Classical logic - Revision history

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	== Examples of classical logics ==		== Examples of classical logics ==
	* [[Aristotle]]'s [[Organon]] introduces his [[Theory]] of syllogisms, which is a logic with a restricted form of judgments: assertions take one of four forms, ''All Ps are Q'', ''Some Ps are Q'', ''No Ps are Q'', and ''Some Ps are not Q''. These judgments find Themselves if two pairs of two dual operators, and each operator is the negation of another, relationships that Aristotle ~~summarised~~ with his ''square of oppositions''. Aristotle explicitly formulated the law of the excluded middle and law of non-contradiction in justifying his system, although These laws cannot be expressed as judgments within the syllogistic framework.		* [[Aristotle]]'s [[Organon]] introduces his [[Theory]] of syllogisms, which is a logic with a restricted form of judgments: assertions take one of four forms, ''All Ps are Q'', ''Some Ps are Q'', ''No Ps are Q'', and ''Some Ps are not Q''. These judgments find Themselves if two pairs of two dual operators, and each operator is the negation of another, relationships that Aristotle summarized with his ''square of oppositions''. Aristotle explicitly formulated the law of the excluded middle and law of non-contradiction in justifying his system, although These laws cannot be expressed as judgments within the syllogistic framework.
	* George Boole's algebraic reformulation of logic, his system of [[Boolean logic]];		* George Boole's algebraic reformulation of logic, his system of [[Boolean logic]];
	* the first-order logic found in Gottlob Frege's [[Begriffsschrift]].		* the first-order logic found in Gottlob Frege's [[Begriffsschrift]].

𝗔𝗿𝗰𝗵𝗮𝗻𝗴𝗲𝗹: Text replacement - "oTher" to "other"

2023-03-13T18:10:19Z

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	# [[Monotonicity of entailment]] and [[Idempotency of entailment]];		# [[Monotonicity of entailment]] and [[Idempotency of entailment]];
	# [[Commutativity of conjunction]];		# [[Commutativity of conjunction]];
	# [[De Morgan duality]]: every [[logical operator]] is dual to ~~anoTher~~;		# [[De Morgan duality]]: every [[logical operator]] is dual to another;

	While not entailed by the preceding conditions, contemporary discussions of classical logic normally only include propositional and first-order logic.<ref>Shapiro, Stewart (2000). Classical Logic. In Stanford Encyclopedia of Philosophy [Web]. Stanford: the Metaphysics Research Lab. Retrieved October 28, 2006, from http://plato.stanford.edu/entries/logic-classical/</ref><ref name="haack">Haack, Susan, (1996). ''Deviant Logic, Fuzzy Logic: Beyond the Formalism''. Chicago: the University of Chicago Press.</ref>		While not entailed by the preceding conditions, contemporary discussions of classical logic normally only include propositional and first-order logic.<ref>Shapiro, Stewart (2000). Classical Logic. In Stanford Encyclopedia of Philosophy [Web]. Stanford: the Metaphysics Research Lab. Retrieved October 28, 2006, from http://plato.stanford.edu/entries/logic-classical/</ref><ref name="haack">Haack, Susan, (1996). ''Deviant Logic, Fuzzy Logic: Beyond the Formalism''. Chicago: the University of Chicago Press.</ref>

	The intended semantics of classical logic is bivalent. With the advent of algebraic logic it became apparent however that classical propositional calculus admits ~~oTher~~ semantics. In Boolean valued semantics (for classical propositional logic), the truth values are the elements of an arbitrary Boolean algebra; "true" corresponds to the maximal element of the algebra, and "false" corresponds to the minimal element. Intermediate elements of the algebra correspond to truth values ~~oTher~~ than "true" and "false". the principle of bivalence holds only when the Boolean algebra is taken to be the two-element algebra, which has no intermediate elements.		The intended semantics of classical logic is bivalent. With the advent of algebraic logic it became apparent however that classical propositional calculus admits other semantics. In Boolean valued semantics (for classical propositional logic), the truth values are the elements of an arbitrary Boolean algebra; "true" corresponds to the maximal element of the algebra, and "false" corresponds to the minimal element. Intermediate elements of the algebra correspond to truth values other than "true" and "false". the principle of bivalence holds only when the Boolean algebra is taken to be the two-element algebra, which has no intermediate elements.

	== Examples of classical logics ==		== Examples of classical logics ==
	* [[Aristotle]]'s [[Organon]] introduces his [[Theory]] of syllogisms, which is a logic with a restricted form of judgments: assertions take one of four forms, ''All Ps are Q'', ''Some Ps are Q'', ''No Ps are Q'', and ''Some Ps are not Q''. These judgments find Themselves if two pairs of two dual operators, and each operator is the negation of ~~anoTher~~, relationships that Aristotle summarised with his ''square of oppositions''. Aristotle explicitly formulated the law of the excluded middle and law of non-contradiction in justifying his system, although These laws cannot be expressed as judgments within the syllogistic framework.		* [[Aristotle]]'s [[Organon]] introduces his [[Theory]] of syllogisms, which is a logic with a restricted form of judgments: assertions take one of four forms, ''All Ps are Q'', ''Some Ps are Q'', ''No Ps are Q'', and ''Some Ps are not Q''. These judgments find Themselves if two pairs of two dual operators, and each operator is the negation of another, relationships that Aristotle summarised with his ''square of oppositions''. Aristotle explicitly formulated the law of the excluded middle and law of non-contradiction in justifying his system, although These laws cannot be expressed as judgments within the syllogistic framework.
	* George Boole's algebraic reformulation of logic, his system of [[Boolean logic]];		* George Boole's algebraic reformulation of logic, his system of [[Boolean logic]];
	* the first-order logic found in Gottlob Frege's [[Begriffsschrift]].		* the first-order logic found in Gottlob Frege's [[Begriffsschrift]].

WikiSysop: Text replacement - " The " to " the "

2023-02-27T09:40:16Z

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Bacchus: Text replacement - "tbe" to "the"

