Changing a semantics: opportunism or courage?

The generalized models for higher-order logics introduced by Leon Henkin, and their multiple offspring over the years, have become a standard tool in many areas of logic. Even so, discussion has persisted about their technical status, and perhaps even their conceptual legitimacy. This paper gives a systematic view of generalized model techniques, discusses what they mean in mathematical and philosophical terms, and presents a few technical themes and results about their role in algebraic representation, calibrating provability, lowering complexity, understanding fixed-point logics, and achieving set-theoretic absoluteness. We also show how thinking about Henkin's approach to semantics of logical systems in this generality can yield new results, dispelling the impression of adhocness. This paper is dedicated to Leon Henkin, a deep logician who has changed the way we all work, while also being an always open, modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and Alonso, E., 201

Modal Fragments of Second-Order Logic

Formaalin logiikan tutkimuskohteina ovat erilaiset muodolliset systeemit eli logiikat, joiden avulla voidaan mm. mekanisoida monenlaisia päättelyprosesseja. Eräs modernin formaalin logiikan keskeisistä tutkimusaiheista on modaalilogiikka, jossa perinteisempää logiikkaa laajennetaan nk. modaliteeteilla. Modaliteettien avulla voidaan luoda mitä erilaisimpia formaaleja systeemejä. Modaalilogiikalla onkin huomattava määrä sovelluksia aina tietojenkäsittelytieteestä ja matematiikan sekä fysiikan perusteista filosofiaan ja kielitieteisiin. Väitöskirja keskittyy modaalilogiikan nk. malliteoriaan. Tutkielmassa luokitellaan erilaisia formaalin logiikan systeemejä perustuen siihen, millaisia ominaisuuksia kyseisten systeemien avulla voidaan ilmaista. Mitä korkeampi ilmaisuvoima formaalilla järjestelmällä on, sitä hitaampaa on järjestelmän avulla suoritettava tietokoneellistettu päättely. Tutkielma käsittelee useita modaalilogiikan systeemejä; painopiste on erittäin korkean ilmaisuvoiman omaavien logiikoiden teoriassa. Tarkastelun kohteena olevat kysymykset liittyvät suoraan muuhun modaalilogiikan alan matemaattiseen tutkimukseen. Tutkielmassa mm. esitetään ratkaisu vuodesta 1983 avoinna olleeseen tekniseen kysymykseen koskien nk. toisen kertaluvun propositionaalisen modaalilogiikan alternaatiohierarkiaa.In this thesis we investigate various fragments of second-order logic that arise naturally in considerations related to modal logic. The focus is on questions related to expressive power. The results in the thesis are reported in four independent but related chapters (Chapters 2, 3, 4 and 5). In Chapter 2 we study second-order propositional modal logic, which is the system obtained by extending ordinary modal logic with second-order quantification of proposition symbols. We show that the alternation hierarchy of this logic is infinite, thereby solving an open problem from the related literature. In Chapter 3 we investigate the expressivity of a range of modal logics extended with existential prenex quantification of accessibility relations and proposition symbols. The principal result of the chapter is that the resulting extension of (a version of) Boolean modal logic can be effectively translated into existential monadic second-order logic. As a corollary we obtain decidability results for multimodal logics over various classes of frames with built-in relations. In Chapter 4 we study the equality-free fragment of existential second-order logic with second-order quantification of function symbols. We show that over directed graphs, the expressivity of the fragment is incomparable with that of first-order logic. We also show that over finite models with a unary relational vocabulary, the fragment is weaker in expressivity than first-order logic. In Chapter 5 we study the extension of polyadic modal logic with unrestricted quantification of accessibility relations and proposition symbols. We obtain a range of results related to various natural fragments of the system. Finally, we establish that this extension of modal logic exactly captures the expressivity of second-order logic

