Some remarks on (weakly) weak modal logics

Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1980

MISCELLANEA LOGICA

Logic and Logical Philosophy, 2019

The present paper studies two approaches to the expressiveness of propositional modal logics based on first-degree entailment logic, FDE. We first consider the basic FDE-based modal logic BK and certain systems in its vicinity, and then turn to some FDE-based modal logics in a richer vocabulary, including modal bilattice logic, MBL. On the one hand, modeltheoretic proofs of the definability of connectives along the lines of [7] and [17] are given for various FDE-based modal logics. On the other hand, building on [10], expressibility is considered in terms of mutual faithful embeddability of one logic into another logic. A distinction is drawn between definitional equivalence, which is defined with respect to a pair of structural translations between two languages, and weak definitional equivalence, which is defined with respect to a weaker notion of translations. Moreover, the definitional equivalence of some FDE-based modal logics is proven, especially the definitional equivalence of MBL and a conservative extension of the logic BK 2 × BK 2 , which underlines the central role played by BK among FDE-based modal logics.

Mathematical Logic, 1990

Logic and Logical Philosophy, 2010

In the present paper 1 we examine very weak modal logics ½, ½, ½, ˼. • , ˼. • +(D), ˼. and some of their versions which are closed under replacement of tautological equivalents (rte-versions). We give semantics for these logics, formulated by means of Kripke style models of the form w, A, V , where w is a «distinguished» world, A is a set of worlds which are alternatives to w, and V is a valuation which for formulae and worlds assigns the truth-vales such that: (i) for all formulae and all worlds, V preserves classical conditions for truth-value operators; (ii) for the world w and any formula ϕ, V (ϕ, w) = 1 iff ∀x∈A V (ϕ, x) = 1; (iii) for other worlds formula ϕ has an arbitrary value. Moreover, for rte-versions of considered logics we must add the following condition: (iv) V (χ, w) = V (χ[ ϕ / ψ ], w), if ϕ and ψ are tautological equivalent. Finally, for ½, ½ and ½ we must add queer models of the form w, V in which: (i) holds and (ii ′) V (ϕ, w) = 0, for any formula ϕ. We prove that considered logics are determined by some classes of above models.

Bulletin of the Section of Logic

In this paper we examine weak logics similar to S0.5[□Φ], where Φ⊆S0·5. We also examine their versions (one of which is S0.5 rte [□Φ]) that are closed under replacement of tautological equivalents (rte). We have that: S0.5 rte [□(K),□(T)]⊊S0·9, S0.5 rte [□(X),□(T)]⊊S1, and in general, if Φ⊆E1, then S0.5 rte [□Φ]⊊S2. In the second part we shall give simplified semantics for these logics, formulated by means of some Kripke-style models. We shall also prove that the logics in question are determined by some classes of these models.

Bulletin of the Section of Logic, 2023

In this paper we shall define semantically some families of propositional modal logics related to the interpretability logic IL. We will introduce the logics BIL and BIL + in the propositional language with a modal operator □ and a binary operator ⇒ such that BIL ⊆ BIL + ⊆ IL. The logic BIL is generated by the relational structures ⟨X, R, N ⟩, called basic frames, where ⟨X, R⟩ is a Kripke frame and ⟨X, N ⟩ is a neighborhood frame. We will prove that the logic BIL + is generated by the basic frames where the binary relation R is definable by the neighborhood relation N and, therefore, the neighborhood semantics is suitable to study the logic BIL + and its extensions. We shall also study some axiomatic extensions of BIL and we will prove that these extensions are sound and complete with respect to a certain classes of basic frames. Finally, we prove that the logic BIL + and some of its extensions are complete respect with the class of neighborhood frames.

2020

This paper from 2008 is the first in a series of three related papers on modal methods in interpretability logics and applications. In this first paper the foundations are laid for later results. These foundations consist of a thorough treatment of a construction method to obtain modal models. This construction method is used to reprove some known results in the area of interpretability like the modal completeness of the logic IL. Next, the method is applied to obtain new results: the modal completeness of the logic IL M_0, and modal completeness of IL( W^*).

"Logic at Work" ed. Ewa Orlowska, pp. 229-234, 1999

Beth's theorem equating implicit and explicit definability is shown to fail in a wide variety of relevant logics. The proof is simple, based on the fact that complements are unique in distributive lattices. It leads to a new and much simpler proof that Craig's interpolation theorem fails for all logics between T and R.

Notre Dame Journal of Formal Logic, 1974

1 This paper is a continuation of the investigations reported in Corcoran and Weaver [1] where two logics j£Π and -CDD, having natural deduction systems based on Lewis's S5, are shown to have the usually desired properties (strong soundness, strong completeness, compactness). As in [1], we desire to treat modal logic as a "clean" natural deduction system with a conceptually meaningful semantics. Here, our investigations are carried out for several S4 based logics. These logics, when regarded as logistic systems (cf. Corcoran [2], p. 154), are seen to be equivalent; but, when regarded as consequence systems (ibid., p. 157), one diverges from the others in a fashion which suggests that two standard measures of semantic complexity may not be linked. Some of the results of [l] are presupposed here and the more obvious definitions will not be repeated in detail.