2023-02-25T07:54:39Z

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← Older revision		Revision as of 22:54, 24 February 2023
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	'''Classical logic''' identifies a class of [[formal logic]] that has been most intensively studied and most widely used. The class is sometimes called '''standard logic''' as well.<ref name="BunninYu2004">{{cite book\|author1=Nicholas Bunnin\|author2=Jiyuan Yu\|title=The Blackwell dictionary of Western philosophy\|url=http://books.google.com/books?id=OskKWI1YA7AC&pg=PA266\|year=2004\|publisher=Wiley-Blackwell\|isbn=978-1-4051-0679-5\|page=266}}</ref><ref name="Gamut1991">{{cite book\|author=L. T. F. Gamut\|title=Logic, language, and meaning, Volume 1: Introduction to Logic\|url=http://books.google.com/books?id=Z0KhywkpolMC&pg=PA156\|year=1991\|publisher=University of Chicago Press\|isbn=978-0-226-28085-1\|pages=156–157}}</ref> ~~tbey~~ are characterised by a number of properties:<ref>Dov Gabbay, (1994). 'Classical vs non-classical logic'. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, (Eds), ''Handbook of Logic in Artificial Intelligence and Logic Programming'', volume 2, chapter 2.6. Oxford University Press.</ref>		'''Classical logic''' identifies a class of [[formal logic]] that has been most intensively studied and most widely used. The class is sometimes called '''standard logic''' as well.<ref name="BunninYu2004">{{cite book\|author1=Nicholas Bunnin\|author2=Jiyuan Yu\|title=The Blackwell dictionary of Western philosophy\|url=http://books.google.com/books?id=OskKWI1YA7AC&pg=PA266\|year=2004\|publisher=Wiley-Blackwell\|isbn=978-1-4051-0679-5\|page=266}}</ref><ref name="Gamut1991">{{cite book\|author=L. T. F. Gamut\|title=Logic, language, and meaning, Volume 1: Introduction to Logic\|url=http://books.google.com/books?id=Z0KhywkpolMC&pg=PA156\|year=1991\|publisher=University of Chicago Press\|isbn=978-0-226-28085-1\|pages=156–157}}</ref> they are characterised by a number of properties:<ref>Dov Gabbay, (1994). 'Classical vs non-classical logic'. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, (Eds), ''Handbook of Logic in Artificial Intelligence and Logic Programming'', volume 2, chapter 2.6. Oxford University Press.</ref>
	# [[Law of The excluded middle]] and [[Double negative elimination]];		# [[Law of The excluded middle]] and [[Double negative elimination]];
	# [[Law of noncontradiction]], and The [[principle of explosion]];		# [[Law of noncontradiction]], and The [[principle of explosion]];

WikiSysop: Text replacement - "the" to "tbe"

2023-02-22T21:44:10Z

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	'''Classical logic''' identifies a class of [[formal logic]] that has been most intensively studied and most widely used. The class is sometimes called '''standard logic''' as well.<ref name="BunninYu2004">{{cite book\|author1=Nicholas Bunnin\|author2=Jiyuan Yu\|title=The Blackwell dictionary of Western philosophy\|url=http://books.google.com/books?id=OskKWI1YA7AC&pg=PA266\|year=2004\|publisher=Wiley-Blackwell\|isbn=978-1-4051-0679-5\|page=266}}</ref><ref name="Gamut1991">{{cite book\|author=L. T. F. Gamut\|title=Logic, language, and meaning, Volume 1: Introduction to Logic\|url=http://books.google.com/books?id=Z0KhywkpolMC&pg=PA156\|year=1991\|publisher=University of Chicago Press\|isbn=978-0-226-28085-1\|pages=156–157}}</ref> ~~they~~ are characterised by a number of properties:<ref>Dov Gabbay, (1994). 'Classical vs non-classical logic'. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, (Eds), ''Handbook of Logic in Artificial Intelligence and Logic Programming'', volume 2, chapter 2.6. Oxford University Press.</ref>		'''Classical logic''' identifies a class of [[formal logic]] that has been most intensively studied and most widely used. The class is sometimes called '''standard logic''' as well.<ref name="BunninYu2004">{{cite book\|author1=Nicholas Bunnin\|author2=Jiyuan Yu\|title=The Blackwell dictionary of Western philosophy\|url=http://books.google.com/books?id=OskKWI1YA7AC&pg=PA266\|year=2004\|publisher=Wiley-Blackwell\|isbn=978-1-4051-0679-5\|page=266}}</ref><ref name="Gamut1991">{{cite book\|author=L. T. F. Gamut\|title=Logic, language, and meaning, Volume 1: Introduction to Logic\|url=http://books.google.com/books?id=Z0KhywkpolMC&pg=PA156\|year=1991\|publisher=University of Chicago Press\|isbn=978-0-226-28085-1\|pages=156–157}}</ref> tbey are characterised by a number of properties:<ref>Dov Gabbay, (1994). 'Classical vs non-classical logic'. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, (Eds), ''Handbook of Logic in Artificial Intelligence and Logic Programming'', volume 2, chapter 2.6. Oxford University Press.</ref>
	# [[Law of The excluded middle]] and [[Double negative elimination]];		# [[Law of The excluded middle]] and [[Double negative elimination]];
	# [[Law of noncontradiction]], and The [[principle of explosion]];		# [[Law of noncontradiction]], and The [[principle of explosion]];

WikiSysop: Text replacement - "tbe" to "the"

2023-02-20T11:04:15Z

Text replacement - "tbe" to "the"

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	'''Classical logic''' identifies a class of [[formal logic]] that has been most intensively studied and most widely used. The class is sometimes called '''standard logic''' as well.<ref name="BunninYu2004">{{cite book\|author1=Nicholas Bunnin\|author2=Jiyuan Yu\|title=The Blackwell dictionary of Western philosophy\|url=http://books.google.com/books?id=OskKWI1YA7AC&pg=PA266\|year=2004\|publisher=Wiley-Blackwell\|isbn=978-1-4051-0679-5\|page=266}}</ref><ref name="Gamut1991">{{cite book\|author=L. T. F. Gamut\|title=Logic, language, and meaning, Volume 1: Introduction to Logic\|url=http://books.google.com/books?id=Z0KhywkpolMC&pg=PA156\|year=1991\|publisher=University of Chicago Press\|isbn=978-0-226-28085-1\|pages=156–157}}</ref> ~~tbey~~ are characterised by a number of properties:<ref>Dov Gabbay, (1994). 'Classical vs non-classical logic'. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, (Eds), ''Handbook of Logic in Artificial Intelligence and Logic Programming'', volume 2, chapter 2.6. Oxford University Press.</ref>		'''Classical logic''' identifies a class of [[formal logic]] that has been most intensively studied and most widely used. The class is sometimes called '''standard logic''' as well.<ref name="BunninYu2004">{{cite book\|author1=Nicholas Bunnin\|author2=Jiyuan Yu\|title=The Blackwell dictionary of Western philosophy\|url=http://books.google.com/books?id=OskKWI1YA7AC&pg=PA266\|year=2004\|publisher=Wiley-Blackwell\|isbn=978-1-4051-0679-5\|page=266}}</ref><ref name="Gamut1991">{{cite book\|author=L. T. F. Gamut\|title=Logic, language, and meaning, Volume 1: Introduction to Logic\|url=http://books.google.com/books?id=Z0KhywkpolMC&pg=PA156\|year=1991\|publisher=University of Chicago Press\|isbn=978-0-226-28085-1\|pages=156–157}}</ref> they are characterised by a number of properties:<ref>Dov Gabbay, (1994). 'Classical vs non-classical logic'. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, (Eds), ''Handbook of Logic in Artificial Intelligence and Logic Programming'', volume 2, chapter 2.6. Oxford University Press.</ref>
	# [[Law of The excluded middle]] and [[Double negative elimination]];		# [[Law of The excluded middle]] and [[Double negative elimination]];
	# [[Law of noncontradiction]], and The [[principle of explosion]];		# [[Law of noncontradiction]], and The [[principle of explosion]];

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	'''Classical logic''' identifies a class of [[formal logic]] that has been most intensively studied and most widely used. ~~tbe~~ class is sometimes called '''standard logic''' as well.<ref name="BunninYu2004">{{cite book\|author1=Nicholas Bunnin\|author2=Jiyuan Yu\|title=The Blackwell dictionary of Western philosophy\|url=http://books.google.com/books?id=OskKWI1YA7AC&pg=PA266\|year=2004\|publisher=Wiley-Blackwell\|isbn=978-1-4051-0679-5\|page=266}}</ref><ref name="Gamut1991">{{cite book\|author=L. T. F. Gamut\|title=Logic, language, and meaning, Volume 1: Introduction to Logic\|url=http://books.google.com/books?id=Z0KhywkpolMC&pg=PA156\|year=1991\|publisher=University of Chicago Press\|isbn=978-0-226-28085-1\|pages=156–157}}</ref> they are characterised by a number of properties:<ref>Dov Gabbay, (1994). 'Classical vs non-classical logic'. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, (Eds), ''Handbook of Logic in Artificial Intelligence and Logic Programming'', volume 2, chapter 2.6. Oxford University Press.</ref>		'''Classical logic''' identifies a class of [[formal logic]] that has been most intensively studied and most widely used. the class is sometimes called '''standard logic''' as well.<ref name="BunninYu2004">{{cite book\|author1=Nicholas Bunnin\|author2=Jiyuan Yu\|title=The Blackwell dictionary of Western philosophy\|url=http://books.google.com/books?id=OskKWI1YA7AC&pg=PA266\|year=2004\|publisher=Wiley-Blackwell\|isbn=978-1-4051-0679-5\|page=266}}</ref><ref name="Gamut1991">{{cite book\|author=L. T. F. Gamut\|title=Logic, language, and meaning, Volume 1: Introduction to Logic\|url=http://books.google.com/books?id=Z0KhywkpolMC&pg=PA156\|year=1991\|publisher=University of Chicago Press\|isbn=978-0-226-28085-1\|pages=156–157}}</ref> they are characterised by a number of properties:<ref>Dov Gabbay, (1994). 'Classical vs non-classical logic'. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, (Eds), ''Handbook of Logic in Artificial Intelligence and Logic Programming'', volume 2, chapter 2.6. Oxford University Press.</ref>
	# [[Law of ~~tbe~~ excluded middle]] and [[Double negative elimination]];		# [[Law of the excluded middle]] and [[Double negative elimination]];
	# [[Law of noncontradiction]], and ~~tbe~~ [[principle of explosion]];		# [[Law of noncontradiction]], and the [[principle of explosion]];
	# [[Monotonicity of entailment]] and [[Idempotency of entailment]];		# [[Monotonicity of entailment]] and [[Idempotency of entailment]];
	# [[Commutativity of conjunction]];		# [[Commutativity of conjunction]];
	# [[De Morgan duality]]: every [[logical operator]] is dual to another;		# [[De Morgan duality]]: every [[logical operator]] is dual to another;

	While not entailed by ~~tbe~~ preceding conditions, contemporary discussions of classical logic normally only include propositional and first-order logic.<ref>Shapiro, Stewart (2000). Classical Logic. In Stanford Encyclopedia of Philosophy [Web]. Stanford: ~~tbe~~ Metaphysics Research Lab. Retrieved October 28, 2006, from http://plato.stanford.edu/entries/logic-classical/</ref><ref name="haack">Haack, Susan, (1996). ''Deviant Logic, Fuzzy Logic: Beyond ~~tbe~~ Formalism''. Chicago: ~~tbe~~ University of Chicago Press.</ref>		While not entailed by the preceding conditions, contemporary discussions of classical logic normally only include propositional and first-order logic.<ref>Shapiro, Stewart (2000). Classical Logic. In Stanford Encyclopedia of Philosophy [Web]. Stanford: the Metaphysics Research Lab. Retrieved October 28, 2006, from http://plato.stanford.edu/entries/logic-classical/</ref><ref name="haack">Haack, Susan, (1996). ''Deviant Logic, Fuzzy Logic: Beyond the Formalism''. Chicago: the University of Chicago Press.</ref>

	The intended semantics of classical logic is bivalent. With ~~tbe~~ advent of algebraic logic it became apparent however that classical propositional calculus admits other semantics. In Boolean valued semantics (for classical propositional logic), ~~tbe~~ truth values are ~~tbe~~ elements of an arbitrary Boolean algebra; "true" corresponds to ~~tbe~~ maximal element of ~~tbe~~ algebra, and "false" corresponds to ~~tbe~~ minimal element. Intermediate elements of ~~tbe~~ algebra correspond to truth values other than "true" and "false". ~~tbe~~ principle of bivalence holds only when ~~tbe~~ Boolean algebra is taken to be ~~tbe~~ two-element algebra, which has no intermediate elements.		The intended semantics of classical logic is bivalent. With the advent of algebraic logic it became apparent however that classical propositional calculus admits other semantics. In Boolean valued semantics (for classical propositional logic), the truth values are the elements of an arbitrary Boolean algebra; "true" corresponds to the maximal element of the algebra, and "false" corresponds to the minimal element. Intermediate elements of the algebra correspond to truth values other than "true" and "false". the principle of bivalence holds only when the Boolean algebra is taken to be the two-element algebra, which has no intermediate elements.

	== Examples of classical logics ==		== Examples of classical logics ==
	* [[Aristotle]]'s [[Organon]] introduces his [[Theory]] of syllogisms, which is a logic with a restricted form of judgments: assertions take one of four forms, ''All Ps are Q'', ''Some Ps are Q'', ''No Ps are Q'', and ''Some Ps are not Q''. These judgments find Themselves if two pairs of two dual operators, and each operator is ~~tbe~~ negation of another, relationships that Aristotle summarized with his ''square of oppositions''. Aristotle explicitly formulated ~~tbe~~ law of ~~tbe~~ excluded middle and law of non-contradiction in justifying his system, although These laws cannot be expressed as judgments within ~~tbe~~ syllogistic framework.		* [[Aristotle]]'s [[Organon]] introduces his [[Theory]] of syllogisms, which is a logic with a restricted form of judgments: assertions take one of four forms, ''All Ps are Q'', ''Some Ps are Q'', ''No Ps are Q'', and ''Some Ps are not Q''. These judgments find Themselves if two pairs of two dual operators, and each operator is the negation of another, relationships that Aristotle summarized with his ''square of oppositions''. Aristotle explicitly formulated the law of the excluded middle and law of non-contradiction in justifying his system, although These laws cannot be expressed as judgments within the syllogistic framework.
	* George Boole's algebraic reformulation of logic, his system of [[Boolean logic]];		* George Boole's algebraic reformulation of logic, his system of [[Boolean logic]];
	* ~~tbe~~ first-order logic found in Gottlob Frege's [[Begriffsschrift]].		* the first-order logic found in Gottlob Frege's [[Begriffsschrift]].

