System L

A proof is a table with 4 columns and unlimited ordered rows. From left to right the columns hold:

1. A set of positive integers, possibly empty
2. A positive integer
3. A well-formed formula (or wff)
4. A set of numbers, possibly empty; a rule; and possibly a reference to another proof

The following is an example:

The second column holds line numbers. The third holds a wff, which is justified by the rule held in the fourth along with auxiliary information about other wffs, possibly in other proofs. The first column represents the line numbers of the assumptions the wff rests on, determined by the application of the cited rule in context. Any line of any valid proof can be converted into a sequent by listing the wffs at the cited lines as the premises and the wff at the line as the conclusion. Analogously, they can be converted into conditionals where the antecedent is a conjunction. These sequents are often listed above the proof, as Modus Tollens is above.

The above proof is a valid one, but proofs don't need to be to conform to the general syntax of the proof system. To guarantee a sequent's validity, however, we must conform to carefully specified rules. The rules can be divided into four groups: the propositional rules (1-10), the predicate rules (11-14), the rules of equality (15-16), and the rule of substitution (17). Adding these groups in order allows one to build a propositional calculus, then a predicate calculus, then a predicate calculus with equality, then a predicate calculus with equality allowing for the derivation of new rules. Some of the propositional calculus rules, like MTT, are superfluous and can be derived as rules from other rules.

1. The Rule of Assumption (A): "A" justifies any wff. The only assumption is its own line number.
2. Modus Ponendo Ponens (MPP): If there are lines a and b previously in the proof containing P→Q and P respectively, "a,b MPP" justifies Q. The assumptions are the collective pool of lines a and b.
3. The Rule of Conditional Proof (CP): If a line with proposition P has an assumption line b with proposition Q, "b,a CP" justifies Q→P. All of a's assumptions aside b are kept.
4. The Rule of Double Negation (DN): "a DN" justifies adding or subtracting two negation symbols from the wff at a line a previously in the proof, making this rule a biconditional. The assumption pool is the one of the line cited.
5. The Rule of ∧-introduction (∧I): If propositions P and Q are at lines a and b, "a,b ∧I" justifies P∧Q. The assumptions are the collective pool of the conjoined propositions.
6. The Rule of ∧-elimination (∧E): If line a is a conjunction P∧Q, one can conclude either P or Q using "a ∧E". The assumptions are line a's. ∧I and ∧E allow for monotonicity of entailment, as when a proposition P is joined with Q with ∧I and separated with ∧E, it retains Q's assumptions.
7. The Rule of ∨-introduction (∨I): For a line a with proposition P one can introduce P∨Q citing "a ∨I". The assumptions are a's.
8. The Rule of ∨-elimination (∨E): For a disjunction P∨Q, if one assumes P and Q and separately comes to the conclusion R from each, then one can conclude R. The rule is cited as "a,b,c,d,e ∨E", where line a has the initial disjunction P∨Q, lines b and d assume P and Q respectively, and lines c and e are R with P and Q in their respective assumption pools. The assumptions are the collective pools of the two lines concluding R minus the lines of P and Q, b and d.
9. Reductio Ad Absurdum (RAA): For a proposition P∧¬P on line a citing an assumption Q on line b, one can cite "b,a RAA" and derive ¬Q from the assumptions of line a aside from b.
10. Modus Tollens (MTT): For propositions P→Q and ¬Q on lines a and b one can cite "a,b MTT" to derive ¬P. The assumptions are those of lines a and b. This is proven from other rules above.
11. Universal Introduction (UI): For a predicate ${\displaystyle Ra}$ on line a, one can cite "a UI" to justify a universal quantification,${\displaystyle (\forall x)Rx}$, provided none of the assumptions on line a have the term ${\displaystyle a}$ in anywhere. The assumptions are those of line a.
12. Universal Elimination (UE): For a universally quantified predicate ${\displaystyle (\forall x)Rx}$ on line a, one can cite "a UE" to justify ${\displaystyle Ra}$. The assumptions are those of line a. UE is a duality with UI in that one can switch between quantified and free variables using these rules.
13. Existential Introduction (EI): For a predicate ${\displaystyle Ra}$ on line a one can cite "a EI" to justify an existential quantification, ${\displaystyle (\exists x)Rx}$. The assumptions are those of line a.
14. Existential Elimination (EE): For an existentially quantified predicate ${\displaystyle (\exists x)Rx}$ on line a, if we assume ${\displaystyle Ra}$ to be true on line b and derive P with it on line c, we can cite "a,b,c EE" to justify P. The term ${\displaystyle a}$ cannot appear in the conclusion P, any of its assumptions aside from line b, or on line a. For this reason EE and EI are in duality, as one can assume ${\displaystyle Ra}$ and use EI to reach a conclusion from ${\displaystyle (\exists x)Rx}$, as the EI will rid the conclusion of the term ${\displaystyle a}$. The assumptions are the assumptions on line a and any on line c aside from b.
15. Equality Introduction (=I): At any point one can introduce ${\displaystyle a=a}$ citing "=I" with no assumptions.
16. Equality Elimination (=E): For propositions ${\displaystyle a=b}$ and P on lines a and b, one can cite "a,b =E" to justify changing any terms ${\displaystyle a}$ terms in P to ${\displaystyle b}$. The assumptions are the pool of a and b.
17. Substitution Instance (SI(S)): For a sequent ${\displaystyle P,Q\vdash R}$ proved in proof X and substitution instances of ${\displaystyle P}$ and ${\displaystyle Q}$ on lines a and b, one can cite "a,b SI(S) X" to justify introducing a substitution instance of ${\displaystyle R}$. The assumptions are those of lines a and b. A derived rule with no assumptions is a theorem, and can be introduced at any time with no assumptions. Some cite that as "TI(S)", for "theorem" instead of "sequent". Additionally, some cite only "SI" or "TI" in either case when a substitution instance isn't needed, as their propositions match the ones of the referenced proof exactly.