System L

System L is a natural deductive logic developed by E.J. Lemmon. Derived from Suppes' method, it represents natural deduction proofs as sequences of justified steps. Both methods are derived from Gentzen's 1934/1935 natural deduction system, in which proofs were presented in tree-diagram form rather than in the tabular form of Suppes and Lemmon. Although the tree-diagram layout has advantages for philosophical and educational purposes, the tabular layout is much more convenient for practical applications.

A similar tabular layout is presented by Kleene. The main difference is that Kleene does not abbreviate the left-hand sides of assertions to line numbers, preferring instead to either give full lists of precedent propositions or alternatively indicate the left-hand sides by bars running down the left of the table to indicate dependencies. However, Kleene's version has the advantage that it is presented, although only very sketchily, within a rigorous framework of metamathematical theory, whereas the books by Suppes and Lemmon are applications of the tabular layout for teaching introductory logic.

The syntax of proof is governed by nine primitive rules:

In system L, a proof has a definition with the following conditions:

If no premise is given, the sequent is called theorem. Therefore, the definition of a theorem in system L is:

An example of the proof of a sequent (Modus Tollendo Tollens in this case):

An example of the proof of a sequent (a theorem in this case):

Each rule of system L has its own requirements for the type of input(s) or entry(es) that it can accept and has its own way of treating and calculating the assumptions used by its inputs.

The historical development of tabular-layout natural deduction systems, which are rule-based, and which indicate antecedent propositions by line numbers (and related methods such as vertical bars or asterisks) includes the following publications.

...
Wikipedia