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onal logical connectives such as displaystyle land and displaystyle lor without enlarging the class of deducible formulas the first four logical axiom schemas allow together with modus ponens for the manipulation of logical connectives p1 ϕ ϕ displaystyle phi to phi p2 ϕ ψ ϕ displaystyle phi to left psi to phi right p3 ϕ ψ ξ ϕ ψ ϕ ξ displaystyle left phi to left psi rightarrow xi right right to left left phi to psi right to left phi to xi right right p4 ϕ ψ ψ ϕ displaystyle left lnot phi to lnot psi right to left psi to phi right the axiom p1 is redundant as it follows from p3 p2 and modus ponens see proof these axioms describe classical propositional logic without axiom p4 we get positive implicational logic minimal logic is achieved either by adding instead the axiom p4m or by defining ϕ displaystyle lnot phi as ϕ displaystyle phi to bot p4m ϕ ψ ϕ ψ ϕ displaystyle left phi to psi right to left left phi to lnot psi right to lnot phi right intuitionistic logic is achieved by adding axioms p4i and p5i to positive implicational logic or by adding axiom p5i to minimal logic both p4i and p5i are theorems of classical propositional logic p4i ϕ ϕ ϕ displaystyle left phi to lnot phi right to lnot phi p5i ϕ ϕ ψ displaystyle lnot phi to left phi to psi right note that these are axiom schemas which represent infinitely many specific instances of axioms for example p1 might represent the particular axiom instance p p displaystyle p to p or it might represent p q p q displaystyle left p to q right to left p to q right the ϕ displaystyle phi is a place where any formula can be placed a variable such as this that ranges over formulae is called a schematic variable with a second rule of uniform substitution us we can change each of these axiom schemas into a single axiom replacing each schematic variable by some propositional variable that isn t mentioned in any axiom to get what we call the substitutional axiomatisation both formalisations have variables but where the one rule axiomatisation has schematic variables that are outside the logic s language the substitutional axiomatisation uses propositional variables that do the same work by expressing the idea of a variable ranging over formulae with a rule that uses substitution us let ϕ p displaystyle phi p be a formula with one or more instances of the propositional variable p displaystyle p and let ψ displaystyle psi be another formula then from ϕ p displaystyle phi p infer ϕ ψ displaystyle phi psi the next three logical axiom schemas provide ways to add manipulate and remove universal quantifiers q5 x ϕ ϕ x t displaystyle forall x left phi right to phi x t where t may be substituted for x in ϕ displaystyle phi q6 x ϕ ψ x ϕ x ψ displaystyle forall x left phi to psi right to left forall x left phi right to forall x left psi right right q7 ϕ x ϕ displaystyle phi to forall x left phi right where x is not free in ϕ displaystyle phi these three additional rules extend the propositional system to axiomatise classical predicate logic likewise these three rules extend system for intuitionistic propositional logic with p1 3 and p4i and p5i to intuitionistic predicate logic universal quantification is often given an alternative axiomatisation using an extra rule of generalisation in which case the rules q6 and q7 are redundant generalization if γ ϕ displaystyle gamma vdash phi and x does not occur free in any formula of γ displaystyle gamma then γ x ϕ displaystyle gamma vdash forall x phi the final axiom schemas are required to work with formulas involving the equality symbol i8 x x displaystyle x x for every variable x i9 x y ϕ z x ϕ z y displaystyle left x y right to left phi z x to phi z y right conservative 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 march 2024 learn how and when to remove this message it is common to include in a hilbert system only axioms for the logical operators implication and negation towards functional completeness given these axioms it is possible to form conservative extensions of the deduction theorem that permit the use of additional connectives these extensions are called conservative because if a formula φ involving new connectives is rewritten as a logically equivalent formula θ involving only negation implication and universal quantification then φ is derivable in the extended system if and only if θ is derivable in the original system when fully extended a hilbert system will resemble more closely a system of natural deduction existential quantification edit introduction x ϕ y ϕ x y displaystyle forall x phi to exists y phi x y elimination x ϕ ψ x ϕ ψ displaystyle forall x phi to psi to exists x phi to psi where x displaystyle x is not a free variable of ψ displaystyle psi conjunction and disjunction edit conjunction introduction and elimination introduction α β α β displaystyle alpha to beta to alpha land beta elimination left α β α displaystyle alpha wedge