Using concepts we can translate the content of modal logic into
ordinary logic. We need only replace the *modal operators* by
*modal functions*. The axioms of modal logic then translate into
ordinary first order axioms. In this section we will treat only *
unquantified modal logic*. The arguments of the modal functions
will not involve quantification although quantification occurs in the
outer logic.

is the proposition that the proposition *Q* is necessary, and
is the proposition that it is possible. To assert necessity
or possibility we must write or . This
can be abbreviated by defining and correspondingly. However, since *nec* is a predicate in the logic
with *t* and *f* as values, cannot be an argument of *nec*
or *Nec*.

Before we even get to modal logic proper we have a decision to
make--shall be considered the same proposition as *Q*, or
is it merely extensionally equivalent? The first is written

and the second

The second follows from the first by substitution of equals for equals, but the converse needn't hold.

In *Meaning and Necessity*, Carnap takes
what amounts to the first alternative,
regarding concepts as L-equivalence classes of expressions. This works
nicely for discussing necessity, but when he wants to discuss knowledge
without assuming that everyone knows Fermat's last theorem if it is true,
he introduces the notion of *intensional* *isomorphism* and has knowledge
operate on the equivalence classes of this relation.

If we choose the first alternative, then we may go on to
identify any two propositions that can be transformed into each other
by Boolean identities. This can be assured by a small collection of
propositional identities like (44) including associative and
distributive laws for conjunction and disjunction, De Morgan's law,
and the laws governing the propositions *T* and *F*. In the second
alternative we will want the extensional forms of the same laws.
When we get to quantification a similar choice will arise, but if we
choose the first alternative, it will be undecideable whether two
expressions denote the same concept. I doubt that considerations of
linguistic usage or usefulness in AI will unequivocally recommend one
alternative, so both will have to be studied.

Actually there are more than two alternatives.
Let *M* be the free algebra
built up from the ``atomic'' concepts by the concept forming function
symbols. If is an equivalence relation on *M* such that

then the set of equivalence classes under may be taken as the set of concepts.

Similar possibilities arise in modal logic. We can choose
between the *conceptual identity*

and the weaker extensional axiom

We will write the rest of our modal axioms in extensional form.

We have

and

yielding a system equivalent to von Wright's T.

S4 is given by adding

and S5 by adding

Actually, there may be no need to commit ourselves to a particular
modal system. We can simultaneously have the functions *NecT*, *Nec*4
and *Nec*5, related by axioms such as

which would seem plausible if we regard S4 as corresponding to provability in some system and S5 as truth in the intended model of the system.

Presumably we shall want to relate necessity and equality by the axiom

Certain of Carnap's proposals translate to the stronger relation

which asserts that two concepts are the same if and only if the equality of what they may denote is necessary.

Tue May 14 16:07:43 PDT 1996