Vacuous Quantification

Today in Semantics 1 we talked about vacuous quantification, looking at how to translate sentences like "Someone is such that Metaphysics is beautiful" into predicate logic.  And now I'm wondering:  how might we translate "Shieva is such that Metaphysics is beautiful" into predicate logic?  I don't want to use a quantifier 'cause (i) intuitively, what we mean doesn't involve one, and (ii) there are relevantly similar sentences where the Free Logic theorist wouldn't like a relevantly similar translation.  Instead, what about just putting 's' where the '$x' would be, at the start of the translation?  (It could occur later in various statements as well, in translations of sentences like "Something is such that Shieva is such that Metaphysics is beautiful".)  I'd have to give some rules and such to go with it, but . . . does this seem fine?  Or am I missing something obvious?

Comments

[chris]

Hi Shieva, you could do this in a straightforward way using lambda calculus. Basically lambda functions as a predicate-forming operator; you could use it to form the predicate 'is such that metaphysics is beautiful' and predicate this property of Sheiva. It would be 'lambda x (metaphysics is beautiful x) Sheiva'. Btw, thanks so much for organizing the MTL conference; it was awesome!!

[Lewis Powell]

Chris, if one generates the predicate that way, would the resulting function 'Lambda x(metaphysics is beautiful x)', just be a constant function?

[chris]

Why so?

[Lewis Powell]

Because it doesn't vary with different values of X. If anyone is such that metaphysics is beautiful, then everyone is such that metaphysics is beautiful.

Otherwise the original quantification wouldn't have been vacuous.

[chris]

It depends on what it takes for something to be such that metaphysics is beautiful. But the question wasn't what it takes for it to be true; it was about how to translate the sentence. As far as I can tell, lambda abstraction does the trick. Is it the translation that you object to?

[Lewis Powell]

I don't object to anything, I just wanted to make sure I was understanding that if we thought that nothing about me contributes to whether or not I am such that metaphysics is beautiful, then the lambda abstraction would express a constant function (either from any X to true, or from any X to false).

[chris]

Oh sorry; I misunderstood. What you say sounds right to me, but the translation seems consistent with someone having a weird view of what it takes to have the propery; one that does not have it that the predicate expresses (or denotes, whatever) a constant function. But I agree with you that some such account seems to be required in order to capture the vacuousness.