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An intuitionistic theory of types
"... An earlier, not yet conclusive, attempt at formulating a theory of this kind was made by Scott 1970. Also related, although less closely, are the type and logic free theories of constructions of Kreisel 1962 and 1965 and Goodman 1970. In its first version, the present theory was based on the strongl ..."
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An earlier, not yet conclusive, attempt at formulating a theory of this kind was made by Scott 1970. Also related, although less closely, are the type and logic free theories of constructions of Kreisel 1962 and 1965 and Goodman 1970. In its first version, the present theory was based on the strongly impredicative axiom that there is a type of all types whatsoever, which is at the same time a type and an object of that type. This axiom had to be abandoned, however, after it was shown to lead to a contradiction by Jean Yves Girard. I am very grateful to him for showing me his paradox. The change that it necessitated is so drastic that my theory no longer contains intuitionistic simple type theory as it originally did. Instead, its proof theoretic strength should be close to that of predicative analysis.
Proof Theory and Meaning
 Handbook of Philosophical Logic, Vol III
, 1986
"... 2. Intermezzo: classical truth and sequent calculi 477 3. Dummett's argument against a truthconditional view on meaning 478 4. Proof as a key concept in meaning theories 485 5. The meaning of the logical constants and the soundess of predicate logic 489 6. Questions of completeness 494 7. The ..."
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2. Intermezzo: classical truth and sequent calculi 477 3. Dummett's argument against a truthconditional view on meaning 478 4. Proof as a key concept in meaning theories 485 5. The meaning of the logical constants and the soundess of predicate logic 489 6. Questions of completeness 494 7. The type theory of MartinLöf 497
Operations on Proofs That Can Be Specified By Means of Modal Logic
"... Explicit modal logic was first sketched by Gödel in [16] as the logic with the atoms "t is a proof of F". The complete axiomatization of the Logic of Proofs LP was found in [4] (see also [6],[7],[18]). In this paper we establish a sort of a functional completeness property of proof polynom ..."
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Explicit modal logic was first sketched by Gödel in [16] as the logic with the atoms "t is a proof of F". The complete axiomatization of the Logic of Proofs LP was found in [4] (see also [6],[7],[18]). In this paper we establish a sort of a functional completeness property of proof polynomials which constitute the system of proof terms in LP. Proof polynomials are built from variables and constants by three operations on proofs: "\Delta" (application), "!" (proof checker), and "+" (choice). Here constants stand for canonical proofs of "simple facts", namely instances of propositional axioms and axioms of LP in a given proof system. We show that every operation on proofs that (i) can be specified in a propositional modal language and (ii) is invariant with respect to the choice of a proof system is realized by a proof polynomial.
Workshop on Realizability Preliminary Version Uniform provability realization of intuitionistic logic, modality and *terms
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The word ’predicative ’ first appeared in Russell’s note On Some Difficulties in the Theory of Transfinite Numbers and Order Types [Rus07]. Paradoxes such
, 2008
"... as Russell’s Paradox show that we cannot form the class {x  φ(x)} for all propositional functions φ(x). Russell proposed we call φ(x) predicative if it defines a class and nonpredicative otherwise, but did not offer a criterion by which we could decide which propositional functions are which. A fi ..."
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as Russell’s Paradox show that we cannot form the class {x  φ(x)} for all propositional functions φ(x). Russell proposed we call φ(x) predicative if it defines a class and nonpredicative otherwise, but did not offer a criterion by which we could decide which propositional functions are which. A first such criterion was offered by Poincare ́ in the third part of his paper Les Mathématiques et la Logique [Poi06]. He proposed the vicious circle principle: “The definitions which ought to be regarded as nonpredicative are those which contain a vicious circle. ” (p. 1063) He indicated by example what he meant by ‘vicious circle’: in both the Richard paradox and the BuraliForti paradox, we define an aggregate E, and make use of E within its own definition. Poincare ́ proposed that definitions involving such a ‘vicious circle ’ are illegitimate: “A definition containing a vicious circle defines nothing. ” [Poi06, p. 1065] Poincare ́ justified this as follows. For a definition to be legitimate, it must be possible to substitute the definiens for the defined term. Recursive definitions
Compositionality, Understanding, and Proofs forthcoming in Mind
, 2008
"... The principle of semantic compositionality, as Jerry Fodor and Ernie Lepore have emphasised, imposes constraints on theories of meaning that are hard to meet by psychological or epistemic accounts. Here I argue that this general tendency is exemplified in Michael Dummett’s account of meaning. On th ..."
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The principle of semantic compositionality, as Jerry Fodor and Ernie Lepore have emphasised, imposes constraints on theories of meaning that are hard to meet by psychological or epistemic accounts. Here I argue that this general tendency is exemplified in Michael Dummett’s account of meaning. On that account, the socalled manifestability requirement has the effect that the speaker who understands a sentence s must be able to tell whether or not s satisfies central semantic conditions. This requirement is not met by truth conditional accounts of meaning. On Dummett’s view, it is met by a proof conditional account: understanding amounts to knowledge of what counts as a proof of a sentence. A speaker is supposed always to be capable of deciding whether or not a given object is a proof of a given sentence she understands. This requirement comes into conflict with compositionality. If meaning is compositionally determined, then all you need for understanding a sentence is what you get from combining your understanding of the