There are just as many F’s as G’s

February 4, 2008

Since Boolos, many people have come to believe that second order plural quantifiers are logical quantifiers. According to Boolos, they are topic neutral, ontologically innocent, and well understood.

One way to cash out topic neutrality would be to say that plural quantifiers are invariant under arbitrary permutations of the domain, but this seems right even if we go for a pretheoretic notion of topic neutrality. One might argue that plural quantification is simply quantification over sets. This would undermine ontological innocence if it where correct. Take for example:

(1) Some cats were making a racket outside

The proposal here being that there is a set of cats such that each cat in the set is making a racket. So according to this analysis, (1) entails the existence of sets. However, (1) doesn’t appear to entail any such thing, indeed the truth of (1) is presumably independent of the ontological status of sets.

Things get even worse the theory when we consider the apparent truism:

(2) There are some sets which are all and only the non self membered sets.

On the current proposal (2) would entail the existence of a set of all non self-membered sets. But as we know, no such set exists. The quantification at hand is irreducibly plural – we must take pains to note that we are quantifying plurally over individuals, not quantifying singularly over pluralities (or sets or classes.)

That plural quantification is well understood is probably the most controversial claim. The logicality of plural quantifiers is partly motivated by the legitimacy of plural quantification in English. This is why, for example, super plural quantification (the analogue of third order quantification, if you like) is not widely considered logical, despite being topic neutral and ontologically innocent (if it even makes sense.)

What does this entail for logic? Well, if we add plural quantifiers to regular first order logic you get a logic which is considerably stronger. For example, the Geach-Kaplan sentence can be used to give a categorical axiomatisation of PA, something which no first order sentence could do.

Ok, nothing new so far. What I wanted to think about, was whether there are any other English expressions which have this property. I.e. are topic neutral, ontologically innocent, well understood and when added to plural logic increase its strength.

The expression I had in mind was:

There are just as many F’s as G’s

First off: it seems to be topic neutral. It is definable in second order logic, so it is certainly invariant under arbitrary permutations of the domain.

One could argue that it’s not ontologically innocent. Someone taking this line might say that it involves full second order quantification over relations, plurally quantifying over pairs, or singularly quantifying over bijections (i.e. sets.) But this doesn’t sound right to me at all, for basically the same reasons as with the plural quantifiers. Consider

(3) There are just as many knifes as forks

(3) doesn’t seem to entail the existence of relations, ordered pairs or functions. Intuitively, (3) will remain true, even if there are no abstract objects. Similarly, (3) is independent of the consistency of ZFC – if set theorists discover an inconsistency somewhere, there will still be as many knifes as forks in my drawer. All (3) depends on is the knifes and the forks, and whether there are as many knifes as forks.

What happens when you add “there are just as many F’s as G’s” to plural logic? That is, when we add the relation xx \approx yy to plural logic, to be given the same truth conditions as the obvious second order formula. The resulting system is somewhere in between plural logic and full second order logic in strength: you cannot define equinumerosity in plural logic, and you cannot emulate full second order quantification over sets using only plural quantifiers and equinumerosity.

The real question is whether this English expression is as well understood as plural quantification. The problem is when F and G are infinite. For example:

(4) There are just as many rationals as naturals

This is a classic example where things go wrong: mathematicians take this sentence to be true, while an ordinary man on the street would take it to be false. Is the mathematician right and the man on the street wrong? Or is the mathematician using (4) in a very stipulative way. I’m inclined to think that the mathematicians use, even if it is different from the way ordinary people talk, is still ontologically innocent (thus sidestepping the issue.) However, this is a difficult claim to back up, since the mathematicians use is grounded heavily on an understanding of the set theoretic definition of equinumerosity in terms of bijections.

Perhaps there are many ways to extend the meaning of “there are just as many F’s as G’s” to infinite cases – this indeterminacy shouldn’t change the point at hand. For even if “there are just as many F’s as G’s” is false whenever F or G is infinite, we can still get a more powerful logic than plural logic, since we can define equinumerosity for finite pluralities.

Either way, it would be interesting to see what other English expressions have these properties?



  1. There’s the obvious ‘$xx$ is (Cantor) finite’. I’m inclined to think this is well understood (and innocent!) even given that the usual definition involves quantification over functions.

  2. Yes, I had thought about that one a little already. It wasn’t clear to me that it was particularly well understood, but maybe I was being a little hasty. I mean, we seem to have a fairly robust intuition about finiteness even before we learn the set theoretic definition. Also, it seems to be definable in terms of “there are an even number of F’s” and “there are an odd number of F’s” – I don’t know why, but for some reason I don’t have the same worries about these two being well understood. I certainly agree that it seems to be ontologically innocent.

  3. […] in the ZF metatheory, I’m not sure what you need for this in the RUW setting.) I argued here that “there are just as many F’s as G’s” is a logical […]

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