Sound and Fury

Signifying nothing

Majority logic I

When I went for an interview at Oxford for my undergrad degree one of the things we had to do was a kind of logic test. It had things like “All men wear hats, some men wear ties” and you were asked what it was possible to conclude from that. Presumably they were looking for something like “Some men wear ties and hats”. One of the questions was “Most men wear ties, most men wear hats”. I said that it wasn’t possible to conclude anything from this. But it was then explained to me that if hat-wearers and tie-wearers are both in a majority, then there must be some who are both: the proper conclusion is that “Some men wear ties and hats”.

“All men” is standing in for the universal quantifier (x) and “Some men” stands for the existential quantifier (Ex). But “Most men” is awkward to put in similar terms without resorting to some sort of logic enriched with predicates relating to numerical proportions. But that sort of “statistical logic” is too complex. The fact that some predicate applies not to all but to a majority of some domain is an easily understood fact that does not need a whole logic of proportions behind it. Let’s make “Most of” a new kind of quantifier: (Mx). What properties does (Mx) have?

• $(Mx)Px \wedge (Mx)Qx \rightarrow (Ex)Px \wedge Qx$

This is the property my interviewer at Oxford was exploiting. If most x’s are P’s and most x’s are Q’s, then some x’s must be both.

• $(x)Px \rightarrow (Mx) Px$
• $(Mx) Px \rightarrow (Ex)Px$

These two properties simply say that if all men wear hats then most men wear hats and if most men wear hats then some men wear hats.

Do we really need a third quantifier here? Is there some way to express “most of” in terms of universal and existential quantifiers? I’m not sure there is. Here’s a first stab. When you say “(Mx)Px” what you are really saying is:

• $[(Ex)Px ] \wedge [(Mx)Qx \rightarrow (Ex)Px \wedge Qx]$

But that’s no good, because that expression still contains an “Mx”. We are kind of going beyond standard predicate logic, but only a little bit. Perhaps Mx shouldn’t be a quantifier, but a property of predicates? But I don’t think that is a particularly satisfying suggestion. A property of predicates would be a third-order entity, and that seems extravagant given the modest goal of formalising the easily understood concept of a majority.

Soon I’ll post again and discuss the “dual” of (Mx): “Only a few men wear hats” “Hat-wearers are in a minority”.

A note on the symbolism: I wanted to use $\forall$ and $\exists$ but I couldn’t think of a way to “invert” the letter “M” in a similar fashion. It shares the vertical symmetry of “A” but when reflected around a horizontal axis as $\forall$ is, you get a “W”, which is no good if the idea is to come up with a new symbol. So I reverted to old-style quantifiers. Suggestions regarding how to “symbolise” majority are most welcome.

Written by Seamus

May 26, 2009 at 10:34 am

Posted in logic, maths, philosophy

Tagged with

4 Responses

1. Your “first stab” at an analysis surprises – not only because it contains “(Mx)”, as you point out, but also because it contains an uninterpreted “Qx”. If Q is meant to be any old predicate, you need to go second-order and preface the second conjunct with “(Q)”.

It’s also worth noting that your remarks on the relationship between (Mx), (x) and (Ex) entail that universally-quantified statements are false on the empty domain, which is controversial.

On the symbolism, why not go with the standard existential and universal quantifiers and leave “(Mx)” as it is? You are introducing it, after all.

Brunellus

May 27, 2009 at 10:29 am

• You’re right about the uninterpreted Qx. Though since I didn’t think it was a viable analysis I didn’t think too hard about it. This I think mitigates my oversight.

As for the empty domain etc, since questions of a (non-trivial) majority only really make sense when you have at least three objects, I was kind of ignoring details like that.

And I kept (x) and (Ex) because it gives a certain uniformity or consistency to the symbolism.