Dialectica Logical Principles

Abstract

Gödel' s Dialectica interpretation was designed to obtain a relative consistency proof for Heyting arithmetic, to be used in conjunction with the double negation interpretation to obtain the consistency of Peano arithmetic. In recent years, proof theoretic transformations (socalled proof interpretations) that are based on Gödel's Dialectica interpretation have been used systematically to extract new content from proofs and so the interpretation has found relevant applications in several areas of mathematics and computer science. Following our previous work on 'Gödel fibrations', we present a (hyper)doctrine characterisation of the Dialectica which corresponds exactly to the logical description of the interpretation. To show that we derive the soundness of the interpretation of the implication connective, as expounded on by Spector and Troelstra, in the categorical model. This requires extra logical principles, going beyond intuitionistic logic, namely Markov's Principle (MP) and the Independence of Premise (IP) principle, as well as some choice. We show how these principles are satisfied in the categorical setting, establishing a tight (internal language) correspondence between the logical system and the categorical framework. This tight correspondence should come handy not only when discussing the applications of the Dialectica already known, like its use to extract computational content from (some) classical theorems (proof mining), its use to help to model specific abstract machines, etc. but also to help devise new applications.

Dialectica Principles via Gödel Doctrines

“…This article is the culmination of various intertwined investigations begun in [34] and [35]. Inspired by Hofstra [14] and Hyland [15], as well as by the work of Maietti and Trotta [32], itself inspired by Trotta [31], we embarked in the programme of expanding the characterisation of the categorical version of the Dialectica Interpretation, to complete the work in Hofstra as far as the characterisation of the Dialectica is concerned and to make sure that all the logical principles involved in the interpretation are represented in the categorical models obtained.…”

“…Taking advantage of the abstract presentation of Hofstra, in previous works [34,35] we introduced an intrinsic presentation of the Dialectica construction via the notion of Gödel fibration and its proof-irrelevant version, namely Gödel doctrines. The key idea is that Gödel fibrations (and doctrines) can be thought of as fibrations generated by some basic elements playing the role of quantifierfree elements.…”

“…The key idea is that Gödel fibrations (and doctrines) can be thought of as fibrations generated by some basic elements playing the role of quantifierfree elements. This categorification of quantifier-free elements is crucial not only to show that the notions of Gödel fibrations introduced in [34] and Dialectica fibrations (as presented in [14]) are mathematically equivalent in the appropriate way, but also to show how Gödel doctrines embody the main logical features of the Dialectica Interpretation [35,36]. While in [34] we presented a reconstruction of where the categorification of concepts came from, in [36] we showed that this categorification worked not only for rules as in [35], but also for the logical principles themselves, which is always more exciting for logicians than for category theorists.…”

“…The main purpose of the work here is to provide a self-contained and complete study of Gödel doctrines, presenting in detail the results developed in [35,36], their connections with [34], and carrying on the analysis of these doctrines. In this paper, we present in detail the notion of doctrine and its logical meaning first, and then we discuss the Dialectica interpretation of the implicational connective and the logical principles involved in the categorical interpretation of this connective.…”

“…We show how these key logical features are satisfied in the categorical setting, establishing a tight correspondence between the logical system and the categorical framework. Our results are built on the categorical presentation of existential and universal-free elements we introduced first in the context of fibrations in [34], and then in the language of doctrines [35]. Having a categorification of such notions is fundamental to properly state logical principles in categorical terms, since both (IP) and (MP) involve quantifier-free formulae.…”