Showing papers on "Normal modal logic published in 1967"

Basic Modal Logic

Abstract: 1. Note the following intuitive equivalency among the two standard alethic modalities (the modes of possibility and necessity): ~Nec φ ≈ Poss ~φ And ~Poss φ ≈ Nec ~φ And note the same basic equivalence among the two standard deontic modalities (the modes of obligation and permission): ~Obl φ ≈ Perm ~φ And ~Perm φ ≈ Obl ~φ In turn, these equivalencies resemble the familiar quantificational equivalencies: ~∀x φ(x) ≈ ∃x ~φ(x) And ~∃x φ(x) ≈ ∀x ~φ(x) 2. Modal logic is meant to capture seeming entailments between such alethic and deontic notions. In basic modal logic we have two new sentential operators. The strong modal operator is symbolized by the box (□), while the weak modal operator is symbolized by the diamond (◊). 3. The syntax for these operators is quite simple: if φ is a wff, then both □φ and ◊φ are both wffs. 4. So what about the semantics for modal logic? Following Kripke, a semantics for modal logic can be understood in terms of a framework of world-models. Here's how it works: (1) We begin with a constellation of distinct \" worlds, \" one of which is often designated as @. This world corresponds to the \" actual \" one, or at least the one from which a sequent or argument is evaluated. (2) These worlds are related to one another by an accessibility relation (which is often diagrammed by arrows). The relation tells us which worlds are \" accessible \" to which. (3) In modal semantics, truth is now intensional. That is, this semantics doesn't assign truth values directly to atomic sentences, but rather assigns truth to an atomic sentence relative to a world.

Essentialism in modal logic

Abstract: s may be interpreted as attributes. To every proposition about an object a there corresponds one or more attributes which ' R. C. Barcan (Marcus), "The identity of individuals in a strict functional calculus of first order," Journal of Symbolic Logic, Vol. XII (1947), pp. 12-15.

Binary Connectives Functionally Complete by Themselves in S5 Modal Logic

Abstract: This paper answers affirmatively the open question of Massey [1] concerning the existence of binary connectives functionally complete by themselves in two-valued truth tabular logic, i.e. in the modal theory S5. Since {∼, ⊃, ◊} is a functionally complete set of connectives (Massey [1, § 4]), the following definitions show that the binary operator ф, the semantics of which is given below, is functionally complete by itself: It is left to the reader to verify, by means of complete sets of truth tables (see Massey [1, §§ 1 and 3]), that the foregoing definitions are correct.