276 research outputs found
An Intuitionistic Formula Hierarchy Based on High-School Identities
We revisit the notion of intuitionistic equivalence and formal proof
representations by adopting the view of formulas as exponential polynomials.
After observing that most of the invertible proof rules of intuitionistic
(minimal) propositional sequent calculi are formula (i.e. sequent) isomorphisms
corresponding to the high-school identities, we show that one can obtain a more
compact variant of a proof system, consisting of non-invertible proof rules
only, and where the invertible proof rules have been replaced by a formula
normalisation procedure.
Moreover, for certain proof systems such as the G4ip sequent calculus of
Vorob'ev, Hudelmaier, and Dyckhoff, it is even possible to see all of the
non-invertible proof rules as strict inequalities between exponential
polynomials; a careful combinatorial treatment is given in order to establish
this fact.
Finally, we extend the exponential polynomial analogy to the first-order
quantifiers, showing that it gives rise to an intuitionistic hierarchy of
formulas, resembling the classical arithmetical hierarchy, and the first one
that classifies formulas while preserving isomorphism
Perspectives for proof unwinding by programming languages techniques
In this chapter, we propose some future directions of work, potentially
beneficial to Mathematics and its foundations, based on the recent import of
methodology from the theory of programming languages into proof theory. This
scientific essay, written for the audience of proof theorists as well as the
working mathematician, is not a survey of the field, but rather a personal view
of the author who hopes that it may inspire future and fellow researchers
A predicative variant of a realizability tripos for the Minimalist Foundation.
open2noHere we present a predicative variant of a realizability tripos validating
the intensional level of the Minimalist Foundation extended with Formal Church
thesis.the file attached contains the whole number of the journal including the mentioned pubblicationopenMaietti, Maria Emilia; Maschio, SamueleMaietti, MARIA EMILIA; Maschio, Samuel
Prenex normalization and the hierarchical classification of formulas
Akama et al. [1] introduced a hierarchical classification of first-order
formulas for a hierarchical prenex normal form theorem in semi-classical
arithmetic. In this paper, we give a justification for the hierarchical
classification in a general context of first-order theories. To this end, we
first formalize the standard transformation procedure for prenex normalization.
Then we show that the classes and introduced in
[1] are exactly the classes induced by and respectively via
the transformation procedure in any first-order theory.Comment: 15 page
General Proof Theory. Celebrating 50 Years of Dag Prawitz's "Natural Deduction". Proceedings of the Conference held in Tübingen, 27-29 November 2015
General proof theory studies how proofs are structured and how they relate to each other, and not primarily what can be proved in particular formal systems. It has been developed within the framework of Gentzen-style proof theory, as well as in categorial proof theory. As Dag Prawitz's monograph "Natural Deduction" (1965) paved the way for this development (he also proposed the term "General Proof Theory"), it is most appropriate to use this topic to celebrate 50 years of this work. The conference took place 27-29 November, 2015 in Tübingen at the Department of Philosophy. The proceedings collect abstracts, slides and papers of the presentations given, as well as contributions from two speakers who were unable to attend
Deductive Systems in Traditional and Modern Logic
The book provides a contemporary view on different aspects of the deductive systems in various types of logics including term logics, propositional logics, logics of refutation, non-Fregean logics, higher order logics and arithmetic
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