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and any schematic variables used by those formulas above it 26 for instance the equivalent of rules 3 and 4 from bostock s definition above is written as follows φ φ φ ψ φ ψ φ ψ φ ψ φ ψ φ ψ φ ψ φ ψ displaystyle frac varphi neg varphi quad frac varphi quad psi varphi lor psi quad frac varphi quad psi varphi land psi quad frac varphi quad psi varphi rightarrow psi quad frac varphi quad psi varphi leftrightarrow psi a different notational convention sees the language s syntax as a categorial grammar with the single category formula denoted by the symbol f displaystyle mathcal f so any elements of the syntax are introduced by categorizations for which the notation is φ f displaystyle varphi mathcal f meaning φ displaystyle varphi is an expression for an object in the category f displaystyle mathcal f 30 the sentence letters then are introduced by categorizations such as p f displaystyle p mathcal f q f displaystyle q mathcal f r f displaystyle r mathcal f and so on 30 the connectives in turn are defined by statements similar to the above but using categorization notation as seen below connectives defined through a categorial grammar 30 conjunction disjunction implication negation a f b f a b f displaystyle frac a mathcal f quad b mathcal f a b mathcal f a f b f a b f displaystyle frac a mathcal f quad b mathcal f vee a b mathcal f a f b f a b f displaystyle frac a mathcal f quad b mathcal f mathord supset a b mathcal f a f a f displaystyle frac a mathcal f neg a mathcal f in the rest of this article the φ f displaystyle varphi mathcal f categorization notation will be used for any gentzen notation statements defining the language s grammar any other statements in gentzen notation will be inferences asserting that a sequent follows rather than that an expression is a well formed formula gentzen style propositional logic edit gentzen style inference rules edit let the propositional language l displaystyle mathcal l be inductively defined as φ p 1 p 2 φ φ φ φ φ φ displaystyle phi p_ 1 p_ 2 dots mid bot mid phi to phi mid phi land phi mid phi lor phi define negation as φ def φ displaystyle neg phi overset text def phi to bot the following is a list of primitive inference rules for natural deduction in propositional logic 31 26 rules for propositional logic introduction rules elimination rules φ ψ φ ψ i displaystyle begin array c varphi qquad psi hline varphi land psi end array land _ i φ ψ φ e displaystyle begin array c varphi land psi hline varphi end array land _ e φ φ ψ i displaystyle begin array c varphi hline varphi lor psi end array lor _ i φ ψ φ u χ ψ v χ χ e u v displaystyle cfrac varphi lor psi quad begin matrix varphi u vdots chi end matrix quad begin matrix psi v vdots chi end matrix chi lor _ e u v φ u ψ φ ψ i u displaystyle begin array c varphi u vdots psi hline varphi to psi end array to _ i u φ φ ψ ψ e displaystyle begin array c varphi qquad varphi to psi hline psi end array to _ e φ e displaystyle begin array c bot hline varphi end array bot _ e 32 φ φ e displaystyle begin array c varphi to bot to bot hline varphi end array neg neg _ e in this table the greek letters φ ψ χ displaystyle varphi psi chi are schemata which range over formulas rather than only over atomic propositions the name of a rule is given to the right of its formula tree for instance the first introduction rule is named i displaystyle land _ i which is short for conjunction introduction minimal logic the natural deduction rules are n d m p c i e i e i e displaystyle nd_ mpc land _ i land _ e lor _ i lor _ e to _ i to _ e without the rules e displaystyle bot _ e and e displaystyle neg neg _ e the system defines minimal logic as discussed by johansson 33 intuitionistic logic the natural deduction rules are n d i p c n d m p c e displaystyle nd_ ipc nd_ mpc cup bot _ e when the rule e displaystyle bot _ e principle of explosion is added to the rules for minimal logic the system defines intuitionistic logic the statement p p displaystyle p to neg neg p is valid already in minimal logic see example 1 below unlike the reverse implication which would entail the law of excluded middle classical logic the natural deduction rules are n d c p c n d i p c e displaystyle nd_ cpc nd_ ipc cup neg neg _ e 32 when all listed natural deduction rules are admitted the system defines classical logic 34 35 36 gentzen style example proofs edit example 1 24 proof within minimal logic of p p displaystyle p to neg neg p goal p p displaystyle p to p to bot to bot proof p v p u e p p p i v i u displaystyle cfrac cfrac p v qquad p to bot u bot to _ e cfrac p to bot to bot p to p to bot to bot to _ i v to _ i u example 2 proof within minimal logic of a b a b displaystyle a to left b to left a land b right right a u b w a b i b a b a b a b i u i w displaystyle