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Logical equivalence and implication

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 tex2html_wrap_inline198


shows that tex2html_wrap_inline198 is equivalent to tex2html_wrap_inline116 . It is easily shown that the statements r and s are equivalent if and only if tex2html_wrap_inline214 is a tautology.

If statements r and s are equivalent we write tex2html_wrap_inline220 . For statements involving three statements, r, s and t, there are 64 different truth tables that can be constructed. In general, there are tex2html_wrap_inline228 truth tables that can be constructed from n statements.

One common error is think that the statements tex2html_wrap_inline116 and its converse tex2html_wrap_inline112 are equivalent, but they are not. The conditional statement tex2html_wrap_inline116 is equivalent to its contrapositive statement tex2html_wrap_inline238 .

We say that r implies s if s is true whenever r is true. If r implies s then we write tex2html_wrap_inline252 . Alternatively, r implies s if and only if the statement tex2html_wrap_inline258 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.  

Peter Williams
Mon Sep 2 15:51:33 PDT 1996