beta to alpha elimination right α β β displaystyle alpha wedge beta to beta disjunction introduction and elimination introduction left α α β displaystyle alpha to alpha vee beta introduction right β α β displaystyle beta to alpha vee beta elimination α γ β γ α β γ displaystyle alpha to gamma to beta to gamma to alpha vee beta to gamma see also edit list of hilbert systems natural deduction sequent calculus notes edit 1 2 máté ruzsa 1997 129 1 2 3 smith peter 2013 02 21 an introduction to gödel s theorems cambridge university press p 10 isbn 978 1 107 02284 3 1 2 3 restall greg 2002 09 11 an introduction to substructural logics routledge pp 73 74 isbn 978 1 135 11131 1 gaifman haim 2002 a hilbert type deductive system for sentential logic completeness and compactness pdf columbia retrieved 2024 08 19 benthem johan van gupta amitabha parikh rohit 2011 04 02 proof computation and agency logic at the crossroads springer science business media p 41 isbn 978 94 007 0080 2 1 2 bacon andrew 2023 09 29 a philosophical introduction to higher order logics taylor francis p 424 isbn 978 1 000 92575 3 eijck jan van 1991 02 26 logics in ai european workshop jelia 90 amsterdam the netherlands september 10 14 1990 proceedings springer science business media p 113 isbn 978 3 540 53686 4 haack susan 1978 07 27 philosophy of logics cambridge university press p 19 isbn 978 0 521 29329 7 1 2 3 bostock david 1997 intermediate logic oxford new york clarendon press oxford university press pp 4 5 8 13 18 19 22 27 29 191 194 isbn 978 0 19 875141 0 lucas j r 2018 10 10 a treatise on time and space routledge p 152 isbn 978 0 429 68517 0 1 2 troelstra a s schwichtenberg h 2000 basic proof theory cambridge tracts in theoretical computer science 2 ed cambridge cambridge university press p 51 doi 10 1017 cbo9781139168717 isbn 978 0 521 77911 1 introduction to logic chapter 4 intrologic stanford edu retrieved 2024 08 16 buss s r 1998 07 09 handbook of proof theory elsevier pp 552 553 isbn 978 0 08 053318 6 1 2 ono hiroakira 2019 08 02 proof theory and algebra in logic springer p 5 isbn 978 981 13 7997 0 eijck jan van 1991 02 26 logics in ai european workshop jelia 90 amsterdam the netherlands september 10 14 1990 proceedings springer science business media p 113 isbn 978 3 540 53686 4 gabbay dov m guenthner franz 2013 03 14 handbook of philosophical logic springer science business media p 201 isbn 978 94 017 0458 8 kute tushar b hilbert systems pdf stony brook university retrieved 21 november 2025 stonybrook chapter 8 hilbert systems pdf stony brook university retrieved 21 november 2025 kute tushar b hilbert systems pdf stony brook university retrieved 21 november 2025 goertzel ben deduction in first order logic pdf goertzel org retrieved 21 november 2025 kute tushar b hilbert systems pdf stony brook university retrieved 21 november 2025 1 2 3 4 5 smullyan raymond m 2014 07 23 a beginner s guide to mathematical logic courier corporation pp 102 103 isbn 978 0 486 49237 7 franks curtis 2023 propositional logic in zalta edward n nodelman uri eds the stanford encyclopedia of philosophy fall 2023 ed metaphysics research lab stanford university retrieved 2024 03 22 1 2 mendelsohn richard l 2005 01 10 the philosophy of gottlob frege cambridge university press p 185 isbn 978 1 139 44403 3 1 2 łukasiewicz jan 1970 jan lukasiewicz selected works north holland p 136 1 2 church alonzo 1996 introduction to mathematical logic princeton university press p 119 isbn 978 0 691 02906 1 1 2 3 4 proof explorer home page metamath us metamath org retrieved 2024 07 02 1 2 cook stephen a reckhow robert a 1979 the relative efficiency of propositional proof systems the journal of symbolic logic 44 1 39 doi 10 2307 2273702 issn 0022 4812 jstor 2273702 walicki michał 2017 introduction to mathematical logic extended ed new jersey world scientific p 126 isbn 978 981 4719 95 7 pudlák pavel buss samuel r 1995 how to lie without being easily convicted and the lengths of proofs in propositional calculus in pacholski leszek tiuryn jerzy eds computer science logic lecture notes in computer science vol 933 berlin heidelberg springer p 152 doi 10 1007 bfb0022253 isbn 978 3 540 49404 1 references edit curry haskell b robert feys 1958 combinatory logic vol i vol 1 amsterdam north holland monk j donald 1976 mathematical logic graduate texts in mathematics berlin new york springer verlag isbn 978 0 387 90170 1 ruzsa imre máté andrás 1997 bevezetés a modern logikába in hungarian budapest osiris kiadó tarski alfred 1990 bizonyítás és igazság in hungarian budapest gondolat it is a hungarian translation of alfred tarski s selected papers on semantic theory of truth david hilbert 1927 the foundations of mathematics translated by stephan bauer menglerberg and dagfinn føllesdal pp 464 479 in van heijenoort jean 1967 from frege to gödel a source book in mathematical logic 1879 1931 3rd printing 1976 ed cambridge ma harvard university press isbn 0 674 32449 8 hilbert s 