cfrac cfrac a u quad b w a land b land _ i cfrac b to left a land b right a to left b to left a land b right right to _ i u to _ i w fitch style propositional logic edit main article fitch notation fitch developed a system of natural deduction which is characterized by linear presentation of the proof instead of presentation as a tree subordinate proofs where assumptions could be opened within a subderivation and discharged later later logicians and educators such as patrick suppes 37 and e j lemmon 38 rebranded fitch s system while they introduced graphical changes such as replacing indentation with vertical bars the underlying structure of fitch style natural deduction remained intact these variations are often referred to as the suppes lemmon format though they are fundamentally based on fitch s original notation suppes lemmon style propositional logic edit main article suppes lemmon notation suppes lemmon style inference rules edit the linear presentation used in fitch and suppes lemmon style proofs with line numbers and vertical alignment assumption sets makes subproofs clearly visible fitch sparingly and cautiously used derived rules suppes lemmon went further and added derived rules to the toolbox of natural deduction rules suppes introduced natural deduction using gentzen style rules 37 he defined negation in terms of contradiction p p displaystyle neg p equiv p to bot he discussed derived rules explicitly though not always distinguishing them clearly from primitive ones in layout his system is close to minimal but allows derived steps for brevity lemmon formalized more derived rules 38 he as well defined negation as implication to falsity p p displaystyle neg p equiv p to bot this is not stated as a formal definition in beginning logic but it is implicitly assumed throughout the system as evidenced by the following use of raa reductio ad absurdum lemmon regularly used raa in the form assume p displaystyle p derive displaystyle bot then conclude p displaystyle neg p this only works if p displaystyle neg p is understood as p displaystyle p to bot proofs involving contradiction lemmon used the fact that from p p displaystyle neg p land p one can derive displaystyle bot this requires treating p displaystyle neg p as p displaystyle p to bot so that modus ponens yields contradiction absence of a primitive rule lemmon did not include a standalone rule for introducing or eliminating instead he derived negation using implication and contradiction in the table below based on lemmon 1978 39 and allen hand 2022 19 lemmon s derived rules are highlighted they can be derived from the non highlighted gentzen rules there are nine primitive rules of proof which are the rule assumption plus four pairs of introduction and elimination rules for the binary connectives and the rules of double negation and reductio ad absurdum of which only one is needed 32 19 disjunctive syllogism can be used as an easier alternative to the proper elimination 19 and mtt is a commonly given rule 39 although it is not primitive 19 list of inference rules rule name alternative names annotation assumption set statement rule of assumptions assumption a the current line number at any stage of the argument introduce a proposition as an assumption of the argument conjunction introduction ampersand introduction conjunction conj 40 m n i the union of the assumption sets at lines m and n from φ displaystyle varphi and ψ displaystyle psi at lines m and n infer φ ψ displaystyle varphi land psi conjunction elimination simplification s ampersand elimination m e the same as at line m from φ ψ displaystyle varphi land psi at line m infer φ displaystyle varphi and ψ displaystyle psi disjunction introduction addition add m i the same as at line m from φ displaystyle varphi at line m infer φ ψ displaystyle varphi lor psi whatever ψ displaystyle psi may be disjunction elimination wedge elimination dilemma dl 40 j k l m n e the lines j k l m n 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 arrow introduction conditional proof cp 40 conditional introduction n i m everything in the assumption set at line n excepting m the line where the antecedent was assumed 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 arrow elimination modus ponendo ponens mpp modus ponens mp 40 conditional elimination m n e the union of the assumption sets at lines m and n from φ ψ displaystyle varphi to psi at line m and φ displaystyle varphi at line n infer ψ displaystyle psi double negation 40 double negation elimination m dn the same as at line m from φ displaystyle neg neg varphi at line m infer φ displaystyle varphi reductio ad absurdum indirect proof ip negation introduction i negation elimination e m n raa k the union of the assumption sets at lines m and n excluding k the denied assumption to simplify the statement