> So? I didn’t ask about boolean logic as much as I asked if the riddle can logically
> have an answer of 2 in the exact language it was written.
Natural language is often ambiguous. Different people have different interpretations of sentences. A single person can have different interpretations of the same sentence at different times. That’s why formal logic (like Boolean logic) was invented, to be able to say things more precisely than can be done in natural language. Just walk away from the argument with the people who claim that no cars is a possible answer to this riddle. There’s no way you’re going to persuade them they’re wrong. It doesn’t matter that the answer forces an unnatural interpretation in your understanding of the sentences of this riddle. It doesn’t matter that they are forcing the sentences into an interpretation that isn’t natural even in their own usual understanding of them. They’re going to insist that they have the correct interpretation.
Well, it’s not false. For the statement to be false, there would have to exist a counterexample; i.e., a Plutonian creature with a non-seven amount of hands. If the statement isn’t false, it’s true, at least under the most common formalization.
Expanding on the point above about universal quantification being vacuously true on empty domains: You are of course free to speak however you want, and obviously there are some conceptual reasons why you feel the urge to do it the way you do. However, taking “all Xs…” to be automatically true over an empty domain would allow for clean correspondences with all the following:
[ol]
[li]“A implies B” being automatically true with a false antecedent A[/li][li]“there exists an X…” being automatically false over an empty domain[/li][li]The empty conjunction being true[/li][li]The empty disjunction being false[/li][li]The empty product being 1[/li][li]The empty sum being 0[/li][li]0^0 being 1 (in contexts where the exponent is understood as a cardinal)[/li][li]a^0 being 1 in general[/li][li]a*0 being 0[/li][/ol]
Those correspondences being by virtue of the following, respectively:
[ol]
[li]We can take “A implies B” equivalent to “All things satisfying A are things satisfying B”, and, conversely, take “All Xes satisfy P” equivalent to “Being an X implies satisfying P”[/li][li]We can take “there does not exist an X satisfying P” equivalent to “All Xes satisfy not P”[/li][li]We can take “All Xs satisfy P” equivalent to “x1 satisfies P and x2 satisfies P and…” where x1, x2, …, are the values over which X ranges[/li][li]From the connection in (3), by taking “not (A or B or …)” equivalent to “(not A) and (not B) and …” (akin to the connection in (2))[/li][li]We can take truth to correspond to positive values and falsehood to correspond to 0, and then take conjunction to correspond to multiplication (of cardinal numbers)[/li][li]From the connection in (5), taking disjunction to correspond to addition[/li][li]From the connection in (5) and that in (1), taking implication to correspond to exponentiation (antecedent corresponding to power, consequent corresponding to base)[/li][li]From the connection in (5), taking a^b to be the b-fold product of a[/li][li]From the connection in (6), taking a*b to be the b-fold sum of a[/li][/ol]
The unity of this web of correspondences is lost (or, at least, damaged) when one demands that universal quantification carries existential import. This is a strong aesthetic point in favor of viewing such restrictions as ad hoc; certainly, formalization and analysis becomes much easier otherwise. But, as always, how you ordinarily speak is up to you, with no right or wrong way. I intend merely to illustrate what naturality can be found in this alternative.
(Of course, I’ve left out other connections between those [e.g., (5) and (6) above also correspond to each other by the connection of addition and multiplication through logarithms and exponentials], as well as other, more abstruse connections to the vacuous truth of empty universal quantification [empty intersections being the appropriate universal collection, there existing a (unique) empty list at every type, the general observations that empty categorical products are terminal objects and empty monoid operations are identity elements…]. I intend merely to give a glimpse of the mathematical resemblances and symmetries which abound, and the corner of that world in the vicinity of this particular issue.)
Yes, this illustrates the enjoyable equivalence of “All Xes satisfy P” and “There does not exist an X such that not P” on the conventional formal view.
