An extended joint consistency theorem for a family of free modal logics with equality

1984 ◽  
Vol 49 (1) ◽  
pp. 174-183 ◽  
Author(s):  
Raymond D. Gumb
Keyword(s):  
Infinite Family ◽  
Modal Logics ◽  
Relational Semantics ◽  
Accessibility Relation ◽  
Barcan Formula ◽  
Distinct Individual ◽  
Individual Variables

In this paper, we establish an extended joint consistency theorem for an infinite family of free modal logics with equality. The extended joint consistency theorem incorporates the Craig and Lyndon interpolation lemmas and the Robinson joint consistency theorem. In part, the theorem states that two theories which are jointly unsatisfiable are separated by a sentence in the vocabulary common to both theories.Our family of free modal logics includes the free versions of I, M, and S4 studied by Leblanc [5, Chapters 8 and 9], supplemented with equality as in [3]. In the relational semantics for these logics, there is no restriction on the accessibility relation in I, while in M(S4) the restriction is reflexivity (refiexivity and transitivity). We say that a restriction on the accessibility relation countenances backward-looping if it implies a sentence of the form ∀x1 …xn(x1Rx2 &…&xn ⊃ xkRxj) (1 ≤ j < k ≤ n ≥ 2), where the xi (1 ≤ i ≤ n) are distinct individual variables. Just as reflexivity and transitivity do not countenance backward-looping, neither do any of the restrictions in our family of free modal logics. (The above terminology is derived from the effect of such restrictions on Kripke tableaux constructions.) The Barcan formula, its converse, the Fitch formula, and the formula T ≠ T′ ⊃ □T ≠ T′ do not hold in our logics.

scholarly journals A Kotas-Style Characterisation of Minimal Discussive Logic

Axioms ◽  
2019 ◽  
Vol 8 (4) ◽  
pp. 108 ◽  
Author(s):  
Krystyna Mruczek-Nasieniewska ◽  
Marek Nasieniewski
Keyword(s):  
Deductive System ◽  
Modal Logics ◽  
Original Model ◽  
Accessibility Relation ◽  
Minimal Logic ◽  
One To One

In this paper, we discuss a version of discussive logic determined by a certain variant of Jaśkowski’s original model of discussion. The obtained system can be treated as the minimal discussive logic. It is determined by frames with serial accessibility relation. As the smallest one, this logic can be treated as a basis which could be extended to richer discussive logics that are obtained by varying accessibility relation and resulting in a lattice of discussive logics. One has to remember that while formulating discussive logics there is no one-to-one determination of discussive logics by modal logics. For example, it is proved that Jaśkowski’s logic D 2 can be expressed by other than S 5 modal logics. In this paper we consider a deductive system for the sketchily described minimal logic. While formulating the deductive system, we apply a method of Kotas that was used to axiomatize D 2 . The obtained system determines a logic D 0 as a set of theses that is contained in D 2 . Moreover, any discussive logic that would be expressed by means of the provided model of discussion would contain D 0 , so it is the smallest discussive logic.

Banishing the rule of substitution for functional variables

1953 ◽  
Vol 18 (3) ◽  
pp. 201-208 ◽  
Author(s):  
Leon Henkin
Keyword(s):  
Functional Calculus ◽  
Standard Interpretation ◽  
Substitution Instance ◽  
Deductive Systems ◽  
Functional Variable ◽  
Distinct Individual ◽  
Individual Variables ◽  
Set Up

Let and be (well-formed) formulas of the functional calculus, let c be an n-adic functional variable, and let a1, …, an be distinct individual variables. Church has defined the metalogical notation to indicate the formula resulting from when each part of of the form c{1, …, n) (such that the occurrence of c is free in ) is replaced by the formula which arises from by replacing every free occurrence of ai by i, i, = 1, …, n. (Here 1, …, n may be any individual variables or constants, not necessarily all distinct.) The notation is not defined for all , , c, a1, …, an, however, but only for those cases (specified in detail by Church) when the resulting formula constitutes a valid substitution instance of the formula according to the standard interpretation of the functional calculus.The full and correct syntactical statement of the conditions (under which this type of substitution is permissible) has proved so arduous, that it seems to have been rendered in error more often than not. Unfortunately the functional calculi are often set up as deductive systems in which this type of substitution occurs as one of the primitive rules of inference, or in one of the axiom schemata. Thus the beginning student who is introduced to the calculi through such a formulation is forced to cope from the outset with details which have proved treacherous even to the initiate. For this reason it is desirable to seek alternative formulations of the functional calculi in which this type of substitution is not mentioned.

A unified completeness theorem for quantified modal logics

2002 ◽  
Vol 67 (4) ◽  
pp. 1483-1510 ◽  
Author(s):  
Giovanna Corsi
Keyword(s):  
Modal Logic ◽  
Completeness Theorem ◽  
Modal Logics ◽  
General Strategy ◽  
Quantified Modal Logic ◽  
Barcan Formula ◽  
Completeness Theorems

AbstractA general strategy for proving completeness theorems for quantified modal logics is provided. Starting from free quantified modal logic K. with or without identity, extensions obtained either by adding the principle of universal instantiation or the converse of the Barcan formula or the Barcan formula are considered and proved complete in a uniform way. Completeness theorems are also shown for systems with the extended Barcan rule as well as for some quantified extensions of the modal logic B. The incompleteness of Q°.B + BF is also proved.

Algorithmic properties of first-order modal logics of finite Kripke frames in restricted languages

2020 ◽  
Vol 30 (7) ◽  
pp. 1305-1329 ◽  
Author(s):  
Mikhail Rybakov ◽  
Dmitry Shkatov
Keyword(s):  
Possible Worlds ◽  
Modal Logics ◽  
Finite Frames ◽  
First Order ◽  
Kripke Frames ◽  
Finite Frame ◽  
Recursively Enumerable ◽  
Finite Set ◽  
Individual Variables ◽  
Natural Classes

Abstract We study the effect of restricting the number of individual variables, as well as the number and arity of predicate letters, in languages of first-order predicate modal logics of finite Kripke frames on the logics’ algorithmic properties. A finite frame is a frame with a finite set of possible worlds. The languages we consider have no constants, function symbols or the equality symbol. We show that most predicate modal logics of natural classes of finite Kripke frames are not recursively enumerable—more precisely, $\varPi ^0_1$-hard—in languages with three individual variables and a single monadic predicate letter. This applies to the logics of finite frames of the predicate extensions of the sublogics of propositional modal logics $\textbf{GL}$, $\textbf{Grz}$ and $\textbf{KTB}$—among them, $\textbf{K}$, $\textbf{T}$, $\textbf{D}$, $\textbf{KB}$, $\textbf{K4}$ and $\textbf{S4}$.

Many-Valued Modal Logics

1991 ◽  
Vol 15 (3-4) ◽  
pp. 235-254
Author(s):  
Melvin C. Fitting
Keyword(s):  
Modal Logics ◽  
Kripke Models ◽  
Sequent Calculi ◽  
Possible World ◽  
Accessibility Relation ◽  
Soundness And Completeness

Two families of many-valued modal logics are investigated. Semantically, one family is characterized using Kripke models that allow formulas to take values in a finite many-valued logic, at each possible world. The second family generalizes this to allow the accessibility relation between worlds also to be many-valued. Gentzen sequent calculi are given for both versions, and soundness and completeness are established.

scholarly journals Pooling Modalities and Pointwise Intersection: Semantics, Expressivity, and Dynamics

2022 ◽  
Author(s):  
Frederik Van De Putte ◽  
Dominik Klein
Keyword(s):  
Expressive Power ◽  
Modal Logics ◽  
Relational Semantics ◽  
Modal Operators

AbstractWe study classical modal logics with pooling modalities, i.e. unary modal operators that allow one to express properties of sets obtained by the pointwise intersection of neighbourhoods. We discuss salient properties of these modalities, situate the logics in the broader area of modal logics (with a particular focus on relational semantics), establish key properties concerning their expressive power, discuss dynamic extensions of these logics and provide reduction axioms for the latter.

RELEVANCE LOGICS AND RELATION ALGEBRAS

2009 ◽  
Vol 2 (1) ◽  
pp. 102-131 ◽  
Author(s):  
KATALIN BIMBÓ ◽  
J. MICHAEL DUNN ◽  
ROGER D. MADDUX
Keyword(s):  
Dynamic Semantics ◽  
Relational Semantics ◽  
Relation Algebras ◽  
Sequent Calculi ◽  
Accessibility Relation ◽  
Cut Rule ◽  
Relevance Logics

Relevance logics are known to be sound and complete for relational semantics with a ternary accessibility relation. This paper investigates the problem of adequacy with respect to special kinds of dynamic semantics (i.e., proper relation algebras and relevant families of relations). We prove several soundness results here. We also prove the completeness of a certain positive fragment of R as well as of the first-degree fragment of relevance logics. These results show that some core ideas are shared between relevance logics and relation algebras. Some details of certain incompleteness results, however, pinpoint where relevance logics and relation algebras diverge. To carry out these semantic investigations, we define a new tableaux formalization and new sequent calculi (with the single cut rule admissible) for various relevance logics.

On sentences which are true of direct unions of algebras

1951 ◽  
Vol 16 (1) ◽  
pp. 14-21 ◽  
Author(s):  
Alfred Horn
Keyword(s):  
Normal Form ◽  
Wide Class ◽  
Conjunctive Normal Form ◽  
Disjunctive Normal Form ◽  
Factor Algebra ◽  
The Matrix ◽  
Distinct Individual ◽  
Direct Union ◽  
Individual Variables ◽  
Image Position

It is well known that certain sentences corresponding to similar algebras are invariant under direct union; that is, are true of the direct union when true of each factor algebra. An axiomatizable class of similar algebras, such as the class of groups, is closed under direct union when each of its axioms is invariant. In this paper we shall determine a wide class of invariant sentences. We shall also be concerned with determining sentences which are true of a direct union provided they are true of some factor algebra. In the case where all the factor algebras are the same, a further result is obtained. In §2 it will be shown that these criteria are the only ones of their kind. Lemma 7 below may be of some independent interest.We adopt the terminology and notation of McKinsey with the exception that the sign · will be used for conjunction. Expressions of the form ∼∊, where ∊ is an equation, will be called inequalities. In accordance with the analogy between conjunction and disjunction with product and sum respectively, we shall call α1, …, αn the terms of the disjunctionand the factors of the conjunctionEvery closed sentence is equivalent to a sentence in prenez normal form,where x1, …, xm distinct individual variables, Q1, …, Qm are quantifiers, and the matrix S is an open sentence in which each of the variables x1, …, xm actually occurs. The sentence S may be written in either disjunctive normal form:where αi,j is either an equation or an inequality, or in conjunctive normal form:.

scholarly journals From 2-Sequents and Linear Nested Sequents to Natural Deduction for Normal Modal Logics

2021 ◽  
Vol 22 (3) ◽  
pp. 1-29
Author(s):  
Simone Martini ◽  
Andrea Masini ◽  
Margherita Zorzi
Keyword(s):  
Natural Deduction ◽  
Modal Logics ◽  
Structural Rule ◽  
Elimination Rule ◽  
Accessibility Relation ◽  
Formal Systems ◽  
Explicit Reference ◽  
Deduction Rules ◽  
Nested Sequents ◽  
Normal Modal Logics

We extend to natural deduction the approach of Linear Nested Sequents and of 2-Sequents. Formulas are decorated with a spatial coordinate, which allows a formulation of formal systems in the original spirit of natural deduction: only one introduction and one elimination rule per connective, no additional (structural) rule, no explicit reference to the accessibility relation of the intended Kripke models. We give systems for the normal modal logics from K to S4. For the intuitionistic versions of the systems, we define proof reduction, and prove proof normalization, thus obtaining a syntactical proof of consistency. For logics K and K4 we use existence predicates (à la Scott) for formulating sound deduction rules.

An incomplete nonnormal extension of S3

1978 ◽  
Vol 43 (2) ◽  
pp. 211-212
Author(s):  
George F. Schumm
Keyword(s):  
Modus Ponens ◽  
Modal Logics ◽  
Relational Semantics ◽  
General Semantics ◽  
Normal Logic ◽  
Enormous Number

Fine [1] and Thomason [4] have recently shown that the familiar relational semantics of Kripke [2] is inadequate for certain normal extensions of T and S4. It is here shown that the more general semantics developed by Kripke in [3] to handle nonnormal modal logics is likewise inadequate for certain of those logics.The interest of incompleteness results, such as those of Fine and Thomason, is of course a function of one's expectations. Define a “normal” logic too broadly and it is not surprising that a given semantics is not adequate for all normal logics. In the case of relational semantics, for example, one would want to require at least that a normal logic contain T, the logic determined by the class of all normal frames, and that it be closed under certain (though perhaps not all) rules of inference which are validity preserving in such frames. The adequacy of that semantics will otherwise be ruled out at the outset.For Kripke a logic is normal if it contains all tautologies, □p→p and □ (p → q)→(□p → □q), and is closed under the rules of substitution, modus ponens and necessitation (from A infer □A). T is the smallest normal logic, and this fact, together with the “naturalness” of the definition and the enormous number of normal logics which have been shown to be complete, made it plausible to suppose that Kripke's original semantics was adequate for all normal logics. That it is not is indeed surprising and would seem to reveal a genuine shortcoming.

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