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Paradox, arithmetic and nontransitive logic
Paradox, arithmetic and nontransitive logic
This dissertation is concerned with motivating, developing and defending nontransitive theories of truth over Peano Arithmetic. Its main goal is to show that such a nontransitive theory of truth is the only theory capable of maintaining all functional roles of the truth predicate: the substitutional and the quantificational roles. By the substitutional roles we mean that the theory ought to prove p iff it proves that p is true and that it proves all instances of the T-schema p iff 'p' is true. A theory fulfils the quantificational role if its axioms governing the truth-predicate are strong enough to mimick as much second-order quantification as possible. Where the literature on classical theories of truth has focused primarily on the fulfilment of the quantificational role, the nonclassical literature is very much obsessed with the substitutional roles. The problem of having a theory of truth fulfilling both the substitutional and quantificational (or already just the full substitutional) role are paradoxes of truth such as the Liar. Where the Liar is a sentence which informally says about itself that it is not true, we can show that it is both true and not true, which typically allows us to conclude any formula whatsoever. This problem is overcome in the current approach by blocking the use of transitivity principles under certain conditions.
Proof Theory, Paradox, Liar, Truth, Arithmetic, Nontransitive Logic
Dittrich, Jonathan Georg
2020
English
Universitätsbibliothek der Ludwig-Maximilians-Universität München
Dittrich, Jonathan Georg (2020): Paradox, arithmetic and nontransitive logic. Dissertation, LMU München: Faculty of Philosophy, Philosophy of Science and the Study of Religion
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Abstract

This dissertation is concerned with motivating, developing and defending nontransitive theories of truth over Peano Arithmetic. Its main goal is to show that such a nontransitive theory of truth is the only theory capable of maintaining all functional roles of the truth predicate: the substitutional and the quantificational roles. By the substitutional roles we mean that the theory ought to prove p iff it proves that p is true and that it proves all instances of the T-schema p iff 'p' is true. A theory fulfils the quantificational role if its axioms governing the truth-predicate are strong enough to mimick as much second-order quantification as possible. Where the literature on classical theories of truth has focused primarily on the fulfilment of the quantificational role, the nonclassical literature is very much obsessed with the substitutional roles. The problem of having a theory of truth fulfilling both the substitutional and quantificational (or already just the full substitutional) role are paradoxes of truth such as the Liar. Where the Liar is a sentence which informally says about itself that it is not true, we can show that it is both true and not true, which typically allows us to conclude any formula whatsoever. This problem is overcome in the current approach by blocking the use of transitivity principles under certain conditions.