Littérature scientifique sur le sujet « Modal logic, modal operator, intuitionistic logic, tense logic »

In this article we shall construct intuitionistic analogues to the main systems of classical tense logic. Since each classical modal logic can be gotten from some tense logic by one of the definitions(i) □ p ≡ p ∧ Gp ∧ Hp, ◇p ≡ p ∨ Fp ∨ Pp; or,(ii) □ p ≡ p ∧ Gp, ◇p = p ∨ Fp(see [5]), we shall find that our intuitionistic tense logics give us analogues to the classical modal logics as well.We shall not here discuss the philosophical issues raised by our logics. Readers interested in the intuitionistic view of time and modality should see [2] for a detailed discussion.In §2 we define the Kripke models for IKt, the intuitionistic analogue to Lemmon's system Kt. We then prove the completeness and decidability of this system (§§3–5). Finally, we extend our results to other sorts of tense logic and to modal logic.In the language of IKt, we have: sentence-letters p, q, r, etc.; the (intuitionistic) connectives ∧, ∨, →, ¬; and unary operators P (“it was the case”), F (it will be the case”), H (“it has always been the case”) and G (“it will always be the case”). Formulas are defined inductively: all sentence-letters are formulas; if X is a formula, so are ¬X, PX, FX, HX, and GX; if X and Y are formulas, so are X ∧ Y, X ∨ Y, and X → Y. We shall see that, in contrast to classical tense logic, F and P cannot be defined in terms of G and H.

AbstractThe Kripke semantics of classical propositional normal modal logic is made algebraic via an embedding of Kripke structures into the larger class of pointed stably supported quantales. This algebraic semantics subsumes the traditional algebraic semantics based on lattices with unary operators, and it suggests natural interpretations of modal logic, of possible interest in the applications, in structures that arise in geometry and analysis, such as foliated manifolds and operator algebras, via topological groupoids and inverse semigroups. We study completeness properties of the quantale based semantics for the systems K, T, K4, S4 and S5, in particular obtaining an axiomatization for S5 which does not use negation or the modal necessity operator. As additional examples we describe intuitionistic propositional modal logic, the logic of programs PDL and the ramified temporal logic CTL.

We present three examples of topological semantics for intuitionistic modal logic with one modal operator □. We show that it is possible to treat neighborhood models, introduced earlier, as topological or multi-topological. From the neighborhood point of view, our method is based on differences between properties of minimal and maximal neighborhoods. Also we propose transformation of multitopological spaces into the neighborhood structures.

Myhill [12] extended the ideas of Shapiro [15], and proposed a system of epistemic set theory IST (based on modal S4 logic) in which the meaning of the necessity operator is taken to be the intuitive provability, as formalized in the system itself. In this setting one works in classical logic, and yet it is possible to make distinctions usually associated with intuitionism, e.g. a constructive existential quantifier can be expressed as (∃x) □ …. This was first confirmed when Goodman [7] proved that Shapiro's epistemic first order arithmetic is conservative over intuitionistic first order arithmetic via an extension of Gödel's modal interpretation [6] of intuitionistic logic.Myhill showed that whenever a sentence □A ∨ □B is provable in IST, then A is provable in IST or B is provable in IST (the disjunction property), and that whenever a sentence ∃x.□A(x) is provable in IST, then so is A(t) for some closed term t (the existence property). He adapted the Friedman slash [4] to epistemic systems.Goodman [8] used Epistemic Replacement to formulate a ZF-like strengthening of IST, and proved that it was a conservative extension of ZF and that it had the disjunction and existence properties. It was then shown in [13] that a slight extension of Goodman's system with the Epistemic Foundation (ZFER, cf. §1) suffices to interpret intuitionistic ZF set theory with Replacement (ZFIR, [10]). This is obtained by extending Gödel's modal interpretation [6] of intuitionistic logic. ZFER still had the properties of Goodman's system mentioned above.

In [6] Gödel observed that intuitionistic propositional logic can be interpreted in Lewis's modal logic (S4). The idea behind this interpretation is to regard the modal operator □ as expressing the epistemic notion of “informal provability”. With the work of Shapiro [12], Myhill [10], Goodman [7], [8], and Ščedrov [11] this simple idea has developed into a successful program of integrating classical and intuitionistic mathematics.There is one question quite central to the above program that has remained open. Namely:Does Ščedrov's extension of the Gödel translation to set theory provide a faithful interpretation of intuitionistic set theory into epistemic set theory?In the present paper we give an affirmative answer to this question.The main ingredient in our proof is the construction of an interpretation of epistemic set theory into intuitionistic set theory which is inverse to the Gödel translation. This is accomplished in two steps. First we observe that Funayama's theorem is constructively provable and apply it to the power set of 1. This provides an embedding of the set of propositions into a complete topological Boolean algebra . Second, in a fashion completely analogous to the construction of Boolean-valued models of classical set theory, we define the -valued universe V(). V() gives a model of epistemic set theory and, since we use a constructive metatheory, this provides an interpretation of epistemic set theory into intuitionistic set theory.

The purpose of this paper is to study logical implications which are much weaker than the implication of intuitionistic logic.In §1 we define the system SI (system of Simple Implication) which is obtained from intuitionistic logic by restricting the inference rules of intuitionistic implication. The implication of the system SI is called the “simple implication” and denoted by ⊃, where the simple implication ⊃ has the following properties:(1) The simple implication ⊃ is much weaker than the usual intuitionistic implication.(2) The simple implication ⊃ can be interpreted by the notion of provability, i.e., we have a very simple semantics for SI so that a sentence A ⊃ B is interpreted as “there exists a proof of B from A”.(3) The full-strength intuitionistic implication ⇒ is definable in a weak second order extension of SI; in other words, it is definable by help of a variant of the weak comprehension schema and the simple implication ⊃. Therefore, though SI is much weaker than the intuitionistic logic, the second order extension of SI is equivalent to the second order extension of the intuitionistic logic.(4) The simple implication is definable in a weak modal logic MI by the use of the modal operator and the intuitionistic implication ⇒ with full strength. More precisely, A ⊃ B is defined as the strict implication of the form ◽(A ⇒ B).In §1, we show (3) and (4). (2) is shown in §2 in a more general setting.Semantics by introduction rules of logical connectives has been studied from various points of view by many authors (e.g. Gentzen [4], Lorentzen [5], Dummett [1], [2], Prawitz [8]. Martin-Löf [7], Maehara [6]). Among them Gentzen (in §§10 and 11 of [4]) introduced such a semantics in order to justify logical inferences and the mathematical induction rule. He observed that all of the inference rules of intuitionistic arithmetic, except for those on implication and negation, are justified by means of his semantics, but justification of the inference rules on implication and negation contains a circular argument for the interpretation by introduction rules, where the natural interpretation of A ⊃ B by ⊃-introduction rule is “there exists a proof of B from A ” (cf. §11 of Gentzen [4]).

In [1] J.-Y. Bèziau formulated a logic called Z. Bèziau’s idea was generalized independently in [6] and [7]. A family of logics to which Z belongs is denoted in [7] by K. In particular; it has been shown in [6] and [7] that there is a correspondence between normal modal logics and logics from the class K. Similar; but only partial results has been obtained also for regular logics (see [8] and [9]). In (Došen; [2]) a logic N has been investigated in the language with negation; implication; conjunction and disjunction by axioms of positive intuitionistic logic; the right-to-left part of the second de Morgan law; and the rules of modus ponens and contraposition. From the semantical point of view the negation used by Došen is the modal operator of impossibility. It is known this operator is a characteristic of the modal interpretation of intuitionistic negation (see [3; p. 300]). In the present paper we consider an extension of N denoted by N+. We will prove that every extension of N+ that is closed under the same rules as N+; corresponds to a regular logic being an extension of the regular deontic logic D21 (see [4] and [13]). The proved correspondence allows to obtain from soundnesscompleteness result for any given regular logic containing D2, similar adequacy theorem for the respective extension of the logic N+.

AbstractBounded commutative residuated lattice ordered monoids (Rℓ-monoids) are a common generalization of, e.g., Heyting algebras and BL-algebras, i.e., algebras of intuitionistic logic and basic fuzzy logic, respectively. Modal operators (special cases of closure operators) on Heyting algebras were studied in [MacNAB, D. S.: Modal operators on Heyting algebras, Algebra Universalis 12 (1981), 5–29] and on MV-algebras in [HARLENDEROVÁ,M.—RACHŮNEK, J.: Modal operators on MV-algebras, Math. Bohem. 131 (2006), 39–48]. In the paper we generalize the notion of a modal operator for general bounded commutative Rℓ-monoids and investigate their properties also for certain derived algebras.

In this paper we analyse logic of false belief in the intuitionistic setting. This logic, studied in its classical version by Steinsvold, Fan, Gilbert and Venturi, describes the following situation: a formula $\varphi$ is not satisfied in a given world, but we still believe in it (or we think that it should be accepted). Another interpretations are also possible: e.g. that we do not accept $\varphi$ but it is imposed on us by a kind of council or advisory board. From the mathematical point of view, the idea is expressed by an adequate form of modal operator $\mathsf{W}$ which is interpreted in relational frames with neighborhoods. We discuss monotonicity of forcing, soundness, completeness and several other issues. We present also some simple systems in which confirmation of previously accepted formula is modelled.

This paper investigates (modal) extensions of Heyting–Brouwer logic, i.e., the logic which results when the dual of implication (alias coimplication) is added to the language of intuitionistic logic. We first develop matrix as well as Kripke style semantics for those logics. Then, by extending the Gödel-embedding of intuitionistic logic into S4 , it is shown that all (modal) extensions of Heyting–Brouwer logic can be embedded into tense logics (with additional modal operators). An extension of the Blok–Esakia-Theorem is proved for this embedding.

Implicative semi-lattices (also known as Brouwerian semi-lattices) are a generalization of Heyting algebras, and have been already well studied both from a logical and an algebraic perspective. In this paper, we consider the variety ISt of the expansions of implicative semi-lattices with tense modal operators, which are algebraic models of the disjunction-free fragment of intuitionistic tense logic. Using methods from algebraic proof theory, we show that the logic of tense implicative semi-lattices has the finite model property. Combining with the finite axiomatizability of the logic, it follows that the logic is decidable.