Linguistics Encyclopedia.pdf

it follows that:
There is a variety of systems of tense logic, which offer interesting insights into the
interplay of tense and quantification, and which augment these tense operators by
studying the complex logical behaviour of temporal indexicals like ‘now’ (see
McCarthur, 1976, Chs. 1–2).
Modal logic was the first extension of classical logic to be developed, initially through
the work of C.I.Lewis (see Lewis, 1918). Like tense logic, it adds non-truth-functional
operators to the simpler logical systems; in modal logic, these operators express the
concepts of possibility and necessity. The concept of possibility is involved in assertions
such as:
It is possible that it will rain tomorrow.
It might rain tomorrow.
It could rain tomorrow.
Necessity is involved in claims like:
Necessarily bachelors are unmarried.
A vixen must be a fox.
Other expressions express these modal notions too.
Just as tense logic formalizes temporal talk by introducing tense operators, so modal
logic employs two operators, ‘L’ and ‘M’, which correspond to ‘It is necessarily the case
that’ and ‘It is possibly the case that’ respectively. The sentences displayed above would
be understood as having the forms ‘M A’ and ‘L A’ respectively. There is an enormous
variety of systems of modal logic, and rather little consensus about which of them capture
the logical behaviour of modal terms from ordinary English. Some of the problems
concern the interplay of modal operators and quantifiers. Others arise out of kinds of
sentences which are very rarely encountered in ordinary conversation—those which
involve several modal operators, some falling within the scope of others. To take a simple
example: if ‘L’ is a sentential operator like negation, then it seems that a sentence of the
form ‘LLLA’ must be well formed. However, we have very few intuitions about the
logical behaviour of sentences which assert that it is necessarily the case that it is
necessarily the case that it is necessarily the case that vixens are foxes. Only philosophers
concerned about the metaphysics of modality are likely to be interested in whether such
statements are true and in what can be inferred from them.
Some principles of inference involving modal notions are uncontroversial. Logicians
in general accept as valid the following inference patterns:
LA, so A.
A-Z 179
For example: vixens are necessarily foxes, so vixens are foxes. If something is
necessarily true then, a fortiori, it is true.