Other nice logical rules lost by giving universal claims existential import:
[ul]
[li]The equivalence of “All Xes satisfy (P implies Q)” with “All (Xes which satisfy P) satisfy Q”[/li][li]The equivalence of “All Xes are Ys” with “All non-Ys are non-Xs”[/li][li]The ability to derive “All Xes are such that A holds” from A[/li][li]The equivalence of “Either all Xes satisfy P or A holds” with “All Xes either satisfy P or are such that A holds”[/li][/ul]
For putative counterexamples, take Xes to be unicorns, Ys to be horned creatures, P to be the property of having exactly one horn, Q to be the property of having at least one horn, and A to be the proposition that 2+2 = 4.
Of course, to be fair, by instead denying this existential import, certain otherwise valid things are lost as well:
[ul]
[li]The ability to derive “There exists an X satisfying P” from “All Xes satisfy P”, of course[/li][li]The ability to derive A from “All Xes are such that A holds”[/li][/ul]
Putative counterexamples as above, except now take A to be the proposition that 2+2 = 5.
Clearly, I would consider the trade-off more than worth it. [Though, to be honest, not so much because of anything here as because of the connections indicated above. Which begs the question, why did I bother writing this post? Ah well…]
Whoops, I didn’t actually give a proper putative counterexample to the first equivalence above. For that, take Xes to be horses in general, rather than only unicorns.
Another nice one which occurred to me, though the thread is dead by my hand now, is the equivalence of “All (X or Y)s satisfy P” with “All Xs satisfy P and all Ys satisfy P” (akin to the equivalence of “(A or B) implies C” with “A implies C and B implies C”).
Is this the shit they are teaching our kids in school?
Anyhoo…the words that stand out to me in the OP are “are Fords”. “are” as in the present form of “to be”. To exist, if I may. The statement implies that Fords exist among the group. So two cannot be the correct answer.
And further, I can’t agree with three as the correct answer. As Ford"s" (plural) imply that I have more than one in my subset. “All but two of my cars are Fords” How would a reasonable person phrase their car inventory with two Jeeps and one Ford? Not that way. The plural “s” on Ford implies a larger subset.
To follow up on Indistiguishable’s example, what about the statement “All horses with one horn are unicorns?” Do unicorns have to exist for the statement to be true?
That can’t possibly be correct. Using the same logic the statement “Some mythical creatures resembling horses, with a single horn in the center of their foreheads are unicorns” is false. Yet that sentence is derived from the dictionary definition of unicorn.
(I’m not comfortable using this kind of example to motivate the view you’re trying to motivate. Because “All horses with one horn are not unicorns” is also true, and “All horses with one horn are reptiles” is even true. For most people, the truth of “All horses with one horn are unicorns” intuitively rests on the fact that unicorn is defined as “a horse with one horn.” So they’ll accept the truth of the statement, but will balk at generalizing from the truth of a statement to a view that any statement predicating something of an empty referent is also true, because for the same reason they acknowledge your statement is true, they’ll tend to balk at agreeing that the other, weirder statements I mentioned are true.
Certainly, you were strictly speaking just trying to show that a referent can be empty yet be such that true predications can be made of it. But really what we’re after is explaining why any predication made of an empty referent is true. And the kind of example you gave will actually present a stumbling block towards that goal.
That is a different example, none of those sets are empty, and barring a language anomaly, “All horses with one horn are reptiles” is simply not true, and it can be proved:
All horses are mammals, by definition
No mammal is a reptile, by definition
Therefore, no horse is a reptile by 1 and 2 above.
There are things with horns, there are reptiles and there are horses. Now just because the intersection of things that have horns and horses is empty, doesn’t mean logical constructions suddenly break down.
