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| {{Expert-subject|mathematics|date=September 2010}}
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| {{Refimprove|date=September 2010}}
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| This article contains a list of sample [[Hilbert system|Hilbert-style]] [[deductive system]]s for [[propositional logic]].
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| ==Classical propositional calculus systems==
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| [[Classical logic|Classical]] propositional calculus is the standard propositional logic. Its intended semantics is [[bivalent logic|bivalent]] and its main property is that it is [[completeness|syntactically complete]], otherwise said that no new axiom not already consequence of the existing axioms can be added without making the logic [[inconsistent]]. Many different equivalent complete axiom systems have been formulated. They differ in the choice of basic [[logical connective|connectives]] used, which in all cases have to be [[functionally complete]] (i.e. able to express by composition all n-ary [[truth tables]]), and in the exact complete choice of axioms over the chosen basis of connectives.
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| ===Implication and negation===
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| The formulations here use implication and negation <math>\{\to,\neg\}</math> as functionally complete set of basic connectives. Every logic system requires at least one non-nullary [[rule of inference]]. Classical propositional calculus typically uses the rule of [[modus ponens]]:
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| :<math>A,A\to B\vdash B.</math>
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| We assume this rule is included in all systems below unless stated otherwise.
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| [[Gottlob Frege|Frege]]'s axiom system:<ref name = "Pro I">Yasuyuki Imai, Kiyoshi Iséki, On axiom systems of propositional calculi, I, Proceedings of the Japan Academy. Volume 41, Number 6 (1965), 436–439.</ref>
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| :<math>A\to(B\to A)</math>
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| :<math>(A\to(B\to C))\to((A\to B)\to(A\to C))</math>
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| :<math>(A\to B)\to(\neg B\to\neg A)</math>
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| :<math>\neg\neg A\to A</math>
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| :<math>A\to\neg\neg A</math>
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| [[David Hilbert|Hilbert]]'s axiom system:<ref name="Pro I"/>
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| :<math>A\to(B\to A)</math>
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| :<math>(A\to(B\to C))\to(B\to(A\to C))</math>
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| :<math>(B\to C)\to((A\to B)\to(A\to C))</math>
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| :<math>A\to(\neg A\to B)</math>
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| :<math>(A\to B)\to((\neg A\to B)\to B)</math>
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| [[Jan Łukasiewicz|Łukasiewicz]]'s axiom systems:<ref name="Pro I"/>
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| *First:
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| ::<math>(A\to B)\to((B\to C)\to(A\to C))</math>
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| ::<math>(\neg A\to A)\to A</math>
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| ::<math>A\to(\neg A\to B)</math>
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| *Second:
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| ::<math>((A\to B)\to C)\to(\neg A\to C)</math>
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| ::<math>((A\to B)\to C)\to(B\to C)</math>
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| ::<math>(\neg A\to C)\to((B\to C)\to((A\to B)\to C))</math>
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| *Third:
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| ::<math>A\to(B\to A)</math>
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| ::<math>(A\to(B\to C))\to((A\to B)\to(A\to C))</math>
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| ::<math>(\neg A\to\neg B)\to(B\to A)</math>
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| *Fourth:{{Citation needed|date=September 2010}}
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| ::<math>(A\to B)\to((B\to C)\to(A\to C))</math>
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| ::<math>A\to(\neg A\to B)</math>
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| ::<math>(\neg A\to B)\to((B\to A)\to A)</math>
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| Łukasiewicz and [[Alfred Tarski|Tarski]]'s axiom system:<ref name = "Pro XIII">Part XIII: Shôtarô Tanaka. On axiom systems of propositional calculi, XIII. Proc. Japan Acad., Volume 41, Number 10 (1965), 904–907.</ref>
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| :<math>[(A\to(B\to A))\to([(\neg C\to(D\to\neg E))\to[(C\to(D\to F))\to((E\to D)\to(E\to F))]]\to G)]\to(H\to G)</math>
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| [[Carew Arthur Meredith|Meredith]]'s axiom system:
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| :<math>((((A\to B)\to(\neg C\to\neg D))\to C)\to E)\to((E\to A)\to(D\to A))</math>
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| [[Elliott Mendelson|Mendelson]]'s axiom system:<ref>Elliott Mendelson, ''Introduction to Mathematical Logic'', Van Nostrand, New York, 1979, p. 31.</ref>
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| :<math>A\to(B\to A)</math>
