We cannot construct any more than 16 truth tables involving two
statements. This is because such a truth table has 4 rows and the truth
value of each row is T or F.
Click on the microscope for a list of 16 inequivalent truth tables.
However, we can certainly construct more than 16 statements
involving two statements. What happens is that many (in fact infinitely
many) statements have identical truth tables. We say that the statements
*r* and *s* are *logically equivalent* if their truth tables are
identical. For example the truth table of

shows that is equivalent to . It is easily
shown that the statements *r* and *s* are equivalent if and only if
is a tautology.

If statements *r* and *s* are equivalent we write .
For statements involving three statements, *r*, *s* and *t*,
there are 64 different truth tables that can be constructed. In general,
there are truth tables that can be constructed from *n* statements.

One common error is think that the statements and its
*converse* are equivalent, but they are not. The
conditional statement is equivalent to its
*contrapositive* statement .

We say that *r* *implies* *s* if *s* is true whenever *r* is true.
If *r* implies *s* then we write . Alternatively, *r* implies
*s* if and only if the statement is a tautology. We say that
*s* is *logically deducible* from *r*. For example, in a mathematical
theorem the hypothesis implies the conclusion or the conclusion is deducible
from the hypothesis.
Click on the microscope for a list of common equivalences and implications.
It is also important to spot equivalent statements when written in English and
not just symbollically.

Mon Sep 2 15:51:33 PDT 1996