This is true, although problems of existential import with the theory which evolved out of this were noticed early on by at least some (Abelard, for example, managed in the early twelfth century to perceive problems with taking particular negations to be both contradictory with universal affirmations and subcontrary with particular affirmations, which are essentially problems of ascribing existential import to universal negations. Though, Abelard’s response was to draw a distinction between “Not all Xs are Ps” and “Some Xs are not Ps”, and, well, look what became of him…).
For any natural number n, the statement “All creatures on Pluto have n hands” is true.
The question that needs answering is, “How many hands do creatures on Pluto have?”
Indistinguishable writes:
> Though, Abelard’s response was to draw a distinction between “Not all Xs are
> Ps” and “Some Xs are not Ps”, and, well, look what became of him…).
Yeah, that’s why I didn’t get my Ph.D. in mathematical logic. I was about to go into the room for my oral defense when my thesis advisor said, “Of course you realize that if you fail to defend your thesis sufficiently well what will happen.” Then I noticed that there was a surgical table in the room and that all of the professors held scapels. I decided to blow off the Ph.D. and get out of there.
There is a contradiction.
Premise) If something is a horse and has a horn it is a reptile.
From Definition) If something is a reptile it is not a horse.
Therefore) If something is a horse and has a horn it is not a horse.
A contradiction, therefore the premise is not true
The statement “All horses with one horn are reptiles” is not a statement of vacuous truth. I think you are confusing “All P are Q” implying “Some P are Q” (which is what the OP’s case is about ), with “All P are Q” implying “If something is a P then it is a Q”, which is generally regarded as true.
It can’t be directly compared to “All but two of my cars are Fords” but rather should be compared with “All but two of my automobiles are not vehicles”, which is a contradiction and you can either treat it as something that cannot be true or false, or simply declare it false.
In the end, it’s a matter of language and context, but I contend that “NOT P” does not make “P IMPLIES NOT P” a true statement.
Would you suggest that “All horses with one horn are things that exist” is also a true statement?
From the perspective being debated here, “If something is a horse and has a horn it is not a horse” is not in itself a contradiction in the sense of being incapable of being true. Rather, this sentence is just a roundabout way of saying “Nothing is actually a horse with a horn”, which can certainly be true. The only contradiction would be between “If something is a horse and has a horn it is not a horse” and “Something is a horse and has a horn” (in which case, we would indeed be put in the inconsistent position of agreeing that there is some actual thing which both is and is not a horse), but it is perfectly consistent to take the former as true as long as one therefore takes the latter as false.
No, I don’t think Frylock is doing that. I think perhaps you are confused as to what the term “vacuous truth” generally means in this context; “All (horses with one horn) are reptiles” is a standard type of vacuous truth: universal quantification over an empty domain. [Instances 2, 3, and/or 4 in that link, depending on how you analyze it]
Let’s ignore the red herring “but two” for a second. “All of my automobiles are not vehicles” is not an intrinsic contradiction, from the perspective being debated, for the same reason as above: it can be true as long as I don’t have any automobiles. Sure, it will also be the case that “All of my automobiles are vehicles”. Combining the two, we might say “All of my automobiles are such that they have inconsistent properties”. This is all perfectly consistent; it just leads us to the inevitable conclusion that I have no automobiles.
Ah. Well, as I said, people’s views on universal quantification may be related to their views on implication. (Or, rather than “views on”, “desired ways of speaking about”). As I pointed out above, though, the position you are taking will require that the truth status of “A implies B” cannot be determined from the truth status of A and B alone. Is that acceptable to you?
Yes, I would. (I’d even say “All Xes exist”, whatever X is, understanding this in an appropriate way) What of it?
I should clarify: Yes, I would, in a certain semi-formal manner of speaking. (I’d say “All unicorns exist” and feel alright about it, because, as far as I’m concerned, this is the same as saying “All the unicorns [that actually exist] exist”, which is fairly tautological. It wouldn’t be, in itself, asserting the existence of any unicorns, just naming properties of all those which do exist)
In ordinary language, however, I’d probably shy away from speaking like that, it being liable to be taken differently, if only through conversational pragmatics. Or, rather, I would say it only with such disambiguation as I am providing now. [E.g., “When I say ‘All unicorns exist’, I mean that ‘Any unicorn you care to point to is one which exists; any object in the world which is a unicorn is an existent object’. I’m not saying all that much; this is simple tautology”]
(Even formally, for that matter, I might not want to speak like that in many contexts, not wanting to take existence as a property of individuals, but rather as a second-order property of concepts [e.g., not ascribing existence to particular unicorns out in the world, but rather to concepts such as “unicorn” or “the grey unicorn whose image haunts my dreams” or whatever themselves]. All the same, in the context of the answer to your question, phrased as it is, yes, I would be willing to say “All horses with one horn are things that exist”.)
To be a contradiction, it would have to be equivalent to “P and not-P” for some P. What do you think is the relevant P here?
Yes, on the formalism we’re discussing, that’s a true sentence. (However I offer the caveat that with that sentence, we’re starting to see why some people have thought you shouldn’t treat existence as a predicate. Because you start to get really strange sounding results like “Anything that doesn’t exist, exists.” But if you’re careful with your interpretation, even that sentence can be seen to be true. What it says, on the formalism we’re discussing, is that whenever you find something in the domain of objects you’re discussing that’s not in the domain of objects you’re discussing, the predicate “exists” is true of that thing. Since of course you’ll never find such an object (the criteria for finding it contain a contradiction), the sentence is true. For since you’ll never find such an object, any such object you do find you can feel free–you will be correct–to apply the predicate “exists” to it.)
Or zero cars. “For all x if x is my car then x is not a ford” is true if there is no x such that x is my car. (Assuming classical logic.) At least I don’t see why this isn’t as reasonable interpretation of the OP puzzle as one that ends up giving you two cars.
I read the OP as the OPer does, but I can’t really pin down any good reason why. I don’t put it down to a resistance to (~pvq :: p->q), nor an intuition about existential import following from universal quantification. An unsubstantiated biographical note, you might say. But note that (unless I have missed it) the only options considered in this thread were 2 or 3 cars, the total lacking of cars was not considered. So it seems there is an intuition about some cars existing (“my cars”) even when people are open to others not existing (the Fords and such).
The reason for the lack of mention of the zero option could just be because it is an unnecessary complication for the issues that are being discussed. But was this the reason?
Or perhaps because I ignored the “but 2 of” clause, whoops. So far I can’t make things work while taking into account the “but 2 of”.