	== Non-classical logics ==		== Non-classical logics ==
	* Computability logic is a semantically constructed formal [[Theory]] of computability, as opposed to classical logic, which is a formal [[Theory]] of truth; integrates and extends classical, linear and intuitionistic logics.		* Computability logic is a semantically constructed formal [[Theory]] of computability, as opposed to classical logic, which is a formal [[Theory]] of truth; integrates and extends classical, linear and intuitionistic logics.
	* Many-valued logic, including fuzzy logic, which rejects ~~tbe~~ law of ~~tbe~~ excluded middle and allows as a truth value any real number between 0 and 1.		* Many-valued logic, including fuzzy logic, which rejects the law of the excluded middle and allows as a truth value any real number between 0 and 1.
	* [[Intuitionistic logic]] rejects ~~tbe~~ law of ~~tbe~~ excluded middle, double negative elimination, and ~~tbe~~ De Morgan's laws;		* [[Intuitionistic logic]] rejects the law of the excluded middle, double negative elimination, and the De Morgan's laws;
	* [[Linear logic]] rejects idempotency of [[Logical consequence\|entailment]] as well;		* [[Linear logic]] rejects idempotency of [[Logical consequence\|entailment]] as well;
	* [[Modal logic]] extends classical logic with non-truth-functional ("modal") operators.		* [[Modal logic]] extends classical logic with non-truth-functional ("modal") operators.
	* Paraconsistent logic (e.g., [[dialeTheism]] and [[relevance logic]]) rejects ~~tbe~~ law of noncontradiction;		* Paraconsistent logic (e.g., [[dialeTheism]] and [[relevance logic]]) rejects the law of noncontradiction;
	* [[Relevance logic]], [[linear logic]], and [[non-monotonic logic]] reject monotonicity of entailment;		* [[Relevance logic]], [[linear logic]], and [[non-monotonic logic]] reject monotonicity of entailment;

	In ''Deviant Logic, Fuzzy Logic: Beyond ~~tbe~~ Formalism'', Susan Haack divided non-classical logics into deviant, quasi-deviant, and extended logics.<ref name="haack" />		In ''Deviant Logic, Fuzzy Logic: Beyond the Formalism'', Susan Haack divided non-classical logics into deviant, quasi-deviant, and extended logics.<ref name="haack" />

	== FurTher reading ==		== FurTher reading ==

← Older revision		Revision as of 08:09, 26 April 2024
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	'''Classical logic''' identifies a class of [[formal logic]] that has been most intensively studied and most widely used. ~~the~~ class is sometimes called '''standard logic''' as well.<ref name="BunninYu2004">{{cite book\|author1=Nicholas Bunnin\|author2=Jiyuan Yu\|title=The Blackwell dictionary of Western philosophy\|url=http://books.google.com/books?id=OskKWI1YA7AC&pg=PA266\|year=2004\|publisher=Wiley-Blackwell\|isbn=978-1-4051-0679-5\|page=266}}</ref><ref name="Gamut1991">{{cite book\|author=L. T. F. Gamut\|title=Logic, language, and meaning, Volume 1: Introduction to Logic\|url=http://books.google.com/books?id=Z0KhywkpolMC&pg=PA156\|year=1991\|publisher=University of Chicago Press\|isbn=978-0-226-28085-1\|pages=156–157}}</ref> they are characterised by a number of properties:<ref>Dov Gabbay, (1994). 'Classical vs non-classical logic'. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, (Eds), ''Handbook of Logic in Artificial Intelligence and Logic Programming'', volume 2, chapter 2.6. Oxford University Press.</ref>		'''Classical logic''' identifies a class of [[formal logic]] that has been most intensively studied and most widely used. tbe class is sometimes called '''standard logic''' as well.<ref name="BunninYu2004">{{cite book\|author1=Nicholas Bunnin\|author2=Jiyuan Yu\|title=The Blackwell dictionary of Western philosophy\|url=http://books.google.com/books?id=OskKWI1YA7AC&pg=PA266\|year=2004\|publisher=Wiley-Blackwell\|isbn=978-1-4051-0679-5\|page=266}}</ref><ref name="Gamut1991">{{cite book\|author=L. T. F. Gamut\|title=Logic, language, and meaning, Volume 1: Introduction to Logic\|url=http://books.google.com/books?id=Z0KhywkpolMC&pg=PA156\|year=1991\|publisher=University of Chicago Press\|isbn=978-0-226-28085-1\|pages=156–157}}</ref> they are characterised by a number of properties:<ref>Dov Gabbay, (1994). 'Classical vs non-classical logic'. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, (Eds), ''Handbook of Logic in Artificial Intelligence and Logic Programming'', volume 2, chapter 2.6. Oxford University Press.</ref>
	# [[Law of ~~the~~ excluded middle]] and [[Double negative elimination]];		# [[Law of tbe excluded middle]] and [[Double negative elimination]];
	# [[Law of noncontradiction]], and ~~the~~ [[principle of explosion]];		# [[Law of noncontradiction]], and tbe [[principle of explosion]];
	# [[Monotonicity of entailment]] and [[Idempotency of entailment]];		# [[Monotonicity of entailment]] and [[Idempotency of entailment]];
	# [[Commutativity of conjunction]];		# [[Commutativity of conjunction]];
	# [[De Morgan duality]]: every [[logical operator]] is dual to another;		# [[De Morgan duality]]: every [[logical operator]] is dual to another;

	While not entailed by ~~the~~ preceding conditions, contemporary discussions of classical logic normally only include propositional and first-order logic.<ref>Shapiro, Stewart (2000). Classical Logic. In Stanford Encyclopedia of Philosophy [Web]. Stanford: ~~the~~ Metaphysics Research Lab. Retrieved October 28, 2006, from http://plato.stanford.edu/entries/logic-classical/</ref><ref name="haack">Haack, Susan, (1996). ''Deviant Logic, Fuzzy Logic: Beyond ~~the~~ Formalism''. Chicago: ~~the~~ University of Chicago Press.</ref>		While not entailed by tbe preceding conditions, contemporary discussions of classical logic normally only include propositional and first-order logic.<ref>Shapiro, Stewart (2000). Classical Logic. In Stanford Encyclopedia of Philosophy [Web]. Stanford: tbe Metaphysics Research Lab. Retrieved October 28, 2006, from http://plato.stanford.edu/entries/logic-classical/</ref><ref name="haack">Haack, Susan, (1996). ''Deviant Logic, Fuzzy Logic: Beyond tbe Formalism''. Chicago: tbe University of Chicago Press.</ref>

	The intended semantics of classical logic is bivalent. With ~~the~~ advent of algebraic logic it became apparent however that classical propositional calculus admits other semantics. In Boolean valued semantics (for classical propositional logic), ~~the~~ truth values are ~~the~~ elements of an arbitrary Boolean algebra; "true" corresponds to ~~the~~ maximal element of ~~the~~ algebra, and "false" corresponds to ~~the~~ minimal element. Intermediate elements of ~~the~~ algebra correspond to truth values other than "true" and "false". ~~the~~ principle of bivalence holds only when ~~the~~ Boolean algebra is taken to be ~~the~~ two-element algebra, which has no intermediate elements.		The intended semantics of classical logic is bivalent. With tbe advent of algebraic logic it became apparent however that classical propositional calculus admits other semantics. In Boolean valued semantics (for classical propositional logic), tbe truth values are tbe elements of an arbitrary Boolean algebra; "true" corresponds to tbe maximal element of tbe algebra, and "false" corresponds to tbe minimal element. Intermediate elements of tbe algebra correspond to truth values other than "true" and "false". tbe principle of bivalence holds only when tbe Boolean algebra is taken to be tbe two-element algebra, which has no intermediate elements.

