Rasiowa–Harrop Disjunction Property

2019

Stud Logica

Self Cite

We study an alternative embedding of IPC into atomic system F whose translation of proofs is based, not on instantiation overflow, but instead on the admissibility of the elimination rules for disjunction and absurdity (where these connectives are defined according to the Russell-Prawitz translation). As compared to the embedding based on instantiation overflow, the alternative embedding works equally well at the levels of provability and preservation of proof identity, but it produces shorter derivations and shorter simulations of reduction sequences. Lambda-terms are employed in the technical development so that the algorithmic content is made explicit, both for the alternative and the original embeddings. The investigation of preservation of proof-reduction steps by the alternative embedding enables the analysis of generation of "administrative" redexes. These are the key, on the one hand, to understand the difference between the two embeddings; on the other hand, to understand whether the final word on the embedding of IPC into atomic system F has been said.

Section: Lhsmentioning

confidence: 94%

Section: Lhsmentioning

confidence: 99%

“…The constructive contents of the proof of this lemma is the proof transformation that underlies instantiation overflow as captured in [5,4]. Def.…”

Section: Lhsmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

A Refined Interpretation of Intuitionistic Logic by Means of Atomic Polymorphism

Santo

2019

Stud Logica

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“…In [2] it was shown that the calculus is expressive enough to embed full intuitionistic propositional logic (IPC). Recent research on F at (see [3,4,6,7]) has revealed some nice proof-theoretical properties of the system. The study in the present paper relies on the following three properties of F at : i) the strong normalization property, ii) the Church-Rosser property, and iii) the subformula property (for normal derivations).…”

Section: Introductionmentioning

confidence: 99%

The computational content of atomic polymorphism

Logic Journal of the IGPL

Vasconcelos

2018

Self Cite

We show that the number-theoretic functions definable in the atomic polymorphic system (${\mathbf{F}}_{\mathbf{at}}$) are exactly the extended polynomials. Two proofs of the above result are presented: one, reducing the functions’ definability problem in ${\mathbf{F}}_{\mathbf{at}}$ to definability in the simply typed lambda calculus ($\lambda ^{\rightarrow }$) and the other, directly adapting Helmut Schwichtenberg’s strategy for definability in $\lambda ^{\rightarrow }$ to the atomic polymorphic setting. The uniformity granted in the polymorphic system, when compared with the simply typed lambda calculus, is emphasized.

Atomic polymorphism and the existence property

Annals of Pure and Applied Logic

2018

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We present a purely proof-theoretic proof of the existence property for the full intuitionistic first-order predicate calculus, via natural deduction, in which commuting conversions are not needed. Such proof illustrates the potential of an atomic polymorphic system with only three generators of formulas-conditional and first and second-order universal quantifiers-as a tool for proof-theoretical studies.