An axiomatic approach to self-referential truth

Friedman, Harvey M.; Sheard, Michael H.

doi:10.1016/0168-0072(87)90073-x

Cited by 199 publications

(140 citation statements)

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“…These two charts are in fact identical to charts 2 and 3 presented in [3] for the classical theories Base T + T-Cons …”

Section: The Main Theoremmentioning

confidence: 60%

“…As it happens, perhaps surprisingly, this does not appear to be the case; most of the inconsistencies are derivable without the use of classical principles. It was observed in [3] that ∃-Inf implies T-Comp over Base T and any set consistent with T-Comp is consistent with ∃-Inf. On the other hand with an intuitionistic base theory the dependency dissolves and ∃-Inf becomes consistent with all subsets of the Optional Axioms, just as ∀-Inf is in the classical setting.…”

Section: The Main Theoremmentioning

confidence: 99%

“…Table 1, below, lists the twelve principles of truth considered by Friedman and Sheard. The next theorem summarises the work of [3]. E. T-Intro, T-Elim, T-Del, T-Cons, ¬T-Intro, ∀-Inf.…”

Section: A Closer Look At the Optional Axiomsmentioning

confidence: 99%

“…logics other than classical or intuitionistic logic, as in these logics "nothing like sustained ordinary reasoning can be carried out" [1, p. 95]. The third direction is exemplified by the work of Friedman and Sheard [3]. The naive notion of truth is removed in favour of twelve principles, referred to as Optional Axioms, each conveying some desirable property of truth.…”

Section: Introductionmentioning

confidence: 99%

“…In [3] a classical base theory, Base T , is used incorporating a truth predicate whose underlying logic is also classical, i.e. Base T ⊢ T( A ∨ ¬A ) for every sentence A where A denotes the Gödel number of A.…”

Section: Introductionmentioning

confidence: 99%

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The Friedman-Sheard programme in intuitionistic logic

Leigh¹,

Rathjen²

2012

J. symb. log.

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“…These two charts are in fact identical to charts 2 and 3 presented in [3] for the classical theories Base T + T-Cons …”

Section: The Main Theoremmentioning

confidence: 60%

Section: The Main Theoremmentioning

confidence: 99%

“…Table 1, below, lists the twelve principles of truth considered by Friedman and Sheard. The next theorem summarises the work of [3]. E. T-Intro, T-Elim, T-Del, T-Cons, ¬T-Intro, ∀-Inf.…”

Section: A Closer Look At the Optional Axiomsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The Friedman-Sheard programme in intuitionistic logic

Leigh¹,

Rathjen²

2012

J. symb. log.

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Notes on Formal Theories of Truth

Cantini¹

1989

Mathematical Logic Qtrly

111

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8 0. Introduction I n this paper we investigate formal systems, which are related to KRIPKE's theory of truth and to its subsequent extensions via four-valued logic (see KRIPKE [22], VISSER [33], WOODRUFF [34]). There are several motivations for an axiomatic study: let us mention a few of them.The cited papers exhibit interpreted languages, which contain their own truth predicates and satisfy a non-classical semantics. The interpretation always produces a pair of sets ( T , F ) ( T = true sentences, F = false sentences), where T and F are closed under and sound with respect to the rules, embodied by the chosen evaluation schema (in short, ( T , P) is the fixed point of a suitable operator). By TARSKI'S theorem, if the basic language has enough self -referential power, either there are sentences in the complement of T u F (the so-called gaps) or there are sentences, which lie in T n F (bhe so-called gluts). Now, if we regard (T, P) "from without", T, F are perfectly good objects; gaps and/or gluts only show the incompleteness and/or the inconsistency of the given evaluation schema, for a language containing its own truth predicate. Therefore, it is quite natural to look for frameworks, in which at least some global facts about gluts, gaps, etc. can be stated and proved. This desiratum yields the first motivation for the present study.The second reason is to be found in the fact that the structure of the fixed points (T, F ) , which is brought out by KRIPKE'S investigation is very rich and mathematically interesting. Hence, it seems worth trying to isolate the common features of all the fixed points.A third argument for axiomatization concerns, as usual, generalization: insofar as we axiomatize KRIPKE'S theory, we are easily led to realize that the duality between consistency (T n F = 0) and completeness (T u P = Sent, the set of all sentences) is a special case of a more general phenomenon. This remark readily provides with a simple axiomatic analogue (see 5 4) of the duality theorem, first observed by WOODRUFF [34], VISSER [33] at a semantical level.Last but not least, we like to mention that in recent years, there has been a growing interest among computer scientists for KRIPKE'S theory and its formal developements, especially in connection with problems arising in logic programming (see for instance PERLIS [25], FITTING [ 171). Presented a t the Logic Seminar, Department of Philosophy, Universith di Firenze in the academic year 1986/87. I wish to thank S. FEFERNAN for kindly making available to me copies of [15], [16]. I am indebted to P. L. MINARI for correcting an earlier fake proof of non-modularity of INT,,,; an argument of his is adapted to the present context and reproduced in 8 with permission. 7 Ztschr. f. math. Logik 98 A. CANTINI As to the choice of the formal framework, there are mainly two ways of formalizing the semantic theories of [22], [34], [33]. The first one considers an extension of classical logic by two unary connectives T , F , which reinterpret the standard logical operators according to...

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Infinitary Contraction-Free Revenge

Fjellstad

2018

Thought: A Journal of Philosophy

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An axiomatic approach to self-referential truth

Cited by 199 publications

References 4 publications

The Friedman-Sheard programme in intuitionistic logic

The Friedman-Sheard programme in intuitionistic logic

Notes on Formal Theories of Truth

Infinitary Contraction-Free Revenge

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