First-order modal logic in the necessary framework of objects

I consider the first-order modal logic which counts as valid those sentences which are true on every interpretation of the non-logical constants. Based on the assumptions that it is necessary what individuals there are and that it is necessary which propositions are necessary, Timothy Williamson has tentatively suggested an argument for the claim that this logic is determined by a possible world structure consisting of an infinite set of individuals and an infinite set of worlds. He notes that only the cardinalities of these sets matters, and that not all pairs of infinite sets determine the same logic. I use so-called two-cardinal theorems from model theory to investigate the space of logics and consequence relations determined by pairs of infinite sets, and show how to eliminate the assumption that worlds are individuals from Williamson’s argument

Lewis meets Brouwer: constructive strict implication

C. I. Lewis invented modern modal logic as a theory of "strict implication". Over the classical propositional calculus one can as well work with the unary box connective. Intuitionistically, however, the strict implication has greater expressive power than the box and allows to make distinctions invisible in the ordinary syntax. In particular, the logic determined by the most popular semantics of intuitionistic K becomes a proper extension of the minimal normal logic of the binary connective. Even an extension of this minimal logic with the "strength" axiom, classically near-trivial, preserves the distinction between the binary and the unary setting. In fact, this distinction and the strong constructive strict implication itself has been also discovered by the functional programming community in their study of "arrows" as contrasted with "idioms". Our particular focus is on arithmetical interpretations of the intuitionistic strict implication in terms of preservativity in extensions of Heyting's Arithmetic.Comment: Our invited contribution to the collection "L.E.J. Brouwer, 50 years later

Dugundji’s theorem revisited

sem informaçãoIn 1940 Dugundji proved that no system between S1 and S5 can be characterized by finite matrices. Dugundji’s result forced the development of alternative semantics, in particular Kripke’s relational semantics. The success of this semantics allowed the creation of a huge family of modal systems. With few adaptations, this semantics can characterize almost the totality of the modal systems developed in the last five decades. This semantics however has some limits. Two results of incompleteness (for the systems KH and VB) showed that not every modal logic can be characterized by Kripke frames. Besides, the creation of non-classical modal logics puts the problem of characterization of finite matrices very far away from the original scope of Dugundji’s result. In this sense, we will show how to update Dugundji’s result in order to make precise the scope and the limits of many-valued matrices as semantic of modal systems. A brief comparison with the useful Chagrov and Zakharyaschev’s criterion of tabularity for modal logics is provided.In 1940 Dugundji proved that no system between S1 and S5 can be characterized by finite matrices. Dugundji’s result forced the development of alternative semantics, in particular Kripke’s relational semantics. The success of this semantics allowed the creation of a huge family of modal systems. With few adaptations, this semantics can characterize almost the totality of the modal systems developed in the last five decades. This semantics however has some limits. Two results of incompleteness (for the systems KH and VB) showed that not every modal logic can be characterized by Kripke frames. Besides, the creation of non-classical modal logics puts the problem of characterization of finite matrices very far away from the original scope of Dugundji’s result. In this sense, we will show how to update Dugundji’s result in order to make precise the scope and the limits of many-valued matrices as semantic of modal systems. A brief comparison with the useful Chagrov and Zakharyaschev’s criterion of tabularity for modal logics is provided.8407422sem informaçãosem informaçãosem informaçãohttp://plato.stanford.edu/archives/win2010/entries/logic-modal-origins/, Ballarin, R.: Modern origins of modal logic. In: The Stanford Encyclopedia of Philosophy, Winter 2010 edition. (2010)Béziau, J.Y., A new four-valued approach to modal logic (2011) Log. Anal, 54 (213), pp. 109-121Bueno-Soler, J., (2009) Multimodalidades anódicas e catódicas: a negação controlada em lógicas multimodais e seu poder expressivo (Anhodic and cathodic multimodalities: controlled negation in multimodal logics and their expressive power, in Portuguese). PhD thesis, Instituto de Filosofia e Ciências Humanas (IFCH), Universidade Estadual de Campinas, , Unicamp, Campinas:Carnielli, W.A., Pizzi, C., Modalities and multimodalities (2008) Logic, Epistemology, and the Unity of Science, vol, p. 12. , Springer-Verlag, New York:Chagrov, A.V., Zakharyaschev, M., Modal logic (1997) Oxford Logic Guides, vol, p. 35. , Oxford University Press, Oxford:Creswell, M.J., Hughes, G.E., (1996) A New Introduction to Modal Logic, , Routledge, London:Dugundji, J., Note on a property of matrices for Lewis and Langford’s calculi of propositions (1940) J. Symb. Log, 5 (4), pp. 150-151Esakia, L., Meskhi, V., Five critical modal systems (1977) Theoria, 43 (1), pp. 52-60Gödel, K.: Eine intepretation des intionistischen Aussagenkalkül. Ergebnisse eines mathematischen Kolloquiums 4, 6–7 (1933) (English translation in [13], pp. 300–303)Gödel, K.: Zur intuitionistischen arithmetik und zahlentheorie. Ergebnisse eines mathematischen Kolloquiums 4, 34–38 (1933) (English translation in [13], pp. 222–225)Gödel, K., Kurt Godel, Collected Works: Publications 1929–1936. Oxford University Press (1986) CaryHenkin, L., Fragments of the proposicional calculus (1949) J. Symb. Log, 14 (1), pp. 42-48Lewis, C.I., Langford, C.H., (1932) Symbolic Logic, , Century, New York:Lemmon, E.J., New foundations for Lewis modal systems (1957) J. Symb. Log, 22 (2), pp. 176-186Lemmon, E.J., Algebraic semantics for modal logics I (1966) J. Symb. Log, 31 (1), pp. 44-65Łukasiewicz,J.: O logice trójwartościowej. Ruch Filozoficzny 5, 170–171 (1920) (English translation in [19] pp. 87–88)Łukasiewicz, J., (1970) Selected Works. Studies in Logic, , North-Holland Publishing Company, London:McKinsey, J.C.C., A reduction in number of the postulates for C. I. Lewis’ system of strict implication (1934) Bull. (New Ser.) Am. Math. Soc, 40, pp. 425-427Magari, R., Representation and duality theory for diagonalizable algebras (1975) Stud. Log, 34 (4), pp. 305-313Scroggs, S.J., Extensions of the Lewis system S5 (1951) J. Symb. Log, 16 (2), pp. 112-120Sobociński, B., Family K of the non-Lewis modal systens. Notre Dame (1964) J. Formal Log, V (4), pp. 313-318Zeman, J.J., Modal Logic: The Lewis Systems. Clarendon Press (1973) U

