Showing papers on "Modal operator published in 1985"

A guide to the modal logics of knowledge and belief: preliminary draft

Abstract: We review and re-examine possible-worlds semantics for propositional logics of knowledge and belief with four particular points of emphasis: (1) we show how general techniques for finding decision procedures and complete axiomatizations apply to models for knowledge and belief, (2) we show how sensitive the difficulty of the decision procedure is to such issues as the choice of modal operators and the axiom system, (3) we discuss how notions of common knowledge and implicit knowledge among a group of agents fit into the possible-worlds framework, and (4) we consider to what extent the possible-worlds approach is a viable one for modelling knowledge and belief. As far as complexity is concerned, we show among other results that while the problem of deciding satisfiability of an S5 formula with one knower is NP-complete, the problem for many knowers is PSPACE-complete. Adding an implicit knowledge operator does not change the complexity substantially, but once a common knowledge operator is added to the language, the problem becomes complete for exponential time.

Graded modalities. I

Abstract: We study a modal system ¯T, that extends the classical (prepositional) modal system T and whose language is provided with modal operators M inn (neN) to be interpreted, in the usual kripkean semantics, as “there are more than n accessible worlds such that...”. We find reasonable axioms for ¯T and we prove for it completeness, compactness and decidability theorems.

Logic of nondeterministic information

Abstract: In the paper we define a class of languages for representation o knowledge in those application areas when a complete information about a domain is not available. In the languages we introduce modal operators determined by accessibility relations depending on parameters.

The consistency of a variant of,Church's Thesis with an axiomatic theory of an epistemic notion

Abstract: In this paper we prove the consistency of a variant of Church's Thesis than can be formulated as a schema in a first order language with a modal operator for intuitive provability. We also conjeture the consistency of a stronger variant.

Epistemic set theory is a conservative extension of intuitionistic set theory

Abstract: In [6] Godel 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 Scedrov [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 Scedrov's extension of the Godel 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 Godel 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.

Reply to Baldwin on De Re Modalities

Abstract: However, Baldwin claims that (io) will not do as an analysis of the proposition in question, and consequently that my approach to de re modalities is 'incompatible with the intended understanding of essentialist claims' (p. 255), and he suggests that the mistake lies in my trying to use a sentential modal operator where only a predicate modal operator can do the job properly. Here I shall try to show that, while (io) may not be the most perspicuous way of analysing Baldwin's essentialist proposition, I was not mistaken in using a sentential modal operator in dealing with de re modalities. Baldwin's argument turns on the issue of how we are to interpret a sentence of the form '(Ax)(OFx)[a]' (the first conjunct of (io) and the negation of its second being sentences of this form). He considers first the proposal (based on a theory of Stalnaker and Thomason) that '(Ax)(oFx)[a]' is true iff the thing which is actually a is F in all possible worlds. But Baldwin points out that this proposal inevitably makes (io)'s first conjunct false, thus failing to do justice to the essentialist claim, since by this account that conjunct 'is true iff the person who is actually Elizabeth II was begotten by George VI in all possible worlds', yet 'neither George VI nor Elizabeth II exist in all possible worlds' (p. 255). He therefore suggests that we consider instead the strategy of 'taking the modal sentence operator in (io) to express "weak necessity". . . i.e. truth only in all those possible worlds in which the things denoted by the terms within the scope of the modal operator exist' (ibid.). But then it seems we fall foul of (i o)'s second conjunct, since 'we want it not to be essential of George VI that he begat Elizabeth II even though in all possible worlds in which she . . exist[s], George VI did beget Elizabeth II' (ibid.). Baldwin goes on to remark: