Showing papers on "Normal modal logic published in 1974"

Semantic Analyses for Dyadic Deontic Logic

Abstract: It ought not to be that you are robbed. A fortiori, it ought not to be that you are robbed and then helped. But you ought to be helped, given that you have been robbed. The robbing excludes the best possibilities that might otherwise have been actualized, and the helping is needed in order to actualize the best of those that remain. Among the possible worlds marred by the robbing, the best of a bad lot are some of those where the robbing is followed by helping.

Reduction of tense logic to modal logic. I

Abstract: It will be shown that propositional tense logic (with the Kripke relational semantics) may be regarded as a fragment of propositional modal logic (again with the Kripke semantics). This paper deals only with model theory. The interpretation of formal systems of tense logic as formal systems of modal logic will be discussed in [6].The languages M and T, of modal and tense logic respectively, each have a countable infinity of propositional variables and the Boolean connectives; in addition, M has the unary operator ⋄ and T has the unary operators F and P. A structure is a pair , or , ⊨ α, if ⊩ (α) = W for every assignment V for . If Γ is a set of formulas of M [T] and α is a formula of M [T], then α is a logical consequence of Γ, or Γ ⊩ α,if α is valid in every model of Γ i.e., in every structure in which all γ ∈ Γ are valid.

Logical consequence in modal logic. II. Some semantic systems for ${\rm S}4$.

Abstract: 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. We consider the logics -Cl, -C2, and -£3. These logics share the same language (^DD) and deductive system (Δ'DD) but each has its own semantics system (Σl, Σ2, Σ3). Σl is an extension of the Kripke [3] semantics for S5 as modified in [l], Σ2 is largely due to Makinson [5], and Σ3 is due to Kripke [4]. -C DD is the usual modal sentential language with D, ~ and 3 as logical constants (see 2 below). Δ'DD (see 3 below), a modification of the natural deduction system given in [l], permits proofs from arbitrary sets of premises. For S a set of sentences and A & sentence, Sv-A means that A is provable from S, i.e., there is a proof (in Δ'DD) of A whose premises are among the members of S. ([-A means S\-A where S is empty.) If S\-A, we sometimes say that the argument (S, A) is demonstrable and when, in addition, S is empty we say that A is provable.

A modal interpretation of three-valued logic

Abstract: It is well-known that the intuitions which led Lukasiewicz to propose his system of three-valued logic were modal in character, having to do with the indeterminacy of contingent future tense propositions.' It is also wellknown that the truth-tables for conjunction and negation which he proposes appear to be inconsistent with any such modal interpretation. Specifically, Lukasiewicz adds to the usual values of truth and falsity a third value, intermediate between these two, which he calls "the possible" (but which is more accurately designated "the contingent"). He argues quite plausibly that when a sentence p has this value, its negation Np should also have it. The process by which he arrives at the table for conjunction is not explicitly stated, but in any event, it has the result that the conjunction Kpq is "possible" when both its arguments are.2 The objection is now immediate: although 'Lukasiewicz will be in Warsaw on Friday' and 'Lukasiewicz will not be in Warsaw on Friday' may both (on Monday) be contingent, the same can surely not be said of 'Lukasiewicz will and will not be in Warsaw on Friday', for the latter is always and necessarily false, on Monday as well as every other day. This objection seems so clear and decisive, that it might be (and often has been) supposed that there is no way at all to interpret Lukasiewicz's system modally.3 Surprisingly, however, this turns out not to be the case. On the contrary, we shall see that there is a translation of his system into the modal system S5, such that exactly those formulas are theses of threevalued logic whose translations are theses of S5. The translation is not without its own peculiarities, however.

Kripke semantics for modal systems including S4.3

Abstract: We present, in terms of Kripke models, the characteristics of all the known extensions of the modal system S4.3. Such a semantic description makes it possible to give a complete picture of the whole class of systems extending S4.3. We obtain answers to some unsolved problems.

A note on universal instantiation in the Stalnaker Thomason conditional logic and M type modal systems

Abstract: In 'A Semantical Analysis of Conditional Logic' (Hereafter SA) Robert Stalnaker and Richmond Thomason provide a possible worlds semantics and an axiomatization for a system of conditional logic which they call CQ.2 In this paper it is shown that the axiom schema which governs universal instantiation for CQ is invalid in the semantics of the system. A new axiom schema which restricts universal instantiation to variables is proposed. The new schema is valid and is strong enough to preserve completeness. It is shown that any valid instantiation to an individual constant is derivable even though the axiom for universal instantiation is restricted to variables. It is also pointed out that this treatment of quantification should work for many modal systems of type M.3 A CQ morphology is a first order predicate calculus with identity and individual constants enriched by a single dyadic modal operator '>' for the Stalnaker conditional. The primitive connectives are '>', '', '~', 'V'. Where A is a formula E A in introduced as a contextual abbreviation for (~A >A). Where A is a formula and x is a variable, 3xA abbreviates ~ Vx ~A. The usual truth functional abbreviations are made. Further