TAUTOLOGY
An expression of propositional logic is a tautology iff its truth table yields nothing but 1's
(i.e., a tautology is true under any circumstance).
E.g. "p or not p" (either it's raining or it's not)
Do the truth table for (p or not p) and you'll see that you end up with nothing but 1's. The
expression "p or not p" is true under any circumstance, so it is a tautology.
CONTRADICTION
An expression of propositional logic is a contradiction iff its truth table yields nothing but
0's (i.e., a contradiction is always false).
E.g. "p and not p" (it's raining and it's not raining)
Do the truth table for "p and not p" and you'll get 0's in your final column. The expression
"p and not p" is never true, so it is a contradiction.
CONTINGENCY
If an expression of propositional logic is not a tautology or a contradiction, it is
contingent (it is sometimes true and sometimes false).
EQUIVALENCE
Two expressions of propositional logic are equivalent iff their truth tables yield the same
values (i.e., they are equivalent iff they are true under exactly the same circumstances).
E.g. "p  q" and "~q  ~p" are equivalent (by the law of contraposition)
Do the truth table for each one and you come out with the same result for both. This
demonstrates that the expressions are equivalent (whenever one is true, the other is true
and whenever one is false, the other is false).