Quantifiers, emptiness, and vacuous truth

[honestrosewater]

I'm learning a bit about the predicate calculus from various online sources so I have a better chance of selecting a good text for a thorough (self-)study.
By "subject class" I mean "x" in $\forall x$. Is it true that the subject class of the universal quantifier can be empty, while the subject class of the existential quantifier cannot be empty?
If so, $$(\forall x [Px \implies (\exists y [whatever])]\ \wedge \neg \exists x [Px]) \implies (\forall x [Px \implies (\exists y [whatever])]\ \mbox{is vacuously true})$$?

Edit: IOW, I'd like to know if I understand how PC handles the problems involving existential import so I can judge how well different texts address the issue. Well, that and I find the issue interesting.

 

[TenaliRaman]

Interesting question...

If **subject class** is actually empty, i believe Px would be false
F->F = T
F->T = T

So u are probably correct. As far as i can recall i never saw any such restrictions on the domain of x.

-- AI

 

[mathwonk]

your statement looks needlessly complicated. I.e. assuming no x satisfies P, implies vacuously ANY statement of form "for all x, (Px implies Q)".

Moreover if you start by assuming the statement "for all x, Px implies Q", as you do, then of course that same statement "for all x, Px implies Q" follows from it, although maybe not vacuously, even without assuming no x satisfies P.

any book will treat these issues adequately. i recoomend an elementary book i used in high school, principles of mathematics, by allendoerfer and oakley.

or you could go to a more formal book, like one by patrick suppes. or willard van orman quine.

 

[honestrosewater]

Interesting question...

If **subject class** is actually empty, i believe Px would be false
F->F = T
F->T = T

So u are probably correct. As far as i can recall i never saw any such restrictions on the domain of x.

-- AI

Yeah, I was carrying over some ideas from catergorical syllogisms (see this article if you're interested).
Now that I've read some more, $$\neg \exists x [Px] \equiv \forall x [\neg Px]$$. So my original statement says, $$\forall x [Px\ ...] \wedge \forall x [\neg Px]$$. That can't be allowable.? I'm not comfortable with the notation (I've seen it a dozen ways), but I think I should have made it three separate statements instead of one. And I guess I meant the subject class to be "Px" not "x". Eh, at least I've decided on my books and should be learning the rules soon. I tend to do better when I know what all the symbols mean too. Thanks.

 

[honestrosewater]

your statement looks needlessly complicated. I.e. assuming no x satisfies P, implies vacuously ANY statement of form "for all x, (Px implies Q)".

Moreover if you start by assuming the statement "for all x, Px implies Q", as you do, then of course that same statement "for all x, Px implies Q" follows from it, although maybe not vacuously, even without assuming no x satisfies P.

I meant something more like: Therefore, "for all x, Px implies Q" is vacuously true. Nevermind, it was a language problem. I usually don't have problems when I know the rules. I admitted from the beginning that I was just learning a little about PC. It's difficult to choose a book otherwise.

any book will treat these issues adequately. i recoomend an elementary book i used in high school, principles of mathematics, by allendoerfer and oakley.

or you could go to a more formal book, like one by patrick suppes. or willard van orman quine.

I decided on Moshe Machover's "Set Theory, Logic, and their Limitations", along with some problem books for set theory and logic.

Edit: Sorry, I was being distracted. I meant to say thanks. Thanks.

 

[mathwonk]

If indeed you meant "for all x, Px implies Q", then my point was this is true becuase you assuemd it was in your hypothesis. you do not need to also assume Px never holds.

so I think what you wanted to say was this: the following is true:


[for all x, not Px] implies [for all x, {Px implies Q}],


no matter what statement Q is, existential or not.

 

[TenaliRaman]

I must say the article was a nice read. I guess empty has always been a problem, not necessarily just in logic.

Number of permutations of O items is 1 ... Show me?
Number of ways i can pick 0 items from n items is 1 ... Duh! How do i do that?
5^4 = 5*5*5*5 , 5^2 = 5*5 , 5^0 = 1 ?
(The last one is not exactly a problem, though at the beginning when i was introduced to the concept of integral exponents, this certainly was a detrimental factor towards my thinking.)

I think its nice to quote this particular line from the article,
One possibility is that logicians previous to the 20th century must have thought that no terms are empty. You see this view referred to frequently as one that others held.[3] But with a few very special exceptions (discussed below) I have been unable to find anyone who held such a view before the nineteenth century. Many authors do not discuss empty terms, but those who do typically take their presence for granted. Explicitly rejecting empty terms was never a mainstream option, even in the nineteenth century.

Interesting isn't it and we think logicians are always picky. :p

Though if we assume that the statements are "vacuously true", then every possible objection raised in that article seems to vanish.(This particular thing is also quoted in the article)
For most of this history, logicians assumed that negative particular propositions ("Some S is not P") are vacuously true if their subjects are empty. This validates the logical laws embodied in the diagram, and preserves the doctrine against modern criticisms.

But i believe that assuming these statements to be vacuously true was a problem to many logicians. Though particularly i am not able to see their view point. For example, let me take one of Buridan's example given.

Every man is a being.
Contrapositive ->
Every non-being is a non-man.

The second statement is claimed as falsehood, since non-being is empty.
So basically they are claiming that if domain is empty the statement given is false by default.

So this one can be added to my previous list of "confusion". And if we allow William of Occam(Ockham) to decide on this , i am sure he would say, "why don't we just say , they are vacuously true and go have a cup of tea"

-- AI

 

[honestrosewater]

Every man is a being.
Contrapositive ->
Every non-being is a non-man.

The second statement is claimed as falsehood, since non-being is empty.
So basically they are claiming that if domain is empty the statement given is false by default.

The statement is false because universal affirmatives are assumed to have existential import (in this version); The statement is claiming that there exists at least one non-being. If universal affirmatives aren't assumed to have existential import, the statement is true.

 

[TenaliRaman]

And what does one mean by existential import, i believe that's equivalent to saying that domain of x is not empty. (Hence the point being made of vacuous truth).

-- AI

 

[honestrosewater]

Sorry, I misread

So basically they are claiming that if domain is empty the statement given is false by default.

as meaning that all statements whose subjects are empty are false by default (which isn't true since O statements were denied existential import and considered true when their subjects were empty (at least in some versions)). (Edit: Eh, perhaps saying they were denied existential import is confusing. Rather, they weren't assumed to have existential import. This whole thread is unnerving.)

And what does one mean by existential import, i believe that's equivalent to saying that domain of x is not empty. (Hence the point being made of vacuous truth).

-- AI

This is where I have problems. In categorical logic (CL), in "All S are P", "S" is (called) the subject and "P" is the predicate. Existential import applied to "S".
In predicate logic (PL), "All S are P" becomes $$[(\forall x)(\neg Sx \vee Px)]$$. Now "x" is (called) the subject, "S" and "P" are predicates, and "Sx" and "Px" are predicate sentences or something similar. So now to what does existential import apply? To me, it seems that CL's "S" is PL's "Sx". So existential import would apply not to "x" but to "Sx".
Also, I've read that in PL when x is empty, in general, $$[(\forall x)(Px)]$$ is true and $$[(\exists x)(Px)]$$ is false. But I have yet to read or figure out how one would say "x is empty" in PL. I suppose x is empty just in case those two conditions are met and non-empty otherwise.
Anyway, I'll remain confused about several things until I learn more about CL, PL, and the translations between them.