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Lectures on the curryhoward isomorphism
, 1998
"... The CurryHoward isomorphism states an amazing correspondence between systems of formal logic as encountered in proof theory and computational calculi as found in type theory. For instance, minimal propositional logic corresponds to simply typed λcalculus, firstorder logic corresponds to dependent ..."
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The CurryHoward isomorphism states an amazing correspondence between systems of formal logic as encountered in proof theory and computational calculi as found in type theory. For instance, minimal propositional logic corresponds to simply typed λcalculus, firstorder logic corresponds to dependent types, secondorder logic corresponds to polymorphic types, etc. The isomorphism has many aspects, even at the syntactic level: formulas correspond to types, proofs correspond to terms, provability corresponds to inhabitation, proof normalization corresponds to term reduction, etc. But there is much more to the isomorphism than this. For instance, it is an old idea—due to Brouwer, Kolmogorov, and Heyting, and later formalized by Kleene’s realizability interpretation—that a constructive proof of an implication is a procedure that transforms proofs of the antecedent into proofs of the succedent; the CurryHoward isomorphism gives syntactic representations of such procedures. These notes give an introduction to parts of proof theory and related
On the complexity of propositional quantification in intuitionistic logic
 Journal of Symbolic Logic
, 1997
"... Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at ..."
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Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at
Propositional quantification in the topological semantics for S4
 Notre Dame Journal of Formal Logic
, 1997
"... quantifiers range over all the sets of possible worlds. S5π+ is decidable and, as Fine and Kripke showed, many of the other systems are recursively isomorphic to secondorder logic. In the present paper I consider the propositionally quantified system that arises from the topological semantics for S ..."
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quantifiers range over all the sets of possible worlds. S5π+ is decidable and, as Fine and Kripke showed, many of the other systems are recursively isomorphic to secondorder logic. In the present paper I consider the propositionally quantified system that arises from the topological semantics for S4, rather than from the Kripke semantics. The topological system, which I dub S4πt, isstrictly weaker than its Kripkean counterpart. I prove here that secondorder arithmetic can be recursively embedded in S4πt. Inthe course of the investigation, I also sketch a proof of Fine’s and Kripke’s results that the Kripkean system S4π+ is recursively isomorphic to secondorder logic. 1Introduction One way to extend a propositional logic to a language with propositional quantifiers is to begin with a semantics for the logic; extract from the semantics a notion of a proposition; and interpret the quantifiers as ranging over the propositions. Thus, Fine [4] extends the Kripke semantics for modal logics to propositionally quantified systems S5π+, S4π+, S4.2π+, and such: given a Kripke frame, the quantifiers range over all sets of possible worlds. S5π+ is decidable ([4] and Kaplan [14]). In later unpublished work, Fine and Kripke independently showed that S4π+, S4.2π+, K4π+, Tπ+, Kπ+, and Bπ+ and others are recursively isomorphic to full secondorder classical logic. (Fine informs me that he later proved this stronger result. Kripke informs me that he too proved this stronger result in the early 1970s. A proof of this result occurs in Kaminski and Tiomkin [13], who use techniques similar to those used in Kremer [16] and to those used below. These techniques do not apply to S4.3π+. But according to Kaminski and Tiomkin, work of Gurevich and Shelah ([9], [10], and [39]) implies that secondorder arithmetic is interpretable in S4.3π+ and furthermore that, under
Completeness of secondorder propositional S4 and H in topological semantics
"... We add propositional quantifiers to the propositional modal logic S4 and to the propositional intuitionsitic logic H, introducing axiom schemes that are the natural analogs to axiom schemes typically used for firstorder quantifiers in classical logic. We show that the resulting logics are sound and ..."
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We add propositional quantifiers to the propositional modal logic S4 and to the propositional intuitionsitic logic H, introducing axiom schemes that are the natural analogs to axiom schemes typically used for firstorder quantifiers in classical logic. We show that the resulting logics are sound and complete for a topological semantics extending, in a natural way, the topological semantics for S4 and for H.