Keith Douglas
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.
Existential Quantifiers
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.”
I’ll be a smartass again, even worse than before:
Your questions are almost entirely fundamentally meaningless as is. But they are good, despite that they don’t lead to answers without considerable further explication, but because they hopefully motivate people to actually get busy and learn modern formal logic by putting in a great deal of effort in the right place.
Now my strong feeling is to NOT study the subject as presented by people who would regard themselves as philosophical logicians—with a very few exceptions. Boolos and Jeffery would be excellent examples of the latter, though their joint book, now updated by the still living John Burgess, is pretty challenging for beginners.
Learn it from mathematical logicians. There certainly are good such very elementary books which do not depend on knowing much modern mathematics at all—only on being comfortable working with symbols. And be very thoughtful about the dichotomy of metalanguage versus formal language. Russell wasn’t, but did a lot for the progress of logic in the early 1900s. No one learns classical mechanics from Newton’s famous book, and you certainly shouldn’t try to learn Logic from Russell and Whitehead’s book, nor from anybody still stuck with his attempted foundation, as many philosophical logicians seem to waste everybody’s time by being so stuck back 100 years ago.
That may even be why some people think that Oppenheimer and Zalta actually contributed something of any value to either
1/ Anselm’s ontological argument supposedly converted into non-modal 1st order logic, or
2/ to anything of the putative use of artificial intelligence in metaphysics there.
This was actually 3 papers published 4 times (the double being in a theological journal!). These are a farce in which their (unknown to them) single premiss for an argument for conclusion C was tantamount to [1=1 implies C]. I use 1=1 as a substitute for another formula, very easily proved to be logically valid.
I know you know what I’m talking about—sorry to some other readers!
I’ve seen Z & O present the work and found it challenging not for the reason suggested (and I think there’s some misrepresentation going on), but because it relies on Z’s metaphysics – which takes a lot of getting used to. That’s why it is not exactly a pure ontological argument – it presupposes a version of mathematical Platonism, and that’s why – as they said in person – the proof is *not* one that Quine would say involves existential commitment to draw its conclusion. It is thus not like the cases I discussed in the column. I don’t buy the Platonism – or the Meinongism for that matter – so I don’t accept the premisses either. One of O & Z is still an atheist – they won’t say which one – and it is for the same reason as the other does not feel reinforced in his belief either – “God is not the number 1!!”
I have learned logic from philosophers of language, from philosophers of science, from proof theorists of both mathematical and philosophical bent (which is a false dichotomy, of course), and from philosophers who for lack of a better phrase I’ll call “Russell scholars”, and have literally dozens of books on the subject. (Including some by computer scientists, set theorists and category theorists, etc.) That is all to say, I don’t find *too* much of difference between them once one gets beyond the introductory (from the philosophy perspective – 100/1000 level courses.) There’s an interesting book – _Three Views of Logic_ which covers some of this. I only wish it was 5 at least – adding the linguistic and psychological views would be interesting too. With that in mind, I do agree with the idea that nobody should use _Principia Mathematica_ as a *contemporary* textbook (though by all means study it for its historical importance!!). Do contemporary philosophers in their presentations get sloppy with use and mention the way R & W famously (now) did? Not in my experience, but there are just so many 100-200 level books that one, so no doubt it happens somewhere.