	== Examples of classical logics ==		== Examples of classical logics ==
	* [[Aristotle]]'s [[Organon]] introduces his [[Theory]] of syllogisms, which is a logic with a restricted form of judgments: assertions take one of four forms, ''All Ps are Q'', ''Some Ps are Q'', ''No Ps are Q'', and ''Some Ps are not Q''. These judgments find Themselves if two pairs of two dual operators, and each operator is ~~the~~ negation of another, relationships that Aristotle summarized with his ''square of oppositions''. Aristotle explicitly formulated ~~the~~ law of ~~the~~ excluded middle and law of non-contradiction in justifying his system, although These laws cannot be expressed as judgments within ~~the~~ syllogistic framework.		* [[Aristotle]]'s [[Organon]] introduces his [[Theory]] of syllogisms, which is a logic with a restricted form of judgments: assertions take one of four forms, ''All Ps are Q'', ''Some Ps are Q'', ''No Ps are Q'', and ''Some Ps are not Q''. These judgments find Themselves if two pairs of two dual operators, and each operator is tbe negation of another, relationships that Aristotle summarized with his ''square of oppositions''. Aristotle explicitly formulated tbe law of tbe excluded middle and law of non-contradiction in justifying his system, although These laws cannot be expressed as judgments within tbe syllogistic framework.
	* George Boole's algebraic reformulation of logic, his system of [[Boolean logic]];		* George Boole's algebraic reformulation of logic, his system of [[Boolean logic]];
	* ~~the~~ first-order logic found in Gottlob Frege's [[Begriffsschrift]].		* tbe first-order logic found in Gottlob Frege's [[Begriffsschrift]].

	== Non-classical logics ==		== Non-classical logics ==
	* Computability logic is a semantically constructed formal [[Theory]] of computability, as opposed to classical logic, which is a formal [[Theory]] of truth; integrates and extends classical, linear and intuitionistic logics.		* Computability logic is a semantically constructed formal [[Theory]] of computability, as opposed to classical logic, which is a formal [[Theory]] of truth; integrates and extends classical, linear and intuitionistic logics.
	* Many-valued logic, including fuzzy logic, which rejects ~~the~~ law of ~~the~~ excluded middle and allows as a truth value any real number between 0 and 1.		* Many-valued logic, including fuzzy logic, which rejects tbe law of tbe excluded middle and allows as a truth value any real number between 0 and 1.
	* [[Intuitionistic logic]] rejects ~~the~~ law of ~~the~~ excluded middle, double negative elimination, and ~~the~~ De Morgan's laws;		* [[Intuitionistic logic]] rejects tbe law of tbe excluded middle, double negative elimination, and tbe De Morgan's laws;
	* [[Linear logic]] rejects idempotency of [[Logical consequence\|entailment]] as well;		* [[Linear logic]] rejects idempotency of [[Logical consequence\|entailment]] as well;
	* [[Modal logic]] extends classical logic with non-truth-functional ("modal") operators.		* [[Modal logic]] extends classical logic with non-truth-functional ("modal") operators.
	* Paraconsistent logic (e.g., [[dialeTheism]] and [[relevance logic]]) rejects ~~the~~ law of noncontradiction;		* Paraconsistent logic (e.g., [[dialeTheism]] and [[relevance logic]]) rejects tbe law of noncontradiction;
	* [[Relevance logic]], [[linear logic]], and [[non-monotonic logic]] reject monotonicity of entailment;		* [[Relevance logic]], [[linear logic]], and [[non-monotonic logic]] reject monotonicity of entailment;

	In ''Deviant Logic, Fuzzy Logic: Beyond ~~the~~ Formalism'', Susan Haack divided non-classical logics into deviant, quasi-deviant, and extended logics.<ref name="haack" />		In ''Deviant Logic, Fuzzy Logic: Beyond tbe Formalism'', Susan Haack divided non-classical logics into deviant, quasi-deviant, and extended logics.<ref name="haack" />

	== FurTher reading ==		== FurTher reading ==

← Older revision		Revision as of 15:51, 12 September 2023
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	'''Classical logic''' identifies a class of [[formal logic]] that has been most intensively studied and most widely used. ~~tbe~~ class is sometimes called '''standard logic''' as well.<ref name="BunninYu2004">{{cite book\|author1=Nicholas Bunnin\|author2=Jiyuan Yu\|title=The Blackwell dictionary of Western philosophy\|url=http://books.google.com/books?id=OskKWI1YA7AC&pg=PA266\|year=2004\|publisher=Wiley-Blackwell\|isbn=978-1-4051-0679-5\|page=266}}</ref><ref name="Gamut1991">{{cite book\|author=L. T. F. Gamut\|title=Logic, language, and meaning, Volume 1: Introduction to Logic\|url=http://books.google.com/books?id=Z0KhywkpolMC&pg=PA156\|year=1991\|publisher=University of Chicago Press\|isbn=978-0-226-28085-1\|pages=156–157}}</ref> ~~tbey~~ are characterised by a number of properties:<ref>Dov Gabbay, (1994). 'Classical vs non-classical logic'. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, (Eds), ''Handbook of Logic in Artificial Intelligence and Logic Programming'', volume 2, chapter 2.6. Oxford University Press.</ref>		'''Classical logic''' identifies a class of [[formal logic]] that has been most intensively studied and most widely used. the class is sometimes called '''standard logic''' as well.<ref name="BunninYu2004">{{cite book\|author1=Nicholas Bunnin\|author2=Jiyuan Yu\|title=The Blackwell dictionary of Western philosophy\|url=http://books.google.com/books?id=OskKWI1YA7AC&pg=PA266\|year=2004\|publisher=Wiley-Blackwell\|isbn=978-1-4051-0679-5\|page=266}}</ref><ref name="Gamut1991">{{cite book\|author=L. T. F. Gamut\|title=Logic, language, and meaning, Volume 1: Introduction to Logic\|url=http://books.google.com/books?id=Z0KhywkpolMC&pg=PA156\|year=1991\|publisher=University of Chicago Press\|isbn=978-0-226-28085-1\|pages=156–157}}</ref> they are characterised by a number of properties:<ref>Dov Gabbay, (1994). 'Classical vs non-classical logic'. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, (Eds), ''Handbook of Logic in Artificial Intelligence and Logic Programming'', volume 2, chapter 2.6. Oxford University Press.</ref>
	# [[Law of ~~tbe~~ excluded middle]] and [[Double negative elimination]];		# [[Law of the excluded middle]] and [[Double negative elimination]];
	# [[Law of noncontradiction]], and ~~tbe~~ [[principle of explosion]];		# [[Law of noncontradiction]], and the [[principle of explosion]];
	# [[Monotonicity of entailment]] and [[Idempotency of entailment]];		# [[Monotonicity of entailment]] and [[Idempotency of entailment]];
	# [[Commutativity of conjunction]];		# [[Commutativity of conjunction]];
	# [[De Morgan duality]]: every [[logical operator]] is dual to ~~anotber~~;		# [[De Morgan duality]]: every [[logical operator]] is dual to another;

