Akira Nakamura. A remark on the truth-value stipulation for the modal system M′. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 13 (1967), pp. 11–14. - Akira Nakamura. On an axiomatic system of modal logic. Zeitschrift für mathematische Logik und Grundlagen der Mathematik vol. 14 (1968), pp. 61–66. - R. A. Bull. On a paper of Akira Nakamura. Zeitschrift für mathematische Logik und Grundlagen der Mathematik vol. 15 (1969), pp. 155–156.

1974 ◽  
Vol 39 (2) ◽  
pp. 351-351
Author(s):  
M. K. Rennie
Keyword(s):  
Modal Logic ◽  
Axiomatic System ◽  
Modal System ◽  
Truth Value

New foundations for Lewis modal systems

1957 ◽  
Vol 22 (2) ◽  
pp. 176-186 ◽  
Author(s):  
E. J. Lemmon
Keyword(s):  
Modal Logic ◽  
Propositional Calculus ◽  
Modal System ◽  
New Foundations ◽  
Image Position ◽  
Classical Propositional Calculus ◽  
Modal Systems

The main aims of this paper are firstly to present new and simpler postulate sets for certain well-known systems of modal logic, and secondly, in the light of these results, to suggest some new or newly formulated calculi, capable of interpretation as systems of epistemic or deontic modalities. The symbolism throughout is that of [9] (see especially Part III, Chapter I). In what follows, by a Lewis modal system is meant a system which (i) contains the full classical propositional calculus, (ii) is contained in the Lewis system S5, (iii) admits of the substitutability of tautologous equivalents, (iv) possesses as theses the four formulae:We shall also say that a system Σ1 is stricter than a system Σ2, if both are Lewis modal systems and Σ1 is contained in Σ2 but Σ2 is not contained in Σ1; and we shall call Σ1absolutely strict, if it possesses an infinity of irreducible modalities. Thus, the five systems of Lewis in [5], S1, S2, S3, S4, and S5, are all Lewis modal systems by this definition; they are in an order of decreasing strictness from S1 to S5; and S1 and S2 alone are absolutely strict.

On an Axiomatic System of Modal Logic

1968 ◽  
Vol 14 (1-5) ◽  
pp. 61-66
Author(s):  
Akira Nakamura
Keyword(s):  
Modal Logic ◽  
Axiomatic System

Provability logic

2018 ◽  
Author(s):  
Albert Visser
Keyword(s):  
Modal Logic ◽  
Formal System ◽  
Formal Theory ◽  
Incompleteness Theorem ◽  
Provability Logic ◽  
Modal System ◽  
Modal Language ◽  
Modal Operators ◽  
Syntactic Approach

Central to Gödel’s second incompleteness theorem is his discovery that, in a sense, a formal system can talk about itself. Provability logic is a branch of modal logic specifically directed at exploring this phenomenon. Consider a sufficiently rich formal theory T. By Gödel’s methods we can construct a predicate in the language of T representing the predicate ‘is formally provable in T’. It turns out that T is able to prove statements of the form - (1) If A is provable in T, then it is provable in T that A is provable in T. In modal logic, predicates such as ‘it is unavoidable that’ or ‘I know that’ are considered as modal operators, that is, as non-truth-functional propositional connectives. In provability logic, ‘is provable in T’ is similarly treated. We write □A for ‘A is provable in T’. This enables us to rephrase (1) as follows: - (1′) □A →□□A. This is a well-known modal principle amenable to study by the methods of modal logic. Provability logic produces manageable systems of modal logic precisely describing all modal principles for □A that T itself can prove. The language of the modal system will be different from the language of the system T under study. Thus the provability logic of T (that is, the insights T has about its own provability predicate as far as visible in the modal language) is decidable and can be studied by finitistic methods. T, in contrast, is highly undecidable. The advantages of provability logic are: (1) it yields a very perspicuous representation of certain arguments in a formal theory T about provability in T; (2) it gives us a great deal of control of the principles for provability in so far as these can be formulated in the modal language at all; (3) it gives us a direct way to compare notions such as knowledge with the notion of formal provability; and (4) it is a fully worked-out syntactic approach to necessity in the sense of Quine.

Grafted frames and S1 -completeness

1999 ◽  
Vol 64 (3) ◽  
pp. 1324-1338 ◽  
Author(s):  
Beihai Zhou
Keyword(s):  
Modal Logic ◽  
Relevance Logic ◽  
Value Assignment ◽  
Combined Logics ◽  
Modal Model ◽  
Truth Value ◽  
Modal Frame ◽  
Relevance Models ◽  
New System ◽  
Relevance Model