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| :<math>(A\to(B\to C))\to((A\to B)\to(A\to C))</math>
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| :<math>(\neg A\to\neg B)\to((\neg A\to B)\to A)</math>
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| [[Bertrand Russell|Russell]]'s axiom system:<ref name="Pro I"/> | |
| :<math>A\to(B\to A)</math>
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| :<math>(A\to B)\to((B\to C)\to(A\to C))</math>
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| :<math>(A\to(B\to C))\to(B\to(A\to C))</math>
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| :<math>\neg\neg A\to A</math>
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| :<math>(A\to\neg A)\to\neg A</math>
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| :<math>(A\to\neg B)\to(B\to\neg A)</math>
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| [[Bolesław Sobociński|Sobociński]]'s axiom systems:<ref name="Pro I"/>
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| *First:
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| :<math>(A\to B)\to(\neg B\to(A\to C))</math>
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| :<math>A\to(B\to(C\to A))</math>
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| :<math>(\neg A\to B)\to((A\to B)\to B)</math>
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| *Second:
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| :<math>\neg A\to(A\to B)</math>
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| :<math>A\to(B\to(C\to A))</math>
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| :<math>(\neg A\to C)\to((B\to C)\to((A\to B)\to C))</math>
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| ===Implication and falsum===
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| Instead of negation, classical logic can also be formulated using the functionally complete set <math>\{\to,\bot\}</math> of connectives.
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| Tarski-[[Paul Bernays|Bernays]]-[[Mordechaj Wajsberg|Wajsberg]] axiom system:
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| :<math>(A\to B)\to((B\to C)\to(A\to C))</math>
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| :<math>A\to(B\to A)</math>
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| :<math>((A\to B)\to A)\to A</math>
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| :<math>\bot\to A</math>
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| [[Alonzo Church|Church]]'s axiom system:
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| :<math>A\to(B\to A)</math>
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| :<math>(A\to(B\to C))\to((A\to B)\to(A\to C))</math>
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| :<math>((A\to\bot)\to\bot)\to A</math>
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| Meredith's axiom systems:
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| *First:<ref name = "Fit">[Fitelson, 2001] [http://fitelson.org/ar.html "New Elegant Axiomatizations of Some Sentential Logics"] by Branden Fitelson</ref><ref>(Computer analysis by Argonne has revealed this to be the shortest single axiom with least variables for propositional calculus).</ref><ref name = "AR">"Some New Results in Logical Calculi Obtained Using Automated Reasoning", Zac Ernst, Ken Harris, & Branden Fitelson, http://www.mcs.anl.gov/research/projects/AR/award-2001/fitelson.pdf</ref>
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| ::<math>((((A\to B)\to(C\to\bot))\to D)\to E)\to((E\to A)\to(C\to A))</math>
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| *Second:<ref name="Fit"/>
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| ::<math>((A\to B)\to((\bot\to C)\to D))\to((D\to A)\to(E\to(F\to A)))</math>
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| ===Negation and disjunction===
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| Instead of implication, classical logic can also be formulated using the functionally complete set <math>\{\neg,\lor\}</math> of connectives. These formulations use the following rule of inference;
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| :<math>A, \neg A\lor B\vdash B.</math>
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| Russell-Bernays axiom system:
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| :<math>\neg (\neg B\lor C)\lor (\neg (A\lor B)\lor (A\lor C))</math>
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| :<math>\neg (A\lor B)\lor (B\lor A)</math>
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| :<math>\neg A\lor (B\lor A)</math>
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| :<math>\neg (A\lor A)\lor A</math>
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| Meredith's axiom systems:<ref>C. Meredith, ''Single axioms for the systems (C, N), (C, 0) and (A, N) of the two-valued propositional calculus'', Journal of Computing Systems, p. 155-164, 1954.</ref>
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| *First:
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| ::<math>\neg (\neg (\neg A\lor B)\lor (C\lor (D\lor E)))\lor (\neg (\neg D\lor A)\lor (C\lor (E\lor A)))</math>
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| *Second:
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| ::<math>\neg (\neg (\neg A\lor B)\lor (C\lor (D\lor E)))\lor (\neg (\neg E\lor D)\lor (C\lor (A\lor D)))</math>
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| *Third:
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| ::<math>\neg (\neg (\neg A\lor B)\lor (C\lor (D\lor E)))\lor (\neg (\neg C\lor A)\lor (E\lor (D\lor A)))</math>
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| Dually, classical propositional logic can be defined using only conjunction and negation.
