Proving unprovability

This paper addresses the question: given some theory T that we accept, is there some natural, generally applicable way of extending T to a theory S that can prove a range of things about what it itself (i.e., S) can prove, including a range of things about what it cannot prove, such as claims to the effect that it cannot prove certain particular sentences (e.g., 0 = 1), or the claim that it is consistent? Typical characterizations of Gödel’s second incompleteness theorem, and its significance, would lead us to believe that the answer is ‘no’. But the present paper explores a positive answer. The general approach is to follow the lead of recent (and not so recent) approaches to truth and the Liar paradox

On a certain fallacy concerning I-am-unprovable sentences

We demonstrate that, in itself and in the absence of extra premises, the following argument scheme is fallacious: The sentence A says about itself that it has a property F, and A does in fact have the property F; therefore A is true. We then examine an argument of this form in the informal introduction of Gödel’s classic (1931) and examine some auxiliary premises which might have been at work in that context. Philosophically significant as it may be, that particular informal argument plays no rôle in Gödel’s technical results. Going deeper into the issue and investigating truth conditions of Gödelian sentences (i.e., those sentences which are provably equivalent to their own unprovability) will provide us with insights regarding the philosophical debate on the truth of Gödelian sentences of systems—a debate which goes back to Dummett (1963)

Explicit fixed-points in provability logic

Smyslem této diplomové práce je prozkoumat explicitní výpoty pevn ých bod v logice dokazatelnosti GL. Vta o pevných bodech zní: Pro kadou modální formuli A(p) v ní kadý výskyt atomu p je vázán modálním operátorem ¤, existuje formule D obsahující pouze výrokové atomy obsaené v A(p), neobsahující výrokový atom p, a taková, e v GL je dokazatelné D ' A(D). Formule D je navíc ur- ena a na dokazatelnou ekvivalenci jednoznan. Nejprve vyslovíme nkolik speciálních pípad vty o pevných bodech a poté podrobnji prozkoumáme vtu v plném znní. Dále ukáeme jednu sémantickou a dv syntaktické konstrukce pevných bod a dokáeme jejich korektnost. V práci se zabýváme také nkterými sloitostními aspekty konstrukce, pedevím uvádíme jednoduché horní odhady délky a modální sloitosti získaných pevných bod.The aim of this diploma thesis is to discuss the explicit calculations of xed-points in provability logic GL. The xed-point theorem reads: For every modal formula A(p) such that each occurrence of p is under the scope of ¤, there is a formula D containing only sentence letters contained in A(p), not containing the sentence letter p, such that GL proves D ' A(D). Moreover, D is unique up to the provable equivalence. Firstly, we establish some special cases of the theorem and then we will look more closely at the full theorem. We show one semantic and two syntactic full xed-point constructions and prove their correctness. We also discuss some complexity aspects connected with the constructions and present basic upper bounds on length and modal depth of the constructed xed-points.Katedra logikyDepartment of LogicFaculty of ArtsFilozofická fakult

Leo Esakia on duality in modal and intuitionistic logics

This volume is dedicated to Leo Esakia's contributions to the theory of modal and intuitionistic systems. Consisting of 10 chapters, written by leading experts, this volume discusses Esakia's original contributions and consequent developments that have helped to shape duality theory for modal and intuitionistic logics and to utilize it to obtain some major results in the area. Beginning with a chapter which explores Esakia duality for S4-algebras, the volume goes on to explore Esakia duality for Heyting algebras and its generalizations to weak Heyting algebras and implicative semilattices. The book also dives into the Blok-Esakia theorem and provides an outline of the intuitionistic modal logic KM which is closely related to the Gödel-Löb provability logic GL. One chapter scrutinizes Esakia's work interpreting modal diamond as the derivative of a topological space within the setting of point-free topology. The final chapter in the volume is dedicated to the derivational semantics of modal logic and other related issues