of the rule the word denial here is used in this way the denial of a formula φ displaystyle varphi that is not a negation is φ displaystyle neg varphi whereas a negation φ displaystyle neg varphi has two denials viz φ displaystyle varphi and φ displaystyle neg neg varphi at lines m and n infer the denial of any assumption appearing in the proof at line k disjunctive syllogism wedge elimination e modus tollendo ponens mtp m n ds the union of the assumption sets at lines m and n from φ ψ displaystyle varphi lor psi at line m and φ displaystyle neg varphi at line n infer ψ displaystyle psi from φ ψ displaystyle varphi lor psi at line m and ψ displaystyle neg psi at line n infer φ displaystyle varphi double arrow introduction biconditional definition df displaystyle leftrightarrow biconditional introduction m n displaystyle leftrightarrow i the union of the assumption sets at lines m and n 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 double arrow elimination biconditional definition df displaystyle leftrightarrow biconditional elimination m displaystyle leftrightarrow e the same as at line m 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 modus tollendo tollens modus tollens mt m n mtt the union of the assumption sets at lines m and n from φ ψ displaystyle varphi to psi at line m and ψ displaystyle neg psi at line n infer φ displaystyle neg varphi suppes lemmon style examples proof edit recall that an example proof was already given when introducing suppes lemmon notation this is a second example example 2 edit p q p p q displaystyle p lor q neg p neg p to neg q vdash bot contradiction suppes lemmon style proof second example assumption set line number formula annotation 1 1 p q displaystyle p lor q a 2 2 p displaystyle neg p a 3 3 p q displaystyle neg p to neg q a 2 3 4 q displaystyle neg q 2 3 e 1 2 3 5 p displaystyle p a subproof 1 2 3 5 6 displaystyle bot 2 5 raa 41 1 2 3 7 q displaystyle q a subproof 1 2 3 7 8 displaystyle bot 4 7 raa 41 1 2 3 9 displaystyle bot 1 5 6 7 8 ve q e d example 3 edit the next derivation proves two theorems lines 1 8 prove within minimal logic m p c p p displaystyle vdash _ mpc neg neg p lor neg p lines 1 9 prove within classical logic c p c p p displaystyle vdash _ cpc p lor neg p goals lines 1 8 m p c p p displaystyle vdash _ mpc p lor p to bot to bot to bot lines 1 9 c p c p p displaystyle vdash _ cpc p lor p to bot p p displaystyle vdash p lor neg p suppes lemmon style proof third example assumption set line number formula annotation 1 1 p p displaystyle p lor p to bot to bot a 1 2 2 p displaystyle p a 1 2 3 p p displaystyle p lor p to bot 2 i 1 2 4 displaystyle bot 1 3 e 1 5 p displaystyle p to bot 2 4 i discharging 2 1 6 p p displaystyle p lor p to bot 5 i 1 7 displaystyle bot 1 6 e 8 p p displaystyle p lor p to bot to bot to bot 1 7 i discharging 1 9 p p displaystyle p lor p to bot 8 dn q e d remark valery glivenko proved the following theorem if φ displaystyle varphi is a propositional formula then φ displaystyle varphi is a classical tautology if and only if φ displaystyle neg neg varphi is an intuitionistic tautology this implies that all classical propositional theorems φ displaystyle varphi can be proved like in this example prove φ displaystyle neg neg varphi within intuitionistic logic i e without e displaystyle neg neg _ e apply e displaystyle neg neg _ e to get φ displaystyle varphi from φ displaystyle neg neg varphi consistency completeness and normal forms edit this section does not cite any sources please help improve this section by adding citations to reliable sources unsourced material may be challenged and removed may 2024 learn how and when to remove this message a theory is said to be consistent if falsehood is not provable from no assumptions and is complete if every theorem or its negation is provable using the inference rules of the logic these are statements about the entire logic and are usually tied to some notion of a model however there are local notions of consistency and completeness that are purely syntactic checks on the inference rules and require no appeals to models the first of these is local consistency also known as local reducibility which says that any derivation containing an introduction of a connective followed immediately by its elimination can be turned into an equivalent derivation without this detour it is a check on the strength of elimination rules they must not be so strong that they include knowledge not already contained in their premises as an example consider conjunctions a u b w a b i a e 1 a u displaystyle begin aligned cfrac cfrac cfrac a u qquad cfrac b w a wedge b wedge _ i a wedge _ e1 end aligned quad rightarrow quad cfrac a u dually local completeness says that the elimination rules are strong enough to decompose a connective into the forms s...