let SentL be a one-place predicate expressing the set of all meta-language codes of sentences of L, and let AtL be a one-place predicate expressing the set of all meta-language codes of atomic sentences of L. Then T (P A) consists of the axioms of P A together with all L+ -instances of the induction axiom above, and the following four axioms: 1. 8x(AtL (x) ! (T r(x) $ T rAtL (x))) 2. 8x(SentL (x) ! (T r(¬. x) $ ¬T r(x))) 3. 8x(SentL (x _. y) ! (T r(x _. y) $ (T r(x) _ T r(y)))) 4. 8x(SentL (8. xy) ! (T r(8. xy) $ 8zT r(sub(y, x, num(z))))) In contrast to T B, the theory T (P A) is not conservative over P A, i.e. there are sentences1 in L which cannot be derived in P A (and T B), but which can be derived in T (P A). Let us again turn to the T r-expansion M+ of M, such that T r is a 1 Among

such sentences is, e.g., a sentence, which represents the consistency of P A, and which, by G¨odels second incompleteness theorem, cannot be derived within P A. For details see Halbach[16].

3.1 Classical Approaches to Truth

35

classical truth predicate in (L+ , M+ ). Then (L+, M+) is often called a “meta-language”2 of LM , as a truth concept of the interpreted language LM can be expressed in (L+ , M+ ), i.e. the T r-scheme is made true by (L+ , M+ ) in case of each sentence of L. Classical theories have been subject to one constant point of criticism. There are many “semantically unproblematic” sentences, which are not in L, and in consequence are not allowed to be in the extension of a classical truth predicate. In case of a liar sentence or a truth teller sentence3 we have the intuition that these sentences are “semantically defective” in some sense, and this seems to justify the resulting exception of the T r-scheme.4 Consider, however, the sentence T r(pT r(p¯2 + ¯2 ⌘ ¯4q)q) of L+. The corresponding informal statement ‘ ‘ ‘ 2 + 2 = 4 ’ is true.’ is true.’ is a perfectly understandable and true statement, and there is nothing “semantically defective” about it. So prima facie this statement should also be derivable in an adequate theory of truth in natural language. But T r(pT r(p¯2 + ¯2 ⌘ ¯4q)q) is not derivable in a classical theory of truth, and in case of our model M+ for classical truth above, we obtain M+ ✏ ¬T r(pT r(p¯2 + ¯2 ⌘ ¯4q)q). In classical approaches one has to introduce another “meta-language”, which contains a classical truth predicate for all sentence of L+ , in order to derive the truth of sentences such as ‘ ‘ ‘ 2 + 2 = 4’ is true.’ is true.’. Let me from now on write ‘L1 ’ instead of ‘L+ ’, ‘S 1 ’ instead of ‘S + ’ and ‘T r1 ’ instead of ‘T r’. The suggestion is then to introduce another one-place predicate T r2 to L1 . Let L2 be the S 1 [ {T r2 }-extension of L1 . Then we let M2 be the S 2 -expansion of M1 , such that the extension of T r2 is assumed to be interpreted by the set of all meta-language codes of the sentences of L1 which are true in (L1 , M1 ), i.e. M2 ✏ T r2 (p'q) i↵ M1 ✏ '. 2 Note

that the relation ‘is a meta-language of’ is not very clearly defined in the literature (cf. Halbach[16]). 3 A truth teller sentence is a fixed point sentence of T r(x), i.e., it is a sentence ⌧ such that M ✏ ⌧ $ T r(p⌧ q). 4 I shall say more on how ‘is semantically problematic’ or ‘is semantically defective’ can be explained in detail in the next section 3.2.

36

Theories of Partial Truth

We then obtain M2 ✏ T r2 (pT r1 (p¯2 + ¯2 ⌘ ¯4q)q). Of course the predication ‘... is true’ can be iterated arbitrarily many finite times. If we want to derive the truth of a sentence corresponding to ‘ ‘ ‘ ‘ 2 + 2 = 4’ is true.’ is true.’ is true.’ we have to step into another meta-language L3 , which is the extension of L2 by a one-place predicate T r3 , and such that T r3 is interpreted by the set of all metalanguage codes of the sentences of L2 which are true in (L2 , M2 ). Repeating this procedure, one is led to a hierarchy of interpreted languages (L1 , M1 ), (L2 , M2 ), (L3 , M3 ), . . ., and truth predicates T r1 , T r2 , T r3 , . . . More precisely, for each n 2 N, let Ln+1 be the language Ln extended by a one-place predicate T rn+1 , and let Mn+1 be an expansion of Mn to a model of Ln+1 . Furthermore, for each sentence ' in Ln , let us assume Mn ✏ T rk (p'q) , there is an i 2 N with i < k  n such that Mi ✏ ', and Mn ✏ ¬T rk (p'q), if ' is no sentence of Lk . We then obtain Mn ✏ T rn (p'q) $ ' for each ' 2 Lm with m, n 2 N and m < n. Following the usual terminology I shall call this resulting hierarchy of interpreted languages the Tarski hierarchy of languages. Halbach[16] has shown how this hierarchy can be extended to index sets containing infinite ordinal numbers. I shall indicate this method in section 5.3. So the proposed solution of classical approaches to the problem of deriving the truth of certain statements that themselves contain a truth predicate is to step into higher “meta-languages”. No such solution will allow for a true sentence containing iterative applications of one and the same truth predicate. This is, however, a commonly accepted feature of the concept of truth in natural languages. For this reason classical approaches are not regarded as a suitable approach to resolve the paradox in natural languages, and classical truth predicates are often criticized for being artificial and unnatural. If we aim at a theory or a structure M+ which represents the concept of truth of natural languages, we therefore have three options: either we find a way for iterative applications of T r, such that at least some

3.2 Iterative Approaches to Truth and Groundedness

37

sentences of the form T r(pT r(p'q)q) become provable in or true in M+ , or secondly, we find an argument which shows that the concept of truth in natural language really does not allow for any true iterative applications of one truth predicate, or thirdly, we find an argument which shows that the concept of truth in natural languages is flawed. There is indeed much hope for a solution along the lines of the first of these three options. In his well known paper Kripke (Kripke[21], 1975) has not only stated a solution that allows for iterative applications of T r, but he thereby inspired many others to develop further such approaches. I shall call such approaches the iterative approaches to truth. They are addressed in the next section.

3.2 Iterative Approaches to Truth and Groundedness A central question for iterative approaches is which iterative applications of the truth predicate should be allowed, and which not. Recall the set , such that L ✓ ✓ L+ from above. It might seem to be a plausible strategy for finding a suitable with \L = 6 ? to put as many sentences into as possible. In other words, one is looking for a set such that the set t

:= { ' $ T r(p'q) : ' 2

and M+ ✏ '}

is a maximal consistent set. As McGee[26] has demonstrated, this requirement is in fact not very useful. There are simply too many di↵erent such sets . Also each of this huge multitude will in general not be axiomatizable5 . The most popular strategy to find a suitable with \L = 6 ? derives from the notion of groundedness, which is frequently employed in the literature on truth and the liar problem. The basic idea of this concept is the following: In natural language a sentence refers to objects, which themselves can be sentences. These sentences again refer to objects, 5 The

reason is that it will be at least of complexity e.g. to Rogers[30]

0. 2

For a definition of ‘

0’ n

confer

38

Theories of Partial Truth

which themselves can be sentences. So each sentence can refer to other sentences, which themselves again refer to other sentences and so on, so that each sentence can indirectly refer to many other sentences. A sentence is called ungrounded, if the “reference-chain” starting from this sentence has no ending, i.e. if each sentence belonging to that chain itself refers to another sentence. Otherwise the sentence is called grounded. The concept of groundedness was first informally introduced by Hans Herzberger in [17] (pp. 147, 148). He explicated the notion of groundedness as follows: Momentarily conceding sense to the notion of aboutness, each sentence has a certain domain. [...] The relation between a sentence S and its domain D(S) is sensitive to some of the same factors that are operative in general set theory. In case some members of D(S) themselves are sentences, they in turn will have their own domains, which collectively can be designated D2 (S): the aggregate of the domains of all sentences in the domain of S. And it can happen that some members of D 2 (S) are sentences, and so on. Any sentence for which this process fails to terminate will be called ‘groundless’. It soon turned out that there are also other ways to explicate groundedness, which deviate slightly from Herzberger’s concept, but which are more advantageous for formulating conditions for the truth-scheme, and which lead to bigger sets . Let from now on S + be the extension of S by a oneplace predicate T r and a one-place predicate Gri (where i 2 {1, 2, 3, 4}), and let L+ be the S +-extension of L. Furthermore, let M+ be any S + expansion of M, such that M+ still meets requirements S1 S4. In what follows I shall present several ways of formally representing ‘is grounded’ in (L+ , M+ ) by stating di↵erent definitions and theories of the predicate symbols Gri (where i 2 {1, 2, 3, 4}) in (L+ , M+ ). The index i in ‘Gri ’ shall indicate the respective definition or theory assumed to formally represent ‘is grounded’. First I show how Herzberger’s initial description of groundedness can be formalized. A typical ungrounded sentence contains a term which denotes the very sentence itself, i.e. it is a self-referential sentence. However not every self-referential sentence should be counted as ungrounded. Let e.g. the one-place predicate R(x) represent the set

3.2 Iterative Approaches to Truth and Groundedness

39

of all meta-language codes of sequences that consist of more than three symbols. Then the self-referential fixed point sentence 'R of R(x), such that M+ ✏ 'R $ R(p'R q), is usually not regarded as ungrounded. This is because R(x) represents no “semantical property” of sentences. The semantic value of R(p'R q) depends only on syntactical properties of 'R . The semantic evaluation process of 'R (which results from following the usual inductive definition6 of ‘M ✏ . . .’) terminates at R(p'R q), i.e., M+ ✏ R(p'R q) can be reduced to some set theoretic expression and does not further depend on M+ ✏ ', as in case of the liar sentence. In general, for any sentence which does not contain any semantic expressions (such as ‘is true’ etc.) the semantic evaluation process (which results from following the usual inductive definition of ‘M ✏ . . .’) terminates. I propose that when defining ‘groundedness’ we should take this into account and not count sentences as 'R above as ungrounded. This distinction is not explicit in Herzberger’s informal description above, but of course he is aware of it, and also accounts for it in his ‘semantic grounding condition’ (cf. Herzberger[17], p. 149). The semantical predicates of (L+ , M+ ) are T r and Gri (i 2 N). I suggest to call a sentence in L+ self-referential, if it contains a term which denotes the very sentence itself and if the very sentence occurs in the scope of a semantical predicate of (L+, M+). A self-referential sentence can be visualized as follows: '1 = . . . T r(p'1 q) . . . '1 occurs in the scope of T r. The dots shall indicate that ‘T r(p'1 q)’ can be a sub-sentence within an arbitrarily complex formula. The existence of such a self-referential sentence is of course a direct consequence of the diagonalization lemma. The following example is a bit more complicated: '1 = . . . T r(p'2 q) . . . where '2 = . . . T r(p'2 q) . . . 6 For

a mathematical characterization of inductive definitions, cf. Moschovakis[27].

40

Theories of Partial Truth

Here '1 directly refers to a self-referential sentence '2 , and thus '1 is ungrounded according to Herzberger informal description. Again the dots shall indicate that T r(p'2 q) can be a sub-sentence within an arbitrarily complex formula. Of course, the dots do not have to represent the same formula in the two lines. Similarly, the following sentences are ungrounded: '1 = . . . T r(p'2 q) . . . where '2 = . . . T r(p'1 q) . . . In both cases the two ungrounded sentences can easily be constructed by the diagonalization lemma (or rather the generalized diagonalization lemma, cf. e.g. Boolos[6], p. 53). Ungrounded sentences can be arbitrarily complex. For each finite number n: '1 = . . . T r(p'2 q) . . . where '2 = . . . T r(p'3 q) . . . where ... 'n = . . . T r(p'n q) . . . Again, we also obtain an ungrounded sentence if the right ‘'n ’ in the last line of the above n equivalence-sentences is replaced by ‘'1 ’. Following Gaifman[11] I shall call the set consisting of these n equivalence-sentences a ‘chain of n sentences’, and the set consisting of such n equivalence-sentences which ends with the right ‘'n ’ replaced by ‘'1 ’ a ‘loop of n sentences’. Nothing speaks against assuming chains of infinitely many sentences as long as they form a recursive set. A first approach to ‘ungroundedness’ is simply to define a sentence in L+ to be ungrounded i↵ it is either selfreferential, or it is an element of a chain of sentences, or it is an element of a loop of sentences. This can be easily formalized in (L+ , M+ ). First we need some additional syntactical predicates to be available in L+ . Assume that L+ is the language, which results from extending S by a one-place

3.2 Iterative Approaches to Truth and Groundedness

41

predicate SentL+ , a two-place predicate SubSentL+ , a two-place predicate SubSemL+ , and a two-place predicate Ref SemL+ in addition to Gr1 and T r. Let L+ be the set of objects d in D, such that d is the meta-language code of a sentence of L+ . Let me assume the following S + -expansion M+ + of M. SentL+ (x) represents the set L+ ✓ D of all S -codes of sentences of L+ . SubSentL+ (x, y) represents the following relation ‘is a sub-sentence of’, in other words it represents the following set: {(d1 , d2 ) : d1 , d2 2 L+ , I(p'1 q) = d1 , I(p'2 q) = d2 , and '1 is a subsequence of '2 }.7 num(x)

For any one-place predicate P (x), let ‘P (x)’ ˙ abbreviate ‘[P (x)]x ’. + SubSemL+ is explicitly defined in L as follows (I have omitted the two outer universal quantifiers over x and y):

h

SubSemL+ (x, y) $ SentL+ (y) ^

h

ii 9z SubSent (z, x) ^ (z ⌘ pT r(y) ˙ q _ z ⌘ pGr1 (y) ˙ q) . L+

M+ ✏ SubSemL+ (p'q, p q), i↵ ' 2 L+ contains either T r(p q) or Gr1 (p q) (with 2 L+ ) as a sub-sentence. Let N be the predicate representing the set of natural numbers. Furthermore, recall the definitions of Seq(x), lh(x) and q(x, y) given on p. 21. Let Ref SemL+ (x, y) be explicitly defined in L+ as follows:

h Ref SemL+ (x, y) $ SentL+ (x) ^ N (y)^ hh 8z Seq(z) ^ lh(z) ⌘ y ^ y > ¯0 ^ ⇣ ⌘ SubSemL+ x, q(z, ¯1) ^

7s

= 2 (di1 , . . . , dim ) is a subsequence of s2 = 2 (dj1 , . . . , djn ) i↵ s1 and s2 are sequences (in the sense represented by Seq) and there is a k 2 N, 1  k  n, such that di1 = djk , di2 = djk+1 , . . . , dim = djk+m 1 .

1

42

Theories of Partial Truth

⇣ ⌘ SubSemL+ q(z, ¯1), q(z, ¯2) ^ . . . ^ ⇣ SubSemL+ q(z, lh(z)

⌘i ¯1), q(z, lh(z)) !

⇣ ⌘ii ¬ 9vSubSemL+ q(z, y), v .

M+ ✏ Ref SemL+ (p'q, n ¯ ), i↵ ' 2 L+ and there are n sentences '1 , '2 , . . . , 'n , such that 'i 2 L+ for all i, 1  i  n, and such that • ' contains either T r(p'1 q) or Gr1 (p'1 q) as a sub-sentence, • '1 contains either T r(p'2 q) or Gr1 (p'2 q) as a sub-sentence, • '2 contains either T r(p'3 q) or Gr1 (p'3 q) as a sub-sentence, . . . • 'n

1

contains either T r(p'n q) or Gr1 (p'n q) as a sub-sentence, and

• 'n contains no sub-sentence that is of the form T r(p q), or of the form Gr1 (p q) (with 2 L+ ). Now we can explicitly define Gr1 in L+ as follows: Definition 3.3 8x(Gr1 (x) $ (SentL+ (x) ^ 9y Ref SemL+ (x, y))).8 Given a groundedness predicate Gr1 , the classical Tarskian inductive characterization of T r from page 34 can be modified as follows: 1. 8x(AtL (x) ! (T r(x) $ T rAtL (x))) 2. 8x((SentL+ (x) ^ Gr1 (x)) ! (T r(pT r(x) ˙ q) $ T r(x))) 3. 8x((SentL+ (x) ^ Gr1 (x)) ! T r(pGr1 (x) ˙ q)) 8 Note

that this explicit definition of Gr1 is not circular, as ‘Gr1 ’ occurs in the definition of SubSemL+ within quotation corners. So not ‘Gr1 ’, but its object language code really occurs in the definition of SubSemL+ .

3.2 Iterative Approaches to Truth and Groundedness

43

4. 8x((SentL+ (x) ^ Gr1 (x)) ! (T r(¬. x) $ ¬T r(x))) 5. 8x((SentL+ (x _. y) ^ Gr1 (x _. y)) ! (T r(x _. y) $ (T r(x) _ T r(y)))) 6. 8x((SentL+ (8. xy) ^ Gr1 (8. xy)) ! (T r(8. xy) $ 8zT r(sub(y, x, num(z))))) 7. 8x(¬Gr1 (x) ! ¬T r(x)) The just given axiom system shall be denoted by ‘ T (P A)Gr ’. Let me give some examples that demonstrate how our definition of Gr1 works and whether it corresponds to Herzberger’s original account: Example 3.1 The liar sentence , such that M+ ✏ $ ¬T r(p q). We have M+ ✏ SubSemL+ (p q, p q), from which we derive M+ ✏ ¬9y Ref SemL+ (p q, y), and thus M+ ✏ ¬ Gr1 (p q). Example 3.2 8x(T r(x) _ T r(¬. x)). According to Herzberger’s account this sentence is ungrounded, as the sentence is about everything in the domain (the quantifier refers to the whole domain), and thus also refers to the very sentence itself. However, according to M+ the sentence will in general be grounded, because of M+ ✏ ¬9ySubSemL+ (p8x(T r(x) _ T r(¬. x))q, y), and M+ ✏ Ref SemL+ (p8x(T r(x) _ T r(¬. x))q, ¯1), and therefore we obtain M+ ✏ Gr1 (p8x(T r(x) _ T r(¬. x))q) (unless x is the object language code of some ungrounded sentence). Example 3.3 Another way of constructing a liar sentence: 2 = 8z(Dg(p8z(Dg(x, z) ! ¬T r(z))q, z) ! ¬T r(z)) where Dg(x, y) formally expresses that the diagonalization of x is equal to y. From the diagonalization lemma and M+ ✏ Dg(p8z(Dg(x, z) ! ¬T r(z))q, p 2 q) ⌘ p8z(Dg(p8z(Dg(x, z) ! ¬T r(z))q, z) ! ¬T r(z))q

44

Theories of Partial Truth

one obtains M+ ✏ 2 $ ¬T r(p 2 q). However, according to definition 3.3 this version of a liar sentence is grounded. We see from examples 3.2 and 3.3 that definition 3.3 of groundedness does not work for sentences that contain universal quantifiers. It is not enough to consider the sub-formula ' of a universal sentence 8x', as definition 3.3 suggests. Let us take into account every instance of 8x', i.e. every sentence [']tx where t is any term, instead. We can do this by replacing SubSemL+ (x, y) in L+ by a two-place predicate SubSemQuL+ , and assume the interpretation of SubSemQuL+ to be given by the following explicit definition:

h SubSemQu (x, y) $ SubSemL+ (x, y) _ L+

h

9z SubSentL+ (z, x) ^ 9uV arL+ (u) ^ 9vF rmL+ (v) ^ z ⌘8. uv ^ h ⇣ ⌘iii 9w T rmL+ (w) ^ SubSemL+ sub(v, u, w), y

Thus M+ ✏ SubSemQuL+ (p'q, p q) i↵ M+ ✏ SubSemL+ (p'q, p q), or ' contains a sub-sentence of the form ‘ 8x'sc ’ (the upper index ‘sc’ is short for ‘scope’), and there is an instance ['sc ]tx of 8x'sc such that SubSemL+ (p['sc ]tx q, p q). Furthermore, we replace Ref SemL+ by the two-place predicate Ref SemQuL+ in L+ , and let Ref SemQuL+ be defined by the explicit definition which results from the explicit definition of Ref SemL+ above, in which each occurrence of ‘SubSemL+ (x, y)’ is replaced by ‘SubSemQuL+ (x, y)’, i.e. Ref SemQuL+ (x, y) is explicitly defined in L+ as follows:

h

Ref SemQuL+ (x, y) $ SentL+ (x) ^ N (y)^ 8z

hh

Seq(z) ^ lh(z) ⌘ y ^ y > ¯0 ^

⇣ ⌘ ¯ SubSemQuL+ x, q(z, 1) ^

3.2 Iterative Approaches to Truth and Groundedness

45

⇣ ⌘ SubSemQuL+ q(z, ¯1), q(z, ¯2) ^ . . . ^ ⇣

SubSemQuL+ q(z, lh(z)

⌘i ¯1), q(z, lh(z)) !

