Reasoning: An Introduction to Logic, Sets, and Functions

Subsection 7.3.7 Back to the Law of the Excluded Middle

Recall that the Law of the Excluded Middle (LEM) says that every logical sentence has a truth value: it must be true or false.

Take another look at the second Law of the Excluded Middle video that we watched back at the time of our discussion of the importance of the LEM is as a theorem-proving tool. Recall that we considered a set of English sentences that appear to challenge the LEM.

The key to understanding why these examples don’t in fact challenge the LEM as a logical tool is that their issues are linguistic. The way to resolve them is with a careful mapping of the English sentences into the language of formal logic.

Consider: 4 and 7 are friends.

The logical predicate Friends isn’t defined on numbers.

Consider: The king of France has red hair.

When we translate sentences with presuppositions into logic, we must make the (implicit) presuppositions explicit.

Consider: The chili is hot.

While English adjectives are often vague, logical predicates cannot be.

Let’s now look at a new example, where we must be careful in how we translate an English adjective into a reasonable logical predicate.

Consider: John was brave.

Suppose that John died peacefully, as an old man, never having faced any situations in which bravery was called for. Could one argue that neither:

Brave(John) nor Brave(John)

is true? If so, we have a challenge to the LEM. But let’s look more closely at how the English word “brave” can reasonably be translated into logic. What should its definition be?

How about:

Brave(x) iff: HasShownBravery(x)

Then, (because he had no opportunity to behave bravely), Brave(John) is true. The LEM is still intact, but this may not seem fair to John.

So how about a definition that describes a personality trait, independent of whether that trait has had a chance to manifest itself:

Brave(x) iff: WouldShowBraveryWheneverNecessary(x)

Now we can say that one of Brave(John) or Brave(John) is true. We’ll just never know which (because John never got a chance to show us). But, again, the LEM is intact. It says only that there is a truth value. Not that we know what it is.

Nifty Aside

This example is due to Michael Dummett.

Finally, let’s consider a class of sentences called Liar Paradox sentences, which appear to pose a substantive threat to the LEM.

Consider this sentence, which we will call S: This sentence is false.

If S is true then, taking it at its word, it is false.

If, on the other hand, S is false, again taking it at its word, it cannot be false.

It appears, then, that S is neither true nor false.

But, again, we must dig deeper. Exactly how shall we represent liar sentences in our logical language? Liar sentences are self-referential: They make a claim about themselves. So we need a way to encode their claims and then assert something about those encoded claims. There is no straightforward way to do that in the first-order predicate logic language that we have defined.

Using other logical mechanisms, however, there are ways to handle Liar sentences. Here are two:

• Define a set of logical languages, arranged in levels. At one level, one can make claims only about sentences at some lower level. Thus no sentence, on any level, can make a claim about itself. (This is the Alfred Tarski solution.)

• Interpret every sentence as implicitly asserting that it, itself, is true (in addition to whatever else it says). So we translate the Liar sentence S, above, as:

This sentence is true ∧ This sentence is false.

The challenge to the LEM has disappeared. That sentence must be false (since every statement of the form P and ¬P is false). (This approach is due to Arthur Prior, among others.)

Big Idea

The Law of the Excluded Middle applies to logical statements, not English ones. And the mapping between English and logic is not always straightforward.