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| ===Sheffer's stroke===
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| Because [[Sheffer's stroke]] (also known as NAND operator) is [[functionally complete]], it can be used to create an entire formulation of propositional calculus. NAND formulations use a rule of inference called [[Jean Nicod|Nicod]]'s modus ponens:
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| :<math>A, A\mid(B\mid C)\vdash C.</math>
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| Nicod's axiom system:<ref name="Fit"/>
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| :<math>(A\mid(B\mid C))\mid[(E\mid(E\mid E))\mid((D\mid B)\mid[(A\mid D)\mid(A\mid D)])]</math>
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| Łukasiewicz's axiom systems:<ref name="Fit"/>
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| *First:
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| ::<math>(A\mid(B\mid C))\mid[(D\mid(D\mid D))\mid((D\mid B)\mid[(A\mid D)\mid(A\mid D)])]</math>
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| *Second:
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| ::<math>(A\mid(B\mid C))\mid[(A\mid(C\mid A))\mid((D\mid B)\mid[(A\mid D)\mid(A\mid D)])]</math>
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| Wajsberg's axiom system:<ref name="Fit"/>
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| :<math>(A\mid(B\mid C))\mid[((D\mid C)\mid[(A\mid D)\mid(A\mid D)])\mid(A\mid(A\mid B))]</math>
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| [[Argonne National Laboratory|Argonne]] axiom systems:<ref name="Fit"/>
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| *First:
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| :<math>(A\mid(B\mid C))\mid[(A\mid(B\mid C))\mid((D\mid C)\mid[(C\mid D)\mid(A\mid D)])]</math>
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| *Second:
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| :<math>(A\mid(B\mid C))\mid[([(B\mid D)\mid(A\mid D)]\mid(D\mid B))\mid((C\mid B)\mid A)]</math><ref>[http://arxiv.org/PS_cache/cs/pdf/0205/0205078v1.pdf , p. 9, A Spectrum of Applications of Automated Reasoning], Larry Wos; arXiv:cs/0205078v1</ref>
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| Computer analysis by Argonne has revealed > 60 additional single axiom systems that can be used to formulate NAND propositional calculus.<ref name="AR"/>
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| ==Implicational propositional calculus==
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| {{Unreferenced section|date=September 2010}}
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| The [[implicational propositional calculus]] is the fragment of the classical propositional calculus which only admits the implication connective. It is not functionally complete (because it lacks the ability to express falsity and negation) but it is however syntactically complete. The implicational calculi below use modus ponens as an inference rule. | |
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| Bernays–Tarski axiom system:
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| :<math>A\to(B\to A)</math>
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| :<math>(A\to B)\to((B\to C)\to(A\to C))</math>
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| :<math>((A\to B)\to A)\to A</math>
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| Łukasiewicz and Tarski's axiom systems:
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| *First:
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| ::<math>[(A\to(B\to A))\to[([((C\to D)\to E)\to F]\to[(D\to F)\to(C\to F)])\to G]]\to G</math>
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| *Second:
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| ::<math>[(A\to B)\to((C\to D)\to E)]\to([F\to((C\to D)\to E)]\to[(A\to F)\to(D\to E)])</math>
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| *Third:
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| ::<math>((A\to B)\to(C\to D))\to(E\to((D\to A)\to(C\to A)))</math>
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| *Fourth:
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| ::<math>((A\to B)\to(C\to D))\to((D\to A)\to(E\to(C\to A)))</math>
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| Łukasiewicz's axiom system:
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| :<math>((A\to B)\to C)\to((C\to A)\to(D\to A))</math>
