Numerical Propositions

Without identity we cannot express propositions about specific quantities in QL. With identity, we can do exactly that. This is important, since with identity QL has the capacity to express truths of mathematics. 

Without introducing identity we have propositions about nothing: "Nothing is funny" is translated as (x)~Fx or as ~(∃x)Fx.  And, of course, we can talk about everything, and say for example, that everything is funny: (x)Fx. With identity we can express propositions such as:

1. There is exactly one pen.

If, using "Px" for "x is a pen" we attempt to translate this as:

(∃x)Px    <-- not correct

we don't capture the meaning of (1), since all we're saying is that there is at least one pen. We need to say that there is at least one pen and at most one pen! How do we do that? We need to say that there is at least one pen, and for anything else, if it's a pen then it is identical to that pen. We translate as follows:

(∃x)(Px & (y)(Py ⊃  x = y))

Read directly, this says that there is an x, such that x is a pen, and for all y if y is a pen then y is identical to x.

How would we translate:

2. There are at least two pens.

Notice that (2) says that there are at least two pens. That's different from saying "There are exactly two pens." and different from "There are at most two pens."  We'll treat all these sentences in order.

Back to sentence (2). The key idea here is that there are two different pens. There's an x which is a pen and a y which is a pen, and x is not identical to y. So we have:

(∃x)(∃y)((Px & Py) &  x ≠ y)

Contrast this with a translation for:

3. There are exactly two pens.

We have two pens, but if anything else is a pen, other than those two things, it is either identical to one or the other. We translate as follows:

(∃x)(∃y)(((Px & Py) &  x ≠ y) & (z)(Pz ⊃ ((z = x) v (z = y)))

This looks complicated, but it's really not. However, it is very important to pay attention to the scope of our quantifiers. Our two quantifiers at the left range over the entire wff. We must make sure that this is unambiguous through the proper use of our grouping indicators.

How do we translate "at most" sentences, such as:

3. There are at most two pens.

Here we need to make it clear that if x or y or both are pens, anything else that is a pen is either x or y. So we have to replace the initial "&" in our last translation with the wedge, and we have to make the quantification universal, because we're not committed to there being two pens. We're just specifying an upper limit.  Thus we have:

(x)(y)((Px & Py)  ⊃ (z)(Pz ⊃ ((z = x) v (z = y))))

Note the similarity to the "exactly" case. The sentence "There are exactly two pens" uses existential quantification and a non-identity clause, but otherwise is syntactically just like the "at most" case. Here we don't use existential quantification, since we're placing an upper limit on the number of pens, without committing ourselves to the existence of any particular number of pens. It can be true that there are at most two pens, when in fact there's actually one.

Exercise: Translating Numerical Propositions