⇣ ⌘ii ¬ 9vSubSemQuL+ q(z, y), v .

Then M+ ✏ Ref SemQuL+ (p'q, n ¯ ), i↵ ' 2 L+ and there are n sentences '1 , '2 , . . . , 'n , such that 'i 2 L+ for all i, 1  i  n and such that • ' contains either T r(p'1 q) or Gr1 (p'1 q) as a sub-sentence, or ' contains a sub-sentence of the form ‘8x'sc ’, and there is an instance ['sc ]tx of 8x'sc such that SubSemL+ (p['sc ]tx q, p q), • '1 contains either T r(p'2 q) or Gr1 (p'2 q) as a sub-sentence, or '1 contains a sub-sentence of the form ‘8x'sc 1 ’, and there is an instance t sc sc t ['sc 1 ]x of 8x'1 such that SubSemL+ (p['1 ]x q, p'2 q), • '2 contains either T r(p'3 q) or Gr1 (p'3 q) as a sub-sentence, or '2 contains a sub-sentence of the form ‘8x'sc 2 ’, and there is an instance sc t sc sc ['2 ]x of 8x'2 such that SubSemL+ (p['2 ]tx q, p'3 q),. . . • 'n 1 contains either T r(p'n q) or Gr1 (p'n q) as a sub-sentence, or 'n 1 contains a sub-sentence of the form ‘8x'sc n 1 ’, and there is an sc t sc t instance ['n 1 ]x of 8x'n 1 such that SubSemL+ (p['sc n 1 ]x q, p'n q), and • 'n contains no sub-sentence that is of the form T r(p q), or of the form Gr1 (p q), and 'n contains no sub-sentence of the form ‘8x'sc n ’, t sc such that there is an instance ['sc n ]x of 8x'n and such that t SubSemL+ (p['sc 2 L+ . n ]x q, p q), where Finally we replace our definition of Gr1 in L+ by the following definition: Definition 3.4 8x(Gr2 (x) $ (Sent(x) ^ 9y Ref SemQuL+ (x, y))). As a consequence of this modified definition 3.4 each quantified sentence of the form 8x', such that T r(x) is a sub-sentence of ', is ungrounded, as we

46

Theories of Partial Truth

can substitute for x the very sentence itself: 'px8x'q contains T r(p8x'q), and thus SubSemQuL+ (p8x'q, p8x'q). Therefore the sentence from example 3.2 as well as the liar sentence from example 3.3 are ungrounded according to definition 3.4. However there are still problems with definition 3.4 as the following example shows: Example 3.4 8x(SentL (x) ! (T r(x) _ ¬T r(x))) (where SentL (x) represents the set of all meta-language codes of sentences in L). According to our current definition 3.4 this sentence is ungrounded. On the other hand it seems to be grounded by Herzberger’s informal explication as it seems to say something only of sentences from L, in other words, it is exclusively ”about” grounded sentences and thus it is itself grounded. At this point it would be helpful to have an accurate definition of the domain of sentences of which a given universal sentence is about. Assume someone would simply propose that the domain a universal sentence 8x(' ! ) is about is the set of sentences ”satisfying” '(x), i.e. the set of sentences that belong to the extension9 of '. This means that 8x(' ! ) is grounded, only if the sentences which are in the extension of ' do not contain a sub-formula T r(x). But 8x(' ! ) is logically equivalent to 8x(¬ ! ¬'). So according to the suggestion just made the domain of 8x(¬ ! ¬') is the set of sentences satisfying ¬ . In case of example 3.4, 8x(¬T r(x) ! ¬SentL (x)) would be ungrounded, while 8x(SentL (x) ! T r(x)) which is logically equivalent, would be grounded. The suggestion is thus useless. I think it is not possible to solve this problem without posing a definition of groundedness that deviates from the original idea explicated above. The sentence from example 3.4 seems to be perfectly understandable and we would rather put it to the sentences that meet the T r-scheme. We thus have conflicting intuitions concerning our notion of groundedness, and indeed many authors (such as e.g. Kripke[21]) have assumed a concept of groundedness which counts sentences as example 3.4 as grounded. I now want to demonstrate how we can derive a new definition of groundedness 9 The

extension of a sentence ' is the subset of D, that is represented by ' according to definition 2.1.

3.2 Iterative Approaches to Truth and Groundedness

47

from definition 3.4 that makes the sentence of example 3.4 grounded. Let us add to L+ a one-place predicate T r0 , such that T r0 is the classical truth predicate for the sentences of L. T r0 can e.g. be characterized by the axiom system T (P A), I have stated in the previous section. We can then replace Ref SemQuL+ (x, y) in L+ by a two-place predicate Ref SemTL+ , the interpretation of which shall be given by the following explicit definition:

h Ref SemTL+ (x, y) $ SentL+ (x) ^ N (y)^ hh 8z Seq(z) ^ lh(z) ⌘ y ^ y > ¯0 ^ SubSemQu

L+

SubSemQuL+

⇣ ⇣ ⇣

⌘ ¯ x, q(z, 1) ^

⌘ ¯ ¯ q(z, 1), q(z, 2) ^ . . . ^

SubSemQuL+ q(z, lh(z)

⌘i ¯1), q(z, lh(z)) !

h ⇣ ⌘ ¬9vSubSemQuL+ q(z, lh(z)), v _

⇣ 9v1 9v2 SentL+ (v1 ) ^ SentL+ (v2 ) ^ q(z, lh(z)) ⌘ v1 _. v2 ^ (T r0 (v1 ) _ T r0 (v2 ))

⌘

_

⇣ 9v1 9v2 V ar(v1 ) ^ F rmL+ (v2 ) ^ q(z, lh(z)) ⌘ 8. v1 v2 ^ 9v3 T r0 (¬. sub(v2 , v1 , num(v3 )))

⌘ iii

.

Let our definition 3.4 of Gr2 in L+ be replaced by the following definition: Definition 3.5 8x(Gr3 (x) $ (SentL+ (x) ^ 9yRef SemTL+ (x, y))). To see how this new groundedness predicate Gr3 works consider another example:

48

Theories of Partial Truth

Example 3.5 _ 2 + 2 ⌘ 4. This sentence is a disjunction of a grounded sentence and an ungrounded sentence. According to each of the definitions 3.3 and 3.4 we have considered so far this sentence is ungrounded. But since M+ ✏ T r0 (p2 + 2 ⌘ 4q) this sentence is grounded according to definition 3.5. Similarly, 8x(SentL (x) ! (T r(x) _ ¬T r(x))) is grounded: Each instance SentL (t) ! (T r(t) _ ¬T r(t)) of 8x(SentL (x) ! (T r(x) _ ¬T r(x))) is grounded, since in case M+ ✏ ¬SentL (t), SentL (t) ! (T r(t) _ ¬T r(t)) is true for the same reasons as example 3.5 is true, and in case M+ ✏ SentL (t), the sentence T r(t) will be grounded, because the sentence which t “denotes” is grounded. So we obtain M+ ✏ Gr3 (p8x(SentL (x) ! (T r(x) _ ¬T r(x)))q). However the following examples suggest that our current definition 3.5 of groundedness can still be improved such that further sentences, which intuitively meet the T r-scheme but are derived to be ungrounded according to definition 3.5, become grounded. Example 3.6 _ T r(p2 + 2 ⌘ 4q). Intuitively this sentence is grounded just as _ 2 + 2 ⌘ 4 is, but according to the above definition 3.5 it is not grounded. Example 3.7 8x(T r(pSentL (x) ˙ q) ! (T r(x) _ ¬T r(x))). Intuitively this sentence is grounded just as 8x(SentL (x) ! (T r(x) _ ¬T r(x))) is, but according to the above definition 3.5 it is not grounded. We can modify definition 3.5 of Ref SemTL+ by simply replacing each occurrence of ‘T r0 ’ by ‘T r’. But note that, because T r is defined in terms of Gr3 , this turns the previously explicit definitions of Ref SemTL+ and Gr3 into an axiom system that simultaneously characterizes Ref SemTL+ , Gr3 and T r. So let Ref SemTL+ be replaced in L+ by a two-place predicate Ref SemT rL+ , and let the interpretation of Ref SemT rL+ be characterized by the following formula:

h

hh Ref SemT rL+ (x, y) $ SentL+ (x) ^ N (y) ^ 8z Seq(z) ^ lh(z) ⌘ y ^

3.2 Iterative Approaches to Truth and Groundedness

49

⇣ ⌘ SubSemQuL+ x, q(z, ¯1) ^ SubSemQuL+

⇣ ⇣

⌘ ¯ ¯ q(z, 1), q(z, 2) ^ . . . ^

SubSemQuL+ q(z, lh(z)

⌘i ¯1), q(z, lh(z)) !

h ⇣ ⌘ ¬9vSubSemQuL+ q(z, lh(z)), v _

⇣ 9v1 9v2 SentL+ (v1 ) ^ SentL+ (v2 ) ^ q(z, lh(z)) ⌘ v1 _. v2 ^ ⌘

(T r(v1 ) _ T r(v2 )) _

⇣ 9v1 9v2 V ar(v1 ) ^ F rmL+ (v2 ) ^ q(z, lh(z)) ⌘ 8. v1 v2 ^ 9v3 T r(¬. sub(v2 , v1 , num(v3 )))

⌘iii

.

Let Gr4 be characterized as follows: Explication 3.1 8x(Gr4 (x) $ (SentL+ (x) ^ (9yRef SemT rL+ (x, y)))). Finally, let T r be characterized by the axiom system T (P A)Gr stated above on page 43. The sentences from examples 3.6 and 3.7 are grounded according to the just stated axioms for Ref SemT rL+ , Gr4 and T r. Fact 3.1 Gr4 satisfies the following axioms: 1. 8x(SentL (x) ! Gr4 (x)) 2. 8x(SentL+ (x) ! ((Gr4 (x) $ Gr4 (pGr4 (x) ˙ q))) 3. 8x(SentL+ (x) ! ((Gr4 (x) $ Gr4 (pT r(x) ˙ q))) 4. 8x(SentL+ (x) ! (Gr4 (x) $ Gr4 (¬. x)))

50

Theories of Partial Truth 5. 8x8y(SentL+ (x) ^ SentL+ (y) ! (Gr4 (x _. y) $ ((Gr4 (x) ^ Gr4 (y)) _ T r(x) _ T r(y)))).

Proof The claim is proved by induction over the complexity of the formulas: • Let M+ ✏ SentL ('). Then M+ ✏ Ref SemT rL+ (p'q, ¯1) and M+ ✏ Gr4 (p'q). • Let ' = Gr4 (p q). Recall that N represents the set of all natural numbers. Then M+ ✏ Gr4 (pGr4 (p q)q) i↵ (by explication 3.1) M+ ✏ 9xN (x) ^ Ref SemT rL+ (pGr4 (p q)q, x) i↵ (since M+ ✏ SubSemQuL+ (p q, q(z, ¯1)) $

SubSemQuL+ (pGr4 (p q)q, q(z, ¯1))

by the definition of SubSemQuL+ on page 44) M+ ✏ 9xN (x) ^ Ref SemT rL+ (p q, x) i↵ M+ ✏ Gr4 (p q). • Let ' = T r(p q). Recall that N represents the set of all natural numbers. Then M+ ✏ Gr4 (pT r(p q)q) i↵ (by explication 3.1) M+ ✏ 9xN (x) ^ Ref SemT rL+ (pT r(p q)q, x) i↵ (since M+ ✏ SubSemQuL+ (p q, q(z, ¯1)) $

SubSemQuL+ (pT r(p q)q, q(z, ¯1))

by the definition of SubSemQuL+ on page 44) M+ ✏ 9xN (x) ^ Ref SemT rL+ (p q, x) i↵ M+ ✏ Gr4 (p q). • Let ' = ¬ . Then M+ ✏ Gr4 (p¬ q) i↵ (by explication 3.1) M+ ✏ 9xN (x) ^ Ref SemT rL+ (p¬ q, x) i↵ (since

3.2 Iterative Approaches to Truth and Groundedness

51

M+ ✏ SubSemQuL+ (p q, q(z, ¯1)) $ SubSemQuL+ (p¬ q, q(z, ¯1)) by the definition of SubSemQuL+ on page 44) M+ ✏ 9xN (x) ^ Ref SemT rL+ (p q, x) i↵ M+ ✏ Gr4 (p q). • Let ' =

1_

2.

Then M+ ✏ Gr4 (p

M+ ✏ 9xN (x) ^ Ref SemT rL+ (p M+ ✏ SubSemQuL+ (p SubSemQuL+ (p

1

_

1

1_

_

2 q)

2 q, x)

1)) 2 q, q(z, ¯

i↵ (by explication 3.1)

i↵ (since

$

1)) ^ SubSemQuL+ (p 2 q, q(z, ¯1)) 1 q, q(z, ¯

by the definition of SubSemQuL+ on page 44) M+ ✏ (9xN (x) ^ Ref SemT rL+ (p Ref SemT rL+ (p

i↵ M+ ✏ (Gr4 (

1 q, x) ^ 9xN (x)^

2 q, x)) _ (T r(p 1 q) _ T r(p 2 q))

1 ) ^ Gr4 ( 2 ))

_ (T r(

1 ) _ T r( 2 )).

Note that the above characterization of Gr4 does not imply:

h

h ⇣ ⌘ 5. 8x8y V arL+ (x) ^ F rmL+ (y) ! Gr4 8. xy $ h ⇣ ⌘ ⇣ ⌘iii 8zGr4 sub(y, x, num(z)) _ 9z¬T r sub(y, x, num(z)) . Assume '(x) is a formula which contains at most free occurrences of the variable x, such that M+ ✏ 8xGr4 (p'(x) ˙ q). In other words, assume that M+ ✏ 8x9yRef SemT rL+ (p'(x) ˙ q, y). But from this, it does not in general follow that M+ ✏ 9y8xRef SemT rL+ (p'(x) ˙ q, y), i.e. M+ ✏ Gr4 (p8x'(x)q). Thus adding 5. to the axioms from fact 3.1 would lead to an essentially stronger groundedness concept. Indeed, as I shall demonstrate below, these axioms together with 5. lead to the concept of groundedness advanced by Kripke[21]. An important feature of Kripke’s characterization of ‘groundedness’, which di↵ers from the one stated by Herzberger, is that “if a sentence asserts, e.g., that all sentences in class C are true, we

52

Theories of Partial Truth

allow it to be false and grounded if one sentence in C is false, irrespective of the groundedness of the other sentences in C.” (cf. Kripke[21], p. 694, footnote 7). Example 3.8 For each n 2 N, let ItT r(p'q, n ¯ ) $ T r(pT r(. . . T r(p'q) . . .)q).10 | {z } n times

Consider the sentence 8x(N (x) ! ItT r(p2 + 2 ⌘ 4q, x)). Then we obtain M+ ✏ ¬Gr4 (p8x(N (x) ! ItT r(p2 + 2 ⌘ 4q, x))q). If we added axiom 5 to explication 3.1, the same sentence would become grounded, as M+ ✏ Gr4 (p8x(N (x) ! ItT r(p2 + 2 ⌘ 4q, x))q) $ 8xGr4 (pN (x) ˙ ! + ItT r(p2+2 ⌘ 4q, x) ˙ q) and certainly M ✏ 8xGr4 (pN (x) ˙ ! ItT r(p2+2 ⌘ 4q, x) ˙ q). Assume we add 5. (from the previous page) to our groundedness concept, and let T r again be characterized by the axiom system T (P A)Gr stated on p. 43. Then T r and Gr are simultaneously characterized by the following axiom system, which I shall denote by ‘ TG ’: 1. 8x(SentL (x) ! Gr(x)) 2. 8x(SentL+ (x) ! ((Gr(x) $ Gr(pGr(x) ˙ q)))) 3. 8x(SentL+ (x) ! ((Gr(x) $ Gr(pT r(x) ˙ q)))) 4. 8x(SentL+ (x) ! (Gr(x) $ Gr(¬. x))) 5. 8x8y(SentL+ (x) ^ SentL+ (y) ! (Gr(x_. y) $ ((Gr(x) ^ Gr(y)) _ T r(x) _ T r(y)))) 6. 8x8y(V arL+ (x) ^ F rmL+ (y) ! (Gr(8. xy) $ (8zGr(sub(y, x, num(z))) _ 9zT r(¬. sub(y, x, num(z)))))) 10 Cf.,

e.g., Shoenfield[32] for an inductive definition of such a predicate.

