All but two of my cars are Fords, all but two of my cars are Toyotas and all but two of my cars are Hondas. How many cars do I have.
The obvious answer is three.
I’m arguing with some folks that are claiming two is an acceptable answer and they are using the rationale that it is fine to have zero Fords according to Boolean logic.
I don’t have a problem with statements like “There are zero Fords in my garage”, but once you separate the amount of cars you have by make and claim that your non-existent cars are Fords, this can’t be correct even with Boolean logic.
Folks are making statements like this:
“I have a technical background and have taken multiple courses on Boolean logic, so I have no problem with the statement “All of my cars are Fords except two,” when the speaker has two cars that are not Fords. It is absolutely a true statement logically. Potentially misleading, but true.”
I think my best refutation to two being an acceptable answer is the following:
If all 'but" two of his cars are Fords, can it be logically possible to have only two cars? No. He is separating two cars from an amount of cars. That amount must logically be greater than 2 or no separation can be made. Claiming that he has 2 cars plus an amount of cars equaling zero that are Fords is illogical.
I have to say, these folks sound a bit loopy. If you have two Toyotas and that’s it, it just isn’t well defined to say what “all” your non-Toyota cars are. It certainly doesn’t make any sense to say they are Fords. They aren’t anything.
Perhaps you can give them a different example and point out the differences. Imagine bob has one brother and no sisters. He says:
(1) “I have no sisters with red hair.”
(2) “All my sisters have brown hair.”
Statement (1) says that the set “sisters with red hair” is empty. That’s perfectly valid. Statement 2 starts with an empty set (“all my sisters”) and claims to operate on the elements of the set – but there’s aren’t any! You may as well try to divide 16 by purple.
From a FOL perspective, it makes perfect sense to say all my sisters have brown hair', even when you have none---the term you are looking for is vacuous truth’.
This is closely tied to the problem of definite descriptions, which has been a hot topic for the past 100 years and probably isn’t going to be resolved anytime soon. See this Stanford site for a detailed introduction, and this old thread for some discussion.
Natural language is whatever it is, and it may well turn out that how some people naturally employ the phrase “all but two” is different from how others do. Arguing from a particular formalization can be a pedantic distraction, if further justification is not given to privilege that formalization over others. “I have a technical background and have taken multiple courses on Boolean logic” is fine for demonstrating why “I” interpret a sentence a particular way, but does not, in itself, present an argument against other interpretations.
That having been said, let us ask nonetheless, what is the most conventional or natural way to formalize “all but two of my Xes are Ys”? I imagine it would be to take this as equivalent to “Precisely two of my Xes are not Ys”. Thus, the OP’s total statement would be equivalent to a suitable formalization of “I have (precisely) two non-Fords, I have two non-Toyotas, and I have two non-Hondas”. Clearly, this would hold true if one had just two cars, both Volkswagens. In a way, this relates to the point Capt. Ridley’s Shooting Party brought up, that “for all” usually encompasses vacuous truth, though I think you can understand it even without worrying about that [the key point would just be the second sentence of this paragraph].
[Side note]I don’t think definite descriptions enters into this in any direct way; it’s not as though, for example, the OP said “The two cars of mine which aren’t Fords are Toyotas”. Where was definite description employed?[/Side note]
Addendum, clarification: One could formalize “All but two of my Xes are Ys” as, not merely “Precisely two of my Xes are not Ys”, but instead, the stronger “Precisely two of my Xes are not Ys and at least one of my Xes is a Y”. However, this seems less clean, and in a way, less natural, to me. Still, if that’s how your speech works, that’s how your speech works. Clearly, for your colleagues, their speech works differently.
Is it really okay to say your sisters have brown hair if you have no sisters? This is more that just misleading and does more than imply that you have sisters; there’s been an actual claim made that you have sisters. If they have brown hair, they exist.
Vacuous truth isn’t about saying “My sisters have brown hair” when you have no sisters. It’s about saying “All my sisters have brown hair” when you have no sisters. [Well, if you read the former as meaning the latter, then it is about the former as well…]
For an example, I am perfectly willing to assert of you that “All of TazMan’s aunts are female”, even though, for all I know, you don’t have any aunts. It doesn’t matter to me, because my use of “all” encompasses vacuous truth. For me (in this context) and many others, to say “All Xes are Ps” is equivalent to saying “There does not exist an X which is not a P” (e.g., “I do not have a sister who lacks brown hair”, “TazMan does not have an aunt who isn’t female”).
It was, but you asked about a formal system, yourself, namely Boolean logic. The answer to the OP rests on whatever translation from ambiguous English into unambiguous FOL you decide to choose, as Indistinguishable points out.
So it’s okay to say all creatures on Pluto have seven hands, and as long as there are no creatures on Pluto that statement is true?
Even so, the riddle states “All but two of my cars are Fords”. Once you separate an amount of cars from your cars based on make, they must exist. You can’t separate zero cars from your cars because those non-existent cars are somehow labeled Fords.
The language of “operate on” may not be that helpful, but at any rate, why not take the Fregean approach of saying statement 2 operates on the set itself, rather than its elements? That seems a more appropriate analysis of most quantifiers.
Yes, that’s the usual convention in formal logic. If you don’t like that convention, that’s fine. You follow a different one. However, the usual convention is cleaner to analyze in some ways and has other reasonable properties arguing in its favor (e.g., I can make the obvious claim that “All red dinosaurs were reptiles” without having to first check whether any dinosaurs were red at all).
Maybe you can’t; I find it extraordinarily easy to do so (language bending to my awesome will like so much putty).
As above, perhaps you should think of of the separation as not acting on individual cars, but on sets of them (though perhaps you shouldn’t think in “acting on” terms at all…). The set of my cars splits into two sets: the set of my Fords and the set of my non-Ford cars. The latter has size 2; the former may well be empty. But this presents no problem; empty sets are perfectly coherent conceptually.
The exact language as it was written is informal, ordinary language, which admits multiple interpretations. The riddle can logically have an answer of 2, on one common, reasonable interpretation. On other interpretations, it will not.
You could, but why would you want to? You’d unnecessarily mislead anyone reading your work, and I’m still not convinced it’s not an illogical statement.
If you had a reason to make such a statement, as in the case of someone asking, it would be more accurate to say. If there were any red dinosaurs, they would have been reptiles also due to our classification system. When you say “all red dinosaurs”, I believe that you have knowledge of the existence of red dinosaurs.
the question comes down to existence vs vacuous statements. does it allow for statements be made about A if A has no elements. unfortunately my mathematical logic books are packed away
Who knows? All kinds of conversations may be in my future; some in contexts where I’d want to say such a thing and leave it at that, some in contexts where I’d only say such a thing with follow-up disambiguation (which are still, however, situations in which I’d aver it), and some where I’d not want to say it at all. But let’s take another example, if it might illustrate contexts in which such speech is natural:
A quasiperfect number is a positive integer n whose proper divisors add up to 2n+1. With a little work, one can show that if n satisfies this property, then n must be odd. Thus, one might naturally say (and most number theorists would) “All quasiperfect numbers are odd”. This despite the fact that no one has ever proved that any quasiperfect numbers actually exist. Is this not a perfectly reasonable way of speaking?
Sorry, this got messed up in the editing process; a quasiperfect number is one whose proper divisors add up to n+1. (2n+1 comes when you also add n [its “improper” divisor] to that)