In classical logic* it is easy to confuse them, but they are different things.
Given two declarative statements to which a truth value can be assigned, an antecedent P (premise) and a consequent C (conclusion), we consider:
P → C (if P then C) a statement with a truth value
P ⇒ C (from P follows C) an inference: C is derivable by assuming P.
The crucial difference lies precisely in this deducibility obtainable in the inference. Instead, the implication P → C is true even when P is false. Therefore it can happen that P is false, C is false, and yet P → C is true. In particular, if P is false, P → C is true whatever C is. For this reason, the implication by itself does not establish (nor prove) the truth of C.
Instead, from P follows C is legitimate only when P is available as a hypothesis in the reasoning: and it is the typical case in which, if P and P → C, then one can conclude C. But if P is false (or is not assumed), the implication is said to be ‘vacuously true’ and it does not authorize any conclusion about C.
If and only if: equivalence as a double inference
The properties of inference make it very useful: a large part of mathematical proofs is based on if and only if, which expresses an equivalence between two statements A and B.
To say:
A if and only if B
means precisely to prove both directions:
A ⇒ B (A is sufficient for B)
B ⇒ A (B is sufficient for A)
When both inferences are valid, A and B are logically equivalent, therefore they have the same truth value.
Classical logic *
This is how one refers to those logical systems (the most used in basic mathematics) in which laws and rules considered traditional hold, such as:
Law of excluded middle: every statement is either true or false (P or not-P)
Non-contradiction: P and not-P cannot both be true
Double negation: negating twice brings one back to the original statement
Explosion (in the presence of contradiction): from a contradiction one can derive any conclusion.‘Non-classical’ logics arise from wanting to give up or modify some of these features. By way of example:
intuitionistic logic: does not accept the law of excluded middle as a general rule
paraconsistent logics: do not accept explosion (they tolerate contradictions without making everything ‘collapse’)
modal logics: add operators such as necessarily / possibly.