3.2 Iterative Approaches to Truth and Groundedness

53

7. 8x(AtL (x) ! (T r(x) $ T rAtL (x))) 8. 8x((SentL+ (x) ^ Gr(x)) ! (T r(pT r(x) ˙ q) $ T r(x))) 9. 8x((SentL+ (x) ^ Gr(x)) ! T r(pGr(x) ˙ q)) 10. 8x((SentL+ (x) ^ Gr(x)) ! (T r(¬. x) $ ¬T r(x))) 11. 8x((SentL+ (x _. y) ^ Gr(x _. y)) ! (T r(x _. y) $ (T r(x) _ T r(y)))) 12. 8x((SentL+ (8. xy) ^ Gr(8. xy)) ! (T r(8. xy) $ 8zT r(sub(y, x, num(z))))) 13. 8x(¬Gr(x) ! ¬T r(x)) I shall now show that TG is proof theoretically equivalent to the well-known Kripke-Feferman theory of truth KF. This also ensures that our theory TG is consistent. The consistency of KF is an immediate consequence of fact 5.7 of chapter five (see p. 96). Models M(E) with M(E) ✏ KF are obtained by extending M by the extension of any fixed point of the operation M (cf. definition 5.1 on p. 91). Let us add to L a new one-place predicate LitL , such that LitL (x) represents the set of all meta-language codes of atomic or negated atomic sentences of L, i.e., the set of all meta-language codes of literals of L. Furthermore, let us add a new one-place predicate T rLitL to L, such that T rLitL (x) represents the set of all true atomic and all true negated atomic formulas of L. To give a definition of T rLitL (x) in L, we have to assume that several predicates and functions, which were already introduced on p. 33, are available in L: First, the one-place predicate ClT rmL (x), which represents the set of all meta-language codes of a closed term of L. Secondly, the function symbol valL , which represents the function that assigns to each meta-language code of a closed term of L the metalanguage code of its semantic value in D according to M. Recall that M ✏ valM (ptq) ⌘ t for each term t 2 T . Furthermore, I assume for each nplace predicate symbol P , a function symbol P., such that P.(x1 , . . . , xn ) ⌘ y

54

Theories of Partial Truth

represents the function that assigns to each n-tuple of meta-language codes of n terms t1 , . . . , tn the meta-language code of P (t1 , . . . , tn ). As I have assumed that M meets S1 S4, all these sets and functions can be defined in (L, M). For the sake of simplicity, let me assume that there are only two predicates in the signature of L, which are ⌘ and an n-place predicate P . It is obvious how the following implicit definition of T rLitL can be extended to arbitrary recursive signatures of L:

h

8x T rLitL (x) $

h

h⇣ ⌘ 9x1 9x2 ClT rmL (x1 ) ^ ClT rmL (x2 ) ^ x ⌘ (x1 ⌘ . x2 ) ! ⇣

⌘i T rLitL (x) $ valL (x1 ) ⌘ valL (x2 ) _

⇣

⌘i T rLitL (x) $ ¬(valL (x1 ) ⌘ valL (x2 )) _

h⇣ ⌘ 9x1 9x2 ClT rmL (x1 ) ^ ClT rmL (x2 ) ^ x ⌘¬. (x1 ⌘ . x2 ) !

9x1 . . . 9xn

h⇣

9x1 . . . 9xn

h⇣

⇣ ⇣

⌘ ClT rmL (x1 ) ^ . . . ^ ClT rmL (xn ) ^ x ⌘ P.(x1 , . . . , xn ) !

⌘i T rLitL (x) $ P (valL (x1 ), . . . , valL (xn )) _

⌘

ClT rmL (x1 ) ^ . . . ^ ClT rmL (xn )) ^ x ⌘¬. P.(x1 , . . . , xn ) !

T rLitL (x) $ ¬P (val(x1 ), . . . , val(xn ))

ii

⌘i

KF is then the following axiom system: 1. 8xLitL (x) ! (T r(x) $ T rLitL (x)) 2. 8x(SentL+ (x) ! (T r(pT r(x) ˙ q) $ T r(x))) 3. 8x(SentL+ (x) ! (T r(p¬T r(x) ˙ q) $ T r(¬. x)))

3.2 Iterative Approaches to Truth and Groundedness

55

4. 8x8y((SentL+ (x) ^ SentL+ (y)) ! (T r(x_. y) $ (T r(x) _ T r(y)))) 5. 8x8y((SentL+ (x) ^ SentL+ (y)) ! (T r(¬. (x_. y)) $ (T r(¬. x) ^ T r(¬. y)))) 6. 8x8y((V arL+ (x) ^ F rmL+ (y)) ! (T r(8. xy) $ 8zT r(sub(y, x, num(z))))) 7. 8x8y((V arL+ x) ^ F rmL+ (y)) ! (T r(¬. 8. xy) $ 9zT r(¬. sub(y, x, num(z))))) 8. 8x(SentL+ (x) ! (T r(x) $ T r(¬. ¬. x))) 9. 8x¬(T r(x) ^ T r(¬. x)) Note that KF contains no groundedness predicate. However we will see by fact 3.2 that a concept of groundedness is “indirectly” captured in the axioms of KF. KF was introduced by Feferman[9] under a di↵erent name ‘Ref (P A)’. Later it was called ‘KF’ after Kripke and Feferman. Meanwhile there are several versions of KF in the literature, which di↵er from another in some minor respects. In Feferman’s original account, there is also a falsity predicate, which is defined simultaneously with the truth predicate. Axiom 9 was originally not included in KF, and this axiom cannot be derived from the other axioms of KF. I shall assume the following definitions of proof-theoretical equivalence and relative interpretability (cf. e.g. Rautenberg[29], p. 200). Definition 3.6 P -relativization, deductive closure and relative interpretability of 1 in 2 . 1. For any formula ' of L and any one-place predicate P of L, let the P -relativization 'P of ' be the result of substituting all sub-formulas of ' that are of the form 8x↵ by 8x(P (x) ! ↵), and of substituting all sub-formulas of ' that are of the form 9x↵ by 9x(P (x) ^ ↵).

56

Theories of Partial Truth 2. For any set S of sentences of L, let the deductive closure Cl(S) of S be the set Cl(S) := {' : S ` '}. 3. Let 1 and 2 be any two theories. Let L1 and L2 be the respective languages of 1 and 2 . Let S1 be the signature of L1 and S2 be the signature of L2 . Then 1 is relatively interpretable in 2 ( 1 / 2 ) i↵ i For each symbol S 2 S1 \S2 , there is an explicit definition of S in L2 . Let the set of all those explicit definitions be denoted by . ii There is a one-place predicate P 2 / S1 [ S2 , and an explicit definition P of P in L2 . iii

P 1

'.

✓ Cl(

2[

[

P ),

where

P 1

= {'P |' 2

1}

Definition 3.7 Proof-theoretical equivalence of theories. 1 and 2 are proof theoretically equivalent i↵ 1 / 2 and

for any formula

2

/

1.

Fact 3.2 TG and KF are proof-theoretically equivalent, more precisely Cl(TG) = Cl(KF [ Gr ), where Gr is the following explicit definition: Gr

= 8x(Gr(x) $ (T r(x) _ T r(¬. x))).

Proof From now on, let ‘KF [

Gr ’

be abbreviated by ‘KF+ ’.

We start with proving the claim KF+ ` TG: 1. We show that KF+ ` 8x(SentL (x) ! Gr(x)) by induction over the structure of formulas. • KF+ ` 8x(LitL (x) ! (T rLitL (x) _ T rLitL (¬. x))), therefore KF+ ` 8x(LitL (x) ! (T r(x) _ T r(¬. x))) because of KF+1.

• Assume KF+ ` SentL (x) ^ x ⌘ y1 _. y2 such that

KF+ ` T r(y1 ) _ T r(¬. y1 ) and KF+ ` T r(y2 ) _ T r(¬. y2 ). KF+ ` 8x8y(T r(x^. x) $ (T r(x) ^ T r(y))), as

3.2 Iterative Approaches to Truth and Groundedness

57

KF+ ` T r(x^. y) $ T r(¬. (¬. x_. ¬. y)),

KF+ ` T r(¬. (¬. x_. ¬. y)) $ (T r(¬. ¬. x) ^ T r(¬. ¬. y)) and KF+ ` (T r(¬. ¬. x) ^ T r(¬. ¬. y)) $ (T r(x) ^ T r(y)).

KF+ ` (T r(y1 ) _ T r(¬. y1 )) ^ (T r(y2 ) _ T r(¬. y2 )) i↵ KF+ ` (T r(y1 ) ^ T r(y2 )) _ (T r(y1 ) ^ T r(¬. y2 ))_

(T r(¬. y1 ) ^ T r(y2 )) _ (T r(¬. y1 ) ^ T r(¬. y2 )) i↵

KF+ ` T r((y1 ^. y2 )_. (y1 ^. ¬. y2 )_. (y1 ^. ¬. y2 )) _ T r(¬. y1 ^¬. y2 ) i↵ KF+ ` T r(y1 _. y2 ) _ T r(¬. (y1 _. y2 )).

• Assume KF+ ` SentL (x) ^ x ⌘¬. (y1 _. y2 ) such that

KF+ ` T r(y1 ) _ T r(¬. y1 ) and KF+ ` T r(y2 ) _ T r(¬. y2 ).

By the previous case KF+ ` T r(y1 _. y2 ) _ T r(¬. (y1 _. y2 )). Thus KF+ ` T r(¬. ¬. (y1 _. y2 )) _ T r(¬. (y1 _. y2 )).

• Assume KF+ ` SentL (x) ^ x ⌘8. yz such that KF+ ` T r(z) _ T r(¬. z). From KF+ ` T r(z_. ¬. z) if follows KF+ ` 8yT r(z_. ¬. z). Therefore KF+ ` T r(8. y(z_. ¬. z)), which implies KF+ ` T r(8. yz_. 9. y¬. z), and thus KF+ ` T r(8. yz) _ T r(9. y¬. z). • Assume KF+ ` SentL (x)^x ⌘¬. 8. yz such that KF+ ` T r(z)_T r(¬. z). By the previous case KF+ ` T r(8. yz) _ T r(¬. 8. yz), and thus KF+ ` T r(¬. ¬. 8. yz) _ T r(¬. 8. yz).

2. We show KF+ ` 8x(SentL+ (x) ! (Gr(x) $ Gr(pGr(x)q))). By Gr , KF+ ` 8x(Gr(pGr(x)q) $ (T r(pT r(x) _ T r(¬. x)q) _ T r(p¬(T r(x) _ T r(¬. x))q))). By axiom 4 of KF+ , we obtain KF+ ` 8xT r(pT r(x) _ T r(¬. x)q) $ 8x(T r(pT r(x)q) _ T r(pT r(¬. x)q)). By axiom 2 of KF+ , we obtain KF+ ` 8x(T r(pT r(x)q) _ T r(pT r(¬. x)q)) $ 8x(T r(x) _ T r(¬. x)). Thus, we obtain KF+ ` 8xT r(pT r(x) _ T r(¬. x)q) $ 8x(T r(x) _ T r(¬. x)). Furthermore, by axiom 5 of KF+ , we obtain

58

Theories of Partial Truth KF+ ` 8xT r(p¬(T r(x)_T r(¬. x))q) $ T r(p¬T r(x)q)^T r(p¬T r(¬. x)q)). By axiom 3 of KF+ , we obtain KF+ ` 8x(T r(p¬T r(x)q) ^ T r(p¬T r(¬. x)q)) $ 8x(T r(¬. x) ^ T r(¬. ¬. x)). By axiom 8 of KF+ , we obtain KF+ ` 8x(T r(¬. x) ^ T r(¬. ¬. x)) $ 8x(T r(¬. x) ^ T r(x)). Thus, we obtain KF+ ` 8xT r(p¬(T r(x) _ T r(¬. x))q) $ 8x(T r(x) ^ T r(¬. x)). Thus, it follows that KF+ ` 8x(T r(pT r(x) _ T r(¬. x)q) _ T r(p¬(T r(x) _ T r(¬. x))q)) $ 8x(T r(x) _ T r(¬. x)). By Gr , we therefore obtain KF+ ` 8x(Gr(pGr(x)q) $ Gr(x)).

3. We show KF+ ` 8x(SentL+ (x) ! (Gr(x) $ Gr(pT r(x)q))). By KF+ ` 8x(SentL+ (x) ! (T r(pT r(x) ˙ q) $ T r(x))) and KF+ ` 8x(SentL+ (x) ! (T r(p¬T r(x) ˙ q) $ T r(¬. x))) we obtain + KF ` SentL+ (x) ! ((T r(x) _ T r(¬. x)) $ (T r(pT r(x) ˙ q) _ T r(p¬T r(x) ˙ q))), i.e. + KF ` SentL+ (x) ! (Gr(x) $ Gr(pT r(x)q)). 4. KF+ ` 8x(SentL+ (x) ! (Gr(x) $ Gr(¬. x))) follows immediately from KF+ ` 8x(SentL+ (x) ! (T r(x) $ T r(¬. ¬. x))). 5. We show KF+ ` 8x8y(SentL+ (x) ^ SentL+ (y) ! (Gr(x_. y) $ ((Gr(x) ^ Gr(y)) _ T r(x) _ T r(y)))). Assume KF+ ` Gr(x_. y). Then KF+ ` T r(x_. y) _ T r(¬. (x_. y)). KF+ ` T r(x_. y) $ (T r(x) _ T r(y)) and KF+ ` T r(¬. (x_. y)) $ (T r(¬. x) ^ T r(¬. y)), therefore KF+ ` Gr(x_. y) ! ((Gr(x) ^ Gr(y)) _ (T r(x) _ T r(y))). Assume KF+ ` ((Gr(x) ^ Gr(y)) _ (T r(x) _ T r(y))). KF+ ` (T r(x) _ T r(y)) ! T r(x_. y) and KF+ ` (Gr(x) ^ Gr(y)) ! ((T r(x) _ T r(¬. x)) ^ (T r(y) _ T r(¬. y))). KF+ ` ((T r(x) _ T r(¬. x) ^ (T r(y) _ T r(¬. y))) $ ((T r(¬. x) ^ T r(¬. y)) _ (T r(x) ^ T r(y))_ (T r(¬. x) ^ T r(y)) _ (T r(x) ^ T r(¬. y)))) implies KF+ ` ((T r(x) _ T r(¬. x) ^ (T r(y) _ T r(¬. y))) !

3.2 Iterative Approaches to Truth and Groundedness

59

((T r(¬. x) ^ T r(¬. y)) _ (T r(x) _ T r(y)))). Therefore KF+ ` ((Gr(x) ^ Gr(y)) _ (T r(x) _ T r(y))) ! ((T r(¬. x) ^ T r(¬. y)) _ (T r(x) _ T r(y))). 6. KF+ ` 8x8y(V arL+ (x) ^ F rmL+ (y) ! (Gr(8. xy) $ (8zGr(sub(y, x, num(z))) _ 9zT r(¬. sub(y, x, num(z)))))) follows immediately from KF+ ` Gr(8. xy) $ (T r(8. xy) _ T r(¬. 8. xy)) $ (8zT r(sub(y, x, num(z))) _ 9zT r(¬. sub(y, x, num(z)))) and KF+ ` (8zGr(sub(y, x, num(z))) _ 9zT r(¬. sub(y, x, num(z)))) $ (8z(T r(sub(y, x, num(z))) _ T r(¬. sub(y, x, num(z))))_ 9zT r(¬. sub(y, x, num(z)))) $ (8zT r(sub(y, x, num(z))) _ 9zT r(¬. sub(y, x, num(z)))_ 9zT r(¬. sub(y, x, num(z)))) $ (8zT r(sub(y, x, num(z))) _ 9zT r(¬. sub(y, x, num(z)))). 7. KF+ ` 8x(AtL (x) ! (T r(x) $ T rAtL (x))) follows immediately from KF+ ` 8xLitL (x) ! (T r(x) $ T rLitL (x)). 8. KF+ ` 8x(AtL (x) ! (T r(x) $ T rAtL (x))) follows immediately from KF+ ` 8xLitL (x) ! (T r(x) $ T rLitL (x)). 9. KF+ ` 8x((SentL+ (x) ^ Gr(x)) ! (T r(pT r(x) ˙ q) $ T r(x))) follows immediately from KF+ ` 8x(SentL+ (x) ! (T r(pT r(x) ˙ q) $ T r(x))). 10. KF+ ` 8x((SentL+ (x_. y)^Gr(x_. y)) ! (T r(x_. y) $ (T r(x)_T r(y)))) follows immediately from KF+ ` 8x8y((SentL+ (x)^SentL+ (y)) ! (T r(x_. y) $ (T r(x)_T r(y)))). 11. KF+ ` 8x((SentL+ (8. xy) ^ Gr(8. xy)) !

60

Theories of Partial Truth (T r(8. xy) $ 8zT r(sub(y, x, num(z))))) follows immediately from KF+ ` 8x8y((V arL+ (x) ^ F rmL+ (y)) ! (T r(8. xy) $ 8zT r(sub(y, x, num(z))))).

12. We show KF+ ` 8x((SentL+ (x) ^ Gr(¬. x)) $ (T r(¬. x) $ ¬T r(x))). Since KF+ ` 8x¬(T r(x) ^ T r(¬. x)), we obtain KF+ ` 8x(T r(¬. ) ! ¬T r(x)). KF+ ` Gr(¬. x) $ (T r(x) _ T r(¬. x)) and KF+ ` T r(¬. x) ! (¬T r(x) ! T r(¬. x)). Furthermore KF+ ` T r(¬. ¬. x) ! T r(x), and thus KF+ ` T r(¬. ¬. x) ! (¬T r(x) ! ') for any formula '. So we have KF+ ` (T r(¬. x) _ T r(¬. ¬. x)) ! (¬T r(x) ! T r(¬. x)). 13. KF+ ` 8x(¬Gr(x) ! ¬T r(x)) follows immediately from KF+ ` Gr(x) $ (T r(x) _ T r(¬. x)). Lets turn to the second claim TG ` KF+ . Note that we do not need the definition Gr of Gr for the following proof. In other words, the set from definition 3.6 is empty, i.e., = ?. 1. TG ` 8xLitL (x) ! (T r(x) $ T rLitL (x)) follows immediately from TG ` 8x(AtL (x) ! (T r(x) $ T rAtL (x))), TG ` 8x(SentL+ (x) ! (Gr(x) $ Gr(¬. x))), and TG ` 8x((SentL+ (x) ^ Gr(x)) ! (T r(¬. x) $ ¬T r(x))). 2. We show TG ` 8x(SentL+ (x) ! (T r(pT r(x) ˙ q) $ T r(x))). Assume TG ` T r(x). We have TG ` T r(x) ! Gr(x) and TG ` Gr(x) ! Gr(pT r(x)q). Furthermore, TG ` Gr(pT r(x)q) ! (T r(pT r(x)q) $ T r(x)), and so TG ` T r(x) ! T r(pT r(x)q). Secondly, assume TG ` T r(pT r(y)q). Then TG ` T r(pT r(x)q) ! Gr(pT r(x)q), and thus, by TG ` Gr(pT r(x)q) ! (T r(pT r(x)q) $ T r(x)), TG ` T r(pT r(x)q) ! T r(x). 3. We show TG ` 8x(SentL+ (x) ! (T r(p¬T r(x) ˙ q) $ T r(¬. x))).

