On a game-theoretic semantics for the Dialectica interpretation of analysis

Treballs Finals del Màster de Lògica Pura i Aplicada, Facultat de Filosofia, Universitat de Barcelona, Curs: 2017-2018, Tutor: Joost J. JoostenGödel's Dialectica interpretation is a tool of practical interest within proof theory. Although it was initially conceived in the realm of Hilbert's program, after Kreisel's fundamental work in the 1950's it has become clear that Dialectica, as well as other popular interpretations, can be used to extract explicit bounds and approximations from classical proofs in analysis. The program that was then started, consisting of using methods of proof theory to analyse and extract new information from classical proofs, is called proof mining. The first extension of the Dialectica interpretation to analysis was achieved by Spector by means of a principle called bar recursion. Recently, Escardó and Oliva presented a new extension using a principle called "product of selection functions", which provides a game-theoretic semantics to the interpreted theorems of analysis. This eases the task of understanding the constructive content and meaning of classical proofs, instead of only extracting quantitative information from them. In this thesis we present the Dialectica interpretation and its extensions to analysis, both using bar recursion and the product of selection functions. A whole chapter is thus devoted to exposing the theory of sequential games by Escardó and Oliva. In their paper "A Constructive Interpretation of Ramsey's Theorem via the Product of Selection Functions", Oliva and Powell gave a constructive proof of the Dialectica interpretation of the Infinite Ramsey Theorem for pairs and two colours using the product of selection functions. This yields an algorithm, which can be understood in game-theoretic terms, computing arbitrarily good approximations to the infinite monochromatic set. In this thesis we revisit this paper, extending all the results for the case of any finite number of colours

A Galois connection between classical and intuitionistic logics. I: Syntax

In a 1985 commentary to his collected works, Kolmogorov remarked that his 1932 paper "was written in hope that with time, the logic of solution of problems [i.e., intuitionistic logic] will become a permanent part of a [standard] course of logic. A unified logical apparatus was intended to be created, which would deal with objects of two types - propositions and problems." We construct such a formal system QHC, which is a conservative extension of both the intuitionistic predicate calculus QH and the classical predicate calculus QC. The only new connectives ? and ! of QHC induce a Galois connection (i.e., a pair of adjoint functors) between the Lindenbaum posets (i.e. the underlying posets of the Lindenbaum algebras) of QH and QC. Kolmogorov's double negation translation of propositions into problems extends to a retraction of QHC onto QH; whereas Goedel's provability translation of problems into modal propositions extends to a retraction of QHC onto its QC+(?!) fragment, identified with the modal logic QS4. The QH+(!?) fragment is an intuitionistic modal logic, whose modality !? is a strict lax modality in the sense of Aczel - and thus resembles the squash/bracket operation in intuitionistic type theories. The axioms of QHC attempt to give a fuller formalization (with respect to the axioms of intuitionistic logic) to the two best known contentual interpretations of intiuitionistic logic: Kolmogorov's problem interpretation (incorporating standard refinements by Heyting and Kreisel) and the proof interpretation by Orlov and Heyting (as clarified by G\"odel). While these two interpretations are often conflated, from the viewpoint of the axioms of QHC neither of them reduces to the other one, although they do overlap.Comment: 47 pages. The paper is rewritten in terms of a formal meta-logic (a simplified version of Isabelle's meta-logic

Semantic and Mathematical Foundations for Intuitionism

Thesis (Ph.D.) - Indiana University, Philosophy, 2013My dissertation concerns the proper foundation for the intuitionistic mathematics whose development began with L.E.J. Brouwer's work in the first half of the 20th Century. It is taken for granted by most philosophers, logicians, and mathematicians interested in foundational questions that intuitionistic mathematics presupposes a special, proof-conditional theory of meaning for mathematical statements. I challenge this commonplace. Classical mathematics is very successful as a coherent body of theories and a tool for practical application. Given this success, a view like Dummett's that attributes a systematic unintelligibility to the statements of classical mathematicians fails to save the relevant phenomena. Furthermore, Dummett's program assumes that his proposed semantics for mathematical language validates all and only the logical truths of intuitionistic logic. In fact, it validates some intuitionistically invalid principles, and given the lack of intuitionistic completeness proofs, there is little reason to think that every intuitionistic logical truth is valid according to his semantics. In light of the failure of Dummett's foundation for intuitionism, I propose and carry out a reexamination of Brouwer's own writings. Brouwer is frequently interpreted as a proto-Dummettian about his own mathematics. This is due to excessive emphasis on some of his more polemical writings and idiosyncratic philosophical views at the expense of his distinctively mathematical work. These polemical writings do not concern mathematical language, and their principal targets are Russell and Hilbert's foundational programs, not the semantic principle of bivalence. The failures of these foundational programs has diminished the importance of Brouwer's philosophical writings, but his work on reconstructing mathematics itself from intuitionistic principles continues to be worth studying. When one studies this work relieved of its philosophical burden, it becomes clear that an intuitionistic mathematician can make sense of her mathematical work and activity without relying on special philosophical or linguistic doctrines. Core intuitionistic results, especially the invalidity of the logical principle tertium non datur, can be demonstrated from basic mathematical principles; these principles, in turn, can be defended in ways akin to the basic axioms of other mathematical theories. I discuss three such principles: Brouwer's Continuity Principle, the Principle of Uniformity, and Constructive Church's Thesis

