Chapter Nine, Tutorial One
An Introduction to Identity and Symbolization

Not to worry! There will be no identity crisis in this chapter. We will not be concerned with personal identity or meaning in life. Identity for the logician is just the familiar mathematician's relationship of equality. Something x is identical to something y just in case they are one and the same thing: x equals y.

We are familiar with arithmetical equality between numbers. For this we use '=', the equal-sign. The same usage is appropriate in logic except that we talk about the identity of things in general, not just numbers, and we will use 'I' as our sign for the identity.

So, for example, instead of writing "4+7=11", we need to write something like


(symbolizing using s: the sum of 4 and 7, e: 11, Ixy: x=y, the only use of 'I' to made in this chapter.) Later in this chapter, we will introduce complex names using functions, to write


where "{f+s}" is the complex name made out of the funny, set brackets to group, names for four and seven, and the function symbol '+'.

But what use can "numerical identity" have outside an arithmetic class? How can this relationship, one which holds only between an object an itself!, have any importance to us in a logic class?

For one thing, we all make use of identity when we have two names for one thing. For example, the highest mountain in the US is often (in the last century) called "Mt. McKinley". But it is more frequently called by one of its native American names: "Denali". So, McKinley is Denali, McKinley=Denali. Two names but one and the same mountain.

English is a bit unclear about expressing identity. Though we often use the word "is" to indicate identity as the example above described:

(*) McKinley is Denali

we may also use "is" differently, to ascribe a property:

(**) McKinley is lofty.

We should be able to see that the first is to be symbolized (using the obvious key) with the identity symbol,

(*)'s symbolization:        Imd

while the second takes a familiar form using a one-place predicate,

(**)'s symbolization:        Lm

When we use the English word "is" to indicate identity, as in (*) we call this the "is" of identity.

When we use "is" to attribute a property and apply a predicate, as in (**), we call it the "is" of predication.

(I'll let the reader make the obvious jokes about what the meaning of the word "is" is.)

Now, which of the following English phrases include the "is" of identity?

  1. Tom is happy.
  2. Carola is a student.
  3. Carola is the best student in class.
  4. Skipper isn't the same person as William Trent.