Linguistics Encyclopedia.pdf

It is not the case that some car in the car park is green.
reflects the scope difference between:
The former asserts that the car park contains at least one non-green car; the second asserts
simply that it does not contain any green cars. If the car park is empty, the first is false
and the second is true. In the first sentence, the negation sign falls within the scope of the
quantifier; in the second case, the scope relation is reversed.
TENSE LOGIC AND MODAL LOGIC
While the logic I have described above may be adequate for expressing the statements of
mathematics and (a controversial claim) natural science, many of the statements of
natural language have greater logical complexity. There are many extensions of this
logical system which attempt to account for the validity of a wider range of arguments.
Tense logic studies arguments which involve tensed statements. In order to simplify a
highly complex subject, I shall discuss only prepositional tense logic, which results from
introducing tense into the propositional calculus. This is normally done by adding tense
operators to the list of logical connectives. Syntactically, ‘It was the case that’ and ‘It will
be the case that’ (‘P’ and ‘F’) are of the same category as negation. The following are
well-formed expressions of tense logic:
PA. It was the case that A.
It is not the case that it will be the case that it was the case that A.
These operators are not truth functional: the present truth value of a sentence occupying
the place marked by A tells us nothing about the truth value of either PA or FA. However,
a number of fundamental logical principles of tense logic can be formulated which
govern our tensed reasoning. For example, if a statement A is true, it follows that:
PFA.
FPA.
Moreover, if it will be the case that it will be the case that A, then it will be the case that
A:
FFA→FA.
More complex examples can be found too. If
PA & PB.
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