	While not entailed by ~~tbe~~ preceding conditions, contemporary discussions of classical logic normally only include propositional and first-order logic.<ref>Shapiro, Stewart (2000). Classical Logic. In Stanford Encyclopedia of Philosophy [Web]. Stanford: ~~tbe~~ Metaphysics Research Lab. Retrieved October 28, 2006, from http://plato.stanford.edu/entries/logic-classical/</ref><ref name="haack">Haack, Susan, (1996). ''Deviant Logic, Fuzzy Logic: Beyond ~~tbe~~ Formalism''. Chicago: ~~tbe~~ University of Chicago Press.</ref>		While not entailed by the preceding conditions, contemporary discussions of classical logic normally only include propositional and first-order logic.<ref>Shapiro, Stewart (2000). Classical Logic. In Stanford Encyclopedia of Philosophy [Web]. Stanford: the Metaphysics Research Lab. Retrieved October 28, 2006, from http://plato.stanford.edu/entries/logic-classical/</ref><ref name="haack">Haack, Susan, (1996). ''Deviant Logic, Fuzzy Logic: Beyond the Formalism''. Chicago: the University of Chicago Press.</ref>

	The intended semantics of classical logic is bivalent. With ~~tbe~~ advent of algebraic logic it became apparent however that classical propositional calculus admits ~~otber~~ semantics. In Boolean valued semantics (for classical propositional logic), ~~tbe~~ truth values are ~~tbe~~ elements of an arbitrary Boolean algebra; "true" corresponds to ~~tbe~~ maximal element of ~~tbe~~ algebra, and "false" corresponds to ~~tbe~~ minimal element. Intermediate elements of ~~tbe~~ algebra correspond to truth values ~~otber~~ than "true" and "false". ~~tbe~~ principle of bivalence holds only when ~~tbe~~ Boolean algebra is taken to be ~~tbe~~ two-element algebra, which has no intermediate elements.		The intended semantics of classical logic is bivalent. With the advent of algebraic logic it became apparent however that classical propositional calculus admits other semantics. In Boolean valued semantics (for classical propositional logic), the truth values are the elements of an arbitrary Boolean algebra; "true" corresponds to the maximal element of the algebra, and "false" corresponds to the minimal element. Intermediate elements of the algebra correspond to truth values other than "true" and "false". the principle of bivalence holds only when the Boolean algebra is taken to be the two-element algebra, which has no intermediate elements.

	== Examples of classical logics ==		== Examples of classical logics ==
	* [[Aristotle]]'s [[Organon]] introduces his [[Theory]] of syllogisms, which is a logic with a restricted form of judgments: assertions take one of four forms, ''All Ps are Q'', ''Some Ps are Q'', ''No Ps are Q'', and ''Some Ps are not Q''. These judgments find Themselves if two pairs of two dual operators, and each operator is ~~tbe~~ negation of ~~anotber~~, relationships that Aristotle summarized with his ''square of oppositions''. Aristotle explicitly formulated ~~tbe~~ law of ~~tbe~~ excluded middle and law of non-contradiction in justifying his system, although These laws cannot be expressed as judgments within ~~tbe~~ syllogistic framework.		* [[Aristotle]]'s [[Organon]] introduces his [[Theory]] of syllogisms, which is a logic with a restricted form of judgments: assertions take one of four forms, ''All Ps are Q'', ''Some Ps are Q'', ''No Ps are Q'', and ''Some Ps are not Q''. These judgments find Themselves if two pairs of two dual operators, and each operator is the negation of another, relationships that Aristotle summarized with his ''square of oppositions''. Aristotle explicitly formulated the law of the excluded middle and law of non-contradiction in justifying his system, although These laws cannot be expressed as judgments within the syllogistic framework.
	* George Boole's algebraic reformulation of logic, his system of [[Boolean logic]];		* George Boole's algebraic reformulation of logic, his system of [[Boolean logic]];
	* ~~tbe~~ first-order logic found in Gottlob Frege's [[Begriffsschrift]].		* the first-order logic found in Gottlob Frege's [[Begriffsschrift]].

	== Non-classical logics ==		== Non-classical logics ==
	* Computability logic is a semantically constructed formal [[Theory]] of computability, as opposed to classical logic, which is a formal [[Theory]] of truth; integrates and extends classical, linear and intuitionistic logics.		* Computability logic is a semantically constructed formal [[Theory]] of computability, as opposed to classical logic, which is a formal [[Theory]] of truth; integrates and extends classical, linear and intuitionistic logics.
	* Many-valued logic, including fuzzy logic, which rejects ~~tbe~~ law of ~~tbe~~ excluded middle and allows as a truth value any real number between 0 and 1.		* Many-valued logic, including fuzzy logic, which rejects the law of the excluded middle and allows as a truth value any real number between 0 and 1.
	* [[Intuitionistic logic]] rejects ~~tbe~~ law of ~~tbe~~ excluded middle, double negative elimination, and ~~tbe~~ De Morgan's laws;		* [[Intuitionistic logic]] rejects the law of the excluded middle, double negative elimination, and the De Morgan's laws;
	* [[Linear logic]] rejects idempotency of [[Logical consequence\|entailment]] as well;		* [[Linear logic]] rejects idempotency of [[Logical consequence\|entailment]] as well;
	* [[Modal logic]] extends classical logic with non-truth-functional ("modal") operators.		* [[Modal logic]] extends classical logic with non-truth-functional ("modal") operators.
	* Paraconsistent logic (e.g., [[dialeTheism]] and [[relevance logic]]) rejects ~~tbe~~ law of noncontradiction;		* Paraconsistent logic (e.g., [[dialeTheism]] and [[relevance logic]]) rejects the law of noncontradiction;
	* [[Relevance logic]], [[linear logic]], and [[non-monotonic logic]] reject monotonicity of entailment;		* [[Relevance logic]], [[linear logic]], and [[non-monotonic logic]] reject monotonicity of entailment;

