Peter van Inwagen has a well-known argument for the univocity of 'exists', in the sense that 'exists' has the same meaning in all circumstances:
1. Numerical terms are univocal.
2. 'Exists' is a numerical term.
C. Therefore, 'exists' is univocal.
Like most of van Inwagen's arguments, this is superficially clever and utterly wrong. Van Inwagen takes (1) to be an obvious truth that no one would deny; as he puts it, if I have thirteen epics and you have thirteen cats, I have the same number of epics as you have cats. Unfortunately, not only is (1) not an obvious truth, it is obviously wrong; numerical terms are not univocal because there are different number systems, and while the number 1 in (say) a rational number system will be similar to, and obviously somehow related to, the number 1 in (say) a surreal number system, you cannot merely assume that they are exactly the same. If we are considering real numbers, -1 has no square root; if we are considering complex numbers, it does. If we are considering rational fractions, 1/0 has no defined meaning; if we modify the number system to conform more closely to projective geometry, 1/0 indicates a point at infinity outside the standard number line. And because of the way mathematics works, if you are getting different properties for the same numerical terms, it is because there is something different about the definition, and therefore the meaning. Numerical terms are not univocal; they are simply relatively easy to relate to each other because in mathematics we are working with definitions within precisely defined, and often extensively explored, number systems. The reason (1) might seem very plausible is because we consider examples like the epics and cats, where we are assuming the same number system (integers) and are counting things that are relevantly the same (sharply defined objects).
Van Inwagen recognizes (2) as something for which he has to argue. His argument goes back to an old idea in analytic philosophy, to Frege, in fact, who argued that affirmation of existence just was denial of the number zero. This has a plausibility to it; if we are trying to guess the number of apples you have and you say that you definitely don't have zero apples, we would usually assume that you have some apples. However, this also fails, and we can see this if we pick a non-counting case of the number of zero. Can we say that denial of the number of zero, when talking about degrees Fahrenheit, is affirmation of existence? Of what would we be affirming existence? If I deny that something is zero degrees Fahrenheit, you might think that this means I am saying that I am affirming the existence of some other degree. But that's not true -- I might be denying that something has temperature at all. I say that the Barbara syllogism is not zero degrees Fahrenheit, which seems to be a necessary truth, I'm not claiming that Barbara syllogism is some other temperature. So what could I could I possibly be affirming the existence of when I deny zero degrees Fahrenheit? You might hold that I'm making some sort of category mistake, but van Inwagen can't do that -- he is arguing that zero, which is a numerical term, has exactly the same meaning whether we are using it to count or to identify a point on a temperature scale, and he is saying that 'exists' means we are denying the number zero.
We should also consider the other direction. If affirmation of existence is just denial of the number of zero, this means that we can never affirm the existence of anything that is zero degrees. We can only (ex hypothesi) affirm the existence of something by denying that its number is zero, so we can never affirm of anything whose number is zero that it exists.
Very obviously the reason why 'affirmation of existence is denial of the number zero' often seems to make sense is that we are really saying that affirmation of having at least one countable existing thing is denial of having zero countable existing things; we are sneaking existence in by way of our normal and everyday counting practices, in which we are always counting existing things. We don't go around counting nonexistent men in the doorway. But we are not always using numbers to count, and not all of our use of numbers has a clear connection with existence.
So numerical terms are not univocal, and existence does not appear to be a numerical term but a presupposition for certain practices when we are using numbers; and it's also the case that 'exists' is not univocal. One reason why it couldn't be is analogous to the point about numerical terms: there are different logical systems and in the ones that make use of some kind of existence, 'exists' cannot mean exactly the same thing because it will have different properties in the different logical systems and will be defined differently in those different logical systems.
It's also a little weird to talk about the univocity of a term like 'exists'. I can obviously use the term 'exists' as a figure of speech or in an extended sense, because I can do that with any term, and one does this, for instance, in existentialism (which talks about existence in a way that definitely doesn't treat it as a numerical term) or when I say things like, "You don't exist to me" or "You exist to me", in which I am definitely not talking about whether I can count you. Terms do not have 'the same meaning' in themselves, but only the same meaning insofar as I use them as having the same meaning; you can't tell whether a term is being used univocally unless you look at how it is actually used. If you just look at all the uses, of course it won't be always used univocally; some of the uses, for instance, will be ironic or hyperbolic. So the only thing that you can mean by saying that 'exists' is univocal is that it keeps the same meaning when you aren't using it with a different meaning.
