Archive for June, 1971

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

Notre Dame Journal of Formal Logic
Volume XVII, Number 4, October 1976


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…

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