The Gödel and the Splitting Translations

When the new research area of logic programming and non-monotonic reasoning emerged at the end of the 1980s, it focused notably on the study of mathematical relations between different non-monotonic formalisms, especially between the semantics of stable models and various non-monotonic modal logics. Given the many and varied embeddings of stable models into systems of modal logic, the modal interpretation of logic programming connectives and rules became the dominant view until well into the new century. Recently, modal interpretations are once again receiving attention in the context of hybrid theories that combine reasoning with non-monotonic rules and ontologies or external knowledge bases. In this talk I explain how familiar embeddings of stable models into modal logics can be seen as special cases of two translations that are very well-known in non-classical logic. They are, first, the translation used by Godel in 1933 to em- ¨ bed Heyting’s intuitionistic logic H into a modal provability logic equivalent to Lewis’s S4; second, the splitting translation, known since the mid-1970s, that allows one to embed extensions of S4 into extensions of the non-reflexive logic, K4. By composing the two translations one can obtain (Goldblatt, 1978) an adequate provability interpretation of H within the Goedel-Loeb logic GL, the system shown by Solovay (1976) to capture precisely the provability predicate of Peano Arithmetic. These two translations and their composition not only apply to monotonic logics extending H and S4, they also apply in several relevant cases to non-monotonic logics built upon such extensions, including equilibrium logic, non-monotonic S4F and autoepistemic logic. The embeddings obtained are not merely faithful and modular, they are based on fully recursive translations applicable to arbitrary logical formulas. Besides providing a uniform picture of some older results in LPNMR, the translations yield a perspective from which some new logics of belief emerge in a natural wa

Computability and analysis: the legacy of Alan Turing

We discuss the legacy of Alan Turing and his impact on computability and analysis.Comment: 49 page

Lewis meets Brouwer: constructive strict implication

C. I. Lewis invented modern modal logic as a theory of "strict implication". Over the classical propositional calculus one can as well work with the unary box connective. Intuitionistically, however, the strict implication has greater expressive power than the box and allows to make distinctions invisible in the ordinary syntax. In particular, the logic determined by the most popular semantics of intuitionistic K becomes a proper extension of the minimal normal logic of the binary connective. Even an extension of this minimal logic with the "strength" axiom, classically near-trivial, preserves the distinction between the binary and the unary setting. In fact, this distinction and the strong constructive strict implication itself has been also discovered by the functional programming community in their study of "arrows" as contrasted with "idioms". Our particular focus is on arithmetical interpretations of the intuitionistic strict implication in terms of preservativity in extensions of Heyting's Arithmetic.Comment: Our invited contribution to the collection "L.E.J. Brouwer, 50 years later

Applications of Intuitionistic Logic in Answer Set Programming

We present some applications of intermediate logics in the field of Answer Set Programming (ASP). A brief, but comprehensive introduction to the answer set semantics, intuitionistic and other intermediate logics is given. Some equivalence notions and their applications are discussed. Some results on intermediate logics are shown, and applied later to prove properties of answer sets. A characterization of answer sets for logic programs with nested expressions is provided in terms of intuitionistic provability, generalizing a recent result given by Pearce. It is known that the answer set semantics for logic programs with nested expressions may select non-minimal models. Minimal models can be very important in some applications, therefore we studied them; in particular we obtain a characterization, in terms of intuitionistic logic, of answer sets which are also minimal models. We show that the logic G3 characterizes the notion of strong equivalence between programs under the semantic induced by these models. Finally we discuss possible applications and consequences of our results. They clearly state interesting links between ASP and intermediate logics, which might bring research in these two areas together.Comment: 30 pages, Under consideration for publication in Theory and Practice of Logic Programmin

An informational view of classical logic

We present an informational view of classical propositional logic that stems from a kind of informational semantics whereby the meaning of a logical operator is specified solely in terms of the information that is actually possessed by an agent. In this view the inferential power of logical agents is naturally bounded by their limited capability of manipulating \u201cvirtual information\u201d, namely information that is not implicitly contained in the data. Although this informational semantics cannot be expressed by any finitely- valued matrix, it can be expressed by a non-deterministic 3-valued matrix that was first introduced by W.V.O. Quine, but ignored by the logical community. Within the general framework presented in [21] we provide an in-depth discussion of this informational semantics and a detailed analysis of a specific infinite hierarchy of tractable approximations to classical propositional logic that is based on it. This hierarchy can be used to model the inferential power of resource-bounded agents and admits of a uniform proof-theoretical characterization that is half-way between a classical version of Natural Deduction and the method of semantic tableaux