The notion of consequentia
Etymologically, ‘consequentia’ suggests a following along. In medieval philosophical literature it was apparently quite proper to say that one concept follows another – e.g., that animal follows man – but more generally consequence was thought of as involving entire propositions.
There are, of course, many different relationships in which propositions can stand to each other. For instance, in a conditional proposition of the form ‘if p then q’ the proposition taking the place of ‘p’ is the antecedent of the conditional proposition, and the proposition taking the place of ‘q’ is the consequent. The relationship between that antecedent and the consequent in a true conditional is called implication by modern logicians. Again, two propositions may be related to each other in such a way that the first cannot be true unless the second is true also; and the relationship between the two propositions in that case is called entailment. Again, two propositions may constitute an argument. In the argument-form ‘p; therefore q’ the proposition taking the place of ‘p’ is the premiss and the proposition taking the place of ‘q’ is the conclusion. To employ an argument is to derive or infer the conclusion from the premiss (or premisses), and so the relationship between those two propositions is called derivation or inference.
Implication, entailment, and inference are all distinct from one another. For instance, a conditional proposition, like other propositions, is accepted or rejected by being classified as true or as false; an argument is neither true nor false but is accepted or rejected as valid or as invalid.