NECESSITY AND
POSSIBILITY
The idea of a statement, proposition, or state of affairs being necessary is a
distinctively philosophical conception, but it is one that is implicit in
ordinary thinking. In metaphysics the most interesting notion is that of
absolute necessity. In this sense, it is necessary that nothing is both
red and not red. And it is necessary that 5+7=12 – indeed all the truths
of logic and mathematics are (presumably) necessary in this sense. This
means that the thing holds or is true in every conceivable situation. If
we are willing to adopt the picturesque terminology of the German philosopher
Gottfried Wilhelm von Leibniz, we may put things in terms of “possible
worlds.” A possible world is a "way things might be or might
have been" -- a coherent, consistent history and catalog of possible facts
[One doesn’t have to take this talk of “possible worlds” too
seriously, at least, not yet.] We may think of a possible world as a
possible story (or book) – a consistent description that contains a
“yes” or “no” answer to every possible question. (It
would have to be a rather long story!)
Then something is (absolutely) necessary if it holds in every possible world
whatsoever. Necessity actually comes in a number of different kinds:
there is such a thing as legal necessity (traffic signs say "Left lane
must turn left") -- if you don't do the thing in question then you are
legally liable to be punished. And there is such a thing as physical or natural
necessity (e.g., two material bodies necessarily attract one another with
such-and-such a force) -- the things' being so follows from a law of
nature. If we want to use the “possible world” terminology,
we might say that something is legally necessary if it obtains in every
legally-permissible possible world (every possible world compatible with the
laws in question) and that it is physically necessary if it obtains in every
physically possible world (every possible world compatible with the laws of
nature).
In this course, we will only be concerned with the strongest of these notions,
absolute necessity. As already explained, we shall say that a statement or
proposition p is (absolutely) necessary if it is true in every possible
world. And we will say that a proposition is possible if it is true is
some possible world. We write "op" to abbreviate "It is necessary
that p" and we write "àp" to abbreviate "It is possible
that p". If we like, we can define necessity in terms of possibility thus:
op =df not-à not-p
[symbolically: ~à~p]
(I.e., “It is necessary that p” means that it is not possibly not
the case that p.)