Different presentations of the principles of logic reflect different approaches to the subject itself. The three kinds of system discussed here treat as fundamental not logical truth, but consequence, the relation holding between the premises and conclusion of a valid argument. They are, however, inspired by different conceptions of this relation. Natural deduction rules are intended to formalize the way in which mathematicians actually reason in their proofs. Tableau systems reflect the semantic conception of consequence; their rules may be interpreted as the systematic search for a counterexample to an argument. Finally, sequent calculi were developed for the sake of their metamathematical properties.
All three systems employ rules rather than axioms. Each logical constant is governed by a pair of rules which do not involve the other constants and are, in some sense, inverse. Take the implication operator ‘→’, for example. In natural deduction, there is an introduction rule for ‘→’ which gives a sufficient condition for inferring an implication, and an elimination rule which gives the strongest conclusion that can be inferred from a premise having the form of an implication. Tableau systems contain a rule which gives a sufficient condition for an implication to be true, and another which gives a sufficient condition for it to be false. A sequent is an array Γ⊢Δ, where Γ and Δ are lists (or sets) of formulas. Sequent calculi have rules for introducing implication on the left of the ‘⊢’ symbol and on the right.
The construction of derivations or tableaus in these systems is often more concise and intuitive than in an axiomatic one, and versions of all three have found their way into introductory logic texts. Furthermore, every natural deduction or sequent derivation can be made more direct by transforming it into a ‘normal form’. In the case of the sequent calculus, this result is known as the cut-elimination theorem. It has been applied extensively in metamathematics, most famously to obtain consistency proofs. The semantic inspiration for the rules of tableau construction suggests a very perspicuous proof of classical completeness, one which can also be adapted to the sequent calculus. The introduction and elimination rules of natural deduction are intuitionistically valid and have suggested an alternative semantics based on a conception of meaning as use. The idea is that the meaning of each logical constant is exhausted by its inferential behaviour and can therefore be characterized by its introduction and elimination rules.
Although the discussion below focuses on intuitionistic and classical first-order logic, various other logics have also been formulated as sequent, natural deduction and even tableau systems: modal logics, for example, relevance logic, infinitary and higher-order logics. There is a gain in understanding the role of the logical constants which comes from formulating introduction and elimination (or left and right) rules for them. Some authors have even suggested that one must be able to do so for an operator to count as logical.
A non-transitive relevant implication corresponding to classical logic consequence
