Section Progress:

28.4 Definite Descriptions: “The”

Definite description: “The so-and-so.” As in:
“The tallest person in the room”;
“The oldest cat that I own”;
“The only child of Alexander the Great.”

In English the word “The” is called the definite article.

A complex noun phrase made with “the” is called a definite description.

Definite descriptions imply uniqueness: there should be exactly one thing fitting the description.

That fact helps us figure out how to translate them into FOL.

For example, say we want to write: “The king of France is bald.”

The logical form of it is: there is exactly one king of France, and he is bald.

This logical analysis of definite descriptions is due to logician Bertrand Russell.

Russell proposed it to solve a puzzle in the philosophy of language, which has to do with the law of the excluded middle.

The law of the excluded middle is Pv~P.It is necessarily true.

But there’s a problem if we apply the law like this:

The God of the Christian Bible is perfect or not perfect.

If that is a law of logic, then we can prove that the Christian God exists from logic alone!

Just imagine doing proof by cases on it.

Proof by cases. Case 1: The God of the Christian Bible is perfect. When the clearly he exists, if he is perfect. Case 2: The God of the Christian Bible is not perfect. When then he has some imperfection, and thus also exists. Either way, God exists. Done.

That proof is correct, as long as we are assuming the original disjunction.

The problem is that that disjunction is not a law of logic, and Russell’s analysis of definite descriptions helps us see why.

First formalize the sentence: “The God of the Christian Bible is perfect.”

∃x(God(x)&∀y(God(y)→x=y)&Perfect(x))

What this analysis shows is that there are two different ways to negate it. We can write:

~∃x(God(x)&∀y(God(y)→x=y)&Perfect(x))

or

∃x(God(x)&∀y(God(y)→x=y)&~Perfect(x))

Now answer this.

If we negate the sentence this way:

∃x(God(x)&∀y(God(y)→x=y)&~Perfect(x))

then the disjunction we assumed is this:

(1) ∃x(God(x)&∀y(God(y)→x=y)&Perfect(x)) v ∃x(God(x)&∀y(God(y)→x=y)&~Perfect(x))

We can put this more succinctly this way:

(2) ∃x(God(x)&∀y(God(y)→x=y)&(Perfect(x)v~Perfect(x))

From that claim, we can indeed prove that God exists. After all, just look at that wide scope existential.

But it is easy to see now that this is not an instance of the law of the excluded middle. Just apply the Truth-Functional Form Algorithm (TFFA) to these sentences, both (1) and (2).

Notice that neither one is Pv~P. Indeed neither one is logically true at all.

The logical truth is this:

∃x(God(x)&∀y(God(y)→x=y)&Perfect(x)) v ~∃x(God(x)&∀y(God(y)→x=y)&Perfect(x))

That does have TFF of Pv~P. But now that the negation is wide scope on the existential in the second disjunct, we can see that it does not entail the existence of God.

The correct logical analysis of definite descriptions therefore allows us to see why that proof of God’s existence doesn’t work.

If you’re interested in learning whether there are any proofs of God’s or gods’ existence that do work, then we hope you’ll take a class in the philosophy of religion!

28.4 Definite Descriptions: “The”