An intensional epistemic logic

One of the fundamental properties inclassical equational reasoning isLeibniz's principle of substitution. Unfortunately, this propertydoes not hold instandard epistemic logic. Furthermore,Herbrand's lifting theorem which isessential to thecompleteness ofresolution andParamodulation in theclassical first order logic (FOL), turns out to be invalid in standard epistemic logic. In particular, unlike classical logic, there is no skolemization normal form for standard epistemic logic. To solve these problems, we introduce anintensional epistemic logic, based on avariation of Kripke's possible-worlds semantics that need not have a constant domain. We show how a weaker notion of substitution through indexed terms can retain the Herbrand theorem. We prove how the logic can yield a satisfibility preserving skolemization form. In particular, we present an intensional principle for unifing indexed terms. Finally, we describe asound andcomplete inference system for a Horn subset of the logic withequality, based onepistemic SLD-resolution.     

An approach to intensional logic

A system of tensed intensional logic excluding iterations of intensions is introduced. Instead of using the type symbols (for ‘sense’), extensional and intensional functor types are distinguished. A peculiarity of the semantics is the general acceptance of value-gaps (including truth-value-gaps): the possible semantic values (extensions) of extensional functors are partial functions. Some advantages of the system (relatively to R. Montague's intensional logic) are briefly indicated. Also, applications for modelling natural languages are illustrated by examples.     

Classical second-order intensional logic with maximal propositions

Conclusion By the standards presented in the Introduction, CMFC2 is deficient on at least one ontological ground: ‘∀’ is a syncategorematic expression and so CMFC2 is not an ideal language. To some there may be an additional difficulty: any two wffs provably equivalent in the classical sense are provably identical. We hope in sequel to present systems free of these difficulties, free either of one or the other, or perhaps both. This work was done with the aid of Canada Council Grant S74-0551-S1.     

Extending Montague's system: a three valued intensional logic

In this note we present a three-valued intensional logic, which is an extension of both Montague's intensional logic and ukasiewicz three-valued logic. Our system is obtained by adapting Gallin's version of intensional logic (see Gallin, D., Intensional and Higher-order Modal Logic). Here we give only the necessary modifications to the latter. An acquaintance with Gallin's work is pressuposed.     

Six-year-olds' difficulties handling intensional contexts

Previous research shows that children have difficulties handling intensional contexts even when they can pass a test of false belief (e.g. Cognition 67 (1998) 287; Cognition 25 (1987) 289). Some authors (Perner, J. (1991). Understanding the representational mind. Cambridge, MA: MIT Press; Cognition 25 (1987) 289) place these difficulties in the linguistic and not the mental representational domain. The experiments reported here examined whether 6-year-old children could answer questions in an intensional context that did not require the explicit verbal characterization of a belief. We replicated previous findings and found that children answered according to their own knowledge in an intensional context. This occurred even though they responded by choosing a picture to insert into a protagonist's thought bubble rather than report the belief verbally. Children could correctly answer questions about the knowledge state of the protagonist and pass a test of false belief. Further experiments ruled out methodological explanations. Experiment 2 showed that the difference in answering according to own knowledge between the false belief and intensional stories is not accounted for by procedural factors in the two types of test. Experiment 3 revealed that children did not answer according to their own knowledge by default. Experiment 4 suggested that answering according to own knowledge was not a result of pictorial salience. Results are discussed in relation to the simulation-theory debate.     

Sets as singularities in the intensional universe

This paper is motivated by the search for a natural and deductively powerful extension of classical set theory. A theory of properties U is developed, based on a system of relevant logic related to RQ. In U the set {a, b, c,...} is identified with the property [x: x=a x=b x=c...]. The universe of all sets V, is identified with the property of being a hereditary set. The main result is that relevant implication collapses to material implication for sentences with quantifiers restricted to V. This demonstrates the naturalness of the system. However, an aparent lack of deductive power leads to the conclusion that the best extension of classical set theory is to be found in intensional theories with the unrestricted comprehension schema based on weak relevant logics. The author has obtained similar collapses of to for these systems.This paper was presented at the 1986 Annual Conference of the Australasian Association of Logic in Auckland, 9–12 July, 1986.