Forward refutation for Gödel-Dummett Logics

We propose a refutation calculus to check the unprovability of a formula in Gödel-Dummett logics. From refutations we can directly extract countermodels for unprovable formulas, moreover the calculus is designed so to support a forward proof-search strategy that can be understood as a top-down construction of a model

Bounding Resource Consumption with Gödel-Dummett Logics

International audienceGödel-Dummett logic LC and its finite approximations LCn are the intermediate logics complete w.r.t. linearly ordered Kripke models. In this paper, we use LCn logics as a tool to bound resource consumption in some process calculi. We introduce a non-deterministic process calculus where the consumption of a particular resource denoted * is explicit and provide an operational semantics which measures the consumption of this resource.We present a linear transformation of a process P into a formula f of LC. We show that the consumption of the resource by P can be bounded by the positive integer n if and only if the formula f admits a counter-model in LCn. Combining this result with our previous results on proof and counter-model construction for LCn, we conclude that bounding resource consumption is (linearly) equivalent to searching counter-models in LCn

Graph-based decision for Gödel-Dummett logics

International audienceWe present a graph-based decision procedure for Gödel-Dummett logics and an algorithm to compute counter-models. A formula is transformed into a conditional bi-colored graph in which we detect some specific cycles and alternating chains using matrix computations. From an instance graph containing no such cycle (resp. no (n+1)-alternating chain) we extract a counter-model in LC (resp. LCn)

Proof theoretic criteria for logical constancy

Logic concerns inference, and some inferences can be distinguished from others by their holding as a matter of logic itself, rather than say empirical factors. These inferences are known as logical consequences and have a special status due to the strong level of confidence they inspire. Given this importance, this dissertation investigates a method of separating the logical from the non-logical. The method used is based on proof theory, and builds on the work of Prawitz, Dummett and Read. Requirements for logicality are developed based on a literature review of common philosophical use of the term, with the key factors being formality, and the absolute generality / topic neutrality of interpretations of logical constants. These requirements are used to generate natural deduction criteria for logical constancy, resulting in the classification of certain predicates, truth functional propositional operators, first order quantifiers, second order quantifiers in sound and complete formal systems using Henkin semantics, and modal operators from the systems K and S5 as logical constants. Semantic tableaux proof systems are also investigated, resulting in the production of semantic tableaux-based criteria for logicality

Aspects of the constructive omega rule within automated deduction

In general, cut elimination holds for arithmetical systems with the w -rule, but not for systems with ordinary induction. Hence in the latter, there is the problem of generalisation, since arbitrary formulae can be cut in. This makes automatic theorem -proving very difficult. An important technique for investigating derivability in formal systems of arithmetic has been to embed such systems into semi- formal systems with the w -rule. This thesis describes the implementation of such a system. Moreover, an important application is presented in the form of a new method of generalisation by means of "guiding proofs" in the stronger system, which sometimes succeeds in producing proofs in the original system when other methods fail

A defence of predicativism as a philosophy of mathematics

A specification of a mathematical object is impredicative if it essentially involves quantification over a domain which includes the object being specified (or sets which contain that object, or similar). The basic worry is that we have no non-circular way of understanding such a specification. Predicativism is the view that mathematics should be limited to the study of objects which can be specified predicatively. There are two parts to predicativism. One is the criticism of the impredicative aspects of classical mathematics. The other is the positive project, begun by Weyl in Das Kontinuum (1918), to reconstruct as much as possible of classical mathematics on the basis of a predicatively acceptable set theory, which accepts only countably infinite objects. This is a revisionary project, and certain parts of mathematics will not be saved. Chapter 2 contains an account of the historical background to the predicativist project. The rigorization of analysis led to Dedekind's and Cantor's theories of the real numbers, which relied on the new notion of abitrary infinite sets; this became a central part of modern classical set theory. Criticism began with Kronecker; continued in the debate about the acceptability of Zermelo's Axiom of Choice; and was somewhat clarified by Poincaré and Russell. In the light of this, chapter 3 examines the formulation of, and motivations behind the predicativist position. Chapter 4 begins the critical task by detailing the epistemological problems with the classical account of the continuum. Explanations of classicism which appeal to second-order logic, set theory, and primitive intuition are examined and are found wanting. Chapter 5 aims to dispell the worry that predicativism might collapses into mathematical intuitionism. I assess some of the arguments for intuitionism, especially the Dummettian argument from indefinite extensibility. I argue that the natural numbers are not indefinitely extensible, and that, although the continuum is, we can nonetheless make some sense of classical quantification over it. We need not reject the Law of Excluded Middle. Chapter 6 begins the positive work by outlining a predicatively acceptable account of mathematical objects which justifies the Vicious Circle Principle. Chapter 7 explores the appropriate shape of formalized predicative mathematics, and the question of just how much mathematics is predicatively acceptable. My conclusion is that all of the mathematics which we need can be predicativistically justified, and that such mathematics is particularly transparent to reason. This calls into question one currently prevalent view of the nature of mathematics, on which mathematics is justified by quasi-empirical means.Supported by the Arts and Humanities Research Council [grant number 111315]

Through and beyond classicality: analyticity, embeddings, infinity

Structural proof theory deals with formal representation of proofs and with the investigation of their properties. This thesis provides an analysis of various non-classical logical systems using proof-theoretic methods. The approach consists in the formulation of analytic calculi for these logics which are then used in order to study their metalogical properties. A specific attention is devoted to studying the connections between classical and non-classical reasoning. In particular, the use of analytic sequent calculi allows one to regain desirable structural properties which are lost in non-classical contexts. In this sense, proof-theoretic versions of embeddings between non-classical logics - both finitary and infinitary - prove to be a useful tool insofar as they build a bridge between different logical regions