Keith’s Conundrums: Existential Quantifiers
Last time I asked about what grounds truth values. To help you in your quest to understand truth makers, I will give some discussion of each of the examples I gave to show how complicated the business is. (Please refer to the earlier column for the items. I haven’t reproduced them here.)
0. This is Newton’s law of gravitation, stated in modern terms and in words rather than as an equation with correspondence rules. The first thing to realize is that this is a universally quantified statement (or so it is under most logical formalizations). In fact, it “quantifies over” gravitational forces, masses, bodies, and distances. So it would appear that all those properties and things (bodies represented by “individuals” in the logical sense) are the truth makers here.
Note that we can be wrong here. One could think that it “takes three to tango” (rather than two) and hence one has to consider how triples behave. Does this require “mathematical objects” to be part of the truth maker — the “squaring function” and the “division function,” for example? I think not. But the traditional theories of reference might well make one think so.
Note also that this is an area of contention — the point for you to consider is whether or not this is necessary. What happens if one does include them, or not, and so forth? I’ve dweled at length on this example, as oftentimes philosophical semantics use “toy” examples from ordinary language rather than the more (logically and metaphysically) complicated examples from science.
1. This one is subjective, but it may well be true. It seems plausible to say that it is like any other proposition if it says of what is that it is and of what is not that it is not, to paraphrase Aristotle. If that’s granted, then presumably my likes are associated with me in some ways (I would have it they are relational properties between my nervous and perhaps endocrine systems and the ice cream). So the truth makers are those, whatever they are.
I do not think it commits one to having “I” as a truth maker except in a deflationary sense. If there are likes, then whatever it is that likes is enough.
2. This one is, in my view, almost exactly like 1. It differs in two ways. One is that there’s no danger of getting hung up about “I.” Another is that “Joan” is perhaps vague — there are lots of people named “Joan,” some of whom may have even lived before there were cellos. If one fixes reference somehow, then it is presumably true or false in virtue of similar truth makers to that of my liking ice cream.
3. This one is quantified, this time with a so-called existential quantifier. (More on this in the puzzle for this month.) So it seems plausible to hold that this works much like 1 and 2, only it is the collection of people that do in fact like anchovy pizza rather than the individual me or Joan of the earlier examples or their bodily systems. Like before, the anchovy pizza is also part of it, otherwise you could not distinguish truth makers for this statement from one of identical logical form.
4. This one is a negative universal. These are tricky — even if one grants that it is 100% true. According to some, it requires all people for its truth maker, at least — understood to mean “this person and that person and that person and still again that person, etc.” But is there need for something else? Some say yes, something that says “and these are all the people” (which sounds circular, yes). Unlike the “this is the church” hand game that bored me as a child, there’s no easy way to “cap” the list here. I’ll leave you to think about this.
5. This one is tricky, and a point of great debate. It is an ethical claim, and the debate would centre around whether or not ethical claims are propositional — i.e., have truth values. In fact, part of the problem with claiming that they are is precisely the difficulty in locating a truth maker for them.
In this case, all might grant that human beings, streets, the negative (handled like in 4) “unarmedness” and such are part of the puzzle. But what then grounds the “wrong”? After all, if one simply enumerates those, there’s nothing to distinguish it being a truth maker for the opposite claim (e.g., “Killing unarmed humans at random on the street is morally obligatory”). Often times the answer is some appeal to human nature and frustration of interests. But this is a programmatic sketch, not an answer. It does not quite answer the question, even if one were to fill it in. It still may well be that one is appealing to other values.
6. This one is like 5, except it might be false, even if one assumes that it is an ethical statement and hence has a truth value. If it is false, then presumably it has no truth maker. Does it thereby have a false maker?
7. This is not a proposition and hence has no truth value and hence has no truth maker to ground it.
8. This is like 7, except it is yet another use of language, one popularized by the great Cracking the Cryptic YouTube channel.
