By Thomas Piecha, Peter Schroeder-Heister

ISBN-10: 3319226851

ISBN-13: 9783319226859

ISBN-10: 331922686X

ISBN-13: 9783319226866

This quantity is the 1st ever assortment dedicated to the sphere of proof-theoretic semantics. Contributions deal with issues together with the systematics of creation and removing principles and proofs of normalization, the categorial characterization of deductions, the relation among Heyting's and Gentzen's methods to which means, knowability paradoxes, proof-theoretic foundations of set thought, Dummett's justification of logical legislation, Kreisel's thought of buildings, paradoxical reasoning, and the defence of version theory.

The box of proof-theoretic semantics has existed for nearly 50 years, however the time period itself was once proposed by way of Schroeder-Heister within the Eighties. Proof-theoretic semantics explains the which means of linguistic expressions mostly and of logical constants specifically by way of the proposal of facts. This quantity emerges from displays on the moment overseas convention on Proof-Theoretic Semantics in Tübingen in 2013, the place contributing authors have been requested to supply a self-contained description and research of an important study query during this zone. The contributions are consultant of the sector and may be of curiosity to logicians, philosophers, and mathematicians alike.

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2, the decidability of the proof relation appears to have a strong pre-theoretical basis in the intuitionists’ desire to view the BHK clauses as providing a decidable proof condition for formulas of arbitrary logical complexity. Although T does not contain any primitive relation symbols itself, a term α can be understood as expressing a binary relation just in case for all pairs of terms s, t, if αst is defined, then αst ≡ or αst ≡ ⊥ may be derived in the theory. e. e. in order to exclude the “third” case that π uv is undefined, we stipulate that it is sufficient to conclude s ≡ t from Δ if this equation is derivable from both the hypotheses Δ, π uv ≡ and also Δ, π uv ≡ ⊥.

After discussing the formulation of the theory itself, we then discuss how it can be used to formalize the BHK interpretation in light of concerns about the impredicativity of intuitionistic implication and Kreisel’s proposed amendments to overcome this. We next reconstruct Goodman’s presentation of a paradox pertaining to a “naive” variant of the theory and discuss the influence this had on its subsequent reception. We conclude by considering various means of responding to this result. Contrary to the received view that the second clause interpretation itself contributes to the paradox, we argue that the inconsistency arises in virtue of an interaction between reflection and internalization principles similar to those employed in Artemov’s Logic of Proofs.

7, 46]) from the early 1980s onward. Two reasons for this appear to be as follows: (1) a “naive” form of the theory was shown by Goodman [16, 17] to be inconsistent in virtue of a “self-referential” antinomy involving constructive provability (we will see below that this is similar in form to what is now known as Montague’s paradox); (2) it was in the context of presenting the Theory of Constructions in which Kreisel first presented a modification to the clauses (P→ ), (P¬ ) and (P∀ ) (which has come to be known as the second clause) which proved to be controversial and has subsequently been excised from modern expositions of the BHK interpretation.

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Advances in Proof-Theoretic Semantics by Thomas Piecha, Peter Schroeder-Heister

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