Biconditional
Biconditional (p -> q)
Read as any of the following:
p if and only if q
q if and only p
If p then q, and conversely
If q then p, and conversely
N.B. If p then q is NOT listed here. It is a
reading of the conditional. Confusing these is a common error among casual
users. These two operators should be studied together so the contrast can be
seen.
In the conditional and biconditional compound statements the p sub-statement is called the antecedent. The q sub-statement is called the consequent. Understand this terminology is helpful here and necessary when working with syllogisms.
The biconditional says that when an antecedent (p) happens (is true) then some other thing (the consequent) (q) will necessarily happen (be true). The biconditional can be defined by taking the reading "If p then q, and conversely," and writing it as "If p then q and If q then p."
Hence (p -> q) ^ (q ->p).
| p | q | p <-> q | Comments |
| T | T | T | The antecedent happens and the consequent also happens so the statement is true since both are true. |
| T | F | F | The antecedent happens but the consequent does not happen, so the statement is false since if one happens (is true) then other must happen (be true). |
| F | T | F | The antecedent does not happen but the consequent does happen, so the statement is false since if one happens (is true) then other must happen (be true). |
| F | F | T | The antecedent does not happen and nor does the consequent so the statement is true since neither happen (are true). |
Derivation
| p | q | (p -> q) | (q ->p) | (p -> q)^(q ->p) |
| T | T | T | T | T |
| T | F | F | T | F |
| F | T | T | F | F |
| F | F | T | T | T |
Proof
| (p -> q)^(q ->p) | p <-> q | ~(p ^ ~ q) <=> p -> q |
| T | T | T |
| F | F | T |
| F | F | T |
| T | T | T |
Since the result is a tautology our derivation and definition are logically equivalent.
