On Oct 11, 12:07 pm, Marshall <marshall.spi...@gmail.com> wrote: |OK. I understood that. This is a redefinition of the symbols |from their more usual meaning, though, isn't it? Or perhaps |an overlay on top of and in addition to the usual meaning. |It seems a confusing terminological approach, but every |human endeavor is full of such.
Well, "more usual" seems to mean essentially "most popular". Do we want to make it a popularity contest? The intuitionist interpretation of the logical connectives and quantifiers is much more natural than it seems at first glance if you're only familiar with classical logic (the "usual").
|Perhaps I could re-ask my question in a slightly different |way? Is it possible for a mathematical structure | |http://en.wikipedia.org/wiki/Structure_(mathematical_logic) | |to exist in which (?x)~Fx is false and (?x)Fx is also false? |It seems to me that it is clearly not possible.
It's intuitionistically valid to reason that if (Ex)Fx is false then (Ax)~Fx is true. So indeed, they cannot both be false.
Intuitionism distinguishes between "they cannot both be false" and "one of them is true". If the two statements are A and B, then "they cannot both be false" is ~(~A & ~B) while "one of them is true is A v B. The former is equivalent to ~~(A v B). The latter implies that there's a way to find one of them that is true.
|Sure, but does it then follow that there is no useful |distinction to be had between knowing how to find |a value x such that Fx and knowing that such a value |exists? It seems to me that such a distinction is a |useful one, in which case the constructivist stricture |seems dubious, unreasonable, anti-pragmatic.
The distinction is still there. You could for example be told by a reliable person that something exists without being told how to find it.
Intuitionism and constructive mathematics generally is more systematic about making distinctions, not less. When you "know that a value exists" classically, you are making some kind of discovery, which you have arranged to describe using "existential" language.
If (as is typically the case) what has happened is that you've found a contradiction arises from assuming that no such value exists, then might it not be more illuminating to say ~(Ax) ~Fx rather than Ex Fx? That strikes me as being a very direct way of stating what you know, as opposed to "one exists but I don't know how to find it", which could be the case if someone else has found out how to find it but only told you that it is possible. The apparent advantage of using classical reasoning is just that you get to "simplify" ~(Ax)~Fx down to (Ex)Fx, but then have this side explanation for why (Ex)Fx has a weaker meaning for you than constructive mathematics gives it. And then you have this additional way of boosting up the meaning of the quantifier by saying, "and I know how to find it". I don't find it obviously easier to go through all those maneuvers just to say either that ~(Ax)~Fx or (Ex)Fx.