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uence is contrasted with semantic consequence 17 which is symbolized with 18 16 in this case the conclusion follows syntactically because natural deduction is a syntactic proof system which assumes inference rules as primitives gentzen s style will be used in much of this article gentzen s discharging annotations used to internalise hypothetical judgments can be avoided by representing proofs as a tree of sequents γ a instead of a tree of judgments that a is true suppes lemmon notation edit many textbooks use suppes lemmon notation 7 so this article will also give that although as of now this is only included for propositional logic and the rest of the coverage is given only in gentzen style a proof laid out in accordance with the suppes lemmon notation style is a sequence of lines containing sentences 19 where each sentence is either an assumption or the result of applying a rule of proof to earlier sentences in the sequence 19 each line of proof is made up of a sentence of proof together with its annotation its assumption set and the current line number 19 the assumption set lists the assumptions on which the given sentence of proof depends which are referenced by the line numbers 19 the annotation specifies which rule of proof was applied and to which earlier lines to yield the current sentence 19 here s an example proof p q q p displaystyle p to q neg q vdash neg p suppes lemmon style proof first example assumption set line number formula wff annotation 1 1 p q displaystyle p to q a 2 2 q displaystyle neg q a 3 3 p displaystyle p a 1 3 4 q displaystyle q 1 3 e 1 2 5 p displaystyle neg p 2 4 raa q e d this proof will become clearer when the inference rules and their appropriate annotations are specified see propositional inference rules suppes lemmon style propositional language syntax edit main article propositional calculus syntax this section defines the formal syntax for a propositional logic language contrasting the common ways of doing so with a gentzen style way of doing so common definition styles edit in classical propositional calculus the formal language l displaystyle mathcal l is usually defined here by recursion as follows 20 each propositional variable is a formula displaystyle bot is a formula if φ displaystyle varphi and ψ displaystyle psi are formulae so are φ ψ displaystyle varphi land psi φ ψ displaystyle varphi lor psi φ ψ displaystyle varphi to psi φ ψ displaystyle varphi leftrightarrow psi nothing else is a formula negation displaystyle neg is defined as implication to falsity ϕ def ϕ displaystyle neg phi overset text def phi to bot where displaystyle bot falsum represents a contradiction or absolute falsehood 21 22 23 24 25 older publications and publications that do not focus on logical systems like minimal intuitionistic or hilbert systems take negation as a primitive logical connective meaning it is assumed as a basic operation and not defined in terms of other connectives 26 27 some authors such as bostock use displaystyle bot and displaystyle top and also define displaystyle neg as primitives 28 29 gentzen style definition edit a syntax definition can also be given using gentzen s tree notation by writing well formed formulas below the inference line 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 ...
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