3.2 Iterative Approaches to Truth and Groundedness

61

By the axioms of TG, we have i TG ` 8x(T r(x) ! Gr(x)), ii TG ` 8x(Gr(¬. x) $ Gr(x)), iii TG ` 8x(Gr(x) $ Gr(pT r(x)q)), iv TG ` 8x(Gr(¬. x) ! (T r(¬. x) $ ¬T r(x))). By the previous case, it follows that v TG ` ¬T r(pT r(x)q) $ ¬T r(x). Assume TG ` T r(¬. x). Thus, by i and iv TG ` T r(¬. x) ! ¬T r(x) and, by v, TG ` T r(¬. x) ! ¬T r(pT r(x)q). Furthermore, by i, ii and iii TG ` T r(¬. x) ! Gr(pT r(x)q) and thus, by ii, TG ` T r(¬. x) ! Gr(p¬T r(x)q). By iv, TG ` Gr(p¬T r(x)q) ! (T r(p¬T r(x)q) $ ¬T r(pT r(x)q)). Therefore we obtain TG ` T r(¬. x) ! T r(p¬T r(x)q). Secondly, assume TG ` T r(p¬T r(x)q). Then, by i, TG ` Gr(p¬T r(x)q) and, by ii and iii, TG ` Gr(¬. x). Thus we can derive TG ` T r(p¬T r(x)q) ! T r(¬. x). 4. TG ` 8x8y((SentL+ (x) ^ SentL+ (y)) ! (T r(x_. y) $ (T r(x) _ T r(y)))) follows immediately from TG ` T r(x_. y) ! Gr(x_. y), TG ` (T r(x) _ T r(y)) ! Gr(x_. y) and TG ` Gr(x_. y) ! (T r(x_. y) $ (T r(x) _ T r(y))). 5. We show TG ` 8x8y((SentL+ (x) ^ SentL+ (y)) ! (T r(¬. (x_. y)) $ (T r(¬. x) ^ T r(¬. y)))). Assume TG ` T r(¬. (x_. y)). We have TG ` T r(¬. (x_. y)) ! Gr(¬. (x_. y)), TG ` Gr(¬. (x_. y)) ! Gr(x_. y) and TG ` Gr(x_. y) $ ((Gr(x) ^ Gr(y)) _ T r(x) _ T r(y)). Since TG ` Gr(¬. (x_. y)) $ (T r(¬. (x_. y)) $ ¬T r((x_. y))), and TG ` (Gr(¬. x)^Gr(¬. y)) $ ((T r(¬. x) $ ¬T r(x))^(T r(¬. y) $ ¬T r(y))) it follows TG ` T r(¬. (x_. y)) ! (Gr(x) ^ Gr(y)).

62

Theories of Partial Truth Thus TG ` T r(¬. (x_. y)) ! (T r(¬. x) ^ T r(¬. y)). Assume TG ` T r(¬. x) ^ T r(¬. y). Then TG ` (T r(¬. x) ^ T r(¬. y)) ! (Gr(¬. x) ^ Gr(¬. y)). Since TG ` Gr(¬. x) $ ¬T r(x) and TG ` Gr(¬. y) $ ¬T r(y)), TG ` (T r(¬. x) ^ T r(¬. y)) ! (¬T r(x) ^ ¬T r(y)). Thus TG ` (T r(¬. x) ^ T r(¬. y)) ! ¬(T r(x) _ T r(y)), from which follows TG ` (T r(¬. x) ^ T r(¬. y)) ! ¬T r(x_. y). By TG ` (Gr(¬. x) ^ Gr(¬. y)) $ (Gr(x) ^ Gr(y)), TG ` (Gr(x) ^ Gr(y)) ! Gr(x_. y) and TG ` Gr(x_. y) $ Gr(¬. (x_. y)), TG ` (T r(¬. x) ^ T r(¬. y)) ! Gr(¬. (x_. y)) and thus TG ` (T r(¬. x) ^ T r(¬. y)) ! (T r(¬. (x_. y)) $ ¬T r(x_. y)). Thus TG ` (T r(¬. x) ^ T r(¬. y)) ! T r(¬. (x_. y)).

6. We show TG ` 8x8y((V arL+ (x) ^ F rmL+ (y)) ! (T r(8. xy) $ 8zT r(sub(y, x, num(z))))). Assume TG ` T r(8. xy). TG ` T r(8. xy) ! Gr(8. xy) and TG ` Gr(8. xy)) $ (T r(8. xy) $ 8zT r(sub(y, x, num(z)))). Thus TG ` T r(8. xy) ! 8zT r(sub(y, x, num(z))). Assume TG ` 8zT r(sub(y, x, num(z))). TG ` 8zT r(sub(y, x, num(z))) ! 8zGr(sub(y, x, num(z))) and TG ` 8zGr(sub(y, x, num(z))) ! Gr(8. xy). Thus since TG ` Gr(8. xy)) $ (T r(8. xy) $ 8zT r(sub(y, x, num(z)))), TG ` 8zT r(sub(y, x, num(z))) ! T r(8. xy). 7. We show TG ` 8x8y((V arL+ (x) ^ F rmL+ (y)) ! (T r(¬. 8. xy) $ 9zT r(¬. sub(y, x, num(z))))). Assume TG ` T r(¬. 8. xy). TG ` T r(¬. 8. xy) ! Gr(¬. 8. xy) and TG ` Gr(¬. 8. xy) ! (T r(¬. 8. xy) $ ¬T r(8. xy)) imply TG ` T r(¬. 8. xy) ! ¬T r(8. xy). Since TG ` ¬T r(8. xy) $ ¬8zT r(sub(y, x, num(z))) (as demonstrated in the previous case), TG ` T r(¬. 8. xy) ! ¬8zT r(sub(y, x, num(z))) and thus TG ` T r(¬. 8. xy) ! ¬9z¬T r(sub(y, x, num(z))). Furthermore, TG ` T r(¬. 8. xy) ! Gr(¬. 8. xy) and TG ` Gr(¬. 8. xy) ! Gr(8. xy).

3.2 Iterative Approaches to Truth and Groundedness Since TG ` (Gr(8. xy) $ (8zGr(sub(y, x, num(z))) _ 9zT r(¬. sub(y, x, num(z))))) we obtain TG ` T r(¬. 8. xy) ! (8zGr(sub(y, x, num(z))) _ 9zT r(¬. sub(y, x, num(z)))). TG ` 8zGr(sub(y, x, num(z))) ! 8zGr(sub(¬. y, x, num(z))) and TG ` Gr(sub(¬. y, x, num(z))) ! (T r(sub(¬. y, x, num(z))) $ ¬T r(sub(y, x, num(z)))). Thus TG ` 8zGr(sub(y, x, num(z))) ! (9zT r(sub(¬. y, x, num(z))) $ 9z¬T r(sub(y, x, num(z)))). Therefore TG ` T r(¬. 8. xy) ! 9zT r(sub(¬. y, x, num(z))). Assume TG ` 9zT r(sub(¬. y, x, num(z))). TG ` T r(sub(¬. y, x, num(z))) ! Gr(sub(¬. y, x, num(z))) and TG ` Gr(sub(¬. y, x, num(z))) ! (T r(sub(¬. y, x, num(z))) $ ¬T r(sub(y, x, num(z)))) imply TG ` T r(sub(¬. y, x, num(z))) ! ¬T r(sub(y, x, num(z))). Thus TG ` 8zT r(sub(¬. y, x, num(z))) ! ¬T r(sub(y, x, num(z))), from which follows TG ` 9zT r(sub(¬. y, x, num(z))) ! 9z¬T r(sub(y, x, num(z))). Furthermore TG ` 9z¬T r(sub(y, x, num(z))) ! ¬8zT r(sub(y, x, num(z))). By the previous case TG ` ¬8zT r(sub(y, x, num(z))) ! ¬T r(8. (sub(y, x, num(z)))) and therefore TG ` 9z¬T r(sub(y, x, num(z))) ! ¬T r(8. (sub(y, x, num(z)))). TG ` Gr(¬. 8. (sub(y, x, num(z)))) ! (T r(¬. 8. (sub(y, x, num(z)))) $ ¬T r(8. (sub(y, x, num(z))))) and TG ` 9zT r(sub(¬. y, x, num(z))) ! Gr(8. (sub(y, x, num(z)))). Thus we obtain TG ` 9zT r(sub(¬. y, x, num(z))) ! T r(¬. 8. (sub(y, x, num(z)))). 8. We show TG ` 8x(SentL+ (x) ! (T r(x) $ T r(¬. ¬. x))). TG ` Gr(¬. x) ! (T r(¬. x) $ ¬T r(x)) and TG ` Gr(¬. ¬. x) ! (T r(¬. ¬. x) $ ¬T r(¬. x)).

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Theories of Partial Truth By TG ` Gr(x) $ Gr(¬. x)) $ Gr(¬. ¬. x), TG ` Gr(x) ! (T r(¬. ¬. x) $ T r(x)). The claim follows from TG ` T r(x) ! Gr(x) and TG ` T r(¬. ¬. x) ! Gr(x).

9. We show TG ` 8x¬(T r(x) ^ T r(¬. x)). I show the claim by showing TG ` 8x(T r(x) ! ¬T r(¬. x)) and TG ` 8x(T r(¬. x) ! ¬T r(x)). First TG ` T r(x) ! Gr(x) and TG ` Gr(x) ! Gr(¬. x), therefore TG ` T r(x) ! Gr(¬. x). Further TG ` Gr(¬. x) $ (T r(¬. x) $ ¬T r(x)) and thus TG ` T r(x) ! ¬T r(¬. x). Secondly, TG ` T r(¬. x) ! Gr(¬. x) and thus TG ` T r(¬. x) $ ¬T r(x). 10. We show TG ` 8x(Gr(x) $ (T r(x) _ T r(¬. x))). TG ` 8x(Gr(x) ! (T r(x) _ T r(¬. x))) follows from TG ` 8x((SentL+ (x) ^ Gr(x)) ! (T r(¬. x) $ ¬T r(x))) and TG ` 8x(T r(x) _ ¬T r(x)). TG ` 8x((T r(x) _ T r(¬. x)) ! Gr(x)) follows from TG ` 8x(¬Gr(x) ! ¬T r(x)) and 8x(SentL+ (x) ! (Gr(x) $ Gr(¬. x))). Example 3.9 8x(T r(x)_¬T r(x)). Since KF 0 T r(p8x(T r(x)_¬T r(x))q) and KF 0 T r(p¬8x(T r(x) _ ¬T r(x))q) (which can be proved by making use of partial models of Kripke’s semantical fixed point approach, which is sketched in section 5.1), it follows that TG 0 Gr(p8x(T r(x) _ ¬T r(x))q). However as 8x(T r(x) _ ¬T r(x)) is a logical truth of predicate logic, it intuitively could be counted as grounded. So TG does not prove all sentences of L+ to be grounded, that one might intuitively view as grounded. A concept of groundedness, which takes into account the sentence of example 3.9 is presented in Leitgeb[22].

4 The Contextual Approach to the Strengthened Liar Problem Theories of partial truth turn out to su↵er from the strengthened liar problem, which was informally described in chapter 1. In section 4.1, I shall formally reconstruct the strengthened liar argument in case of formal languages (which were introduced in section 2.1) and their partial truth predicates (which were introduced in the last chapter). I shall also illustrate the formal conditions of the strengthened liar problem. Subsequently, in section 4.2, I shall clarify how contextual approaches to the strengthened liar problem are formally realized. In section 4.3, I shall present the contextual approaches of Burge, Parsons and Barwise and Etchementy. A fundamental question for any contextual approach is to find out at which position of the strengthened liar argument the interpretation of the partial truth predicate changes due to a semantic shift. I shall demonstrate that the contextual approaches, which I present in section 4.3, agree on this question in case of my formal reconstruction of the strengthened liar problem.

4.1 The Strengthened Liar Problem I shall now present two formal reconstructions of the strengthened liar problem. For the rest of this chapter, we again assume that LM is a first order language of predicate logic, that meets the conditions S1 S4, which were explicated in section 2.2. Furthermore, we assume that T r and Gr are predicates of L, such that M ✏ TG.1 Consider a liar sentence, i.e., a fixed point sentence of the one-place formula ¬T r(x). Thus M ✏ p q ⌘ p¬T r(p q)q, and by substitution of 1 Note

that I now write ‘L’ instead of ‘L+ ’.

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identities M ✏ $ ¬T r(p q). In chapter one, I have informally demonstrated that some semantical statements about the strengthened liar sentence cannot be expressed in the same (informal) language, to which the liar sentence belongs, but only in a more comprehensive meta-language. E.g. we cannot express that the liar sentence is not true in the sense of a strong negation, or that the strengthened liar sentence is semantically defective, in the sense given by the truth table on page 12. This fact also holds in case of our formal interpreted language (L, M) and our theory TG. We can prove that our theory TG for T r and Gr cannot meet the following rules (otherwise it would be inconsistent): TG ` ¬T r(p'q) i↵ TG 0 T r(p'q), and TG ` ¬Gr(p'q) i↵ TG 0 T r(p'q) and TG 0 T r(p¬'q). This means that ¬T r does not express ‘not true’ in the sense of a strong negation, but it expresses a weaker concept. Accordingly ¬Gr does not express ‘semantical defectiveness’ in the sense stated on page 12, but it expresses a weaker concept of ‘semantical defectiveness’. Our first formal version of the strengthened liar argument has following premises: P1 M ✏

$ ¬T r(p q)

P2 M ✏ ¬Gr(p q) P3 M ✏ T r(p¬Gr(p q)q) P1 expresses that is a liar sentence. P2 and P3 express certain semantic statements about , or, more precisely, P2 expresses that is ungrounded, and P3 expresses the truth of this semantic diagnosis about . Let us briefly recapitulate the status of these premises. By the diagonalization lemma (see lemma 2.1 on p. 23), which holds for all languages of first order predicate logic with identity LM that meet the conditions S1 S4, we can construct a strengthened liar sentence such that M ✏

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$ ¬T r(p q). Thus we obtain the first premise P1. The second premise P2 holds for all models M with M ✏ TG. Although this seems to be very plausible, let me demonstrate how to prove P2. Proof of P2. Since M ✏ TG, if follows that M ✏ KF (by fact 3.2). Let M ✏ T r(p q). Then, by substitution of identities, M ✏ T r(p¬T r(p q)q). Therefore, by axiom 3 of KF, M ✏ T r(p¬ q). Furthermore, by axiom 12 of TG, M ✏ Gr(p q). Therefore, by axiom 9 of TG, M ✏ ¬T r(p q), which contradicts to M ✏ T r(p q). Secondly, let M ✏ T r(p¬ q). Then, by substitution of identities, M ✏ T r(p¬¬T r(p q)q). Therefore, by axiom 3 of KF, M ✏ ¬T r(p¬T r(p q)q), and, by a second application of axiom 3 of KF, M ✏ ¬T r(p¬ q), which contradicts to M ✏ T r(p¬ q). Also the third premise P3 seems to be reasonable. However, the truth of the semantic diagnosis ¬Gr(p q) about , in other words, T r(p¬Gr(p q)q), cannot be true in M. As I shall now show, assuming premises P2 and P3 simultaneously leads to a contradiction. From P3 and axiom 13 of TG we conclude C1 M ✏ Gr(p¬Gr(p q)q). By C1 and axiom 4 of TG, we obtain C2 M ✏ Gr(pGr(p q)q). By C2 and axiom 2 of TG, we obtain C3 M ✏ Gr(p q). From P2 and C3 we obtain

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The Contextual Approach to the Strengthened Liar Problem C4 M ✏ Gr(p q) ^ ¬Gr(p q)

Thus, given our reasoning from P1 to C4 has been correct, assuming that according to our model M, is semantically defective and it is true that is semantically defective leads to a contradiction C4. We obtain the second formal reconstruction of the strengthened liar argument by the following premises: P10 M ✏

0

$ ¬T r(p 0 q)

P20 M ✏ ¬T r(p 0 q) P30 M ✏ T r(p¬T r(p 0 q)q) Note that the only di↵erence to our first formal reconstruction of the strengthened liar argument is that we have ‘¬T r(p 0 q)’ instead of ‘¬Gr(p 0 q)’ in the second premise P20 , and ‘T r(p¬T r(p 0 q)q)’ instead of ‘T r(p¬Gr(p 0 q)q)’ in the third premise P30 . From P30 it follows, by substitution of identities, that C10 M ✏ T r(p 0 q). From P20 and C10 we obtain C20 M ✏ T r(p 0 q) ^ ¬T r(p 0 q) Thus, given our reasoning from P10 to C20 has been correct, assuming that according to our model M, 0 is not true and it is true that 0 is not true leads to a contradiction C20 . Our two formal reconstructions of the strengthened liar argument show that no semantical diagnosis about strengthened liar sentence together with a sentence expressing the truth of this semantical diagnosis can be true in our model M. This is a disadvantage for M and TG (and KF, respectively).

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4.2 The Contextual Approach So far, not too many contextual approaches (and approaches in general) to the strengthened liar problem have been put forward. A problem in the current literature of contextual approaches is the diversity of logicomathematical formalisms that are applied by the authors. They use di↵erent methods to represent a context in a formal language, di↵erent signatures, and also di↵erent ways to describe the proposed theory of truth. Furthermore, they assume di↵erent standards of formal justification of their theories, i.e. they di↵er in whether any proofs for the existence, consistency or axiomatizability of their theories are given. A main open problem is to find out what the essential commonalities of and di↵erences between currently existing contextual approaches are. To provide the means for a comparative study of contextual approaches one first needs to fix a common formal representation that encompasses all contextual approaches. In this section, I shall present such a formal representation. There are several strategies to evade the strengthened liar argument which directly derive from the formal representation, that I have stated in the previous section.2 Besides premises P1, P2 and P3, we have used the axioms of TG.3 Thus the basic strategies that arise from the structure of the argument are to reject at least one of the premises, or to reject at least one of the axioms of TG. Rejecting any of the premises means to ignore the whole problem. The fact M ✏ $ ¬T r(p q) cannot be rejected without rejecting either the weak conditions of the diagonalization lemma (recall S1 S4 which were introduced in section 2.2 on p. 29) or the rule of substitution of identities. If we doubt that we can derive (the truth of) any semantic statement about in our theory, we admit expressive limitations that can obviously not be found in the case of natural languages. So, as I have already argued in chapter one (p. 12), we are confronted with the 2 Of

course not all assumptions that have been made are displayed within my formal representation of the strengthened liar argument. I have made the argument just clear enough in order to make it reveal the strengthened liar problem. It is disputable whether it is necessary to make all assumptions explicit to solve the strengthened liar problem, and it is also doubtful whether this is actually possible. 3 Note that P1 has only been used in the second version of the formal representation of the strengthened liar argument.

