450 research outputs found
η-conversions of IPC implemented in atomic F
It is known that the β-conversions of the full intuitionistic propositional calculus (IPC) translate into βη-conversions of the atomic polymorphic calculus Fat. Since Fat enjoys the property of strong normalization for βη-conversions, an alternative proof of strong normalization for IPC considering β-conversions can be derived. In the present article, we improve the previous result by analysing the translation of the η-conversions of the latter calculus into a technical variant of the former system (the atomic polymorphic calculus Fat^∧_at). In fact, from the strong normalization of Fat^∧_at we can derive the strong normalization of the full intuitionistic propositional calculus considering all the standard (β and η) conversions.info:eu-repo/semantics/publishedVersio
Rasiowa–Harrop disjunction property
We show that there is a purely proof-theoretic proof of the Rasiowa–Harrop disjunction property for the full intuitionistic propositional calculus (IPC), via natural deduction, in which commuting conversions are not needed. Such proof is based on a sound and faithful embedding of IPC into an atomic polymorphic system. This result strengthens a homologous result for the disjunction property of IPC (presented in a recent paper co-authored with Fernando Ferreira) and answers a question then posed by Pierluigi Minari.info:eu-repo/semantics/publishedVersio
A herbrandized functional interpretation of classical first-order logic
We introduce a new typed combinatory calculus with a type constructor that, to each type σ, associates the star type σ^∗ of the nonempty finite subsets of elements of type σ. We prove that this calculus enjoys the properties of strong normalization and confluence. With the aid of this star combinatory calculus, we define a functional interpretation of first-order predicate logic and prove a corresponding soundness theorem. It is seen that each theorem of classical first-order logic is connected with certain formulas which are tautological in character. As a corollary, we reprove Herbrand’s theorem on the extraction of terms from classically provable existential statements.info:eu-repo/semantics/publishedVersio
Elementary Proof of Strong Normalization for Atomic F
We give an elementary proof (in the sense that it is formalizable in Peano arithmetic) of the strong normalization of the atomic polymorphic calculus Fₐₜ (a predicative restriction of Girard’s system F)
On Various Negative Translations
Several proof translations of classical mathematics into intuitionistic
mathematics have been proposed in the literature over the past century. These
are normally referred to as negative translations or double-negation
translations. Among those, the most commonly cited are translations due to
Kolmogorov, Godel, Gentzen, Kuroda and Krivine (in chronological order). In
this paper we propose a framework for explaining how these different
translations are related to each other. More precisely, we define a notion of a
(modular) simplification starting from Kolmogorov translation, which leads to a
partial order between different negative translations. In this derived
ordering, Kuroda and Krivine are minimal elements. Two new minimal translations
are introduced, with Godel and Gentzen translations sitting in between
Kolmogorov and one of these new translations.Comment: In Proceedings CL&C 2010, arXiv:1101.520
Analysis in weak systems
The authors survey and comment their work on weak analysis. They describe the basic set-up of analysis in a feasible second-order theory and consider the impact of adding to it various forms of weak Konig's lemma. A brief discussion of the Baire categoricity theorem follows. It is then considered a strengthening of feasibility obtained (fundamentally) by the addition of a counting axiom and showed how it is possible to develop Riemann integration in the stronger system. The paper finishes with three questions in weak analysis.info:eu-repo/semantics/publishedVersio
Atomic polymorphism and the existence property
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.The author acknowledges the support of Fundação para a Ciência e a Tecnologia
[UID/MAT/04561/2013, UID/CEC/00408/2013 and grant SFRH/BPD/93278/2013] and is also
grateful to Centro de Matemática, Aplicações Fundamentais e Investigação Operacional and LargeScale Informatics Systems Laboratory.info:eu-repo/semantics/publishedVersio
Interpretability in Robinson's Q
Edward Nelson published in 1986 a book defending an extreme formalist view of
mathematics according to which there is an impassable barrier in the totality of exponentiation.
On the positive side, Nelson embarks on a program of investigating how much mathematics can
be interpreted in Raphael Robinson’s theory of arithmetic Q. In the shadow of this program,
some very nice logical investigations and results were produced by a number of people, not only
regarding what can be interpreted in Q but also what cannot be so interpreted. We explain some
of these results and rely on them to discuss Nelson’s position.info:eu-repo/semantics/publishedVersio
The faithfulness of atomic polymorphism
It is known that the full intuitionistic propositional calculus can be embedded into the atomic polymorphic system Fat, a calculus with only two connectives: the
conditional and the second-order universal quantifier. The embedding uses a translation
of formulas due to Prawitz and relies on the so-called property of instantiation overflow.
In this paper, we show that the previous embedding is faithful i.e., if a translated formula is derivable in Fat, then the original formula is already derivable in the propositional calculus.info:eu-repo/semantics/publishedVersio
Techniques in weak analysis for conservation results
We review and describe the main techniques for setting up systems of weak
analysis, i.e. formal systems of second-order arithmetic related to subexponential
classes of computational complexity. These involve techniques of proof theory
(e.g., Herbrand’s theorem and the cut-elimination theorem) and model theoretic
techniques like forcing. The techniques are illustrated for the particular case of
polytime computability. We also include a brief section where we list the known
results in weak analysis.info:eu-repo/semantics/publishedVersio
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