On the proof theory of the intermediate logic MH

We give a Natural Deduction formulation of an adaptation of Gödel’s functional (Dialectica) interpretation to the extraction of (more) efficient programs from (classical) proofs. We adapt Jørgensen’s formulation of pure Dialectica translation by eliminating his “Contraction Lemma” and allowing free variables in the extracted terms (which is more suitable in a Natural Deduction setting). We also adapt Berger’s uniform existential and universal quantifiers to the Dialectica-extraction context. The use of such quantifiers without computational meaning permits the identification and isolation of contraction formulas which would otherwise be redundantly included in the pure-Dialectica extracted terms. In the end we sketch the possible combination of our refinement of Gödel’s Dialectica interpretation with its adaptation to the extraction of bounds due to Kohlenbach into a light monotone functional interpretation.KeywordsProgram extraction from (classical) proofsComplexity of extracted programsBerger’s uniform quantifiersGödel’s Functional interpretationProof-Carrying CodeProof Mining

A semantical proof of Craig's interpolation theorem for the intuitionistic predicate logic and some intermediate prepositional logics will be given. Our proof is an extension of Henkin's method developed in [4]. It will clarify the relation between the interpolation theorem and Robinson's consistency theorem for these logics and will enable us to give a uniform way of proving the interpolation theorem for them.

The aim of this chapter is to show how the answer to a Datalog program can be computed. In order to achieve a computational approach to Datalog, in Sect. 7.1 we develop the proof theory of this language. We show how new facts can be inferred from an extensional database by use of a Datalog program. We then demonstrate that our proof-theoretic method is sound and complete, i.e., that it computes exactly what it should according to the model-theoretic semantics of Datalog. The proof theory of Datalog leads to a first algorithm “INFER” for processing Datalog programs. This algorithm iteratively computes new facts from facts that have already been established by using a forward chaining technique.

A constructive proof is provided for the claim that classical first-order logic admits of a natural deduction formulation featuring the subformula property.

In this paper we suggest reading a constructive necessity of a formula $$\Box A$$ as internalizing a notion of constructive truth of A a proof within a deductive system I and validity of A a proof under an interpretation $$[\![{A}]\!]_{J}$$ within some system J. An example of such a relation is provided by the simply typed lambda calculus as I and its implementation in SK combinators as J. We utilize justification logic to axiomatize the notion of validity-under-interpretation and, hence, treat a semantical notion in a purely proof-theoretic manner. We present the system in Gentzen-style natural deduction formulation and provide reduction and expansion rules for the $$\Box $$ connective. Finally, we add proof-terms and proof-term equalities to obtain a corresponding calculus Jcalc $$^{-}$$ that can be viewed as an extension of the Curry---Howard isomorphism with justifications. We provide standard metatheoretic results and suggest a programming language interpretation in languages with foreign function interfaces FFIs.

We define what we call 'possibilistic intermediate logic (PIL)'; we present results analogous to those of the well-known intermediate logic, such as a deduction theorem, a generalised version of the deduction theorem, a cut rule, a weak version of a refutation theorem, a substitution theorem and Glivenko's theorem. Also, we present a definition for 'possibilistic safe beliefs'. This definition allows us to establish a relation between safe beliefs, as presented on 'applications of intuitionistic logic in answer set programming' by Osorio et al., and our version for possibilistic intermediate logic (PILX). We also present a characterisation of possibilistic safe beliefs for possibilistic normal logic programs.

This volume is dedicated to Prof. Dag Prawitz and his outstanding contributions to philosophical and mathematical logic. Prawitz's eminent contributions to structural proof theory, or general proof theory, as he calls it, and inference-based meaning theories have been extremely influential in the development of modern proof theory and anti-realistic semantics. In particular, Prawitz is the main author on natural deduction in addition to Gerhard Gentzen, who defined natural deduction in his PhD thesis published in 1934. The book opens with an introductory paper that surveys Prawitz's numerous contributions to proof theory and proof-theoretic semantics and puts his work into a somewhat broader perspective, both historically and systematically. Chapters include either in-depth studies of certain aspects of Dag Prawitz's work or address open research problems that are concerned with core issues in structural proof theory and range from philosophical essays to papers of a mathematical nature. Investigations into the necessity of thought and the theory of grounds and computational justifications as well as an examination of Prawitz's conception of the validity of inferences in the light of three dogmas of proof-theoretic semantics are included. More formal papers deal with the constructive behaviour of fragments of classical logic and fragments of the modal logic S4 among other topics. In addition, there are chapters about inversion principles, normalization of proofs, and the notion of proof-theoretic harmony and other areas of a more mathematical persuasion. Dag Prawitz also writes a chapter in which he explains his current views on the epistemic dimension of proofs and addresses the question why some inferences succeed in conferring evidence on their conclusions when applied to premises for which one already possesses evidence.

