The First-Order Hypothetical Logic of Proofs

The Propositional Logic of Proofs (LP) is a modal logic in which the modality □A is revisited as [​[t]​]​A , t being an expression that bears witness to the validity of A . It enjoys arithmetical soundness and completeness, can realize all S4 theorems and is capable of reflecting its own proofs ( ⊢A implies ⊢[​[t]​]A , for some t ). A presentation of first-order LP has recently been proposed, FOLP, which enjoys arithmetical soundness and has an exact provability semantics. A key notion in this presentation is how free variables are dealt with in a formula of the form [​[t]​]​A(i) . We revisit this notion in the setting of a Natural Deduction presentation and propose a Curry–Howard correspondence for FOLP. A term assignment is provided and a proof of strong normalization is given.Fil: Steren, Gabriela. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; ArgentinaFil: Bonelli, Eduardo Augusto. Universidad Nacional de Quilmes. Departamento de Ciencia y Tecnología; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Proceedings of the Conference on Hypothetical Reasoning, 23-24 August 2014, Tübingen

Hypothetical reasoning or reasoning under assumptions is a key concept of logic, philosophy of science and mathematics. The Conference on Hypothetical Reasoning focussed on its logical aspects, such as assumption-based calculi and their proof theory, logical consequence from a proof-theoretic or model-theoretic point of view, logics of conditionals, proof systems, structure of assumption-based proofs, hypotheses in proof-theoretic semantics, notions of implication, substructural logics, hypotheses in categorial logic, logical aspects of scientific explanation, hypothetical reasoning in mathematics and reasoning from definitions and axioms. The conference took place 23–24 August, 2014 in Tübingen at the Department of Philosophy, in conjunction with ESSLLI 2014. The proceedings collect abstracts, slides and papers of the presentations given

Constructive Provability Logic

We present constructive provability logic, an intuitionstic modal logic that validates the L\"ob rule of G\"odel and L\"ob's provability logic by permitting logical reflection over provability. Two distinct variants of this logic, CPL and CPL*, are presented in natural deduction and sequent calculus forms which are then shown to be equivalent. In addition, we discuss the use of constructive provability logic to justify stratified negation in logic programming within an intuitionstic and structural proof theory.Comment: Extended version of IMLA 2011 submission of the same titl

Revisiting Dummett's Proof-Theoretic Justification Procedures

Dummett’s justification procedures are revisited. They are used as background for the discussion of some conceptual and technical issues in proof-theoretic semantics, especially the role played by assumptions in proof-theoretic definitions of validity