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| ==Intuitionistic and intermediate logics==
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| [[Intuitionistic logic]] is a subsystem of classical logic. It is commonly formulated with <math>\{\to,\land,\lor,\bot\}</math> as the set of (functionally complete) basic connectives. It is not syntactically complete since it lacks [[excluded middle]] A∨¬A or [[Peirce's law]] ((A→B)→A)→A which can be added without making the logic inconsistent. It has modus ponens as inference rule, and the following axioms:
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| :<math>A\to(B\to A)</math>
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| :<math>(A\to(B\to C))\to((A\to B)\to(A\to C))</math>
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| :<math>(A\land B)\to A</math>
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| :<math>(A\land B)\to B</math>
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| :<math>A\to(B\to(A\land B))</math>
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| :<math>A\to(A\lor B)</math>
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| :<math>B\to(A\lor B)</math>
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| :<math>(A\to C)\to((B\to C)\to((A\lor B)\to C))</math>
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| :<math>\bot\to A</math>
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| Alternatively, intuitionistic logic may be axiomatized using <math>\{\to,\land,\lor,\neg\}</math> as the set of basic connectives, replacing the last axiom with
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| :<math>(A\to\neg A)\to\neg A</math>
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| :<math>\neg A\to(A\to B)</math>
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| [[Intermediate logics]] are in between intuitionistic logic and classical logic. Here are a few intermediate logics:
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| * Jankov logic (KC) is an extension of intuitionistic logic, which can be axiomatized by the intuitionistic axiom system plus the axiom<ref name="CZ">A. Chagrov, M. Zakharyaschev, ''Modal logic'', Oxford University Press, 1997.</ref>
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| :<math>\neg A\lor\neg\neg A.</math>
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| * Gödel–Dummett logic (LC) can be axiomatized over intuitionistic logic by adding the axiom<ref name="CZ"/>
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| :<math>(A\to B)\lor(B\to A).</math>
| |
| | |
| ==Positive implicational calculus==
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| The positive implicational calculus is the implicational fragment of intuitionistic logic. The calculi below use modus ponens as an inference rule.
| |
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| Łukasiewicz's axiom system:
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| :<math>A\to(B\to A)</math>
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| :<math>(A\to(B\to C))\to((A\to B)\to(A\to C))</math>
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| Meredith's axiom systems:
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| *First:
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| ::<math>E\to((A\to B)\to(((D\to A)\to(B\to C))\to(A\to C)))</math>
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| *Second:
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| ::<math>A\to(B\to A)</math>
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| ::<math>(A\to B)\to((A\to(B\to C))\to(A\to C))</math>
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| *Third:
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| ::<math>((A\to B)\to C)\to(D\to((B\to(C\to E))\to(B\to E)))</math><ref>C. Meredith, ''A single axiom of positive logic'', Journal of Computing Systems, p. 169-170, 1954.</ref>
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| Hilbert's axiom systems:
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| *First:
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| ::<math>(A\to(A\to B))\to(A\to B)</math>
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| ::<math>(B\to C)\to((A\to B)\to(A\to C))</math>
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| ::<math>(A\to(B\to C))\to(B\to(A\to C))</math>
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| ::<math>A\to(B\to A)</math>
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| *Second:
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| ::<math>(A\to(A\to B))\to(A\to B)</math>
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| ::<math>(A\to B)\to((B\to C)\to(A\to C))</math>
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| ::<math>A\to(B\to A)</math>
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| *Third:
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| ::<math>A\to A</math>
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| ::<math>(A\to B)\to((B\to C)\to(A\to C))</math>
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| ::<math>(B\to C)\to((A\to B)\to(A\to C))</math>
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| ::<math>(A\to(A\to B))\to(A\to B)</math>
| |
| | |
| ==Positive propositional calculus==
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| Positive propositional calculus is the fragment of intuitionistic logic using only the (non functionally complete) connectives <math>\{\to,\land,\lor\}</math>. It can be axiomatized by any of the above mentioned calculi for positive implicational calculus together with the axioms
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| :<math>(A\land B)\to A</math>
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| :<math>(A\land B)\to B</math>
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| :<math>A\to(B\to(A\land B))</math>
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| :<math>A\to(A\lor B)</math>
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| :<math>B\to(A\lor B)</math>
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| :<math>(A\to C)\to((B\to C)\to((A\lor B)\to C))</math>
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| Optionally, we may also include the connective <math>\leftrightarrow</math> and the axioms
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| :<math>(A\leftrightarrow B)\to(A\to B)</math>
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| :<math>(A\leftrightarrow B)\to(B\to A)</math>
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| :<math>(A\to B)\to((B\to A)\to(A\leftrightarrow B))</math>
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| [[Ingebrigt Johansson|Johansson]]'s [[minimal logic]] can be axiomatized by any of the axiom systems for positive propositional calculus and expanding its language with the nullary connective <math>\bot</math>, with no additional axiom schemas. Alternatively, it can also be axiomatized in the language <math>\{\to,\land,\lor,\neg\}</math> by expanding the positive propositional calculus with the axiom
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| :<math>(A\to\neg B)\to(B\to\neg A)</math>
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| or the pair of axioms
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| :<math>(A\to B)\to(\neg B\to\neg A)</math>
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| :<math>A\to\neg\neg A</math>
| |
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| Intuitionistic logic in language with negation can be axiomatized over the positive calculus by the pair of axioms
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| :<math>(A\to\neg B)\to(B\to\neg A)</math>
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| :<math>\neg A\to(A\to B)</math>
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| or the pair of axioms<ref name="Hackstaff">L. H. Hackstaff, ''Systems of Formal Logic'', Springer, 1966.</ref>
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| :<math>(A\to\neg A)\to\neg A</math>
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| :<math>\neg A\to(A\to B)</math>
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| Classical logic in the language <math>\{\to,\land,\lor,\neg\}</math> can be obtained from the positive propositional calculus by adding the axiom
| |
| :<math>(\neg A\to\neg B)\to(B\to A)</math>
| |
| or the pair of axioms
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| :<math>(A\to\neg B)\to(B\to\neg A)</math>
| |
| :<math>\neg\neg A\to A</math>
| |
| | |
| Fitch calculus takes any of the axiom systems for positive propositional calculus and adds the axioms<ref name="Hackstaff"/>
| |
| :<math>\neg A\to(A\to B)</math>
| |
| :<math>A\leftrightarrow\neg\neg A</math>
| |
| :<math>\neg(A\lor B)\leftrightarrow(\neg A\land\neg B)</math>
| |
| :<math>\neg(A\land B)\leftrightarrow(\neg A\lor\neg B)</math>
| |
| Note that the first and third axioms are also valid in intuitionistic logic.