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unable-to-say problem. TG has been motivated in section 3.2. As I have argued there, the axioms of TG display reasonable concepts of truth and groundedness. However, the great majority of philosophers4 dealing with the strengthened liar problem do not pursue any of the just mentioned strategies. In fact the most philosophers dealing with the strengthened liar problem favor a contextual approach. The contextual strategy is to agree to each conclusion of the revenge argument, but to show that some equivocal, context-sensitive element has been overlooked which reveals that C4 and C20 are fact is no instances of a contradiction. In natural languages a context is always related to a concrete utterance of a sentence. Therefore the concept of a context belongs to the domain of pragmatics. According to conventional linguistic theories, the context, together with semantical factors, determines how an utterance is interpreted. In case of a formal language however, there is no theory of pragmatics, as the sentences of formal languages are not uttered. So there is nothing like e.g. a social context, as suggested by certain linguistic theories, which contributes to the interpretation of a sentence. Contextual approaches to the strengthened liar problem are not committed to any particular pragmatic theory. Accordingly, proponents of contextual approaches to the strengthened liar problem say little or nothing about what kind of context-shift they have in mind. What they analyze is in fact a semantic shift. In formal languages the interpretation of a sentence is completely determined by a semantic model. We use a meta-language to describe how a sentence is interpreted by this model and to state the truth of a sentence in a model. There are three options to implement a semantic shift. The first option is to change the model, by which the sentences of the formal language are interpreted.56 4 Exceptions

are Skyrms[35] and McDonald[25]. to models, one could also use di↵erent theories to formally represent a contextual approach to the strengthened liar problem. If the interpretation of the symbols of the formal language considered is given by a theory, then switching that theory will account for the semantic shift. 6 Glanzberg (cf. Glanzberg[12]) even refers to models as ‘contexts’. But this does not mean that he identifies models with contexts, or that contexts can be reduced to models. It is only to stress that models correspond to contexts in some way which (as Glanzberg[12] argues) is linguistically well motivated. I shall however not discuss the 5 Alternatively

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71

The second option is to change the formal representations of the natural language sentences. In case of the strengthened liar problem one can e.g. assume that ‘is true’ is formally represented by two (or even more) di↵erent predicate constants T r1 and T r2 , and that ‘is grounded’ is formally represented by two (or even more) di↵erent predicate constants Gr1 and Gr2 . The semantic shift then results from switching from T r1 (Gr1 ) to T r2 (Gr2 ) at a particular position in the formal strengthened liar argument. Another possible version of this second option is to represent ‘is true’ by a two-place predicate T r(x, y) and implement the semantic shift by switching between di↵erent instances at the second coordinate. The third option is simply the combination of the first and the second option, i.e., several models, which each provide interpretations for several predicate constants T r1 , T r2 , Gr1 , Gr2 , . . . are assumed in order to implement the semantic shift in the informal strengthened liar argument. Current contextual approaches favor the second and third option. Since current contextual approaches assume that the interpretation of ‘is true’ changes within one and the same sentence, several di↵erent predicate constants, which represent ‘is true’, have to be available in one model. We will see that most of the contextual approaches that are introduced here assume an infinite hierarchy of partial (iterative) truth predicates and an infinite hierarchy of models. For each of these models, there is still one distinguished truth predicate, which stands in the special relation to the model of ‘being the truth predicate of that model’. The other truth predicates are not the truth predicates of that model, but they “coincide”7 with the truth predicates of other models. So, speaking loosely, we can refer in one context to the set of all sentences that are true in another context. A central question for contextual approaches is of course at which point the semantic shifts take place in our informal strengthened liar reasoning and, respectively, in the above strengthened liar argument. I will now show how this is answered in existing contextual approaches. The basic components of contextual reconstructions are two models M1 and M2 , and two truth predicates T r1 and T r2 . M1 is a {T r 1 }-expansion of some model M, often called the ground model. M2 is a {T r2 }-expansion of M1 . The 7I

question of the nature of this correspondence here. will clarify what ‘coincide’ means later.

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first premise could then e.g. be reformulated as follows8 : P1 CR M1 ✏

$ ¬T r1 (p q)

But of course there are also three other possibilities of how to formally represent the first premise, in which M and T r are associated with other indices. The same is the case for the other premises and conclusions. The only condition that should be met by contextual reconstructions is that the conclusions C4 and C20 should be no instances of a contradiction. In the current literature there is basically only one proposal for inserting indices to the items of the strengthened liar argument. The majority of contextual approaches (e.g. Parsons[28], Burge[8] and Glanzberg[12]) suggest to assume the following “location” of the semantic shift:9 P1 CR M1 ✏

$ ¬T r1 (p q)

P2 CR M2 ✏ ¬Gr1 (p q) P3 CR M2 ✏ T r2 (p¬Gr1 (p q)q) The analogous suggestion, in case of our second formal version of the strengthened liar argument, is the following: P10 CR M1 ✏

$ ¬T r1 (p q)

P20 CR M2 ✏ ¬T r1 (p q) P30 CR M2 ✏ T r2 (p¬T r 1 (p q)q) From now on, let both of these contextual reconstruction schemes be denoted by ‘CR1’, and let ‘P1 CR1 ’, ‘P2 CR1 ’ and ‘P3 CR1 ’ denote the first, second and third premises of CR1. So P1 CR1 = P1 CR = P10 CR , and 8 The

subscript ‘CR’ stands for ‘contextual reconstruction’. the contextual approach assumes only one model (and no hierarchy of models) then ‘M1 ’ and ‘M2 ’ can be replaced by ‘M’.

9 If

4.3 Three Instances of CR1

73

either P2 CR1 = P2 CR and P3 CR1 = P3 CR or P2 CR1 = P20 CR and P3 CR1 = P30 CR . Let me indicate CR1 as follows: P1 CR1 M1 ✏

$ ¬T r1 (p q)

P2 CR1 M2 ✏ ¬Gr1 (p q) or M2 ✏ ¬T r1 (p q) P3 CR1 M2 ✏ T r2 (p¬Gr1 (p q)q) or M2 ✏ T r2 (p¬T r1 (p q)q) The intuitive idea behind CR1 is that the strengthened liar sentence is not true and not semantically determined in the initial model, but that it is true in the second model. In the next sections we will meet three different ways to motivate this idea. To this end, I will present three instances of CR1 that can be found in the literature, the contextual approaches of Burge, Parsons and Barwise and Etchemendy. Note that of course the Tarski hierarchy of languages I have introduced in section 3.1. provides an instance for CR1 as well. Some authors (such as e.g. Burge[8]) have suggested a contextual interpretation of the Tarski hierarchy. However, this interpretation has not raised much attention since the Tarskian truth predicates T r1 , T r2 , . . . do not allow for any iterative applications. This is a serious weakness in comparison to other partial truth predicates. Tarski himself did neither consider his hierarchical approach as a model for natural languages, nor did he ever speak of any context shift occurring in the liar argument. Thus the Tarski hierarchy of languages is usually not counted as a contextual approach. Further instances for CR1 are the contextual approaches of Gaifman[11], Simmons[33] and Glanzberg[12].

4.3 Three Instances of CR1 1. Burge’s approach: Burge[8] proposes to attach natural numbers as subscripts to the truth predicate true in order to indicate the contextually shifting extension of true. The relevant context associated with a sentence occurrence in general

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consists of certain implicatures that are pragmatically associated with that sentence occurrence. Burge presents the revenge liar reasoning in three steps (a), (b) and (c). In step (a) we start with a strengthened liar sentence denoted by ‘(1)’: step (a): (1) i↵ (1) is not truei . Implicature: (1) is evaluated by the truei -scheme. Thereby, i is an arbitrary natural number, and the truei -scheme is the scheme ‘S ’ is truei i↵ S.10 In the next step (b) this implicature is canceled which, according to Burge, is forced by foregoing naive liar-reasoning (see chapter one) which follows from assuming the unrestricted truei -scheme. The consequence of this cancellation is, according to Burge, that (1) is regarded as a pathological sentence and therefore our intuitive conclusion is that (1) is not truei : step (b): (1) is not truei . The implicature of step (a) is canceled. In the last step (c), the implicature of step (a) is replaced by the new implicature that (1) is evaluated according to the truek -scheme (where k is any natural number greater than i). Furthermore, we have concluded in step (b) that (1) is not truei , and this is just what (1) is saying. Thus, as Burge argues, our next conclusion is that (1) is truek : step (c): (1) is truek . Implicature: (1) is evaluated by the truek -scheme. Burge describes three di↵erent theories, which he calls ‘constructions’, for his truth predicates truei . The first two constructions C1 and C2 are 10 S

is any sentence and ‘S ’ is a name of S.

4.3 Three Instances of CR1

75

mainly stated to motivate the third construction C3. Burge considers C3 to be the most useful system as it is probably closest to intuition (cf. [8], p. 190, paragraph 3). Furthermore, C3 is the most liberal system (cf. Burge[8] p. 188, last paragraph) as it derives intuitively non-pathological sentences that the other two systems declare as pathological. As Burge states (cf. Burge[8], p. 185, last paragraph), the basic idea behind all of the three constructions is to define a notion of a pathologicali sentence and then to claim that pathologicali sentences are not truei and to assert all and only instances of the truei -scheme obtained from substituting sentences that are not pathologicali . If k > i, a sentence that is pathologicali may be non-pathologicalk and truek . I shall only present the most interesting third construction C3. Burge assumes a language of first order predicate logic LM to which he adds for each i 2 N • a two-place satisfaction predicate Sati (x, y) and • a one-place predicate variable Ri (y) for ‘y is rootedi ’. To make Burge’s system better comparable with other theories I shall reformulate C3 with truth predicate “variables” T ri (x) (for each i 2 N). It should be noted that by speaking of predicate “variables” Burge means that the numerals substituted for i do not mark new predicate constants, but are contextual applications of one single indexical truth predicate (cf. Burge[8], p. 191). But formally T ri (x) is no variable, as it is not possible to quantify into the index i of T r (cf. Burge[8], p. 192, second paragraph). Actually there is no hint in Burge’s text that T ri (x) should formally be treated in any other way than ordinary predicate symbols of first order logic. I shall also add a one-place predicate constant Senti (x) for each i 2 N to the language, which represent the set of all meta-language codes of sentences with object language code x that contain only occurrences of T rk (x) and Rk (y) such that k  i. There are other small deviations in my reformulation of Burge’s C3, which do not make much di↵erence, but make the system better comparable to other theories stated here. So let (my reformulation of) C3 be the theory that results from the following axiom schemata and rule:

76

The Contextual Approach to the Strengthened Liar Problem

Burge’s construction C3: 1. Sentk (p'q) ! Ri (p'q) for all sentences ' and for all k < i, i, k 2 N; 2. Ri (p'q) ! (Ri (pT ri (p'q)q) ^ Ri (pRi (p'q)q)) for all sentences ' and for all i 2 N; 3. Ri (p'q) ! Ri (p¬'q) for all sentences ' and for all i 2 N; 4. (Ri (p'q) ^ Ri (p q)) _ (T ri (p'q) _ T ri (p q)) ! Ri (p' _ q) for all sentences ', and for all i 2 N; 5. 8yRi (sub(p'q, x, num(y)) _ 9y T ri (sub(p¬'q, x, num(y)))) ! Ri (p8x'q) for all sentences ', and for all i 2 N; 6. For any sentence ', if Ri (p'q) 2 C3 for any i 2 N, then Ri (p'q) is derivable from the axioms 1. 5., together with the axioms of a suitable system of first order arithmetic and the axioms of a suitable system of predicate logic, and the modus ponens rule; 7. ¬Ri (p'q) ! ¬T ri (p'q) for all sentences ' and for all i 2 N; 8. Ri (p'q) ! (T ri (p'q) $ ') for all sentences ' and for all i 2 N. As Burge also claims of C3, to account for the above steps (a) (c) and furthermore that C3 meets the law of excluded middle (i.e. for each sentence ', either T ri (p'q) or ¬T ri (p'q) is in C3, cf. Burge[8] p. 190, paragraph 4), C3 should be supplemented by the following rule that I shall denote ‘6*’: 6*. If for some sentence ', Ri (p'q) is not derivable from the axioms 1. 5., together with the axioms of a suitable system of first order arithmetic and the axioms of a suitable system of predicate logic and the modus ponens rule, then ¬Ri (p'q) 2 C3. Let us assume that

is a sentence such that

4.3 Three Instances of CR1

P1 Bu C3 `

77

$ ¬T r1 (p q).

Then premise P2 Bu , i.e., P2 Bu C3 ` ¬R1 (p q) can be verified by rule 6* and by proving, by induction over the complexity of formulas, that R1 (p q) cannot be derived from C3. Premise P3 Bu , i.e. P3 Bu C3 ` T r2 (p¬R1 (p q)q) is proved by using axioms 1 and 8 of C3 and P2 Bu . C3 is thus an instance for CR1. Unfortunately the technical side of Burge’s proposal is not sufficiently elaborated. Burge does not give any strict existence- and uniqueness-proof for C3, and also no proof for the consistency of C3. In addition, it would be interesting to know whether the axiom schemata 1 5, 7 and 8 and rule 6 could be replaced by axioms. Furthermore, some facts about C3 are stated without proof, e.g. that the law of excluded middle is met by C3 (cf. Burge[8] p. 188, paragraph 3). 2. Parsons’ approach: A very di↵erent formal representation of the strengthened liar problem is proposed by Parsons[28]. Parsons aims at an account that stresses the analogies of the paradoxes of set theory with that of semantics, and that makes plausible an interpretation of natural language in terms of some hierarchy such as Tarski’s language levels (cf. Parsons[28], p. 10, paragraph 2). He gives two kinds of such representations, both within a first order language L of predicate logic interpreted by an ordinary extensional semantics, which is determined by a model M = (D, I) (consisting of a set D called the universe of discourse and an assignment I) and which has the means to describe its own syntax (in the sense stated in the second chapter). The second representation is mainly the same as my above

78

The Contextual Approach to the Strengthened Liar Problem

contextual reconstruction scheme CR1. The basic strategy of his first representation is to focus on quantifiers and their interactions with semantic predicates. The interpretation of a quantifier is determined by the universe of discourse. The universe, and the assignment of extensions to the semantic predicates can change from context to context, and it is this sort of change that resolves the strengthened liar problem according to Parsons’ first representation. No quantifiers occur in the above strengthened liar sentence . However Parsons suggests to focus on another formal representation of a strengthened liar sentence which contains quantifiers, and which is based on the idea that sentences should be replaced by propositions as bearers of truth11 . Parsons introduces propositions as new types of individuals to the domain D, and he assumes his language to contain a one-place predicate P rop, such that P rop(x) represents the set of all propositions in D. Furthermore, he introduces a two-place predicate Exp in L, such that Exp(x, y) represents an expression relation between sentences and propositions in D. If x denotes the object language code of a sentence ' and y denotes a proposition p, then M ✏ Exp(x, y) i↵ ' expresses the proposition p. Furthermore, T r(x) represents the set of all true propositions in D. Instead of Tarski’s T r-scheme the following modified scheme is assumed for each sentence ' of L: Tr-Prop M ✏ 8x((P rop(x) ^ Exp(p'q, x) ! (T r(x) $ '))) Then one can construct a new strengthened liar sentence as follows: P1 Pa1 M ✏

Pa

$ ¬9x(P rop(x) ^ Exp(p

P a q, x) ^ T r(x))

We can now derive that M ✏ ¬9x(P rop(x) ^ Exp(p Assume, for reductio, that M ✏ 9x(P rop(x) ^ Exp(p 11 This

P a q, x)).

P a q, x))

as follows:

Then, by Tr-Prop,

move is frequently discussed in the literature on the simple liar paradox. The unanimous view is that it does not help to eliminate the problem, as another strengthened liar sentence can be formed.

4.3 Three Instances of CR1 M✏

Pa

! 9x(P rop(x) ^ Exp(p

79

P a q, x) ^ T r(x)).

However by P1-Prop M✏

Pa

$ ¬9x(P rop(x) ^ Exp(p

Thus M ✏ ¬ M✏¬

Pa

P a.

P a q, x) ^ T r(x)).

On the other hand

! 9x(P rop(x) ^ Exp(p

P a q, x) ^ T r(x)).

But because Tr-Prop M ✏ 8x((P rop(x) ^ Exp(p

P a q, x) ^ ¬ P a )

from which it follows that M ✏ ¬¬ dum, M ✏ ¬9x(P rop(x) ^ Exp(p

P a.

! ¬T r(x)),

Thus, by reductio ad absur-

P a q, x)).

We have proved in M that P a does not express a proposition, which intuitively seems to be a correct semantical statement about P a . It would therefore be desirable to also be able to prove the truth of this semantical diagnosis in M. This means that we would like to derive: M ✏ 9x(P rop(x) ^ Exp(p¬9x(P rop(x) ^ Exp(p

P a q, x))q, x) ^ T r(x)).

So, according to Parsons’ first representation we can derive the following premises, which correspond to the premises of CR1: P1 Pa1 M ✏

Pa

$ ¬9x(P rop(x) ^ Exp(p

P2 Pa1 M ✏ ¬9x(P rop(x) ^ Exp(p

P a q, x) ^ T r(x))

P a q, x))

P3 Pa1 M ✏ 9x(P rop(x) ^ Exp(p¬9x(P rop(x) ^ Exp(p

P a q, x))q, x) ^ T r(x))

80

The Contextual Approach to the Strengthened Liar Problem

Parsons does not explicitly mention the axiom system he uses for deriving a contradiction. I think it is useful to provide some more details at this point in order to see the parallels to Burge’s account and the above general scheme. The following axioms are analogous to axioms 4 and 2 of TG: NegProp M ✏ 9x(P rop(x) ^ Exp(p'q, x)) $ 9x(P rop(x)^ Exp(p¬'q, x)). ItProp M ✏ 9x(P rop(x) ^ Exp(p'q, x)) $ 9y(P rop(y) ^ Exp(p9x(P rop(x) ^ Exp(p'q, x))q, y)). By a similar argument to that given on p. 66, we can derive a contradiction from these assumptions. From P3 Pa1 , we obtain (by predicate logic) C1 Pa1 M ✏ 9x(P rop(x) ^ Exp(p¬9x(P rop(x) ^ Exp(p

P a q, x))q, x)).

By C1 Pa1 and NegProp, we get C2 Pa1 M ✏ 9x(P rop(x) ^ Exp(p9x(P rop(x) ^ Exp(p

P a q, x))q, x)).

By C2 Pa1 and ItProp, we obtain C3 Pa1 M ✏ 9x(P rop(x) ^ Exp(p

P a q, x)).

Thus, by C3 Pa1 and P2 Pa1: C4 Pa1 M ✏ 9x(P rop(x) ^ Exp(p ¬9x(P rop(x) ^ Exp(p

P a q, x))^

P a q, x)).

