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Text of the page (random words):
ced 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 suitable for its introduction rule again for conjunctions a b u a b u a e 1 a b u b e 2 a b i displaystyle cfrac a wedge b u quad rightarrow quad begin aligned cfrac cfrac cfrac a wedge b u a wedge _ e1 qquad cfrac cfrac a wedge b u b wedge _ e2 a wedge b wedge _ i end aligned these notions correspond exactly to β reduction beta reduction and η conversion eta conversion in the lambda calculus using the curry howard isomorphism by local completeness we see that every derivation can be converted to an equivalent derivation where the principal connective is introduced in fact if the entire derivation obeys this ordering of eliminations followed by introductions then it is said to be normal in a normal derivation all eliminations happen above introductions in most logics every derivation has an equivalent normal derivation called a normal form the existence of normal forms is generally hard to prove using natural deduction alone though such accounts do exist in the literature most notably by dag prawitz in 1961 42 it is much easier to show this indirectly by means of a cut free sequent calculus presentation first and higher order extensions 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 summary of first order system the logic of the earlier section is an example of a single sorted logic i e a logic with a single kind of object propositions many extensions of this simple framework have been proposed in this section we will extend it with a second sort of individuals or terms more precisely we will add a new category term denoted t displaystyle mathcal t we shall fix a countable set v displaystyle v of variables another countable set f displaystyle f of function symbols and construct terms with the following formation rules v v v t var f displaystyle frac v in v v mathcal t hbox var _ f and f f t 1 t t 2 t t n t f t 1 t 2 t n t app f displaystyle frac f in f qquad t_ 1 mathcal t qquad t_ 2 mathcal t qquad cdots qquad t_ n mathcal t f t_ 1 t_ 2 cdots t_ n mathcal t hbox app _ f for propositions we consider a third countable set p of predicates and define atomic predicates over terms with the following formation rule ϕ p t 1 t t 2 t t n t ϕ t 1 t 2 t n f pred f displaystyle frac phi in p qquad t_ 1 mathcal t qquad t_ 2 mathcal t qquad cdots qquad t_ n mathcal t phi t_ 1 t_ 2 cdots t_ n mathcal f hbox pred _ f the first two rules of formation provide a definition of a term that is effectively the same as that defined in term algebra and model theory although the focus of those fields of study is quite different from natural deduction the third rule of formation effectively defines an atomic formula as in first order logic and again in model theory to these are added a pair of formation rules defining the notation for quantified propositions one for universal and existential quantification x v a f x a f f x v a f x a f f displaystyle frac x in v qquad a mathcal f forall x a mathcal f forall _ f qquad qquad frac x in v qquad a mathcal f exists x a mathcal f exists _ f the universal quantifier has the introduction and elimination rules a t u a x a x a i u a x a t t t x a e displaystyle cfrac begin array c cfrac a mathcal t text u vdots a x a end array forall x a forall _ i u a qquad qquad frac forall x a qquad t mathcal t t x a forall _ e the existential quantifier has the introduction and elimination rules t x a x a i a t u a x a v x a c c e a u v displaystyle frac t x a exists x a exists _ i qquad qquad cfrac begin array cc underbrace cfrac a mathcal t hbox u quad cfrac a x a hbox v vdots exists x a quad c end array c exists _ e a u v in these rules the notation t x a stands for the substitution of t for every visible instance of x in a avoiding capture 43 as before the superscripts on the name stand for the components that are discharged the term a cannot occur in the conclusion of i such terms are known as eigenvariables or parameters and the hypotheses named u and v in e are localised to the second premise in a hypothetical derivation although the propositional logic of earlier sections was decidable adding the quantifiers makes the logic undecidable so far the quantified extensions are first order they distinguish propositions from the kinds of objects quantified over higher order logic takes a different approach and has only a single sort of propositions the quantifiers have as the domain of quantification the very same sort of propositions as reflected in the formation rules p f u a f p a f f u p f u a f p a f f u displaystyle cfrac begin matrix cfrac p mathcal f hbox u vdots a mathcal f end matrix forall p a mathcal f forall _ f u qquad qquad cfrac begin matrix cfrac p mathcal f hbox u vdots a mathcal f end matrix exists p a mathcal f exists _ f u a discussion of the introduction and elimination forms for higher order logic is beyond the scope of this article it is possible to be in between first order and higher order logics for example second order logic has two kinds of propositions one kind quantifying over terms and the second kind quantifying over propositions of the first kind proofs and type theory edit this section does not cite any sources 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