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Replacing Modal Operators by Modal Functions

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.

tex2html_wrap_inline1207 is the proposition that the proposition Q is necessary, and tex2html_wrap_inline1211 is the proposition that it is possible. To assert necessity or possibility we must write tex2html_wrap_inline1213 or tex2html_wrap_inline1215 . This can be abbreviated by defining tex2html_wrap_inline1217 and tex2html_wrap_inline1219 correspondingly. However, since nec is a predicate in the logic with t and f as values, tex2html_wrap_inline1227 cannot be an argument of nec or Nec.

Before we even get to modal logic proper we have a decision to make--shall tex2html_wrap_inline1233 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 tex2html_wrap_inline1247 is an equivalence relation on M such that


then the set of equivalence classes under tex2html_wrap_inline1247 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




yielding a system equivalent to von Wright's T.gif

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, Nec4 and Nec5, 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.

next up previous
Next: More Philosophical Examples-Mostly Well Up: FIRST ORDER THEORIES OF Previous: Relations between Knowing What

John McCarthy
Sun Mar 10 22:57:10 PST 1996