Tutorial Seven
Quantifiers and Variables

We have been developing a symbolic language. This language is often called "Predicate Logic". In this tutorial we add quantifers and variables to predicate logic's lexicon. And at this point, we give the language a name: "PL".

The most important new feature of our language PL is quantification. We will be able to symbolize quantity terms like "some" and "all". The following examples will help you see why this is valuable and at the same time give an idea of symbolization technique.

PL was first explicitly used for mathematics. It has nothing in particular to do with mathematics, but mathematical logic was the starting point for predicate logic and a good place for us to begin: the examples should be quite familiar to you.

Let's suppose that we are just talking about positive integer numbers. You know, the counting numbers: 1, 2, 3, 4 and so forth. We may then...

Take 'a' as a PL name for 1, 'b' for 2, and so forth.

Of course we'll quickly run out of names unless we subscript. But we'll only need to name the first few numbers anyway.

Also, let's use 'L' to represent the "less than" relationship and 'E' to represent the predicate "is even".

We already know how to symbolize some basic arithmetic: to say "1<2" in PL we write:

Lab

To symbolize "Two is even but one is not" we write:

Eb&~Ea

Thinking about the numbers will help us see how to quantify. For example, the English

There is an even number less than three

means that there is at least one thing x, which is even and less than three. In other words:

(*) There is an x, x is even and x is less than three.

Bear with me! There's a reason to go through this example involving the variable x. Let's begin translating into PL; the above comes to:

(**) There is an x such that: Ex & Lxc

Right? (**) is a direct translation into a hybrid form -- half English, half PL. Make sure you see this. And the "there is an x that ..." just means "there is some number making ... true".

OK? So, (**) is true because there is a number x (namely the number two!) which is even and less than 3.

The only thing new here is that (**) involves the quantity phrase "there is an x such that". In PL we will abbreviate this phrase as "(%x)". The backward-E means "there is" or "some" or "at least one".

So, "there is an even number less than three" can be symbolized as

(%x)(Ex&Lxc)

I.e., there is a number x, x is even and x is less than 3.

We use the parentheses around 'Ex&Lxc' so that the quantifier '%' can be seen to apply to that whole phrase.

Again you should mostly think of reading the quantifier phrase "(%x)" as "there is at least one thing x such that...". Here it means, "there is at least one thing making 'Ex&Lxc' true." And yes there is; in fact there is exactly one counting number which is both even and less than three.

Now, how would you symbolize...

There is at least one even number which is less than both 4 and 6?

Click on the best symbolization from those below...

(Remember, 'd' is a name for 4 and 'f' for 6.)