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| | |
| ==Equivalential calculus==
| |
| Equivalential calculus is the subsystem of classical propositional calculus that only allows the (functionally incomplete) [[biconditional|equivalence]] connective, denoted here as <math>\equiv</math><!-- because if I had to write \leftrightarrow for every single one of them, I would go mad -->. The rule of inference used in these systems is as follows:
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| :<math>A,A\equiv B\vdash B</math>
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| Iséki's axiom system:<ref>Kiyoshi Iséki, On axiom systems of propositional calculi, XV, Proceedings of the Japan Academy. Volume 42, Number 3 (1966), 217–220.</ref>
| |
| :<math>((A\equiv C)\equiv(B\equiv A))\equiv(C\equiv B)</math>
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| :<math>(A\equiv(B\equiv C))\equiv((A\equiv B)\equiv C)</math>
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| Iséki-Arai axiom system:<ref name = "XVII">Yoshinari Arai, On axiom systems of propositional calculi, XVII, Proceedings of the Japan Academy. Volume 42, Number 4 (1966), 351–354.</ref>
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| :<math>A\equiv A</math>
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| :<math>(A\equiv B)\equiv(B\equiv A)</math>
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| :<math>(A\equiv B)\equiv((B\equiv C)\equiv(A\equiv C))</math>
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| Arai's axiom systems;
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| *First:
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| ::<math>(A\equiv(B\equiv C))\equiv((A\equiv B)\equiv C)</math>
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| ::<math>((A\equiv C)\equiv(B\equiv A))\equiv(C\equiv B)</math>
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| *Second:
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| ::<math>(A\equiv B)\equiv(B\equiv A)</math>
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| ::<math>((A\equiv C)\equiv(B\equiv A))\equiv(C\equiv B)</math>
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| Łukasiewicz's axiom systems:<ref name = "XCB">[http://www.mcs.anl.gov/uploads/cels/papers/P966.pdf XCB, the Last of the Shortest Single Axioms for the Classical Equivalential Calculus], LARRY WOS, DOLPH ULRICH,
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| BRANDEN FITELSON; arXiv:cs/0211015v1</ref>
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| *First:
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| ::<math>(A\equiv B)\equiv((C\equiv B)\equiv(A\equiv C))</math>
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| *Second:
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| ::<math>(A\equiv B)\equiv((A\equiv C)\equiv(C\equiv B))</math>
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| *Third:
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| ::<math>(A\equiv B)\equiv((C\equiv A)\equiv(B\equiv C))</math>
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| Meredith's axiom systems:<ref name="XCB"/>
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| *First:
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| ::<math>((A\equiv B)\equiv C)\equiv(B\equiv(C\equiv A))</math>
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| *Second:
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| ::<math>A\equiv((B\equiv(A\equiv C))\equiv(C\equiv B))</math>
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| *Third:
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| ::<math>(A\equiv(B\equiv C))\equiv(C\equiv(A\equiv B))</math>
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| *Fourth:
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| ::<math>(A\equiv B)\equiv(C\equiv((B\equiv C)\equiv A))</math>
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| *Fifth:
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| ::<math>(A\equiv B)\equiv(C\equiv((C\equiv B)\equiv A))</math>
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| *Sixth:
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| ::<math>((A\equiv(B\equiv C))\equiv C)\equiv(B\equiv A)</math>
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| *Seventh:
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| ::<math>((A\equiv(B\equiv C))\equiv B)\equiv(C\equiv A)</math>
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| [[John Arnold Kalman|Kalman]]'s axiom system:<ref name="XCB"/>
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| ::<math>A\equiv((B\equiv(C\equiv A))\equiv(C\equiv B))</math>
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| [[S. Winker|Winker]]'s axiom systems:<ref name="XCB"/>
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| *First:
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| ::<math>A\equiv((B\equiv C)\equiv((A\equiv C)\equiv B))</math>
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| *Second:
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| ::<math>A\equiv((B\equiv C)\equiv((C\equiv A)\equiv B))</math>
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| XCB axiom system:<ref name="XCB"/>
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| ::<math>A\equiv(((A\equiv B)\equiv(C\equiv B))\equiv C)</math>
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| ==References==
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| {{Reflist}}
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| {{DEFAULTSORT:List Of Logic Systems}}
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| [[Category:Logical calculi|Logic systems, List of]]
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| [[Category:Systems of formal logic|Logic systems, List of]]
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| [[Category:Propositional calculus|Logic systems, List of]]
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| [[Category:Mathematical logic|Logic systems, List of]]
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| [[Category:Philosophy-related lists|Logic systems]]
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