Seldom Seen Slim in a characteristic back-to-the-camera pose evaluates the shooting skills of the man we call 'Doc' (in allusion to Doc Holliday). Slim writes:
Whilst I'm mulling over your thoughts on souls and salvation, here's a trifle you might agree with.
You write "There are many examples of the use of tautological sentences to express non-tautological propositions."
Indeed, my favorite ordinary language example is the use of the double identity "a=a and b=b" to assert that a and b are quite different (in some salient respect under discussion), and to imply that the listener is rashly ignoring this obvious fact!
"Why did she do that?" Men are men and women are women.
"The hell they are!" does not reject the identities, but the salient difference.
An exercise I used to give to my (brighter) logic students was to formalize what "men are men and women are women" is trying to assert in the Predicate Calculus.
……………
'Men are men and women are women,' which appears to be a conjunction of two tautologies and thus a tautology, is, however, typically used to express the non-tautological proposition that men and women are different as Slim suggests. The idea is not that each man is numerically different from each woman, but that there are properties had by men, but not by women, which render men and women qualitatively different. So perhaps we can symbolize the intended non-tautological proposition as follows using second-order predicate logic:
There is a property P such that for every x and every y, if x is a man, then x has P, and if y is a woman, then y does not have P. Symbolically:
(EP)(x)(y)[(Mx –>Px) & (Wy –>~Px)]
where 'E' is the existential quantifier, 'x' and 'y' are individual variables, 'M' and 'W' are predicate constants, 'P' is a predicate variable, '&' is the sign for truth-functional conjunction, and '–>' denotes the material or Philonian conditional.
But on second thought, this doesn't seem right. For when we say that men are men and women women, we do not mean that there is one particular property that all men have and all women lack that renders them qualitatively different; what we we mean is that there are some properties which render them different, allowing that these could be different properties for different men and women. To illustrate: consider a universe consisting of just two men and two women: Al, Bill, Carla, and Diana. The property of having lousy social skills might be had by both Al and Bill and lacked by both Carla and Diana. But it might also be that there are two properties, the property of being ornery and the property of being highly unemotional such that Al is highly unemotional but not ornery and Bill is ornery but not highly unemotional while neither of the ladies has either. In that case there would be no one propery that distinguishes the men from the women.
So let's try:
(x)(y)[(Mx & Wy –> (EP) (Px & ~Py)]
What say you, Slim?