	In ''Deviant Logic, Fuzzy Logic: Beyond ~~tbe~~ Formalism'', Susan Haack divided non-classical logics into deviant, quasi-deviant, and extended logics.<ref name="haack" />		In ''Deviant Logic, Fuzzy Logic: Beyond the Formalism'', Susan Haack divided non-classical logics into deviant, quasi-deviant, and extended logics.<ref name="haack" />

	== FurTher reading ==		== FurTher reading ==

← Older revision		Revision as of 13:24, 8 September 2023
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	'''Classical logic''' identifies a class of [[formal logic]] that has been most intensively studied and most widely used. ~~the~~ class is sometimes called '''standard logic''' as well.<ref name="BunninYu2004">{{cite book\|author1=Nicholas Bunnin\|author2=Jiyuan Yu\|title=The Blackwell dictionary of Western philosophy\|url=http://books.google.com/books?id=OskKWI1YA7AC&pg=PA266\|year=2004\|publisher=Wiley-Blackwell\|isbn=978-1-4051-0679-5\|page=266}}</ref><ref name="Gamut1991">{{cite book\|author=L. T. F. Gamut\|title=Logic, language, and meaning, Volume 1: Introduction to Logic\|url=http://books.google.com/books?id=Z0KhywkpolMC&pg=PA156\|year=1991\|publisher=University of Chicago Press\|isbn=978-0-226-28085-1\|pages=156–157}}</ref> ~~they~~ are characterised by a number of properties:<ref>Dov Gabbay, (1994). 'Classical vs non-classical logic'. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, (Eds), ''Handbook of Logic in Artificial Intelligence and Logic Programming'', volume 2, chapter 2.6. Oxford University Press.</ref>		'''Classical logic''' identifies a class of [[formal logic]] that has been most intensively studied and most widely used. tbe class is sometimes called '''standard logic''' as well.<ref name="BunninYu2004">{{cite book\|author1=Nicholas Bunnin\|author2=Jiyuan Yu\|title=The Blackwell dictionary of Western philosophy\|url=http://books.google.com/books?id=OskKWI1YA7AC&pg=PA266\|year=2004\|publisher=Wiley-Blackwell\|isbn=978-1-4051-0679-5\|page=266}}</ref><ref name="Gamut1991">{{cite book\|author=L. T. F. Gamut\|title=Logic, language, and meaning, Volume 1: Introduction to Logic\|url=http://books.google.com/books?id=Z0KhywkpolMC&pg=PA156\|year=1991\|publisher=University of Chicago Press\|isbn=978-0-226-28085-1\|pages=156–157}}</ref> tbey are characterised by a number of properties:<ref>Dov Gabbay, (1994). 'Classical vs non-classical logic'. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, (Eds), ''Handbook of Logic in Artificial Intelligence and Logic Programming'', volume 2, chapter 2.6. Oxford University Press.</ref>
	# [[Law of ~~the~~ excluded middle]] and [[Double negative elimination]];		# [[Law of tbe excluded middle]] and [[Double negative elimination]];
	# [[Law of noncontradiction]], and ~~the~~ [[principle of explosion]];		# [[Law of noncontradiction]], and tbe [[principle of explosion]];
	# [[Monotonicity of entailment]] and [[Idempotency of entailment]];		# [[Monotonicity of entailment]] and [[Idempotency of entailment]];
	# [[Commutativity of conjunction]];		# [[Commutativity of conjunction]];
	# [[De Morgan duality]]: every [[logical operator]] is dual to ~~another~~;		# [[De Morgan duality]]: every [[logical operator]] is dual to anotber;

	While not entailed by ~~the~~ preceding conditions, contemporary discussions of classical logic normally only include propositional and first-order logic.<ref>Shapiro, Stewart (2000). Classical Logic. In Stanford Encyclopedia of Philosophy [Web]. Stanford: ~~the~~ Metaphysics Research Lab. Retrieved October 28, 2006, from http://plato.stanford.edu/entries/logic-classical/</ref><ref name="haack">Haack, Susan, (1996). ''Deviant Logic, Fuzzy Logic: Beyond ~~the~~ Formalism''. Chicago: ~~the~~ University of Chicago Press.</ref>		While not entailed by tbe preceding conditions, contemporary discussions of classical logic normally only include propositional and first-order logic.<ref>Shapiro, Stewart (2000). Classical Logic. In Stanford Encyclopedia of Philosophy [Web]. Stanford: tbe Metaphysics Research Lab. Retrieved October 28, 2006, from http://plato.stanford.edu/entries/logic-classical/</ref><ref name="haack">Haack, Susan, (1996). ''Deviant Logic, Fuzzy Logic: Beyond tbe Formalism''. Chicago: tbe University of Chicago Press.</ref>

	The intended semantics of classical logic is bivalent. With ~~the~~ advent of algebraic logic it became apparent however that classical propositional calculus admits ~~other~~ semantics. In Boolean valued semantics (for classical propositional logic), ~~the~~ truth values are ~~the~~ elements of an arbitrary Boolean algebra; "true" corresponds to ~~the~~ maximal element of ~~the~~ algebra, and "false" corresponds to ~~the~~ minimal element. Intermediate elements of ~~the~~ algebra correspond to truth values ~~other~~ than "true" and "false". ~~the~~ principle of bivalence holds only when ~~the~~ Boolean algebra is taken to be ~~the~~ two-element algebra, which has no intermediate elements.		The intended semantics of classical logic is bivalent. With tbe advent of algebraic logic it became apparent however that classical propositional calculus admits otber semantics. In Boolean valued semantics (for classical propositional logic), tbe truth values are tbe elements of an arbitrary Boolean algebra; "true" corresponds to tbe maximal element of tbe algebra, and "false" corresponds to tbe minimal element. Intermediate elements of tbe algebra correspond to truth values otber than "true" and "false". tbe principle of bivalence holds only when tbe Boolean algebra is taken to be tbe two-element algebra, which has no intermediate elements.

	== Examples of classical logics ==		== Examples of classical logics ==
	* [[Aristotle]]'s [[Organon]] introduces his [[Theory]] of syllogisms, which is a logic with a restricted form of judgments: assertions take one of four forms, ''All Ps are Q'', ''Some Ps are Q'', ''No Ps are Q'', and ''Some Ps are not Q''. These judgments find Themselves if two pairs of two dual operators, and each operator is ~~the~~ negation of ~~another~~, relationships that Aristotle summarized with his ''square of oppositions''. Aristotle explicitly formulated ~~the~~ law of ~~the~~ excluded middle and law of non-contradiction in justifying his system, although These laws cannot be expressed as judgments within ~~the~~ syllogistic framework.		* [[Aristotle]]'s [[Organon]] introduces his [[Theory]] of syllogisms, which is a logic with a restricted form of judgments: assertions take one of four forms, ''All Ps are Q'', ''Some Ps are Q'', ''No Ps are Q'', and ''Some Ps are not Q''. These judgments find Themselves if two pairs of two dual operators, and each operator is tbe negation of anotber, relationships that Aristotle summarized with his ''square of oppositions''. Aristotle explicitly formulated tbe law of tbe excluded middle and law of non-contradiction in justifying his system, although These laws cannot be expressed as judgments within tbe syllogistic framework.
	* George Boole's algebraic reformulation of logic, his system of [[Boolean logic]];		* George Boole's algebraic reformulation of logic, his system of [[Boolean logic]];
	* ~~the~~ first-order logic found in Gottlob Frege's [[Begriffsschrift]].		* tbe first-order logic found in Gottlob Frege's [[Begriffsschrift]].

