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ines m and n infer the denial of any assumption appearing in the proof at line k 39 double arrow introduction 39 biconditional definition df 111 biconditional introduction m n i 39 the union of the assumption sets at lines m and n 39 from φ ψ displaystyle varphi to psi and ψ φ displaystyle psi to varphi at lines m and n infer φ ψ displaystyle varphi leftrightarrow psi 39 double arrow elimination 39 biconditional definition df 111 biconditional elimination m e 39 the same as at line m 39 from φ ψ displaystyle varphi leftrightarrow psi at line m infer either φ ψ displaystyle varphi to psi or ψ φ displaystyle psi to varphi 39 double negation 111 113 double negation elimination m dn 111 the same as at line m 111 from φ displaystyle varphi at line m infer φ displaystyle varphi 111 modus tollendo tollens 111 modus tollens mt 113 m n mtt 111 the union of the assumption sets at lines m and n 111 from φ ψ displaystyle varphi to psi at line m and ψ displaystyle psi at line n infer φ displaystyle varphi 111 natural deduction proof example edit the proof below 39 derives p displaystyle p from p q displaystyle p to q and q displaystyle q using only mpp and raa which shows that mtt is not a primitive rule since it can be derived from those two other rules derivation of mtt from mpp and raa assumption set line number sentence of proof annotation 1 1 p q displaystyle p to q a 2 2 q displaystyle q a 3 3 p displaystyle p a 1 3 4 q displaystyle q 1 3 e 1 2 5 p displaystyle p 2 4 raa syntactic proof via axioms edit main article hilbert system it is possible to perform proofs axiomatically which means that certain tautologies are taken as self evident and various others are deduced from them using modus ponens as an inference rule as well as a rule of substitution which permits replacing any well formed formula with any substitution instance of it 114 alternatively one uses axiom schemas instead of axioms and no rule of substitution is used 114 this section gives the axioms of some historically notable axiomatic systems for propositional logic for more examples as well as metalogical theorems that are specific to such axiomatic systems such as their completeness and consistency see the article axiomatic system logic frege s begriffsschrift edit although axiomatic proof has been used since the famous ancient greek textbook euclid s elements of geometry in propositional logic it dates back to gottlob frege s 1879 begriffsschrift 38 114 frege s system used only implication and negation as connectives 2 it had six axioms 114 115 116 proposition 1 a b a displaystyle a to b to a proposition 2 c b a c b c a displaystyle c to b to a to c to b to c to a proposition 8 d b a b d a displaystyle d to b to a to b to d to a proposition 28 b a a b displaystyle b to a to neg a to neg b proposition 31 a a displaystyle neg neg a to a proposition 41 a a displaystyle a to neg neg a these were used by frege together with modus ponens and a rule of substitution which was used but never precisely stated to yield a complete and consistent axiomatization of classical truth functional propositional logic 115 łukasiewicz s p 2 edit jan łukasiewicz showed that in frege s system the third axiom is superfluous since it can be derived from the preceding two axioms and that the last three axioms can be replaced by the single sentence c c n p n q c q p displaystyle ccnpnqcqp 116 which taken out of łukasiewicz s polish notation into modern notation means p q q p displaystyle neg p rightarrow neg q rightarrow q rightarrow p hence łukasiewicz is credited 114 with this system of three axioms p q p displaystyle p to q to p p q r p q p r displaystyle p to q to r to p to q to p to r p q q p displaystyle neg p to neg q to q to p just like frege s system this system uses a substitution rule and uses modus ponens as an inference rule 114 the exact same system was given with an explicit substitution rule by alonzo church 117 who referred to it as the system p 2 117 118 and helped popularize it 118 schematic form of p 2 edit one may avoid using the rule of substitution by giving the axioms in schematic form using them to generate an infinite set of axioms hence using greek letters to represent schemata metalogical variables that may stand for any well formed formulas the axioms are given as 38 118 φ ψ φ displaystyle varphi to psi to varphi φ ψ χ φ ψ φ χ displaystyle varphi to psi to chi to varphi to psi to varphi to chi φ ψ ψ φ displaystyle neg varphi to neg psi to psi to varphi the schematic version of p 2 is attributed to john von neumann 114 and is used in the metamath set mm formal proof database 118 it has also been attributed to hilbert 119 and named h displaystyle mathcal h in this context 119 proof example in p 2 edit as an example a proof of a a displaystyle a to a in p 2 is given below first the axioms are given names a1 p q p displaystyle p to q to p a2 p q r p q p r displaystyle p to q to r to p to q to p to r a3 p q q p displaystyle neg p to neg q