AbstractA grafted frame is a new kind of frame which combines a modal frame and some relevance frames. A grafted model consists of a grafted frame and a truth-value assignment. In this paper, the grafted frame and the grafted model are constructed and used to show the completeness of S1. The implications of S1-completeness are discussed.A grafted frame does not combine two kinds of frames simply by putting relations defined in the components together. That is, the resulting grafted frame is not in the form of <W,R,R′>, or more generally, in the form of <W, R, R′,R″>,…>, which consists of a non-empty set with several relations defined on it.1 Rather, it resembles the construction of fibering proposed by D. M. Gabbay and M. Finger (see [4] and [3]). On a grafted frame, some modal worlds, which belong to the initial modal frame, are attached by some relevance frames.However, these two semantics have important differences. Consider the combined semantics involving semantics of relevance logic and modal logic. A fibred model and a grafted model proposed in this paper differ in the following respects. First, a fibred model is constructed from a class of modal models and a class of relevance models. A grafted model consists of a grafted frame and a truth-value assignment, where the grafted frame is constructed from a modal frame and some relevance frames, and the assignment is a union of a modal truth-value assignment VM and some relevance truth-value assignments VR. VM (VR) defined in this paper is not the same as the assignment contained in a modal (relevance) model. Second, in a fibred model each relevance world is associated (or fibred) with a modal model and each modal world with a relevance model.2 To be the grafted frame on which a grafted model is based, it is enough to have some modal worlds attached by some relevance frames. Moreover, no relevance world is associated with a modal frame in the grafted frame. Third, fibred models are intended to provide an appropriate semantics to combined logics. Grafted frames and grafted models are inspired to characterize S1, which, containing only one modality □, is not a combined logic. It is shown in this paper that S1 is sort of a meta-logic of the intersection of S0.4 and F, where S0.4, a new system proposed in this paper, is in turn a meta-logic of the relevance logic.

scholarly journals Meta-inferences and Supervaluationism

2021 ◽  
Author(s):  
Luca Incurvati ◽  
Julian J. Schlöder
Keyword(s):  
Modal Logic ◽  
Deductive Reasoning ◽  
Higher Order ◽  
Proof System ◽  
Inference Rules ◽  
Consequence Relation ◽  
Normal Modal Logic ◽  
First Order ◽  
Truth Value ◽  
Crucial Ingredient

AbstractMany classically valid meta-inferences fail in a standard supervaluationist framework. This allegedly prevents supervaluationism from offering an account of good deductive reasoning. We provide a proof system for supervaluationist logic which includes supervaluationistically acceptable versions of the classical meta-inferences. The proof system emerges naturally by thinking of truth as licensing assertion, falsity as licensing negative assertion and lack of truth-value as licensing rejection and weak assertion. Moreover, the proof system respects well-known criteria for the admissibility of inference rules. Thus, supervaluationists can provide an account of good deductive reasoning. Our proof system moreover brings to light how one can revise the standard supervaluationist framework to make room for higher-order vagueness. We prove that the resulting logic is sound and complete with respect to the consequence relation that preserves truth in a model of the non-normal modal logic NT. Finally, we extend our approach to a first-order setting and show that supervaluationism can treat vagueness in the same way at every order. The failure of conditional proof and other meta-inferences is a crucial ingredient in this treatment and hence should be embraced, not lamented.

An infinitary axiomatization of dynamic topological logic

2020 ◽  
Author(s):  
Somayeh Chopoghloo ◽  
Morteza Moniri
Keyword(s):  
Continuous Function ◽  
Modal Logic ◽  
Topological Space ◽  
Axiomatic System ◽  
Proof System ◽  
Strong Completeness ◽  
Alexandrov Spaces ◽  
Modal Language ◽  
Derivation Rule ◽  
Dynamic Topological Logic

Abstract Dynamic topological logic ($\textsf{DTL}$) is a multi-modal logic that was introduced for reasoning about dynamic topological systems, i.e. structures of the form $\langle{\mathfrak{X}, f}\rangle $, where $\mathfrak{X}$ is a topological space and $f$ is a continuous function on it. The problem of finding a complete and natural axiomatization for this logic in the original tri-modal language has been open for more than one decade. In this paper, we give a natural axiomatization of $\textsf{DTL}$ and prove its strong completeness with respect to the class of all dynamic topological systems. Our proof system is infinitary in the sense that it contains an infinitary derivation rule with countably many premises and one conclusion. It should be remarked that $\textsf{DTL}$ is semantically non-compact, so no finitary proof system for this logic could be strongly complete. Moreover, we provide an infinitary axiomatic system for the logic ${\textsf{DTL}}_{\mathcal{A}}$, i.e. the $\textsf{DTL}$ of Alexandrov spaces, and show that it is strongly complete with respect to the class of all dynamical systems based on Alexandrov spaces.

Filtering unification and most general unifiers in modal logic

2004 ◽  
Vol 69 (3) ◽  
pp. 879-906 ◽  
Author(s):  
Silvio Ghilardi ◽  
Lorenzo Sacchetti
Keyword(s):  
Modal Logic ◽  
Point Of View ◽  
Modal System ◽  
Algebraic Point ◽  
Natural Extensions ◽  
Excluded Middle

Abstract.We characterize (both from a syntactic and an algebraic point of view) the normal K4-logics for which unification is filtering. We also give a sufficient semantic criterion for existence of most general unifiers, covering natural extensions of K4.2+ (i.e., of the modal system obtained from K4 by adding to it, as a further axiom schemata, the modal translation of the weak excluded middle principle).

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