LESSON 14: Expressing Identity

Assignment 14

The last new topic for the course is how we make use of the idea that one individual is the same as another. For this we have another symbol with the equals sign (and we do not use it in parenthesis, even though we can negate the relationship by using the tilde in front).

Let's imagine we want to say the individual you know as Al and the individual I know as Bob are really the same person. We'd use a=b, and we have an inference rule that if there is just a single person anything we say is true of one named individual has to be true of the other: a=b, Fa |- Fb LL (for Leibniz's Law)

The most common use is to express the number of individuals we are talking about (at least for very small numbers). Suppose we want to say that there is more than one person who is a student. We'd use

Ex(Sx & ~x=y) Suppose we want to say there is just one person who is a student (and the existential quantifier by itself does not carry this thought) . We'd use Ex(Sx & Ay(Sy -> y=x)) And, finally, maybe we want to make the point that there are only two students. ExEy(Sx & Sy & ~x=y & Az(Sz -> (z=x v z=y))) This is certainly a very complicated way of doing things, and it shows the overall limitations of trying to express even fairly simple ideas from our natural languages.

A classic example of using such expressions is in what we call definite descriptions. Imagine we want to express the idea that the mayor of California is a student, which we know is false. We do not want to just say Ex(Mxc & Sx), since it leaves it open just what is supposed to be false (are we saying that there is a mayor but he is not a student?) and so might suggest the wrong thing. The solution proposed by Bertrand Russell in his discussion of definite descriptions is to have something more like this: Ex(Mxc & Ay(Mxc -> y=x & Sx))

Go on to Assignment 14.