Continuum of normal extensions of the modal logic of provability with the interpolation property

1990 ◽  
Vol 30 (6) ◽  
pp. 935-944
Author(s):  
L. L. Maksimova
2013 ◽  
Vol 19 ◽  
pp. 39-50
Author(s):  
А.С. Карпенко

In this paper von Wright’s truth-logic T__ is considered. It seems that it is a De Morgan four-valued logic DM4 (or Belnap’s four-valued logic) with endomorphism e2. In connection with this many other issues are discussed: twin truth operators, a truth-logic with endomorphism g (or logic Tr), the lattice of extensions of DM4, modal logic V2, Craig interpolation property, von Wright–Segerberg’s tense logic W, and so on.


2003 ◽  
Vol 68 (2) ◽  
pp. 463-480 ◽  
Author(s):  
Patrick Blackburn ◽  
Maarten Marx

AbstractCraig's interpolation lemma (if φ → ψ is valid, then φ → θ and θ → ψ are valid, for θ a formula constructed using only primitive symbols which occur both in φ and ψ) fails for many propositional and first order modal logics. The interpolation property is often regarded as a sign of well-matched syntax and semantics. Hybrid logicians claim that modal logic is missing important syntactic machinery, namely tools for referring to worlds, and that adding such machinery solves many technical problems. The paper presents strong evidence for this claim by defining interpolation algorithms for both propositional and first order hybrid logic. These algorithms produce interpolants for the hybrid logic of every elementary class of frames satisfying the property that a frame is in the class if and only if all its point-generated subframes are in the class. In addition, on the class of all frames, the basic algorithm is conservative: on purely modal input it computes interpolants in which the hybrid syntactic machinery does not occur.


2019 ◽  
Vol 29 (5) ◽  
pp. 785-802
Author(s):  
Giovanna D’Agostino

Abstract In this paper we consider modal team logic, a generalization of classical modal logic in which it is possible to describe dependence phenomena between data. We prove that most known fragments of full modal team logic allow the elimination of the so called ‘existential bisimulation quantifiers’, where the existence of a certain set is required only modulo bisimulation (i.e. not in the model itself but possibly in a bisimilar model). As a consequence, we prove that these fragments enjoy the uniform interpolation property.


2020 ◽  
pp. 1-29
Author(s):  
YURY SAVATEEV ◽  
DANIYAR SHAMKANOV

Abstract We present a sequent calculus for the Grzegorczyk modal logic $\mathsf {Grz}$ allowing cyclic and other non-well-founded proofs and obtain the cut-elimination theorem for it by constructing a continuous cut-elimination mapping acting on these proofs. As an application, we establish the Lyndon interpolation property for the logic $\mathsf {Grz}$ proof-theoretically.


2019 ◽  
Vol 48 (1) ◽  
Author(s):  
Mitio Takano

The modal logic S4.2 is S4 with the additional axiom ◊□A ⊃ □◊A. In this article, the sequent calculus GS4.2 for this logic is presented, and by imposing an appropriate restriction on the application of the cut-rule, it is shown that, every GS4.2-provable sequent S has a GS4.2-proof such that every formula occurring in it is either a subformula of some formula in S, or the formula □¬□B or ¬□B, where □B occurs in the scope of some occurrence of □ in some formula of S. These are just the K5-subformulas of some formula in S which were introduced by us to show the modied subformula property for the modal logics K5 and K5D (Bull Sect Logic 30(2): 115–122, 2001). Some corollaries including the interpolation property for S4.2 follow from this. By slightly modifying the proof, the finite model property also follows.


Author(s):  
Richard Patterson
Keyword(s):  

Author(s):  
Brian F. Chellas
Keyword(s):  

2019 ◽  
Vol 28 (1) ◽  
pp. 19-27
Author(s):  
Ja. O. Petik

The connection of the modern psychology and formal systems remains an important direction of research. This paper is centered on philosophical problems surrounding relations between mental and logic. Main attention is given to philosophy of logic but certain ideas are introduced that can be incorporated into the practical philosophical logic. The definition and properties of basic modal logic and descending ones which are used in study of mental activity are in view. The defining role of philosophical interpretation of modality for the particular formal system used for research in the field of psychological states of agents is postulated. Different semantics of modal logic are studied. The hypothesis about the connection of research in cognitive psychology (semantics of brain activity) and formal systems connected to research of psychological states is stated.


2007 ◽  
Vol 50 (1) ◽  
pp. 7-27
Author(s):  
Gerald Massey

Contending that the quest for a logic of scientific discovery was prematurely abandoned, the author lays down eight phenomena that such a logic or theory must explain: the banality of scientific discovery; the trainability of scientists; the high incidence of simultaneous discoveries; the ubiquity of relative novices; the fact of scientific genius; the barrenness of isolated workers; the incommensurability of concepts of successive theories; and the quasi-incorporation of old concepts, objects, and methods in successor theories, The author then presents a new theory or logic of discovery according to which discoveries are the termini of "tweak paths" generated when scientists "tinker" with the laws, concepts, methods, and instruments of a given theory. Tinkering and tweaking are illustrated by examples from many-valued and modal logic and from Darwinian biology. Through the history of planetary discovery, the accidental role played by luck or good fortune in some discoveries is explored, but the author emphasizes that in a deep sense serendipity is an in eliminable feature of all scientific discovery because scientists never know m advance whether their tweaks will lead to dead ends or to positive developments. The author's new theory of scientific discovery is shown to account for all eight explananda, ft also reveals science to be a more egalitarian enterprise than the traditional view of scientific discovery as ultimately inexplicable depicts it.


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