9. Here the first tricky bit lies in discovering the referent to “everyone.” It is also ambiguous — does everyone love some lover or other, or is there some one lover that everyone loves? Settling the ambiguity is necessary. This seems to suggest that only well formed propositions (rather than statements, which can be ambiguous) can have truth makers. Once that’s settled, it may well be false, in which case it has no truth maker. But note that sharpening out the ambiguity is needed in order to test and hence to know this.
10. This is a statement, and is almost true, sometimes. Some small proportion of liquid water (roughly one part in 10 million under usual conditions) is at any given time H3O+ and even larger ions. What effect does this have? Arguably none. Like before, it seems that since the statement is false, it has no truth maker.
But to many this seems too quick. It seems plausible to suggest that there are truth makers for statements which are partially true. Something in the world is almost as the statement has it, to whatever degree. If that’s so, it seems plausible to say that hydrogen, oxygen, and water everywhere are the truth makers. But this requires a theory of partial truth — something that is very contentious in the literature.
11. This one returns us to our discussion about mathematics. Does this require “1” to exist to have a truth maker? What about “the equals,” as Plato would have put it? (In our terms, equality is a relation, not a thing, so this is not exactly felicitous as an expression.) Factualism (or realism) about mathematics seems to require it — or some set theoretic surrogate, perhaps.
12. There’s a twist to this. It might look like it is false, even if one grants that mathematical statements have truth makers. However, there’s another problem. By saying that you’ve assumed that I was talking about “ordinary” arithmetic, not (say) arithmetic modulo 2, which assembly language programmers and computer architects will know and love. Is this another trap of ambiguity? Perhaps.
Understanding “ordinary language” uses of mathematics is an interesting part of the debate over mathematical realism. Note that if one assumes that what I meant was actually “true” in modulo 2 arithmetic, one has to presumably distinguish one of the items from its “ordinary” counterpart, and find another truth maker. This gets tricky. There’s a position in the literature that allows for all “set theoretic universes” to be the scope of an expanded set theory (not just, say, the cumulative hierarchy familiar to some) but I find it very difficult to state the position in realist terms. As a fictionalist, I’m OK with it, even if this proposal runs into a weird sort of inconsistency similar to that alluded to by P. Grim in his underappreciated The Incomplete Universe.
13. This one is easy.
14. And this one is very, very, very hard. There is a quantification over “statements” here, so presumably every statement is the truth maker here. There are some people who think that this actually has a very simple truth maker — literally everything. But that’s not quite right. And in either case, how does one argue with Brouwer, who thought that this statement was false? Can one simply point to everything and claim of it that it applies? Arguably not, since that, in fact, begs the question: maybe. I leave this to people with much better metalogical brains than I.
On to this month’s conundrum. We are taught in sophomore-level logic that “Zeus and Hera are Greek gods” can be rendered “There is something called Zeus and something called Hera and each of those is a Greek god.” In turn we are told to formalize these uses of “is” using the “existential quantifier.” It is here that the trouble begins. Does this not suggest that the statement is in fact claiming that there are Greek gods? Are we in danger of proving the existence of gods by logic alone?
Kant famously thought the way out was to hold that “existence is not a predicate,” and that that solves the problem. The quantifier is not a predicate, after all, and the name is just unfortunate. Yet we also have closer-to-our-day people like Quine claiming that we “commit to the existence of something” by “quantifying over it.”
Who is right? Kant seems to make an ad hoc move (is it only by the consequence that we know, supposedly, that existence is not a predicate); Quine agrees that there is no existence predicate yet he says we still make “existential commitments” with quantifiers. Yet he also is an atheist (in general, not just about Greek gods). Presumably he has to say “Zeus is a thunderbolt thrower” is false, as is “Zeus is not a thunderbolt thrower.”
Your mission this week, should you choose to accept it, is to contribute to your self-understanding (and maybe that of others) on how to resolve this puzzle. Hint: Formalize, without commitment to its truth or falsity: “Some Greek gods exist, but they are really space aliens.”