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Conditional
Conditional (p -> q)
Read as any of the following:
If p then q
q, if p
p, only if q
All p are q
N.B. p if and only if q is NOT listed here. It is a
reading of the biconditional. 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 conditional says that when an antecedent (p) happens (is true) then some other thing (the consequent) (q) will also happen (be true). This is the opposite of saying the antecedent (p) must occur and the consequent (q) must not occur.
Hence ~(p ^ ~ q).
| p | q | p -> q | Comments |
| T | T | T | The antecedent happens and the consequent also happens so the statement is true. |
| T | F | F | The antecedent happens but the consequent does not happen |
| F | T | T | The antecedent does not happen, yet the consequent does happen. The statement is true be cause we did NOT say that consequent would happen ONLY if the antecedent happened. (That would be a biconditional.) |
| F | F | T | The antecedent does not happen and nor does the consequent so the statement is true. |
Derivation
| p | q | ~q | p ^ ~q | ~(p ^ ~ q) |
| T | T | F | F | T |
| T | F | T | T | F |
| F | T | F | F | T |
| F | F | T | F | T |
Proof
| ~(p ^ ~ q) | p -> q | ~(p ^ ~ q) <=> p -> q |
| T | T | T |
| F | F | T |
| T | T | T |
| T | T | T |
Since the result is a tautology our derivation and definition are logically equivalent.