Parsons suggests two ways to implement a semantic shift in his reformulated strengthened liar argument. His first proposal is to assume shifting universes of discourse by which the existential quantifiers occurring in premises P1 Pa1 P3 Pa1 are interpreted (cf. Parsons[28], p. 18, paragraph

4.3 Three Instances of CR1

81

4). Parsons speaks of an ‘ontological lack’ of the first model12 with respect to the second model (cf. Parsons[28], p. 23, paragraph 2). One could, e.g., interpret the first two premises as ranging over an initial domain D 1 and the third premise as ranging over a second, more comprehensive domain D2 , D 2 ◆ D1 . At this point, Parsons skips some technical complications that arise from assuming two di↵erent domains. To employ di↵erent domains in general means to employ di↵erent assignments of individuals to the symbols of the language. Although Parsons does not explicitly mention how the two interpretation functions I 1 and I 2 belonging to the two di↵erent domains D1 and D 2 are related, it is obvious what he has in mind. Namely that for all constants c, I 1 (c) = I 2 (c), and for all predicate and function symbols s, I 1 (s) ✓ I 2 (s). Contrary to the first existential quantifier occurring in premise P3 Pa1 the existential quantifier occurring in ¬9x(P rop(x) ^ Exp(p P a q, x)) (the object language code of which occurs in P3 Pa1 ) shall range over the original domain D 1 . This points to a minor technically difficulty for Parsons contextual reformulation of premise P3 Pa1 above. In the new model M2 = (D 2 , I 2 ) we have to refer to the sentence ¬9x(P rop(x) ^ Exp(p P aq, x)) in the way as it was interpreted according to the old model. This could e.g. be realized by introducing a new predicate Dom1 to L2 , such that I 1 (Dom1 ) = D1 : P3⇤Pa1

(D 2 , I 2 )

h ✏ 9x P rop(x)

^ ⇣ Exp x, p¬9x(Dom1 (x) ^ P rop(x) ^ Exp(p ^ i T r(x)

⌘ P a q, x))q

The second way to implement the semantic into the strengthened liar argument is simply to assume that the extension of the expression relation Exp is shifting. Whereas in the first model no pair consisting of the liar sentence and any proposition is in the extension of Exp, in the second model there is such a pair. The proposition expressed by the liar sentence in the second model could already be available in the first model, i.e., it could be 12 Actually,

Parsons uses the term ‘theory’ here, but this would derange my current terminology.

82

The Contextual Approach to the Strengthened Liar Problem

an element of the domain of the first model. So in the above representation M1 = (D1 , I1 ) would in general only shift to M2 = (D1 , I2 ). The interpretation function changes with respect to what extension is assigned to the expression relation. Parsons describes this situation as follows: ‘The interpretation does not capture its own relation of expression.’ (cf. Parsons[28], p. 26, paragraph 1). Parsons second proposal to implement the semantic shift thus leads to the following reformulation of the strengthened liar argument: P1 Pa2 M1 ✏

1 Pa

$ ¬9x(P rop(x) ^ Exp1 (p

P2 Pa2 M2 ✏ ¬9x(P rop(x) ^ Exp1 (p

1 q, x) ^ T r 1 (x)) Pa

1 q, x)) Pa

P3 Pa2 M2 ✏ 9x(P rop(x) ^

Exp2 (p¬9x(P rop(x) ^ Exp1 (p

1 q, x))q, x) Pa

^ T r2 (x))

Also Parsons omits many technical details of his account. The proposed theories are described on an informal meta linguistic level, sometimes they are even incompletely stated. Questions of axiomatizability and consistency are not considered. 3. Barwise’s and Etchemendy’s approach: According to Barwise and Etchemendy, what is crucial to meet the strengthened liar problem is to discern between sentences and propositions expressed by sentences. A sentence can express di↵erent propositions depending on the “situation” in which the sentence is uttered. Barwise and Etchemendy adopt the notion of a situation from situation semantics (see Barwise and Perry[4]). Accordingly Barwise and Etchemendy introduce a formal language LP rop , which ‘gives us mechanisms for expressing propositions’ and which also ‘gives us mechanisms for referring to [...] propositions’ (cf. Barwise and Etchemendy[3], p. 31). The basic vocabulary of LP rop consists of certain constants and certain predicates among which there is

4.3 Three Instances of CR1

83

a one-place predicate T r. The semantics of LP rop is formalized within a language of set theory LSemP rop , i.e. the terms of LSemP rop all refer to sets or ‘atoms’ (i.e. urelements) of a certain domain of sets.13 Therefore Barwise’s and Etchemendy’s contextual approach is stated in a very specific language, and it is not so obvious how it is related to the reformulation of the strengthened liar argument in LM stated in section 4.1. In what follows I shall stick to Barwise’s and Etchemendy’s specific formalism. However, as it has been demonstrated by Koons[20] and Brendel[7], Barwise’s and Etchemendy’s account (i.e. their interpreted language) can as well be translated14 to and reformulated within a conventional first order language of predicate logic such as LM of section 4.1, which does not contain any special terms that refer to propositions or situations. To define the notions ‘proposition’ and ‘situation’, Barwise and Etchemendy put forward a rigorous set-theoretic formalism that is based on Peter Aczel’s[1] alternative set theory ZFC/AFA. ZFC/AFA consists of all axioms of ZFC except that the axiom of foundation15 is replaced by the so-called anti-foundation axiom AFA, which is a strong form of the negation of the axiom of foundation. This axiom system is ‘tailor-made’ (cf. Barwise and Etchemendy[3]. p. 35) to structurally represent a concept of circular propositions (cf. Barwise and Etchemendy[3], p. 34), which Barwise and Etchemendy consider to be the kinds of objects “expressed” by liar sentences. In their formalism propositions are certain kinds of sets, and circular proposition are certain kinds of non-well founded sets, i.e. sets that violate the foundation axiom. Let V be the domain of sets by which LSemP rop is interpreted. The sets in V are formed according to the axioms of ZFC/AFA. Among the urelements of V are the constants and predicate symbols of LP rop . Within LSemP rop the formulas of LP rop (to be more accurate, the meta-language codes of these formulas) can be 13 In

order to discern Barwise’s and Etchemendy’s language from the conventional formal language LM I have used in section 4.1, I refer to Barwise’s and Etchemendy’s language by ‘LP rop ’. This terminology, as well as the term ‘LSemP rop ’, do not correspond to Barwise’s and Etchemendy’s original terminology. 14 Koons[20] has shown that there is a homomorphism between a certain subclass of the semantics of Barwise and Etchemendy and the semantics developed by Burge[8]. 15 The axiom of foundation asserts that 2 is well-founded, i.e.: any nonempty collection Y of sets has a member y 2 Y which is disjoint from Y .

84

The Contextual Approach to the Strengthened Liar Problem

related to propositions (to be more accurate, to terms of LSemP rop that denote propositions in V ). The propositions in V are the truth bearers, that is they are assigned truth values, ‘true’ or ‘false’, by a certain inductive definition (cf. Barwise and Etchemendy[3], p. 127). Barwise’s and Etchemendy’s set theoretic representation of a proposition16 is inspired by a concept of propositions that has been advanced by Austin[2]. Barwise and Etchemendy define the following types of sets: the ‘propositions’, the ‘states of a↵airs’ and the ‘situations’. For the sake of simplicity, let me assume the following notation: I shall use ‘c1 ’, ‘c2 ’, . . . as constants of LSemP rop that refer to the constants of LP rop , I shall use ‘p1 ’, ‘p2 ’, . . . as constants of LSemP rop that refer to propositions in V , and I shall use ‘s1 ’, ‘s2 ’, . . . as constants of LSemP rop that refer to situations in V . Furthermore, I shall use ‘t1 ’, ‘t2 ’, . . . as arbitrary terms of LSemP rop , and ‘P1 ’, ‘P2 ’, . . . as constants of LSemP rop that refer to the predicate symbols of LP rop . For any x, y 2 V , let {x; y} denote the ordered pair consisting of the set x and the set y. Barwise and Etchemendy now define the following types of sets: Definition 4.1 The ‘states of a↵airs’, the ‘atomic types’, the ‘types’, the ‘situations’ and the ‘propositions’. 1. A state of a↵airs is a set of the form hP, t1 , . . . , tn , vi, where P is an atom in V denoting an n-place predicate symbol of LP rop , and t1 , . . . , tn are n terms of LSemP rop and v is either 0 or 1. Let Soa be the set of all states of a↵airs. 2. Any subset of Soa is called a situation. Let Sit be the set of all situations. 3. An atomic proposition is a pair p = {s; } where s 2 Sit is a situation and 2 Soa. Let AtP rop be the set of all atomic propositions. 16 In

fact, Barwise and Etchemendy introduce two di↵erent set theoretic representations of propositions, the Russellian propositions and the Austinian propositions. I shall only take into account the Austinian propositions. Barwise and Etchemendy ultimately argue that only the Austinian propositions can help to get rid of the strengthened liar problem.

4.3 Three Instances of CR1

85

Barwise and Etchemendy assume that there exists a so-called atomic type, denoted ‘[P, t1 , . . . , tn , v]’, to each state of a↵airs hP, t1 , . . . , tn , vi. This atomic type is ontologically just another set, which stands in a certain relation to hP, t1 , . . . , tn , vi.17 In addition, Barwise and Etchemendy introduce a “negation”-operation ‘¬’ on V that maps sets to other sets in V , such that di↵erent sets are always mapped to di↵erent sets in V . Furthermore Barwise and Etchemendy introduce a generalized “and”-operation ‘^’ on V and a generalized “or”-operation ‘_’ on V that map sets of a finite number of sets in V to other sets in V , such that di↵erent sets of sets are always mapped to di↵erent sets in V . Exactly how these operators are defined is not essential (cf. Barwise and Etchemendy[3], p. 62). For any set z, I shall denote the result of the first of these operations by ‘[¬z]’. For any finite set X of sets from V , let ‘[^X]’ and ‘[_X]’ denote the results of the second and the third operation. Next Barwise and Etchemendy define the set of types as follows: Definition 4.2 The ‘types’. Let the set T ype of types be the smallest set that is closed under the following rules: 1. All atomic types are types. 2. If x is any type, then also [¬x] is a type. 3. If X is any finite subset of types, then also [_X], [^X] are types. A proposition {s; T } is then a pair consisting of a situation s 2 Sit and a type T 2 T ype. s is also called the situation the proposition {s; T } is about. The situation a proposition is about is also the situation with respect to which the symbols contained in the type of the propositions are interpreted. Next truth values are assigned to each proposition by the following rules (cf. Barwise and Etchemendy[3], p. 127): 17 Barwise

and Etchemendy do not give any details of this relation. In fact it is not relevant for their account to formally discern between states of a↵airs and atomic types. Nevertheless Barwise and Etchemendy keep this distinction in order to be as close as possible to Austin’s concept of propositions.

86

The Contextual Approach to the Strengthened Liar Problem

Definition 4.3 True and false propositions. 1. Each proposition is either true or false. 2. Each atomic proposition {s; } is true i↵

2 s.

3. For any {s; ¬T } 2 P rop, {s; ¬T } is true i↵ {s; T } is not true. 4. For any {s; ^X} 2 P rop, {s; ^X} is true i↵ {s; T } is true for all T 2 X. 5. For any {s; _X} 2 P rop, {s; _X} is true i↵ {s; T } is true for some T 2 X. If hP, t1 , . . . , tn , vi is a state of a↵airs, then the state of a↵airs hP, t1 , . . . , tn , v 0 i, where v 0 = 0 i↵ v = 1, is called the dual state of a↵airs of hP, t1 , . . . , tn , vi. Barwise and Etchemendy restrict their investigations to a certain class of situations that meet certain coherence conditions, the so-called actual situations: Definition 4.4 BE-models and actual situations. A BE-model M is a collection of states of a↵airs which satisfies the following conditions: • No state of a↵airs and its dual are in M. • If hT r, p; 1i 2 M, then p is true. • If hT r, p; 0i 2 M, then p is false. A situation s is actual in a model M i↵ s ✓ M. For each situation s there are two di↵erent liar propositions about s: first, the assertive liar proposition fs such that fs = {s; [T r, fs , 0]}, which corresponds to a simple liar sentence. Second, the denial liar proposition ds such that ds = ¬{s; [T r, ds , 1]}, which corresponds to a strengthened liar sentence. The existence of both propositions follows from the axioms of ZFC/AFA. For each actual situation s (in some partial model M) it can

4.3 Three Instances of CR1

87

then be shown that the assertive liar proposition about s is false, and the denial proposition about s is true (Barwise and Etchemendy[3], p. 132 and p. 167). For our current purposes, let us concentrate on the denial liar proposition only. Consider the situation s0 = s [ {[T r, ds , 1]}. Since s has been assumed to be an actual situation (in some partial model M), also s0 is an actual situation (in some partial model), and the proposition p = {s0 ; [T r, ¬{s; [T r, ds , 1]}, 1]} about s0 , which expresses the truth of ds , is true. This shows that Barwise’s and Etchemendy’s account belongs to the contextual reconstruction scheme CR1. There is a small but for our purposes inessential di↵erence between the expressive means of LSemP rop and the interpreted language LM of section 4.1 (see also Koons[20]). In LSemP rop the occurrences of truth predicates in the type of a proposition can only be interpreted which respect to one situation the situation the proposition is about. In the informal meta-language of LM one also has expressions such as ‘M2 ✏ T r1 (x)’. So each model Mn also contains all “global” truth predicates T r1 , T r2 , . . . , T rn 1 of the preceding models M0 , . . . , Mn 1 . However, a proposition {s; T } of LSemP rop can indirectly refer to situations s0 that are di↵erent from s, as T might be of the form [P, t1 , . . . , tn , v], where some of the terms t1 , . . . , tn are themselves propositions that are about situations s0 6= s. This feature is enough to reformulate the strengthened liar argument according to CR1. The premises of CR1 correspond to the following three premises in Barwise’s and Etchemendy’s account: P1 BE ds = ¬{s; [T r, ds , 1]} P2 BE ¬{s; [T r, ds , 1]} is true P3 BE {s0 ; [T r, ¬{s; [T r, ds , 1]}, 1]} is true Note that all three premises belong to LSemP rop . The situation s corresponds to M1 , and the situation s0 corresponds to M2 . Each occurrence of ‘T r’ in the type of a proposition about s corresponds to T r1 of (L1 , M1 ), and each occurrence of ‘T r’ in the type of a proposition about s0 corresponds to T r2 of (L2 , M2 ).

88

The Contextual Approach to the Strengthened Liar Problem

The three contextual approaches, which I have just presented, considerably di↵er with respect to the formal apparatus that is applied. However, they all share the same essential idea, which I have formally represented by CR1: The strengthened liar sentence is semantically defective in a certain initial model, and it is true in a second model that, in a certain sense, is more comprehensive than the initial model.

5 Contextual Approaches by Iterating Partial Truth Predicates In section 5.1, I show how an instance of CR1 can be obtained by an iteration of Kripke’s account. In section 5.2, I propose an alternative contextual reconstruction scheme CR2, in which a di↵erent position of the semantic shift is assumed. I will illustrate how an instance of CR2 can be obtained by a minor variation of the iteration of Kripke’s account. We will see that both reformulations lead to hierarchies of models and truth predicates. I shall indicate how the range of this hierarchy can be extended to the class of ordinal numbers in section 5.3. Finally, in section 5.4, I analyze a very serious objection against contextual approaches, namely the objection that they are unnatural.

5.1 A Hierarchy of Models According to CR1 In this section, I will present a hierarchy of models that meets CR1. More precisely, I will confine myself to the second formal version of CR1, i.e., the argument starting with the following premises: P1 CR1 M1 ✏

$ ¬T r1 (p q)

P2 CR1 M2 ✏ ¬T r1 (p q) P3 CR1 M2 ✏ T r2 (p¬T r1 (p q)q). By applying the explicit definition Gr = 8x(Gr(x) $ (T r(x) _ T r(¬. x))) from section 3.2, we can easily verify that the hierarchy of models presented

90

Contextual Approaches by Iterating Partial Truth Predicates

in this section also meets the first formal version of CR1. As we will see, the hierarchy of models presented in this section results from an iterative application of Kripke’s fixed point-construction of a truth-extension and the corresponding theory KF. I will first demonstrate how (the second formal version of) CR1 can be formalized with the help of Kripke[21]’s semantic fixed point approach. A detailed presentation of Kripke’s account can e.g. be found in McGee[26]. In this section, it is assumed that LM is a typical language of first order predicate logic with identity which meets S1 S4, as presented in chapter two. Let us extend L by a one-place predicate T r, and let the resulting language be denoted by L+ . The interpretation of the new symbol T r shall be given by a so-called partial interpretation, i.e. a pair (E, A) consisting of two disjoint subsets E and A of D, the domain of discourse. E is called ‘the extension of T r’, and A is called ‘the anti-extension of T r’. Some (meta-language codes of) sentences in D are neither in E, nor in A. This indicates that it is undefined whether the predicate T r applies to these sentences (cf. Kripke[21], p. 718, paragraph 2). Let a partial interpretation function I be an interpretation function such that at least one predicate constant is assigned to a partial interpretation. Let a partial model be any model with a partial interpretation function. Further let an expansion of M to a partial model, such that T r is partially interpreted by (E, A), be denoted by ‘M(E, A)’. Kripke next defines a three-valued valuation function V alM(E,A) , according to which there are three truth values, the two classical ones t and f , and a third value n for ‘neither true, nor false’. V alM(E,A) (T r(t1 )) = n shall obtain i↵ V alM(E,A) (t1 ) is neither in the extension, nor in the anti-extension of T r. Compound formulas are evaluated by V alM(E,A) , according to the strong Kleene scheme (the strong Kleene scheme was introduced by Kleene[19], p. 332 340), can be visualized by the following truth-tables:

¬ ' f t t f n n

( )

'^

t f n

t t f n (') f f f f n n f n

( )

'_

t f n

t t t t (') f t f n n t n n

5.1 A Hierarchy of Models According to CR1

( )

'!

t f n

t t f n (') f t t t n t n n

91

( )

'$

t f n

t t f n (') f f t n n n n n

Next, Kripke introduces the so-called strong Kleene jump operator M . Recall that L+ is the set of objects d in D, such that d is the meta-language code of a sentence of L+ . Definition 5.1 The strong Kleene jump operator M . M (E, A) := (E 0 , A0 ) such that E 0 = {d 2 D : d 2 L+ , I(p'q) = d, V alM(E,A) (') = t}

A0 = {d 2 D : d 2 L+ , I(p'q) = d, V alM(E,A) (') = f }[ {d 2 D : d 2 / L+ } A pair (E, A) is called a fixed point of M i↵ E = M (E) and A = M (A). If (E, A) is a fixed point of M , then E = {d 2 D : d 2 L+ , I(p'q) = d, V alM(E,A) (') = t}, and A = {d 2 D : d 2 L+ , I(p'q) = d, V alM(E,A) (') = f }[ {d 2 D : d 2 / L+ }.

