Construction of Satisfaction Classes for Nonstandard Models

Kotlarski, Henryk; Krajewski, S.; Lachlan, A. H.

doi:10.4153/cmb-1981-045-3

Cited by 65 publications

(76 citation statements)

References 6 publications

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“…More specifically, Krajewski [11] employed the framework of satisfaction classes over base theories formulated in relational languages as in this paper, however, the later series of papers [10], [16], and [17] all used the framework of truth classes over Peano arithmetic formulated in a relational language, augmented with 'domain constants'. Later, Kaye [9] developed the theory of satisfaction classes over models of PA in languages incorporating function symbols; his work was extended by Engström [3] to truth classes over models of PA in functional languages.…”

Section: Truth Classesmentioning

confidence: 99%

“…The question whether the analogue of (2) holds for PA remained open until the appearance of the joint work [10] of Kotlarski, Krajewski, and Lachlan in 1981, in which the rather exotic proof-theoretic technology of 'M-logic' (an infinitary logical system based on a nonstandard model M), was invented to construct 'truth classes' over countable recursively saturated models of PA.…”

Section: Introductionmentioning

confidence: 99%

“…10 Here α[t : e] is the assignment obtained by redefining the value of α at the variable t to be e if t ∈ Dom(α); note that α[t : e] = α if t / ∈ Dom(α). 11 More specifically, first define…”

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confidence: 99%

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New Constructions of Satisfaction Classes

Enayat

Visser²

2015

Unifying the Philosophy of Truth

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Section: Truth Classesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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New Constructions of Satisfaction Classes

Enayat

Visser²

2015

Unifying the Philosophy of Truth

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“…There are several ways of defining truth for nonstandard sentences. Here we use (a slight modification of) the notion of a satisfaction class (see Kaye [24] and Kotlarski [25] for overviews).…”

Section: Is the Usual Liar Sentence Withmentioning

confidence: 99%

“…According to results by Kotlarski, Krajewski and Lachlan [25,28] and assuming that M is a countable nonstandard model of PA, there is a satisfaction class for (M, S 1 ) if and only if (M, S 1 ) is recursively saturated. Every countable recursively saturated model has uncountable many satisfaction classes.…”

Section: Is the Usual Liar Sentence Withmentioning

confidence: 99%

Possible-Worlds Semantics for Modal Notions Conceived as Predicates

Halbach

Leitgeb

Welch

2003

Journal of Philosophical Logic

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Abstract. If is conceived as an operator, i.e., an expression that gives applied to a formula another formula, the expressive power of the language is severely restricted when compared to a language where is conceived as a predicate, i.e., an expression that yields a formula if it is applied to a term. This consideration favours the predicate approach. The predicate view, however, is threatened mainly by two problems: Some obvious predicate systems are inconsistent, and possible-worlds semantics for predicates of sentences has not been developed very far. By introducing possible-worlds semantics for the language of arithmetic plus the unary predicate , we tackle both problems. Given a frame W, R consisting of a set W of worlds and a binary relation R on W, we investigate whether we can interpret at every world in such a way that A holds at a world w ∈ W if and only if A holds at every world v ∈ W such that wRv. The arithmetical vocabulary is interpreted by the standard model at every world. Several 'paradoxes' (like Montague's Theorem, Gödel's Second Incompleteness Theorem, McGee's Theorem on the ω-inconsistency of certain truth theories etc.) show that many frames, e.g., reflexive frames, do not allow for such an interpretation. We present sufficient and necessary conditions for the existence of a suitable interpretation of at any world. Sound and complete semi-formal systems, corresponding to the modal systems K and K4, for the class of all possible-worlds models for predicates and all transitive possible-worlds models are presented. We apply our account also to nonstandard models of arithmetic and other languages than the language of arithmetic.

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Bounded Induction and Satisfaction Classes

Kotlarski¹

1986

Mathematical Logic Qtrly

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Zeitseiir. j . iIi(bth. Logik titid Grutzdlugeii d. X a l h . lid. JZ, S . SJl-624 ( 1 8 8 6 ) H. KOTLARSKI T h e o r e m 1.4. Kaufmann's model M admits no full satisfaction class (cf. SCHMERL [12] RATAJCZYK [lo] studied satisfaction classes from another point of view. He gave a combinatorial axiomatisation of the arithmetical part of the theory PA(#) = PA + S is a full satisfaction class + induction in L,, u {a}, and gave an independent sentence in the Paris-Harrington style.It should be noticed that a weaker notion, t,hat of a partial satisfaction class was applied to study recursively saturated models of PA, see SCHMERL [12], KIRBY, MCALOON and MURAWSKI [l] and KOTLARSKI [4].The notion of a full satisfaction class is a single sentence in L,, u ( 8 ) . The aim of this paper is to study models of the theory do-PA(#) = PA + S is a full satisfaction class + do-induction in L,,u ( S } .We hope that a t least some of the ideas developed below ill shed some light on problems about do-PA, see PARIS-WILKIE [9].for the definition and construction of a Kaufmann model). Finite axiomatisability I n this section we shall prove T h e o r e m 2.1. Ao-PA(S) is finitely axiomatisable.Let PAbe the theory in L,, consisting of recursive definitions of addition and multiplication and the definition of inequality. It is well known that PAis sufficiently strong to represent all the recursive functions (SHOENFIELD [13]), FO to arithmetise the language. Thus PAallows us to state the definition of a full satisfaction class.Let T be the following theory:

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Construction of Satisfaction Classes for Nonstandard Models

Cited by 65 publications

References 6 publications

New Constructions of Satisfaction Classes

New Constructions of Satisfaction Classes

Possible-Worlds Semantics for Modal Notions Conceived as Predicates

Bounded Induction and Satisfaction Classes

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