Kripke’s Modal Logic: A Historical Study

In a very short time Saul Kripke provided a suitable and rigorous semantics for different axiomatic modal systems and established a series of related results. Many key ideas were already in the air in the late Fifties, but it was Kripkean articles’ merit to system atically introduce comprehensive devices and solutions. Later on, the spreading of possible-worlds semantics massively changed the approach to modal logic, which enormously increased in popularity after that. Since Kripke’s work in modal logic is central to the development of the discipline, the aim of this essay is to present the fundamental results published between 1959 and 1965. Indeed, it was in such a brief and early phase of his career that Kripke was able to conceive the main novelties that would become central to the subsequent academic debates about modality. Here, their presentation will follow the original historical progressive introduction. Particular attention will be given to the interconnection between articles, their similarities in structure and the unified analysis produced by means of them. It actually appears quite impressive that, already in 1959, Kripke seemed to have planned all the developments he would present, one after the other, in the following years. First, an overview of the background where Kripke’s ideas start to rise is given. Then, each text’s results are individually briefly analysed

On the Logic of Belief and Propositional Quantification

We consider extending the modal logic KD45, commonly taken as the baseline system for belief, with propositional quantifiers that can be used to formalize natural language sentences such as “everything I believe is true” or “there is some-thing that I neither believe nor disbelieve.” Our main results are axiomatizations of the logics with propositional quantifiers of natural classes of complete Boolean algebras with an operator (BAOs) validating KD45. Among them is the class of complete, atomic, and completely multiplicative BAOs validating KD45. Hence, by duality, we also cover the usual method of adding propositional quantifiers to normal modal logics by considering their classes of Kripke frames. In addition, we obtain decidability for all the concrete logics we discuss

Revisiting Epistemic Logic with Names

This paper revisits the multi-agent epistemic logic presented in [10], where agents and sets of agents are replaced by abstract, intensional “names”. We make three contributions. First, we study its model theory, providing adequate notions of bisimulation and frame morphisms, and use them to study the logic’s expressive power and definability. Second, we show that the logic has a natural neighborhood semantics, which in turn allows to show that the axiomatization in [10] does not rely on possibly controversial introspective properties of knowledge. Finally, we extend the logic with common and distributed knowledge operators, and provide a sound and complete axiomatization for each of these extensions. These results together put the original epistemic logic with names in a more modern context and opens the door for a logical analysis of epistemic phenomena where group membership is uncertain or variable

Parainconsistency, or inconsistency tamed, investigated and exploited

Our aim in the paper is, firstly, to discuss several answers to the question, and secondly, and more importantly to provide a proper frames to explain and to exploit inconsistencies. The framework which will force inconsistencies to work in a positive way, i.e., to enlarge and to deep our understanding of problems involved

G\"odel's Notre Dame Course

This is a companion to a paper by the authors entitled "G\"odel's natural deduction", which presented and made comments about the natural deduction system in G\"odel's unpublished notes for the elementary logic course he gave at the University of Notre Dame in 1939. In that earlier paper, which was itself a companion to a paper that examined the links between some philosophical views ascribed to G\"odel and general proof theory, one can find a brief summary of G\"odel's notes for the Notre Dame course. In order to put the earlier paper in proper perspective, a more complete summary of these interesting notes, with comments concerning them, is given here.Comment: 18 pages. minor additions, arXiv admin note: text overlap with arXiv:1604.0307