2.5: Negation

In chapter 1, we considered as an example the sentence,

The Earth is not the center of the universe.

At first glance, such a sentence might appear to be fundamentally unlike a conditional.  It does not contain two sentences, but only one.  There is a “not” in the sentence, but it is not connecting two sentences.  However, we can still think of this sentence as being constructed with a truth functional connective, if we are willing to accept that this sentence is equivalent to the following sentence.

It is not the case that the Earth is the center of the universe.

If this sentence is equivalent to the one above, then we can treat “It is not the case” as a truth functional connective.  It is traditional to replace this cumbersome English phrase with a single symbol, “¬”.   Then, mixing our propositional logic with English, we would have

¬The Earth is the center of the universe.

And if we let W be a sentence in our language that has the meaning The Earth is the center of the universe, we would write

¬W

This connective is called “negation”.  Its syntax is:  if Φ is a sentence, then

¬Φ

is a sentence.  We call such a sentence a “negation sentence”.

The semantics of a negation sentence is also obvious, and is given by the following truth table.

Φ ¬Φ
T F
F T

To deny a true sentence is to speak a falsehood.  To deny a false sentence is to say something true.

Our syntax always is recursive.  This means that syntactic rules can be applied repeatedly, to the product of the rule.  In other words, our syntax tells us that if P is a sentence, then ¬P is a sentence.  But now note that the same rule applies again:  if ¬P is a sentence, then ¬¬P is a sentence.  And so on.  Similarly, if P and Q are sentences, the syntax for the conditional tells us that (P→Q) is a sentence.  But then so is ¬(P→Q), and so is (¬(P→Q) → (P→Q)).  And so on.  If we have just a single atomic sentence, our recursive syntax will allow us to form infinitely many different sentences with negation and the conditional.