The logical structure underlying temporal references in natural language, The University of Texas at Austin, Computer Sciences Department.

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* Notre Dame Journal of Formal Logic*

Volume XVII, Number 4, October 1976

NDJFAM

A LOGIC FOR UNKNOWN OUTCOMES

BERTRAM BRUCE

1 Introduction

In computer question answering and problem solving

programs many of the questions of modal and tense logics appear as

practical design problems. One problem of particular interest appears

when we allow events to have the truth value “unknown”, a natural value to

assign to some events which occur at other times than the present.

However, allowing a third value is not as simple as it seems. Suppose that

statements P and Q each have the truth value “unknown”. What values

should be assigned to {PΛQ)? If (PvQ) is necessary, it should have the

value “true”, otherwise it has the value “unknown”. The “modal”

composition of truth values cannot be achieved in a three (“true”,

“unknown”, “false”) valued truth functional logic. In fact, as shown by

Dugundji [l], no finite valued truth functional logic can be given the modal

interpretation. Consequently, semantic analysis of most modal systems

must be quasi-truth-functional or involve infinite matrices or both. For

example, Kripke [2] introduces the concept of a set of “possible worlds”

with a model which assigns to each well formed formula (wff) a set of truth

values, one for each world. If the set of worlds is infinite then each wff

will have an infinite sequence for its value. Furthermore, the composition

of truth values is not strictly truth-functional since it depends on the

“possibility” relation between worlds. Another example is the infinite

product logic, πC2, where C2 is the classical two-valued propositional

calculus [5]. In this logic wffs again have sequence for their values. These

sequences can be viewed as the value a wff takes over time [3] and thus

provide a link between modal logic and tense logic. A final example, out of

many others, is the probabilistic approach as discussed by Rescher [4], [5].

He shows that assigning a probability to each wff and applying certain

minimal features of a probability calculus yields a set of tautologies

equivalent to the theorems of S5. Here again the logic is infinite valued and

quasi-truth-functional in the compositions.

With a concern for computer applications such as question answering it

seems appropriate to discuss yet another approach, which appears to have

a simpler (though non-truth functional) decision procedure while requiring…