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Re: [ontolog-forum] A Question About Logic

To: Thomas Johnston <tmj44p@xxxxxxx>, "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Pat Hayes <phayes@xxxxxxx>
Date: Tue, 13 Oct 2015 00:42:48 -0500
Message-id: <9DDB8FE8-1A8F-4DB9-91F2-6CB39390899A@xxxxxxx>

On Oct 12, 2015, at 4:00 PM, Thomas Johnston <tmj44p@xxxxxxx> wrote:    (01)

> 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.    (02)

I will do my best.
> 
> In predicate logic, universal quantification does not involve ontological 
>commitment, whereas existential quantification (as the name suggests) does.    (03)

Right. Although this is perhaps slightly misleading, in that the logic itself 
makes an ontological commitment of a minimal sort, viz. that something exists. 
There is a basic semantic assumption underlying predicate logic that the 
universe is non-empty. (Without this assumption, the universal would not imply 
the existential.)    (04)

> 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.)    (05)

I confess I do not understand the question. This was not "decided"; it follows 
simply from the meanings of the words involved. What does it mean to say that 
some expression "foo" carries ontological commitment? Surely what this *means* 
is that "foo" entails something of the form "There exists...". (Right? What 
else could it mean?)  So of course  "There exists...." carries ontological 
committment: how could it be otherwise?     (06)

OK, consider the other one, why does "All x..." not carry ontological 
committment? Well, it would be most peculiar if it did, for one can use this 
form to assert that something does not exist, as in Ux( ~Unicorn(x) ). So if 
the universal carried ontological import, then this would entail that something 
exists: but what would that existent be? (A non-unicorn?) 
Another way to look at it is to use deMorgans law, and note that 
Ux ~ Unicorn(x)
is equivalent to
~Ex Unicorn(x)
which is a direct statement of NON-existence, so can hardly be said to have an 
existential committment.     (07)

> 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.    (08)

? To me it seems completely and blatantly obvious, not strange at all. It 
follows from the very idea of negation. If "foo" carries any kind of import, 
then its negation "~foo" should fail to carry that same import. That is what we 
mean by denial, expressed by the negative logical form.     (09)

> 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.    (010)

Your asserting this commitment does have something to do with negation, 
however. When you negate the expression of your existential claim, it is no 
longer an existential claim: it is in fact the denial of an existential claim. 
Which is to say, it is a statement of non-existence.     (011)

> Note, too, that Aristotle's square of opposition didn't have this strange 
>feature.    (012)

Sure it did.    (013)

> 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".    (014)

And the negation of the falsehood "Some unicorns are green" - false because it 
makes a false existential committment - is "No unicorns are green", which is 
true because there are no unicorns (of any color.)  The negation does not make 
the ontological committment that the opposite does, just as in predicate logic.     (015)

> I suspect that this strange feature, of negation having ontological import,    (016)

Whoa. Negation has no ontological import as such, but indeed a denial of an 
existential does deny existence. This is not strange, but natural and obvious. 
How could it be otherwise?    (017)

> 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.    (018)

I don't see any special connection with Frege here, but then I don't see that 
there is anything that needs to be explained. 
> 
> 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.    (019)

I would not trust any logic in which the negation of P carries the same 
ontological commitment as P, for any P. Perhaps you could sketch how the 
semantics of this logic would work?    (020)

Pat Hayes    (021)

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