That is, E is the set of all (meta-language codes of) sentences that are true in M(E, A), and A is the set of all (meta-language codes of) sentences that are false in M(E, A), together with the (meta-language codes of) nonsentences in L+. For each sentence ' of L+ , let us define that M(E, A) ✏ ' i↵ V alM(E,A) (') = t, and M(E, A) 2 ' i↵ V alM(E,A) (') 6= t. Because of • V alM(E,A) (T r(p'q)) = t i↵ I(p'q) 2 E i↵ V alM(E,A) (') = t, and • V alM(E,A) (T r(p'q)) = f i↵ I(p'q) 2 A i↵ V alM(E,A) (') = f ,

92

Contextual Approaches by Iterating Partial Truth Predicates

for any sentence ' of L+ and for any fixed point (E, A), we obtain Fact 5.1 If (E, A) is a fixed point, then 1. V alM(E,A) (T r(p'q)) = V alM(E,A) (') for all sentences ' of L+ , and 2. M(E, A) ✏ T r(p'q) if, and only if M(E, A) ✏ ' for all sentences ' of L+ . The strong Kleene scheme for ‘$’ is given by the following table ( )

'$

t f n

t t f n (') f f t n n n n n It cannot be replaced by any other stronger1 evaluation scheme as this would result in another liar paradox (cf. Gupta[14]). As either V alM(E,A) (') = t or V alM(E,A) (') = f , for any sentence ' of L, and for any fixed point (E, A), we obtain Fact 5.2 If (E, A) is a fixed point, then M(E, A) ✏ T r(p'q) $ ' for all sentences ' of L. Fact 5.2 does not hold for all formulas ' of L+ . Exceptions are all formulas that n is assigned to, according to V alM(E,A) . Martin and Woodru↵[24] have proved the existence of maximal fixed points by employing Zorn’s lemma. Kripke constructed the least fixed point by the following procedure. (E0 , A0 ) := (;, ;); If ↵ is a successor ordinal, ↵ = 1 By

+ 1, then (E↵ , A↵ ) is defined by

a ‘stronger’ scheme, I mean an evaluation scheme which assigns more sentences to a classical truth value, i.e. to t or f , than the strong Kleene scheme for ‘$’ does.

5.1 A Hierarchy of Models According to CR1

93

(E↵ , A↵ ) := M ((E , A )). If ↵ is a limit ordinal, then (E↵ , A↵ ) is defined by (E↵ , A↵ ) :=

S

0;

where I n+1 is the interpretation function that belongs to (Mn , E). Proof The claim is proved by induction over n. If n = 0, the claim follows from the first part of theorem 4.3 on p. 93 in McGee[26]. Let n > 0 and assume the claim has been proved for each m  n. Let Mn+1 := (Mn , E). Direction 1: We show (Mn , E) ✏ KF0n+1 ) 0Mn (E, A) = (E, A), where A = {I n+1 (p'q) : I n+1 (p¬'q) 2 E and I n+1 (p'q) 2 Ln+1 \Ln }[ {I n+1 (p'q) : I n+1 (p¬'q) 2 / E and I n+1 (p'q) 2 Ln }[ {d 2 D : d 2 / Ln+1 } if n > 0.

It is enough to show 0Mn+1 (E, A) ✓ (E, A) (the other direction was assumed). To do this, we will show the following four facts by induction over formulas. For all sentences ' 2 Ln+1 1. If ' 2 Ln+1 \Ln and V al(Mn ,(E,A)) (') = t, then I n+1 (p'q) 2 E. 2. If ' 2 Ln and V alMn (T rn (p'q)) = t, then I n+1 (p'q) 2 E. 3. If ' 2 Ln+1 \Ln and V al(Mn ,(E,A)) (') = f , then I n+1 (p¬'q) 2 E. 4. If ' 2 Ln and V alMn (T rn (p'q)) 6= t, then I n+1 (p'q) 2 / E. (If d is no sentence, then it will be in 0Mn+1 (A) as well as in A.) We start with 2. and 4.: ad 2. Let ' 2 Ln and V alMn (T rn (p'q)) = t. Thus, by Mn+1 ✏ KF0n+1 , V alMn+1 (T rn+1 (p'q)) = t, and so ' 2 E.

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Contextual Approaches by Iterating Partial Truth Predicates

ad 4. Let ' 2 Ln and V alMn (') 6= t. By the induction hypothesis for each m  n, E m = {d 2 D : d 2 Lm \Lm 1 , I m (p'q) = d, V alMm (') = t} [ E m 1 . Thus ' 2 / E m and so we have Mm ✏ ¬T rm(p'q). By Mm+1 ✏ KF0m+1 , it follows that Mm+1 ✏ ¬T rm+1 (p'q) and so ' 2 / E m+1 . Furthermore, for all m  n, Mm ✏ ¬T rm (p'q) and so ' 2 / E m. Thus '2 / E n. 1. and 3. are shown by simultaneous induction over formulas: Let ' be an atomic formula in Ln+1 , i.e. ' = T rn+1 (p q) for some Then

.

ad 1. V alMn+1 (T rn+1 (p q)) = t ) Mn+1 ✏ T rn+1 (p q) ) (by Mn+1 ✏ KF0n+1 ) Mn+1 ✏ T r n+1 (pT rn+1 (p q)q) ) T rn+1 (p q) 2 E. ad 3. V alMn+1 (T rn+1 (p q)) = f ) I n+1 (p q) 2 Ln+1 \Ln and ¬ 2 E, or I n+1 (p q) 2 Ln and 2 / E n . If ¬ 2 E and I n+1 (p q) 2 Ln+1 \Ln then (by Mn+1 ✏ KF0n+1 ) Mn+1 ✏ T rn+1 (p¬T rn+1 (p q)q), and thus ¬T rn+1 (p q) 2 E. If 2 Ln and 2 / E n then Mn ✏ ¬T rn (p q), and thus (by Mn+1 ✏ KF0n+1 ) Mn+1 ✏ ¬T rn+1 (pT rn+1 (p q)q), i.e. ¬T rn+1 (p q) 2 E. Let ' be a negated atomic formula in Ln+1 , i.e. ' = ¬T rn+1 (p q). Then • V alMn+1 (¬T rn+1 (p q)) = t ) V alMn+1 (T rn+1 (p q)) = f ) (by I n+1 (pT rn+1 (p q)q) 2 Ln+1 \Ln and the induction hypothesis) ¬T rn+1 (p q) 2 E, i.e. ' 2 E. • V alMn+1 (¬T rn+1 (p q)) = f ) V alMn+1 (T rn+1 (p q)) = t ) (by I n+1 (pT rn+1 (p q)q) 2 Ln+1 \Ln and the induction hypothesis) T rn+1 (p q) 2 E. By Mn+1 ✏ KF0n+1 , it follows that Mn+1 ✏ T rn+1 (p¬¬T rn+1 (p q)q) and thus ¬' 2 E. Let ' =

1

_

2

for some formulas

1

and

2

of Ln+1 . Then

• V alMn+1 ( 1 _ 2 ) = t , V alMn+1 ( 1 ) = t or V alMn+1 ( 2 ) = t , (by the induction hypothesis) 1 2 E or 2 2 E , Mn+1 ✏ T rn+1 (p 1 q) or Mn+1 ✏ T rn+1 (p 2 q) , Mn+1 ✏ T rn+1 (p 1 q) _ T rn+1 (p 2 q) , Mn+1 ✏ T rn+1 (p 1 _ 2 q) , 1 _ 2 2 E.

5.2 A Hierarchy of Models According to CR2

109

• V alMn+1 ( 1 _ 2 ) = f , V alMn+1 ( 1 ) = f and V alMn+1 ( 2 ) = f , (by the induction hypothesis) ¬ 1 2 E and ¬ 2 2 E , Mn+1 ✏ T rn+1 (p¬ 1 q) and Mn+1 ✏ T r n+1 (p¬ 2 q) , Mn+1 ✏ T rn+1 (p¬ 1 q) ^ T rn+1 (p¬ 2 q) , Mn+1 ✏ T r n+1 (p¬( 1 _ 2 )q) , ¬( 1 _ 2 ) 2 E. Let ' = ¬(

1

_

2)

for some formulas

1

and

• V alMn+1 (¬( 1 _ 2 )) = t , V alMn+1 ( case) ¬( 1 _ 2 ) 2 E.

2

in Ln+1 . Then

1 _ 2)

= f ) (by the previous

• V alMn+1 (¬( 1 _ 2 )) = f , V alMn+1 ( 1 _ 2 ) = t ) (by the previous case) 1 _ 2 2 E ) (by Mn+1 ✏ KF0n+1 ) ¬¬( 1 _ 2 ) 2 E. Let ' = 8x where is a formula of Ln+1 . To make the formulas more legible I shall abbreviate ‘sub(p'q, x, y)’ by ‘p[']yx q’, for each sentence ' and variables x and y. Then • V al(Mn ,(E,An+1 )) (8x ) = t , ¯ for all d 2 D, V al(Mn ,(E,An+1 )) ([ ]dx ) = t ) (by the induction hypothesis) ¯ ¯ for all d 2 D : [ ]dx 2 E , for all d 2 D, Mn+1 ✏ T rn+1 (p[ ]dx q) num(z) , Mn+1 ✏ 8zT rn+1 (p[ ]x q) , (as Mn+1 ✏ KF0n+1 ) Mn+1 ✏ T rn+1 (p8x q) , T rn+1 (p8x q) 2 E. • V al(Mn ,(E,An+1 )) (8x ) = f , ¯ there is a d 2 D: V al(Mn ,(E,An+1 )) ([ ]dx ) = f ) (by the induction hypothesis) ¯ there is a d 2 D : [¬ ]dx 2 E , there is a d 2 D, ¯ num(z) Mn+1 ✏ T rn+1 (p[¬ ]dx q) , Mn+1 ✏ 9zT rn+1 (p[¬ ]x q) , 0 n+1 n+1 n+1 (as M ✏ KFn+1 ) M ✏ T r (p¬8x q) , ¬T rn+1 (p8x q) 2 E. The case where ' = ¬8x and to the previous case.

is a formula of Ln+1 is proved analogously

Direction 2: We show 0Mn+1 (E, A) = (E, A) ) (Mn , E) ✏ KF0n+1 . Recall

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Contextual Approaches by Iterating Partial Truth Predicates

that 0Mn+1 (E, A) = (E, A) i↵ E = {d 2 D : d 2 Ln+1 \Ln , I n+1 (p'q) = d, V alMn (E,A) (') = t} [ {d 2 D : d 2 Ln , I n (p'q) = d, V alMn (T rn (p'q)) = t}, and A = {d 2 D : d 2 Ln+1 \Ln , I n+1 (p'q) = d, V alMn (E,A) (') = f } [ {d 2 D : d 2 Ln , I n (p'q) = d, V alMn (T rn (p'q)) 6= t} [ {d 2 D : d 2 / Ln+1 }. I show that (M n , E) ✏ KF0n+1 by showing the following ten items: 1. (Mn , E) ✏ 8x(SentLn (x) ! (T rn+1 (x) $ T rn (x))); 2. (Mn , E) ✏ 8x(SentLn+1 (x) ! (T rn+1 (pT r n+1 (x) ˙ q) $ T rn+1 (x))); 3. (Mn , E) ✏ 8x((SentLn+1 (x) ^ ¬SentLn (x)) !

(T r n+1 (p¬T rn+1 (x) ˙ q) $ T rn+1 (¬. x)));

4. (Mn , E) ✏ 8x(SentLn (x) ! (T rn+1 (p¬T rn+1 (x) ˙ q) $ ¬T rn (x))); 5. (Mn , E) ✏ 8x8y(SentLn+1 (x_. y) !

(T r n+1 (x_. y) $ (T rn+1 (x) _ T rn+1 (y))));

6. (Mn , E) ✏ 8x8y8z(SentLn+1 (¬. (x_. y)) !

(T r n+1 (¬. (x_. y)) $ (T rn+1 (¬. x) ^ T rn+1 (¬. y))));

7. (Mn , E) ✏ 8x8y((V arLn+1 (x) ^ F rmLn+1 (y)) !

(T r n+1 (9. xy) $ 9zT rn+1 (sub(y, x, num(z)))));

8. (Mn , E) ✏ 8x8y((V arLn+1 (x) ^ F rmLn+1 (y)) !

(T r n+1 (¬. 9. xy) $ 8zT rn+1 (¬. sub(y, x, num(z)))));

9. (Mn , E) ✏ 8x8y((SentLn+1 (x) ^ x ⌘ ¬. ¬. y) ! (T r n+1 (x) $ T rn+1 (y)));

10. (Mn , E) ✏ 8x(SentLn+1 (x) ! ¬(T rn+1 (x) ^ T rn+1 (¬. x))).

5.2 A Hierarchy of Models According to CR2

111

Since I have assumed that for each object d, there is a closed term d¯ denoting d (that is, Mn (E) meets condition S2 on p. 29), I will assume universal ¯ sentences to be interpreted as follows: (Mn , E) ✏ 8x' i↵ (Mn , E) ✏ [']dx for each d 2 D. ¯ and (Mn , E) ✏ ad 1. For any d 2 D, assume (Mn , E) ✏ SentLn (d) ¯ T rn+1 (d). Then there is a ' of Ln , such that I n (p'q) = d and I n (p'q) 2 E. ¯ = t, and thus we obtain the left-to-right direcTherefore V alMn (T rn (d)) tion (Mn , E) ✏ 8x(SentLn (x) ! (T rn+1 (x) ! T rn (x))). ¯ and (Mn , E) ✏ For the other direction, assume (Mn , E) ✏ SentLn (d) ¯ Then there is a sentence ' of Ln , such that I n (p'q) = d and T rn (d). ¯ V alMn (T rn (p'q)) = t. Therefore d 2 E n+1 , and thus (Mn , E) ✏ T rn+1 (d). So we obtain the right-to-left direction (Mn , E) ✏ 8x(SentLn (x) ! (T rn (x) ! T rn+1 (x))). ¯ , V alMn (E,A) (T rn+1 (d)) ¯ =t ad 2. For any d 2 D, (Mn , E) ✏ T r n+1 (d) ¯ 2 E , (Mn , E) ✏ T rn+1 (pT rn+1 (d) ¯ q). Thus (Mn , E) ✏ , T rn+1 (d) 8x(SentLn+1 (x) ! (T rn+1 (pT rn+1 (x) ˙ q) $ T rn+1 (x))). ¯ ^ ¬SentLn (d), ¯ and ad 3. For any d 2 D, assume (Mn , E) ✏ SentLn+1 (d) ¯ (Mn , E) ✏ T rn+1 (¬. d). Then there is a ', such that I n+1 (p¬'q) 2 Ln+1 \Ln and I n+1 (p¬'q) 2 E, and thus V alMn (E,A) (¬') = t. Therefore, I n+1 (p'q) 2 A, and so we obtain V alMn (E,A) (T rn+1 (p'q)) = f , i.e. V alMn (E,A) (¬T rn+1 (p'q)) = t. ¯ q). So we get ¬T rn+1 (p'q) 2 E, and hence (Mn , E) ✏ T rn+1 (p¬T rn (d) This proves the right-to-left direction, i.e. (Mn , E) ✏ 8x((SentLn+1 (x) ^ ¬SentLn (x)) ! ((T rn+1 (¬. x) ! T rn+1 (p¬T rn+1 (x) ˙ q))). ¯ q), and let For the other direction, assume (Mn , E) ✏ T rn+1 (p¬T rn (d) ' be a sentence, such that I n+1 (p'q) = d and I n+1 (p'q) 2 Ln+1 \Ln . Then ¬T rn+1 (p'q) 2 E, and so V alMn (E,A) (¬T rn+1 (p'q)) = t, which is equivalent to V alMn (E,A) (T rn+1 (p'q)) = f . So I n+1 (p'q) 2 A and, since ¯ from I n+1 (p'q) 2 / SentLn , I n+1 (p¬'q) 2 E. Thus (Mn , E) ✏ T rn+1 (¬. d), which we obtain (Mn , E) ✏ 8x((SentLn+1 (x) ^ ¬SentLn (x)) ! (T rn+1 (p¬T rn+1 (x) ˙ q) ! T rn+1 (¬. x))).

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Contextual Approaches by Iterating Partial Truth Predicates

¯ ^ T rn+1 (¬T rn+1 (pd¯q)). ad 4. Assume (Mn , E) ✏ SentLn (d) It follows that I n+1 (p¬T rn+1 (pd¯q)q) 2 E and therefore ¯ = t. Thus V alMn (E,A) (T rn+1 (d)) ¯ = f , and so V alMn (E,A) (¬T rn+1 (d)) ¯ 2 A. From (Mn , E) ✏ SentLn (d), ¯ it then follows V alMn (T r n (d)) ¯ 6= I n+1 (d) ¯ = t. And thus Mn (E, A) ✏ ¬T rn (d). ¯ This t, that is V alMn (¬T rn (d)) proves (Mn , E) ✏ 8x(SentLn (x) ! (T rn+1 (p¬T rn+1 (x) ˙ q) ! ¬T rn (x))) and so we have the left-to-right direction. ¯ and I n+1 (d) ¯ 2 Ln implies I n+1 (d) ¯ 2 (Mn , E) ✏ ¬T rn (d) / E n . Thus ¯ 6= t, and therefore I n+1 (d) ¯ 2 A. So V alMn (E,A) (T rn+1 (d)) ¯ V alMn (T r n (d)) ¯ = t, and so I n+1 (p¬T rn+1 (d) ¯ q) 2 E. = f , i.e. V alMn (E,A) (¬T rn+1 (d)) ¯ q)) = t, i.e. Thus V alMn (E,A) (T rn+1 (p¬T rn+1 (d) (Mn , E) ✏ ¯ q). This proves the right-to-left direction (Mn , E) ✏ T rn+1 (p¬T rn+1 (d) 8x(SentLn (x) ! (¬T rn (x) ! T rn+1 (p¬T rn+1 (x) ˙ q))). ¯ and (Mn , E) ✏ d¯ ⌘ p'q. ad 5. Assume (Mn , E) ✏ SentLn+1 (d) Then either I n+1 (p'q) 2 Ln , or I n+1 (p'q) 2 Ln+1 \Ln . In case that I n+1 (p'q) 2 Ln+1 \Ln , (Mn , E) ✏ T rn+1 (p¬¬'q) , ¬¬' 2 E , V al(Mn (E,A))(¬¬') = t , V al(Mn ,E) (') = t , ' 2 E , (Mn , E) ✏ ¯ T rn+1 (p'q) , (Mn , E) ✏ T rn+1 (d). In case that I n+1 (p'q) 2 Ln : (Mn , E) ✏ T rn+1 (p¬¬'q) , ¬¬' 2 E , V alMn (T r n (p¬¬'q)) = t , (induction hypothesis) V alMn (T rn (p'q)) = t , V al(Mn ,E) (T rn (p'q)) = t , V al(Mn ,E) (T rn+1 (p'q)) = t (by 1.) , ¯ (Mn , E) ✏ T rn+1 (d). ad 6. Assume (Mn , E) ✏ SentLn+1 (d¯1 _. d¯2 ). Then there are two sentences of Ln+1 , such that I n+1 (p'1 q) = d1 , I n+1 (p'2 q) = d2 . (Mn , E) ✏ T rn+1 (p'1 _ '2 q) i↵ I n+1 (p'1 _ '2 q) 2 E. From I n+1 (p'1 _ '2 q) 2 E, it follows either V alMn (E,A) ('1 _ '2 ) = t (in case '1 _ '2 is no sentence of Ln ), or V alMn (T rn (p'1 _ '2 q)) = t (in case '1 _ '2 is a sentence of Ln ). In the first case we have V alMn (E,A) ('1 _ '2 ) = t , V alMn (E,A) ('1 ) = t or V alMn (E,A) ('2 ) = t , '1 2 E or '2 2 E (and '1 or '2 is no sentence of Ln )) , (Mn , E) ✏ T rn+1 (p'1 q) or (Mn , E) ✏ T rn+1 (p'2 q) , (Mn , E) ✏ T rn+1 (p'1 q) _ T r n+1 (p'2 q).