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.

One task of metamathematics is to relate suggestive but nonelementary modeltheoretic concepts to more elementary proof-theoretic concepts, thereby opening up modeltheoretic problems to proof-theoretic methods of attack. Herbrand's Theorem (see [8] or also [9], vol. 2) or Gentzen's Extended Hauptsatz (see [5] or also [10]) was first used along these lines by Beth [1]. Using a modified version he showed that for all first-order systems a certain modeltheoretic notion of definability coincides with a certain proof theoretic notion. In the present paper the Herbrand-Gentzen Theorem will be applied to generalize Beth's results from primitive predicate symbols to arbitrary formulas and terms.This may be interpreted as showing that (apart from some relatively minor exceptions which will be made apparent below) the expressive power of each first-order system is rounded out, or the system is functionally complete, in the following sense: Any functional relationship which obtains between concepts that are expressible in the system is itself expressible and provable in the system.A second application is concerned with the hierarchy of second-order formulas. A certain relationship is shown to hold between first-order formulas and those second-order formulas which are of the form (∃T1)…(∃Tk)A or (T1)…(Tk)A with A being a first-order formula. Modeltheoretically this can be regarded as a relationship between the class AC and the class PC⊿ of sets of models, investigated by Tarski in [12] and [13].

The conception of the very first decision procedures for first-order sentences brought about the need for giving citizenship to function symbols (e.g. Skolem functions). We argue that a closer look at proof procedures for first-order sentences with equality brings about the need for introducing (function) symbols for rewrites. This is appropriately done via the framework of labelled natural deduction which allows to formulate a proof theory for the “logical connective” of propositional equality. The basic idea is that when analysing an equality sentence into (i) proof conditions (introduction) and (ii) immediate consequences (elimination), it becomes clear that we need to bring in identifiers (i.e. function symbols) for sequences of rewrites, and this is what we claim is the missing entity in P. Martin-Löf’s equality types (both intensional and extensional). What we end up with is a formulation of what appears to be a middle ground solution to the ‘intensional’ versus ‘extensional’ dichotomy which permeates most of the work on characterizing propositional equality in natural deduction style. (Part of this material was presented at the Logical Methods in the Humanities Seminar, Stanford University, and the authors would like to thank Solomon Feferman and Grigori Mints for their comments and suggestions.)

This is a companion to a paper by the authors entitled “Godel on deduction”, which examined the links between some philosophical views ascribed to Godel and general proof theory. When writing that other paper, the authors were not acquainted with a system of natural deduction that Godel presented with the help of Gentzen’s sequents, which amounts to Jaśkowski’s natural deduction system of 1934, and which may be found in Godel’s unpublished notes for the elementary logic course he gave in 1939 at the University of Notre Dame. Here one finds a presentation of this system of Godel accompanied by a brief reexamination in the light of the notes of some points concerning his interest in sequents made in the preceding paper. This is preceded by a brief summary of Godel’s Notre Dame course, and is followed by comments concerning Godel’s natural deduction system.

Logic of agreement: foundations, semantic system and proof theory

14 Hybrid logics

The Incredible Proof Machine is an easy and fun to use program to conduct formal proofs. It employs a novel, intuitive proof representation based on port graphs, which is akin to, but even more natural than, natural deduction. In particular, we describe a way to determine the scope of local assumptions and variables implicitly. Our practical classroom experience backs these claims.

A structure which generalizes formulas by including substitution terms is used to represent proofs in classical logic. These structures, called expansion trees, can be most easily understood as describing a tautologous substitution instance of a theorem. They also provide a computationally useful representation of classical proofs as first-class values. As values they are compact and can easily be manipulated and transformed. For example, we present an explicit transformations between expansion tree proofs and cut-free sequential proofs. A theorem prover which represents proofs using expansion trees can use this transformation to present its proofs in more human-readable form. Also a very simple computation on expansion trees can transform them into Craig-style linear reasoning and into interpolants when they exist. We have chosen a sublogic of the Simple Theory of Types for our classical logic because it elegantly represents substitutions at all finite types through the use of the typed λ-calculus. Since all the proof-theoretic results we shall study depend heavily on properties of substitutions, using this logic has allowed us to strengthen and extend prior results: we are able to prove a strengthen form of the firstorder interpolation theorem as well as provide a correct description of Skolem functions and the Herbrand Universe. The latter are not straightforward generalization of their first-order definitions.