Logic

The study and process of the principles of valid reasoning, inference, and proof, concerned with whether the reasons presented for a claim – if those reasons were true – would justify accepting the claim.

A branch of philosophy in which rules are explicitly stated for deriving correct conclusions, and exploring legitimate and erroneous inference and argument – the ‘science’ which describes and evaluates such arguments.

More generally: a system or instance of reasoning or inference;

sensible thought, argument, or actions taken;

the relationship between particular events, situations, or objects, etc., and the ensuing consequences; 

means of convincing or proving (something, e.g., a proposition).

Illogical

Not following or reaching conclusions which are not in accordance with such rules of logic;

also, by extension, unreasonableness or perversity in one’s thought processes, behaviour, or responses.

(see also: induction)