5.2 A Hierarchy of Models According to CR2

113

In the second case we have V alMn (T r n (p'1 _ '2 q)) = t , (induction hypothesis) V alMn (T rn (p'1 q)_T rn (p'2 q)) = t , V alMn (T rn (p'1 q)) = t or V alMn (T rn (p'2 q)) = t , (by the definition of E) I n+1 (p'1 q) 2 E or I n+1 (p'2 q) 2 E , (Mn , E) ✏ T rn+1 (p'1 q) or (Mn , E) ✏ T rn+1 (p'2 q) , (Mn , E) ✏ T rn+1 (p'1 q) _ T r n+1 (p'2 q). Thus I n+1 (p'1 _ '2 q) 2 E is equivalent to (Mn , E) ✏ T rn+1 (p'1 q) _ T rn+1 (p'2 q). Therefore, we have shown that (Mn , E) ✏ T rn+1 (p'1 _ '2 q) i↵ (Mn , E) ✏ T rn+1 (p'1 q) _ T rn+1 (p'2 q), i.e. (Mn , E) ✏ T rn+1 (d¯1 _. d¯2 ) i↵ (Mn , E) ✏ T rn+1 (d¯1 ) _ T rn+1 (d¯2 ). ad 7. Assume (Mn , E) ✏ SentLn+1 (¬. (d¯1 _. d¯2 )). Then there are two sentences '1 and '2 of Ln+1 , such that I n+1 (p'1 q) = d1 , I n+1 (p'2 q) = d2 . Either I n+1 (p'1 q) 2 Ln+1 \Ln and I n+1 (p'2 q) 2 Ln+1 \Ln , or I n+1 (p'1 q) 2 Ln and I n+1 (p'2 q) 2 Ln , or (without loss of generality) I n+1 (p'1 q) 2 Ln+1 \Ln and I n+1 (p'2 q) 2 Ln . In case that I n+1 (p'1 q) 2 Ln+1 \Ln and I n+1 (p'2 q) 2 Ln+1 \Ln : (Mn , E) ✏ T rn+1 (p¬'1 q) ^ T rn+1 (p¬'2 q) , I n+1 (p¬'1 q) 2 E and I n+1 (p¬'2 q) 2 E , V alMn (E,A) (¬'1 ) = t and V alMn (E,A) (¬'2 ) = t , V alMn (E,A) (¬'1 ^¬'2 ) = t , V alMn (E,A) (¬('1 _'2 )) = t , I n+1 (p¬('1 _ '2 )q) 2 E , (Mn , E) ✏ T rn+1 (p¬('1 _ '2 )q). In case that I n+1 (p'1 q) 2 Ln and I n+1 (p'2 q) 2 Ln : (Mn , E) ✏ T rn+1 (p¬'1 q)^T rn+1 (p¬'2 q) , I n+1 (p'1 q) 2 E and I n+1 (p'2 q) 2 E , Mn ✏ T rn (p'1 q) and Mn ✏ T rn (p'2 q) , Mn ✏ T rn (p'1 q) ^ T rn (p'2 q) , (induction hypothesis) Mn ✏ T rn (p¬('1 _ '2 )q) , I n+1 (p¬('1 _ '2 )q) 2 E , (Mn , E) ✏ T rn+1 (¬('1 _ '2 )q). In case that I n+1 (p'1 q) 2 Ln+1 \Ln and I n+1 (p'2 q) 2 Ln : (Mn , E) ✏ T rn+1 (p¬('1 _ '2 )q) , (because of 3.) (Mn , E) ✏ T rn+1 (p¬T rn+1 (p'1 _ '2 q)q) , I n+1 (p¬T rn+1 (p'1 _'2 q)q) 2 E , V alMn (E, A)(¬T rn+1 (p'1 _ '2 q)) = t , (by strong Kleene logic6 ) V alMn (E, A)(T rn+1 (p'1 _'2 q)) = f , I n+1 (p'1 _ '2 q) 2 A , V alMn (E, A)('1 _'2 ) = f , (by strong Kleene logic) V alMn (E, A)('1 ) = f and V alMn (E, A)('2 ) = f , (by strong Kleene logic) V alMn (E, A)(¬'1 ) = t and V alMn (E, A)(¬'2 ) = t. 6 Recall

pp. 90-91 of section 5.1.

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We continue as follows: V alMn (E, A)(¬'1 ) = t , I n+1 (p¬'1 q) 2 E , (Mn , E) ✏ T rn+1 (p¬'1 q). Concerning '2 , either I n+1 (p'2 q) 2 L0 or there is a k, 0 < k  n, such that I n+1 (p'2 q) 2 Lk \Lk 1 . In case that I n+1 (p'2 q) 2 L0 , V alMn (E, A)(¬'2 ) = t , M1 ✏ T r1 (p¬'2 q) , (because of 1.) Mn+1 ✏ T rn+1 (p¬'2 q). In case that I n+1 (p'2 q) 2 Lk \Lk 1 , V alMk (¬'2 ) = t , Mk ✏ T rk (p¬'2 q) , (because of 1.) Mn+1 ✏ T rn+1 (p¬'2 q). Therefore, in case that I n+1 (p'1 q) 2 Ln+1 \Ln and I n+1 (p'2 q) 2 Ln , (Mn , E) ✏ T rn+1 (p¬('1 _ '2 )q) , Mn+1 ✏ T rn+1 (p¬'1 q) ^ T rn+1 (p¬'2 q). Thus (Mn , E) ✏ SentLn+1 (¬. (d¯1 _. d¯2 )) ! (T rn+1 (¬. (d¯1 _. d¯2 )) $ (T rn+1 (¬. d¯1 ) ^ T rn+1 (¬. d¯1 ))). ad 8. Assume (Mn , E) ✏ V arLn+1 (d¯1 ) ^ F rmLn+1 (d¯2 ), and let I n+1 (pxq) = d1 and I n+1 (p'q) = d2 . Then either (Mn , E) ✏ F rmLn+1 (p'q) ^ ¬F rmLn (p'q) or (Mn , E) ✏ F rmLn (p'q). In case that (Mn , E) ✏ F rmLn+1 (p'q) ^ ¬F rmLn (p'q), (Mn , E) ✏ T rn+1 (p9x'q) , I n+1 (p9x'q) 2 E , V al(Mn ,E) (9x') = t ¯ , there is a d 2 D, such that V al(Mn ,E) ([']dx ) = t , there is a d 2 D, ¯ such that I n+1 (p[']dx q) 2 E , there is a d 2 D, such that (Mn , E) ✏ ¯ T rn+1 (p[']dx q) , (Mn , E) ✏ 9zT rn+1 (sub(p'q, x, num(z))).

In case that (Mn , E) ✏ F rmLn (p'q), (Mn , E) ✏ T rn+1 (p9x'q) , 9x' 2 E , V alMn (T rn (p9x'q)) = t , (by the induction hypothesis) V alMn (9zT rn (sub(p'q, x, num(z)))) = t , there is a d 2 D, such that ¯ V alMn (T r n (p[']dx q)) = t , (because of 1.) there is a d 2 D, such ¯ that V al(Mn ,E) (T rn+1 (p[']dx q)) = t , (Mn , E) ✏ 9zT rn+1 (sub(p'q, x, num(z))). Thus, in both cases it holds that (Mn , E) ✏ T r n+1 (p9x'q) $ 9zT rn+1 (sub(p'q, x, num(z))), in other words (Mn , E) ✏ T r n+1 (9. d¯1 d¯2 ) $ 9zT rn+1 (sub(d¯2 , d¯1 , num(z))). ad 9. Assume (Mn , E) ✏ V arLn+1 (d¯1 ) ^ F rmLn+1 (d¯2 ), and let I n+1 (pxq) = d1 and I n+1 (p'q) = d2 . Then either (Mn , E) ✏ F rmLn+1 (p'q) ^ ¬F rmLn (p'q) or (Mn , E) ✏ F rmLn (p'q). In case that (Mn , E) ✏ F rmLn+1 (p'q) ^ ¬F rmLn (p'q), (Mn , E) ✏ T rn+1 (p¬9x'q) , I n+1 (p¬9x'q) 2 E , V al(Mn ,E) (¬9x') = ¯ ¯ t , for all d 2 D, V al(Mn ,E) ([¬']dx ) = t , for all d 2 D,I n+1 (p[¬']dx ) 2 E

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¯

, for all d 2 D, (Mn , E) ✏ T rn+1 (p[¬']dx q) , (Mn , E) ✏ 8zT rn+1 (sub(p¬'q, x, num(z))). In case that (Mn , E) ✏ F rmLn (p'q), (Mn , E) ✏ T rn+1 (p¬9x'q) , I n+1 (p¬9x'q) 2 E , V alMn (T rn (p¬9x'q)) = t , (by the induction hypothesis) V alMn (8zT rn (sub(p¬'q, x, num(z)))) = t , for all d 2 D, ¯ V alMn (T rn (p[¬']dx q)) = t , (because of axiom 1 of KF’n + 1 on p. 104) ¯ for all d 2 D, V al(Mn ,E) (T rn+1 (p[¬']dx q)) = t , (Mn , E) ✏ 8zT rn+1 (sub(p¬'q, x, num(z))). Thus, in both cases we obtain (Mn , E) ✏ T rn+1 (p¬9x'q) $ 8zT rn+1 (sub(p¬'q, x, num(z))), in other words (Mn , E) ✏ T rn+1 (9. d¯1 d¯2 ) $ 8zT rn+1 (sub(¬. d¯2 , d¯1 , num(z))). ¯ It follows that there is a senad 10. Assume (Mn , E) ✏ SentLn+1 (d). tence ' such that (Mn , E) ✏ d¯ ⌘ p'q, and either I n+1 (p'q) 2 Ln+1 \Ln or I n+1 (p'q) 2 Ln . In the first case V al(Mn ,E) (') 6= V al(Mn ,E) (¬'), and so (Mn , E) ✏ ¬(T rn+1 (p'q) ^ T rn+1 (p¬'q)). In the second case V alMn (T rn (p'q)) 6= V alMn (T rn (p¬'q)), as by the induction hypothesis it holds that not both I n+1 (p'q) 2 E n and I n+1 (p¬'q) 2 E n is the case. Therefore we again obtain (Mn , E) ✏ ¬(T rn+1 (p'q) ^ T r n+1 (p¬'q)) The iterated truth predicates T rn are characterized in the next fact. Fact 5.11 For each m n 0: 1. (Diagnosis of the strengthened liar). Mm ✏ ¬T rm (p n q). 2. (T rn is a partial truth predicate). Mn ✏ SentL0 (p'q) ! (T rn (p'q) $ '). 3. (Condition for T rn -scheme). Mn ✏ Grn (p'q) ! (T rn (p'q) $ '), where it is assumed that Mn ✏ 8xGrn (x) $ (SentLn (x) ^ (T rn (x) _ T rn (¬. x))). Proof: Proof of 1.: M1 ✏ ¬T r1 (p 1 q) (see section 5.1). For each n 1, Mn ✏ KF0n implies Mn ✏ T rn (p'q) $ T r1 (p'q), for each sentence ' of L1 . Therefore Mn ✏ ¬T rn (p 1 q), for each n 1. Furthermore, let m > n 1. Since Mn = Mn 1 (En0 lf ), where En0 lf is the least fixed point of KF0n ,

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Mn ✏ ¬T rn (p n q), for all n 1. Since Mm ✏ KF0m, it follows that Mm ✏ T rm(p'q) $ T rn (p'q), for each sentence ' of Ln . We therefore obtain Mm ✏ ¬T rm(p n q). Proof of 2.: By fact 5.6, we have M1 ✏ SentL0 (p'q) ! (T r 1 (p'q) $ '), and therefore Mn ✏ SentL0 (p'q) ! (T r1 (p'q) $ '). Because of Mn ✏ KF0n , it follows that Mn ✏ T rn (p'q) $ T r1 (p'q), for each sentence ' of L1 , and thus Mn ✏ SentL0 (p'q) ! (T rn (p'q) $ '). Proof of 3.: The third claim is proved by induction over formulas. It is enough to prove the case where ' = ¬ , as all other cases follow directly from the axioms of KFn0 . If Mn ✏ Grn (p¬ q), then Mn ✏ Grn (p q), and thus, by the induction hypothesis, Mn ✏ T rn (p q) $ . Therefore Mn ✏ Grn (p¬ q) ! (¬T rn (p q) $ ¬ ), and so it is enough to show that Mn ✏ Grn (p¬ q) ! (T rn (p¬ q) $ ¬T rn (p q)). Mn ✏ Grn (p¬ q) ! (¬T rn (p¬ q) ! T rn (p¬ q)) follows from the definition of Grn . Mn ✏ Grn (p¬ q) ! (T rn (p q) ! ¬T rn (p q)) follows from axiom 10 of KF0n . Thus Mn ✏ Grn (p¬ q) ! (T rn (p¬ q) $ ¬ ). In contrast to the hierarchy of models in section 5.1, we do not obtain Mm ✏ SentLn (p'q) ! (T rm (p'q) $ '), for each m, n 2 N, m > n. But we obtain the following result, which is reveals the philosophical di↵erence between KFn and KF0n : Fact 5.12 For all n m 1 2 N and for any sentence ' of Lm , Mn ✏ T rn (p'q) i↵ Mm ✏ T rm(p'q). In other words: for any sentence ' of Lm , either Mn ✏ T rn (p'q), for any n m 1, or Mn ✏ ¬T rn (p'q) for any n m.7 Definition 5.3 Let ' be any sentence. Then • Let the two sentences T rn (p'q) and ¬T rn (p'q) be called the semantic diagnoses of ' on level n. • Let the set of semantic meta-diagnoses of ' on level n be the smallest set S that contains the semantic diagnoses of ' on level n, and such 7 The

same holds if T r is replaced by Gr.

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that T rn (p q) 2 S and ¬T rn (p q) 2 S, whenever 2 S. Furthermore, let any element of S be called a semantic meta-diagnosis of ' on level n. • Let 'n be a semantic meta-diagnosis of ' on level n, and let 'n be a semantic meta-diagnosis of ' on level m. Then let 'n be of the same type as 'm , i↵ 'n is the formula, which results from 'm by replacing each occurrence of ‘T rm ’ by ‘T rn ’. Thus from fact 5.12, it follows that, if ' is a sentence of Lm then for each n m, the semantic diagnosis of ' on level n, which is true in Mn , is of the same type as the semantic diagnosis of ' on level m, which is true in Mm . However, there can be semantic meta-diagnoses of ' on level n that are true in Mn , but that are not of the same type as any semantic meta-diagnosis of ' on level m that is true in Mn . E.g., we have M2 ✏ T r2 (p¬T r2 (p 1 q)q), but M1 ✏ ¬T r1 (p¬T r1 (p 1 q)q). What is the philosophical di↵erence between KFn and KF0n ? According to KF02 , the original liar sentence 1 is semantically defective and not true in the second model M2 . In contrast, according to KF2 , the original liar sentence 1 is true in the second model M2 . This initially seems to be unexpected, as our first intuitive diagnosis of a liar sentence, which is formally expressed in premise P2, is that a liar sentence is semantically defective and thus not true. Let us briefly compare KFn and KF0n with respect to their philosophical plausibility. In section 4.3, we have seen di↵erent attempts to explain why, in the end, a liar sentence turns out to be true. Let us consider the following informal strengthened liar sentence token (S1 ): (S1 ) This very sentence is not true. By the informal strengthened liar reasoning (cf. p. 10), (S1 ) is true i↵ (S1 ) is not true. Proponents of CR1 conclude from this that (S1 ) is not true in the sense of what is expressed by ‘is true’ occurring in the original liar sentence token (S1 ), and furthermore that (S1 ) is true in another “reflective” sense of ‘is true’. Let us visualize these di↵erent senses of ‘is true’ by writing ‘trues ’ if it is read in the original sense, and ‘truer ’ if it is read

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in the “reflective” sense. We then have the following informal strengthened liar sentence token: (S2 ) This very sentence is not trues . The diagnoses concerning (S2 ) of proponents of CR1 are the following tokens: (D1 ) (S2 ) is not trues . (D2 ) (S2 ) is truer . In this informal variant of CR1 it is completely inessential whether (S2 ) (literally) says of itself that it is not trues , or whether it (literally) says of itself that it does not exhibit any other property as long as only (S2 ) does not (literally) say of itself that it is not truer . So ‘is trues ’ is treated as just any other non-semantical predicate. Summarizing, in their informal theory of truth, proponents of CR1 do not postulate any connection between the two truth predicates ‘is trues ’ and ‘is truer ’. This aspect of CR1, however, does not have to be accepted. One could e.g. claim that whenever a sentence has been established as trues (or as not trues ), it will also be truer (or not truer ). This is what is implicitly done in the axiom systems KF0n , more precisely in the first axiom of KF0n . By fact 5.12, if ' is a sentence of Lm then for each n m, the semantic diagnosis of ' on level n, which is true in Mn , is of the same type as the semantic diagnosis of ' on level m, which is true in Mm . In case of KFn we have Mn ✏ ¬T rn (p n q), but Mn+1 ✏ T rn+1 (p n q) for each liar sentence n . In this sense the truth predicates of the hierarchy resulting from KF0 n are conceptually more coherent than the truth predicates of the hierarchy resulting from KFn . The main disadvantage of KF0n compared to KFn is that fact 5.8 (3.) does not hold for the models of KF0n . Thus, there is no guarantee for the T rn+1 -schemes to hold for all sentences of Ln .

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5.3 Extending the Hierarchies to Transfinite Ordinal Numbers Up to now, we have defined a model Mn and a truth predicate T rn with extension E n for each natural number n 2 N. To each model Mn , we have a successor model Mn+1 , which is an expansion of Mn , and which can be used to truthfully express semantical statements that cannot be truthfully expressed in Mn . Let us define M