1927 based on an earlier 1925 foundations lecture pp 367 392 presents his 17 axioms axioms of implication 1 4 axioms about and v 5 10 axioms of negation 11 12 his logical ε axiom 13 axioms of equality 14 15 and axioms of number 16 17 along with the other necessary elements of his formalist proof theory e g induction axioms recursion axioms etc he also offers up a spirited defense against l e j brouwer s intuitionism also see hermann weyl s 1927 comments and rebuttal pp 480 484 paul bernay s 1927 appendix to hilbert s lecture pp 485 489 and luitzen egbertus jan brouwer s 1927 response pp 490 495 kleene stephen cole 1952 introduction to metamathematics 10th impression with 1971 corrections ed amsterdam ny north holland publishing company isbn 0 7204 2103 9 cite book isbn date incompatibility help see in particular chapter iv formal system pp 69 85 wherein kleene presents subchapters 16 formal symbols 17 formation rules 18 free and bound variables including substitution 19 transformation rules e g modus ponens and from these he presents 21 postulates 18 axioms and 3 immediate consequence relations divided as follows postulates for the propostional calculus 1 8 additional postulates for the predicate calculus 9 12 and additional postulates for number theory 13 21 external links edit gaifman haim a hilbert type deductive system for sentential logic completeness and compactness pdf farmer w m propositional logic pdf it describes among others a specific hilbert style proof system that is restricted to propositional calculus v t e mathematical logic general axiom list cardinality first order logic formal proof formal semantics foundations of mathematics information theory lemma logical consequence model theorem theory type theory theorems list paradoxes gödel s completeness incompleteness theorems tarski s undefinability banach tarski paradox cantor s theorem paradox diagonal argument compactness halting problem lindström s löwenheim skolem russell s paradox logics traditional classical logic logical truth tautology proposition inference logical equivalence consistency equiconsistency argument soundness validity syllogism square of opposition venn diagram propositional boolean algebra boolean functions logical connectives propositional calculus propositional formula truth tables many valued logic 3 finite predicate first order list second order monadic higher order fixed point free quantifiers predicate monadic predicate calculus set theory set hereditary class ur element ordinal number extensionality forcing relation equivalence partition set operations intersection union complement cartesian product power set identities types of sets countable uncountable empty inhabited singleton finite infinite transitive ultrafilter recursive fuzzy universal universe constructible grothendieck von neumann maps cardinality function map domain codomain image in sur bi jection schröder bernstein theorem isomorphism gödel numbering enumeration large cardinal inaccessible aleph number operation binary theories zermelo fraenkel axiom of choice continuum hypothesis general kripke platek morse kelley naive new foundations tarski grothendieck von neumann bernays gödel ackermann constructive formal systems list language syntax alphabet arity automata axiom schema expression ground extension by definition conservative relation formation rule grammar formula atomic closed ground open free bound variable language metalanguage logical connective predicate functional variable propositional variable proof quantifier rank sentence atomic spectrum signature string substitution symbol function logical constant non logical variable term theory list example axiomatic systems list of true arithmetic peano second order elementary function primitive recursive robinson skolem of the real numbers tarski s axiomatization of boolean algebras canonical minimal axioms of geometry euclidean elements hilbert s tarski s non euclidean principia mathematica proof theory formal proof natural deduction logical consequence rule of inference sequent calculus theorem systems axiomatic deductive hilbert list complete theory independence from zfc proof of impossibility ordinal analysis reverse mathematics self verifying theories model theory interpretation function of models model atomic equivalence finite prime saturated spectrum submodel non standard model of non standard arithmetic diagram elementary categorical theory model complete theory satisfiability semantics of logic strength theories of truth semantic tarski s kripke s t schema transfer principle truth predicate truth value type ultraproduct validity computability theory church encoding church turing thesis computably enumerable computable function computable set decision problem decidable undecidable p np p versus np problem kolmogorov complexity lambda calculus primitive recursive function recursion recursive set turing machine type theory related abstract logic algebraic logic a...