Oct 10, 2015. A Question About Logic.
I have a question about logic, which I hope the logicians in this group can help me with.
In predicate logic, universal quantification does not involve ontological commitment, whereas existential quantification (as the name suggests) does.
To illustrate:
"All dogs are renates" is formalized as "If anything is a dog, it is a renate".
In notation: Ux(Dx --> Rx)
"Some dogs are friendly" doesn't require translation, and in notation is: Ex(Dx & Fx).
(I use "U" for the universal quantifier, "E" for the existential quantifier, "-->" for material implication, "~" for not, "&" for conjunction, and "<--->"
for the metalogical operator of being equivalent to.)
So Ex(Dx & Fx) says: "There exists a dog such that it is friendly." "There exists"; in other words, an ontological commitment to the existence of at
least one dog.
But Ux(Dx --> Rx) says: "If anything is a dog, it is a renate". No ontological commitment here.
This is all very familiar, of course. But here's a question: why, in the formalization of predicate logic, was it decided that "Some X" would carry
ontological commitment whereas "All X" would not? (I think the question has been asked and answered before, but I don't recall what the answer is.)
Now let's move on to the deMorgan's equivalences, in which the negation of a universal quantification is an existential one, and vice versa. In notation:
~Ux(Dx --> Rx) <---> Ex(Dx & ~Rx)
~Ex(Dx & Fx) <---> Ux(Dx --> ~Fx)
In English: "It is not the case that if something is a dog, then it is a renate" is equivalent to "There exists something that is a dog and is not a
renate". And: "It is not the case that there exists a dog which is friendly" is equivalent to "If something is a dog, then it is not friendly".
I've worked with deMorgan equivalences for so long that they seem intuitively right to me. But now notice something: negation creates and removes ontological
commitment. And this seems really strange. Why should negation do this? My being ontologically committed to something doesn't have anything to do with negation; it's simply the _expression_ of my belief that the world contains something, of such-and-such a type.
Note, too, that Aristotle's square of opposition didn't have this strange feature. The negation of "All dogs are renates" is simply "Some dog is not
a renate", and the negation of "Some dogs are friendly" is simply "No dogs are friendly".
I suspect that this strange feature, of negation having ontological import, has something to do with Frege's meta-logical interpretation of properties
(predicates) as sets, i.e. as purely extensional objects. But I don't know, and that's what I asking about.
I'd also like to know if there are formal logics which do not impute this extravagant power of ontological commitment / de-commitment to the negation
operator in predicate logics.
As recent earlier comments have indicated, I'm currently on the track of semantics, primarily of the cognitive variety, and definitely including the
diachronic variety. And so this question is well-off that track. It came up as I was (re-)reading a book which, although it is 25 years old, I highly recommend:
Meaning and Grammar: an Introduction to Semantics
(MIT, 1990), by Gennaro Chierchia and Sally McConnell-Ginet
What dates this book is that it is heavily influenced by Chomsky who, at the time of the book's publication, had left behind (i) transformational-generative
grammar, (ii) extended transformational-generative grammar (the result of the "linguistics wars"), and was in either his (iii) principles and parameters incarnation, or (iv) his X-bar theory incarnation, or somewhere between the two.
But the book is at least as deeply indebted to "west coast" semantics, i.e. the Montague program, and, as it seems to me, the Chomskyean associations
do not run deep enough to tie this work to any of Chomsky's later repudiated positions.