to q to p and the proof is as follows a b a a displaystyle a to b to a to a instance of a1 a b a a a b a a a displaystyle a to b to a to a to a to b to a to a to a instance of a2 a b a a a displaystyle a to b to a to a to a from 1 and 2 by modus ponens a b a displaystyle a to b to a instance of a1 a a displaystyle a to a from 4 and 3 by modus ponens solvers edit one notable difference between propositional calculus and predicate calculus is that satisfiability of a propositional formula is decidable 120 81 deciding satisfiability of propositional logic formulas is an np complete problem however practical methods exist e g dpll algorithm 1962 chaff algorithm 2001 that are very fast for many useful cases recent work has extended the sat solver algorithms to work with propositions containing arithmetic expressions these are the smt solvers see also edit philosophy portal higher logical levels edit first order logic second order propositional logic second order logic higher order logic related topics edit boolean algebra logic boolean algebra structure boolean algebra topics boolean domain boolean function boolean valued function categorical logic combinational logic combinatory logic conceptual graph disjunctive syllogism entitative graph equational logic existential graph implicational propositional calculus intuitionistic propositional calculus jean buridan laws of form list of logic symbols logical graph logical nor logical value mathematical logic operation mathematics paul of venice peirce s law peter of spain author propositional formula symmetric difference tautology rule of inference truth function truth table walter burley william of sherwood notes edit many sources write this with a definite article as the propositional calculus while others just call it propositional calculus with no article zeroth order logic is sometimes used to denote a quantifier free predicate logic that is propositional logic extended with functions relations and constants 6 for propositional logic the formal language used is a propositional language not to be confused with the formal language s alphabet see all possible connectives on truth functional propositional logic with some of their properties the or both makes it clear 35 that it s a logical disjunction not an exclusive or which is more common in english the set of premises may be the empty set 38 39 an argument from an empty set of premises is valid if and only if the conclusion is a tautology 38 39 the turnstile for syntactic consequence is of lower precedence than the comma which represents premise combination which in turn is of lower precedence than the arrow used for material implication so no parentheses are needed to interpret this formula 45 a very general and abstract syntax is given here following the notation in the sep 2 but including the third definition which is very commonly given explicitly by other sources such as gillon 15 bostock 38 allen hand 39 and many others as noted elsewhere in the article languages variously compose their set of atomic propositional variables from uppercase or lowercase letters often focusing on p p q q and r r with or without subscript numerals and in their set of connectives they may include either the full set of five typical connectives displaystyle neg land lor to leftrightarrow or any of the truth functionally complete subsets of it and of course they may also use any of the notational variants of these connectives note that the phrase principle of composition has referred to other things in other contexts and even in the context of logic since bertrand russell used it to refer to the principle that a proposition which implies each of two propositions implies them both 53 the name interpretation is used by some authors and the name case by other authors this article will be indifferent and use either since it is collaboratively edited and there is no consensus about which terminology to adopt a truth functionally complete set of connectives 2 is also called simply functionally complete or adequate for truth functional logic 40 or expressively adequate 78 or simply adequate 40 78 see a table of all 16 bivalent truth functions some of these definitions use the word interpretation and speak of sentences formulas being true or false under it and some will use the word case and speak of sentences formulas being true or false in it published reliable sources wp rs have used both kinds of terminological convention although usually a given author will use only one of them since this article is collaboratively edited and there is no consensus about which convention to use these variations in terminology have been left standing conventionally φ displaystyle models varphi with nothing to the left of the turnstile is used to symbolize a tautology it may be interpreted as saying that φ displaystyle varphi is a semantic consequence of the empty set of formulae i e φ displaystyle models varphi but with the empty brackets 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