Sequent - Intuitive Meaning

Intuitive Meaning

The intuitive meaning of the sequent is that under the assumption of Γ the conclusion of Σ is provable. Classically, the formulae on the left of the turnstile can be interpreted conjunctively while the formulae on the right can be considered as a disjunction. This means that, when all formulae in Γ hold, then at least one formula in Σ also has to be true. If the succedent is empty, this is interpreted as falsity, i.e. means that Γ proves falsity and is thus inconsistent. On the other hand an empty antecedent is assumed to be true, i.e., means that Σ follows without any assumptions, i.e., it is always true (as a disjunction). A sequent of this form, with Γ empty, is known as a logical assertion.

Of course, other intuitive explanations are possible, which are classically equivalent. For example, can be read as asserting that it cannot be the case that every formula in Γ is true and every formula in Σ is false (this is related to the double-negation interpretations of classical into intuitionistic logic, such as Glivenko's theorem).

In any case, these intuitive readings are only pedagogical. Since formal proofs in proof theory are purely syntactic, the meaning of (the derivation of) a sequent is only given by the properties of the calculus that provides the actual rules of inference.

Barring any contradictions in the technically precise definition above we can describe sequents in their introductory logical form. represents a set of assumptions that we begin our logical process with, for example "Socrates is a man" and "All men are mortal". The represents a logical conclusion that follows under these premises. For example "Socrates is mortal" follows from a reasonable formalization of the above points and we could expect to see it on the side of the turnstile. In this sense, means the process of reasoning, or "therefore" in English.