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noted more striking to the look and feel of a proof however is the variation in notation styles the gentzen notation which was covered earlier for a short argument can actually be stacked to produce large tree shaped natural deduction proofs 44 16 not to be confused with truth trees which is another name for analytic tableaux 72 there is also a style due to stanisław jaśkowski where the formulas in the proof are written inside various nested boxes 44 and there is a simplification of jaśkowski s style due to fredric fitch fitch notation where the boxes are simplified to simple horizontal lines beneath the introductions of suppositions and vertical lines to the left of the lines that are under the supposition 44 lastly there is the only notation style which will actually be used in this article which is due to patrick suppes 44 but was much popularized by e j lemmon and benson mates 112 this method has the advantage that graphically it is the least intensive to produce and display which made it a natural choice for the editor who wrote this part of the article who did not understand the complex latex commands that would be required to produce proofs in the other methods a proof then laid out in accordance with the suppes lemmon notation style 44 is a sequence of lines containing sentences 39 where each sentence is either an assumption or the result of applying a rule of proof to earlier sentences in the sequence 39 each line of proof is made up of a sentence of proof together with its annotation its assumption set and the current line number 39 the assumption set lists the assumptions on which the given sentence of proof depends which are referenced by the line numbers 39 the annotation specifies which rule of proof was applied and to which earlier lines to yield the current sentence 39 see the natural deduction proof example inference rules edit natural deduction inference rules due ultimately to gentzen are given below 111 there are ten primitive rules of proof which are the rule assumption plus four pairs of introduction and elimination rules for the binary connectives and the rule reductio ad adbsurdum 39 disjunctive syllogism can be used as an easier alternative to the proper elimination 39 and mtt and dn are commonly given rules 111 although they are not primitive 39 list of inference rules rule name alternative names annotation assumption set statement rule of assumptions 111 assumption 39 a 111 39 the current line number 39 at any stage of the argument introduce a proposition as an assumption of the argument 111 39 conjunction introduction ampersand introduction 111 39 conjunction conj 39 113 m n i 39 111 the union of the assumption sets at lines m and n 39 from φ displaystyle varphi and ψ displaystyle psi at lines m and n infer φ ψ displaystyle varphi psi 111 39 conjunction elimination simplification s 39 ampersand elimination 111 39 m e 39 111 the same as at line m 39 from φ ψ displaystyle varphi psi at line m infer φ displaystyle varphi and ψ displaystyle psi 39 111 disjunction introduction 111 addition add 39 m i 39 111 the same as at line m 39 from φ displaystyle varphi at line m infer φ ψ displaystyle varphi lor psi whatever ψ displaystyle psi may be 39 111 disjunction elimination wedge elimination 111 dilemma dl 113 j k l m n e 111 the lines j k l m n 111 from φ ψ displaystyle varphi lor psi at line j and an assumption of φ displaystyle varphi at line k and a derivation of χ displaystyle chi from φ displaystyle varphi at line l and an assumption of ψ displaystyle psi at line m and a derivation of χ displaystyle chi from ψ displaystyle psi at line n infer χ displaystyle chi 111 disjunctive syllogism wedge elimination e 39 modus tollendo ponens mtp 39 m n ds 39 the union of the assumption sets at lines m and n 39 from φ ψ displaystyle varphi lor psi at line m and φ displaystyle varphi at line n infer ψ displaystyle psi from φ ψ displaystyle varphi lor psi at line m and ψ displaystyle psi at line n infer φ displaystyle varphi 39 arrow elimination 39 modus ponendo ponens mpp 111 39 modus ponens mp 113 39 conditional elimination m n e 39 111 the union of the assumption sets at lines m and n 39 from φ ψ displaystyle varphi to psi at line m and φ displaystyle varphi at line n infer ψ displaystyle psi 39 arrow introduction 39 conditional proof cp 113 111 39 conditional introduction n i m 39 111 everything in the assumption set at line n excepting m the line where the antecedent was assumed 39 from ψ displaystyle psi at line n following from the assumption of φ displaystyle varphi at line m infer φ ψ displaystyle varphi to psi 39 reductio ad absurdum 111 indirect proof ip 39 negation introduction i 39 negation elimination e 39 m n raa k 39 the union of the assumption sets at lines m and n excluding k the denied assumption 39 from a sentence and its denial p at lines 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 omitted for simplicity 38 which is just the same as to say that it is a tautology i e that there is no interpretation under which it is false 38 to simplify the stateme...
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