Algorithmic correspondence and completeness in modal logic

Abstract This thesis takes an algorithmic perspective on the correspondence between modal and hybrid logics on the one hand, and first-order logic on the other. The canonicity of formulae, and by implication the completeness of logics, is simultaneously treated. Modal formulae define second-order conditions on frames which, in some cases, are equiv- alently reducible to first-order conditions. Modal formulae for which the latter is possible are called elementary. As is well known, it is algorithmically undecidable whether a given modal formula defines a first-order frame condition or not. Hence, any attempt at delineating the class of elementary modal formulae by means of a decidable criterium can only consti- tute an approximation of this class. Syntactically specified such approximations include the classes of Sahlqvist and inductive formulae. The approximations we consider take the form of algorithms. We develop an algorithm called SQEMA, which computes first-order frame equivalents for modal formulae, by first transforming them into pure formulae in a reversive hybrid language. It is shown that this algorithm subsumes the classes of Sahlqvist and inductive formulae, and that all formulae on which it succeeds are d-persistent (canonical), and hence axiomatize complete normal modal logics. SQEMA is extended to polyadic languages, and it is shown that this extension succeeds on all polyadic inductive formulae. The canonicity result is also transferred. SQEMA is next extended to hybrid languages. Persistence results with respect to discrete general frames are obtained for certain of these extensions. The notion of persistence with respect to strongly descriptive general frames is investigated, and some syntactic sufficient conditions for such persistence are obtained. SQEMA is adapted to guarantee the persistence with respect to strongly descriptive frames of the hybrid formulae on which it succeeds, and hence the completeness of the hybrid logics axiomatized with these formulae. New syntactic classes of elementary and canonical hybrid formulae are obtained. Semantic extensions of SQEMA are obtained by replacing the syntactic criterium of nega- tive/positive polarity, used to determine the applicability of a certain transformation rule, by its semantic correlate—monotonicity. In order to guarantee the canonicity of the formulae on which the thus extended algorithm succeeds, syntactically correct equivalents for monotone formulae are needed. Different version of Lyndon’s monotonicity theorem, which guarantee the existence of these equivalents, are proved. Constructive versions of these theorems are also obtained by means of techniques based on bisimulation quantifiers. Via the standard second-order translation, the modal elementarity problem can be at- tacked with any second-order quantifier elimination algorithm. Our treatment of this ap- proach takes the form of a study of the DLS-algorithm. We partially characterize the for- mulae on which DLS succeeds in terms of syntactic criteria. It is shown that DLS succeeds in reducing all Sahlqvist and inductive formulae, and that all modal formulae in a single propositional variable on which it succeeds are canonical

Modal Logics for Nominal Transition Systems

We define a uniform semantic substrate for a wide variety of process calculi where states and action labels can be from arbitrary nominal sets. A Hennessy-Milner logic for these systems is introduced, and proved adequate for bisimulation equivalence. A main novelty is the use of finitely supported infinite conjunctions. We show how to treat different bisimulation variants such as early, late and open in a systematic way, and make substantial comparisons with related work. The main definitions and theorems have been formalized in Nominal Isabelle

The Probability of the Truth on a Truth Table

In symbolic (or mathematical) logic the truth table is used to establish the truth or falsity (falsehood) of both simple and compound statements (or arguments). However, the use of the term “truth table” falsely suggests that all arguments through the table are true. It is known that the table contains both true and false arguments. So the use of “truth table” is indiscriminate. Consequently, this study was focused on solving this problem by finding out the probability of having a true argument associated with every one of the four binary proposition connectives- “and”, “double implication” “inclusive v”, single implication”  used in the arguments. The obtained probabilities are ordered as 1/4, ½ and ¾ respectfully for “and”, “double implication”, and (“inclusive v” and “single implication”). So the “truth tables” are discriminately decomposed into “falsehood tables”, “neutral tables” and “truth tables” at probabilities of ¼, ½ and ¾ respectively. These probabilities are independent of the number of statements, n, greater than unity

Canonical varieties with no canonical axiomatisation

Accepted versio

The principle of analyticity of logic : a philosophical and formal Perspective

The subject of the present work is the principle of analyticity of logic. In order for the question \u2018Is logic analytic?\u2019 to make sense and before trying to \ufb01nd an answer to this problem, it is obviously necessary to specify two preliminary issues, namely, the meaning of the term \u2018analytic\u2019 and the meaning of the term \u2018logic\u2019. The former issue is somehow justi\ufb01ed and expected: after all, analyticity represents one of the philosophical concepts par excellence and, as such, it has been at the core of a lively debate throughout the history of the discipline. But, despite possible appearances to the contrary, the second issue is probably more decisive than the former in determining the answer to the initial question: both the contents and the philosophical conceptions of logic play a fundamental role in the study of the epistemological status of this discipline. We could even say that the clari\ufb01cation of the concepts of analyticity and of logic constitutes in itself the decision on the analyticity of logic. This thesis studies the principle of analyticity of logic through two di\ufb00erent, but related, methodologies, which individuate the two main parts of the work: the former o\ufb00ers a historical and philosophical reconstruction of the problem; the latter proposes two formal characterizations of the analytic-synthetic distinction. The reconstruction of the \ufb01rst part does not presume to be exhaustive and is restricted to the theories of the following philosophers: Kant, Bolzano, Frege and Hintikka. The material has been chosen according to the following criteria. First, this work aims at showing the \u2018historical\u2019 nature of the principle of analyticity of logic, which has a certain genealogy and a precise starting point. Although after the Vienna Circle this tenet has been taken for granted, there are many and signi\ufb01cant conceptions that criticize it. Theories holding that logic is either not analytic or synthetic are the main characters of our reconstruction. This explains, for example, why we have dedicated great attention to Bolzano, while leaving little margin to the logical empiricist movement, despite the fact that analyticity is probably more fundamental for the latter\u2019s thought than for the former\u2019s philosophical construction. As a result of this choice, theories of meaning and their connection to analyticity are completely overlooked, since they belong to the logical empiricists\u2019 interpretation of the analytic-synthetic distinction. In other words, the principle of analyticity of logic and the philosophers arguing for it are taken as a critical target, but the true focus is on the varieties of reactions against them. [...