Certainly reasonable. I suspect that many people’s ordinary language treatment of universal quantification actually does correspond to their treatment of implication, with their position on vacuous truth over empty domains in the former corresponding to their position on vacuous truth over false antecedents in the latter. Just as an accurate model of the latter may accurately require wrapping material implication under a suitable modality of necessity, so may an accurate formal model of the former require wrapping universal quantification, as traditionally formally analyzed, under the same kind of modality (thus, “All horses with one horn are unicorns” will be true because it holds necessarily, but “All horses with one horn are reptiles” will not, because it presumably holds only contingently).
But, my own intent in the last several posts has not been to motivate the view “You must approach universal quantification this way” (which I have explicitly disavowed), but merely “You should see why it is not as manifestly ridiculous as you apparently think, and is actually even useful for many purposes, to approach universal quantification this way; it is glib to dismiss this out of hand as untenable”. I cannot speak for Lemur866’s motivations, but I suspect they are similar; the point is not to attack alternative ways of speaking, but merely to defend this one against automatic rejection.
With that in mind, once one admits the plausibility of reasonably construing “All horses with one horn are unicorns” as true, enough of a foothold presents itself as to allow argument defending reasonably construing all vacuous universal quantification as true. (As might be expected from above, a natural argument proceeds the same way as one might for defending the material conditional) To wit:
A very attractive, perhaps even intuitive, principle is that of (a very weak) truth-functionality: that the truth or falsehood of a large statement should depend only on which of its constituent “atomic” claims are true. That is, in this particular case, that the truth or falsehood of “All Xs have property P” should depend only on which particular Xs have property P. But, as far as horses with one horn go, the collection of those which are unicorns is precisely the same as the collection of those which are reptiles; empty, in both cases. Thus, if we are to hew to this principle, we must take “All horses with one horn are unicorns” and “All horses with one horn are reptiles” as equally true or false. At this point, if one is inclined to accept the former as reasonably considered true, one hopefully can understand why the latter, perceived as equivalent on this account, can also be reasonably considered true.
You’ve correctly demonstrated that no horse is a reptile. This is not, however, necessarily in opposition to the claim that “All horses with one horn are reptiles”; indeed, those who would make that claim would also make the claim that “All horses with one horn are non-reptiles”. And even “All horses with one horn are both reptiles and non-reptiles” [this being, in such a way of speaking, simply a roundabout way of saying “There aren’t any horses with one horn”].
The idea that “All Xs have property P” is in conflict with “No Xs have property P” is just a holdover from the idea that universal claims have existential import; if one abandons this existential import, there is no longer any contradiction between the two.
Indistinguishable has given a wonderful account of why most people with formal logical training these days would think it very natural to apply “all Fords are …” to cases with no Fords.
Historically, however, those who studied logic formally would not have agreed with us. In Aristotle’s system, the syllogistic forms Darapti and Bramantip, among others, implicitly assume that “For all X, Y” implies “For some X, Y”.
So, those who share the OP’s intuitions can take comfort in the knowledge that, up to 200 years ago or so, the most illustrious logicians in history would have agreed with them.
Er, anyway, but even if one doesn’t make that leap to the general case, still, one should recognize that such metaphysical argument as that " ‘All Xs…’ can never be true if there aren’t any Xs, because…" instantly founders once one is prepared to accept “All horses with one horn are unicorns” as true. One needn’t thus accept vacuous truth in general, but one can no longer reject specific instances out of hand either.
I could also have said “All horses with one horn are manticores.”
This is exactly the kind of balking I was talking about in my post.
To answer your argument, I’ll point out that these two statements
No horse is a reptile
All horses with one horn are reptiles
are actually logically compatible. (When interpreted using the formalism we’re discussing in this thread.)
For if there are no horses with one horn, then it remains true that no horse is a reptile even if every horse with one horn is a reptile.
Put another way, there is no contradiction between the statement “all horses with one horn are reptiles” and the statement “there is nothing which is both a horse and a reptile.” For if there are no horses with one horn, then even though all horses with one horn are reptiles, since there are no horses with one horn, it can still be true that there are no horses that are reptiles.
