Monday, January 30, 2006

Quidquid latine dictum sit, altum videtur

I argued (commenting in Alan Rhoda's weblog here) that

(1) It is possible that there are unicorns.

does not imply

(2) There are unicorns

Alan disputes this, arguing that (1) can be formalised

(3) (∃w)(Ww & (∃x)(xEw & Ux))

But how do we translate this formalisation? If it translates 'it is possible that there are unicorns' we still can't infer that there are unicorns. The 'that' clause acts, as it is designed to, to protect us from any inference as to the truth of the statement embedded in the clause (just as it protects us from inferring 'there are unicorns' from 'Jack thinks that there are unicorns'. If, on the other hand, we translate it as

(3') There is a possible-world, and there are unicorns, such that the unicorns are in that world

then this does logically imply there are unicorns. But that is because the translation strips out the 'that' clause. Which begs the question. If we are allowed to translate 'it is possible there are unicorns', which does not imply there are unicorns, by a sentence that contains 'there are unicorns' as a logical component, and which does for that reason imply there are unicorns, then Alan has pulled off the trick. But are we allowed to?

Whatever is said in formal logic, seems deep.

1 comment:

Alan Rhoda said...

Just a quick comment, Ocham. First, I presume that you meant (2) to read "There are possible unicorns." Second, I don't dispute that (1) does not imply (2). What I dispute is the charge that I'm affirming (2) in the sense that (2) would be ordinarily understood.

As ordinarily construed, (2) says that there exist (in the domain of actuals) such things as 'possible unicorns'. I reject that as misleading. What I want to say is that in certain abstract conceptual contexts (e.g., mathematical logic, possible worlds semantics, etc.) we can use (∃x) to quantify over possibilia. Hence "there is", in the sense represented by (∃x), does not always imply "is actual".

For example, a mathematician might say "That polynomial has three imaginary roots", thereby quantifying over complex numbers. But nobody, so far as I am aware, thinks that there can in reality be complex quantities of anything (e.g., 1+3i potatoes). The quantification is over conceptual or intentional objects, not real objects.