Identity and Non-Identity Propositions

We introduce two symbols, "=" and "≠ " to express identity and non-identity respectively. The simplest identity and non-identity propositions are about individuals, abbreviated by individual constants. So we can express:

1. Fred is Mr. T.

where we abbreviate "Fred" with "a" and "Mr. T." with "b" as:

a = b

The negation of (1) is:

a ≠  b

We can also use quantifiers and individual variables. For example, we can express:

2. Someone is identical to Fred.

as:

(∃x)(x = a)

and we can say more complex things, like:

3. Everything identical to Ignat is bubbly.

which can be translated as:

(x)((x = a) ⊃ Bx)

where "Bx" is "x is bubbly" and "a" is "Ignat". Notice that this is a universal affirmative categorical proposition.

It's important to note that not every "is" is the "is" of identity. When we say "Fred is tall" we're not making an identity claim. That this is so should be obvious from the fact that "tall" is not a name.

Here's a translation exercise to get you in the swing of things with identity.