Logic Tutorial, Part 5.
Logical Inference

(c) 2017 by Barton Paul Levenson

Here's a fourth logical symbol, to be seen much less often than the other three:

≡ ("is logically equivalent to")

It applies whenever any two atomic statements (a and b, p and q, x and y, or whatever) always have the same truth value. If, for all possible values, the following truth table applies:

Then we can write a ≡ b. This can be used to show that double negations can be replaced by a simple atomic statement, or in other words, ¬¬a ≡ a.

Copying Waner and Costenoble (2001), while a double negative is okay to use in French--"Ceci n'est pas une pipe"--or colloquial English--"This ain't no pipe"--it doesn't hold in formal logic. In logic, a double negative negates itself out of existence.

We now have all the tools we need to write our first "law of inference," one of the rules of operation by which you can get a new premise from an old one logically:

If you construct a truth table for each side of this equation (and it would be a fairly big table), you can see that each side of the equivalence sign is logically equivalent to the other. In English, we have just said "a and b are not both true," and shown that it is logically equivalent to "either a is false or b is false."

De Morgan's Second Law is similar:

De Morgan's laws are the equivalent of a "distributive rule" for negation in formal logic.

Some rules apply only to compound statements. For instance, a tautology is always true whatever the truth values of its variables--for examine, "a ∨ ¬a". Construct a truth table for this expression to see what I mean.

A contradiction is always false whatever the value of its variables--for example, "a ∧ ¬a". Again, check it with a truth table.

A tautology is always true. A contradiction is never true.

The latter statement is more important than I can possibly say. Unless we accept the "law of noncontradiction:"

a is not not-a

then no formal logic is possible. If we do not accept this law, we cannot argue logically--or rather, we can make any wild statement we want, and can never prove it's either true or false. If you do not accept the law of non-contradiction, stop reading this web site now. It will be useless to you.

For those interested in logic, I continue. The following logical equivalences apply to every possible statement, whether atomic or compound: