Final answer:
A conditional statement 'If P, then Q' and its contrapositive 'If not Q, then not P' are logically equivalent. The converse 'If Q, then P', inverse 'If not P, then not Q', and biconditional 'P if and only if Q' statements are not necessarily logically equivalent to the original conditional. A counterexample disproves a statement by providing a case where the statement does not hold true.
Step-by-step explanation:
To determine logical equivalences between a conditional statement, its converse, and other related statements, let's define what these statements are. A conditional statement is structured as 'If P, then Q' where P is the antecedent and Q is the consequent. The converse flips the components: 'If Q, then P'. The contrapositive negates and flips: 'If not Q, then not P'. Lastly, the biconditional combines the conditional and its converse: 'P if and only if Q'.
To illustrate, let's take the conditional statement 'If you study, then you will pass the exam'. The converse would be 'If you pass the exam, then you studied'. The contrapositive is 'If you do not pass the exam, then you did not study', and the biconditional is 'You study if and only if you pass the exam'.
The inverse negates both components of the conditional: 'If not P, then not Q'. However, the inverse not logically equivalent to the conditional, whereas the contrapositive is. Using our example, the inverse would be 'If you do not study, then you will not pass the exam'.
The law of noncontradiction states that a statement and its negation cannot both be true simultaneously. The law of the excluded middle says that for any proposition, either that proposition is true, or its negation is true. These laws together imply that for any given statement, there must be a definite truth value without a third possibility.
A counterexample is an example that disproves a statement, particularly in the context of universal statements or conditionals. If a statement claims something is true for all cases, a single counterexample where the statement is false is enough to invalidate the entire claim.