	== Non-classical logics ==		== Non-classical logics ==
	* Computability logic is a semantically constructed formal [[Theory]] of computability, as opposed to classical logic, which is a formal [[Theory]] of truth; integrates and extends classical, linear and intuitionistic logics.		* Computability logic is a semantically constructed formal [[Theory]] of computability, as opposed to classical logic, which is a formal [[Theory]] of truth; integrates and extends classical, linear and intuitionistic logics.
	* Many-valued logic, including fuzzy logic, which rejects ~~the~~ law of ~~the~~ excluded middle and allows as a truth value any real number between 0 and 1.		* Many-valued logic, including fuzzy logic, which rejects tbe law of tbe excluded middle and allows as a truth value any real number between 0 and 1.
	* [[Intuitionistic logic]] rejects ~~the~~ law of ~~the~~ excluded middle, double negative elimination, and ~~the~~ De Morgan's laws;		* [[Intuitionistic logic]] rejects tbe law of tbe excluded middle, double negative elimination, and tbe De Morgan's laws;
	* [[Linear logic]] rejects idempotency of [[Logical consequence\|entailment]] as well;		* [[Linear logic]] rejects idempotency of [[Logical consequence\|entailment]] as well;
	* [[Modal logic]] extends classical logic with non-truth-functional ("modal") operators.		* [[Modal logic]] extends classical logic with non-truth-functional ("modal") operators.
	* Paraconsistent logic (e.g., [[dialeTheism]] and [[relevance logic]]) rejects ~~the~~ law of noncontradiction;		* Paraconsistent logic (e.g., [[dialeTheism]] and [[relevance logic]]) rejects tbe law of noncontradiction;
	* [[Relevance logic]], [[linear logic]], and [[non-monotonic logic]] reject monotonicity of entailment;		* [[Relevance logic]], [[linear logic]], and [[non-monotonic logic]] reject monotonicity of entailment;

	In ''Deviant Logic, Fuzzy Logic: Beyond ~~the~~ Formalism'', Susan Haack divided non-classical logics into deviant, quasi-deviant, and extended logics.<ref name="haack" />		In ''Deviant Logic, Fuzzy Logic: Beyond tbe Formalism'', Susan Haack divided non-classical logics into deviant, quasi-deviant, and extended logics.<ref name="haack" />

	== FurTher reading ==		== FurTher reading ==

← Older revision		Revision as of 00:40, 27 February 2023
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	'''Classical logic''' identifies a class of [[formal logic]] that has been most intensively studied and most widely used. ~~The~~ class is sometimes called '''standard logic''' as well.<ref name="BunninYu2004">{{cite book\|author1=Nicholas Bunnin\|author2=Jiyuan Yu\|title=The Blackwell dictionary of Western philosophy\|url=http://books.google.com/books?id=OskKWI1YA7AC&pg=PA266\|year=2004\|publisher=Wiley-Blackwell\|isbn=978-1-4051-0679-5\|page=266}}</ref><ref name="Gamut1991">{{cite book\|author=L. T. F. Gamut\|title=Logic, language, and meaning, Volume 1: Introduction to Logic\|url=http://books.google.com/books?id=Z0KhywkpolMC&pg=PA156\|year=1991\|publisher=University of Chicago Press\|isbn=978-0-226-28085-1\|pages=156–157}}</ref> they are characterised by a number of properties:<ref>Dov Gabbay, (1994). 'Classical vs non-classical logic'. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, (Eds), ''Handbook of Logic in Artificial Intelligence and Logic Programming'', volume 2, chapter 2.6. Oxford University Press.</ref>		'''Classical logic''' identifies a class of [[formal logic]] that has been most intensively studied and most widely used. the class is sometimes called '''standard logic''' as well.<ref name="BunninYu2004">{{cite book\|author1=Nicholas Bunnin\|author2=Jiyuan Yu\|title=The Blackwell dictionary of Western philosophy\|url=http://books.google.com/books?id=OskKWI1YA7AC&pg=PA266\|year=2004\|publisher=Wiley-Blackwell\|isbn=978-1-4051-0679-5\|page=266}}</ref><ref name="Gamut1991">{{cite book\|author=L. T. F. Gamut\|title=Logic, language, and meaning, Volume 1: Introduction to Logic\|url=http://books.google.com/books?id=Z0KhywkpolMC&pg=PA156\|year=1991\|publisher=University of Chicago Press\|isbn=978-0-226-28085-1\|pages=156–157}}</ref> they are characterised by a number of properties:<ref>Dov Gabbay, (1994). 'Classical vs non-classical logic'. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, (Eds), ''Handbook of Logic in Artificial Intelligence and Logic Programming'', volume 2, chapter 2.6. Oxford University Press.</ref>
	# [[Law of ~~The~~ excluded middle]] and [[Double negative elimination]];		# [[Law of the excluded middle]] and [[Double negative elimination]];
	# [[Law of noncontradiction]], and ~~The~~ [[principle of explosion]];		# [[Law of noncontradiction]], and the [[principle of explosion]];
	# [[Monotonicity of entailment]] and [[Idempotency of entailment]];		# [[Monotonicity of entailment]] and [[Idempotency of entailment]];
	# [[Commutativity of conjunction]];		# [[Commutativity of conjunction]];
	# [[De Morgan duality]]: every [[logical operator]] is dual to anoTher;		# [[De Morgan duality]]: every [[logical operator]] is dual to anoTher;

	While not entailed by ~~The~~ preceding conditions, contemporary discussions of classical logic normally only include propositional and first-order logic.<ref>Shapiro, Stewart (2000). Classical Logic. In Stanford Encyclopedia of Philosophy [Web]. Stanford: ~~The~~ Metaphysics Research Lab. Retrieved October 28, 2006, from http://plato.stanford.edu/entries/logic-classical/</ref><ref name="haack">Haack, Susan, (1996). ''Deviant Logic, Fuzzy Logic: Beyond ~~The~~ Formalism''. Chicago: ~~The~~ University of Chicago Press.</ref>		While not entailed by the preceding conditions, contemporary discussions of classical logic normally only include propositional and first-order logic.<ref>Shapiro, Stewart (2000). Classical Logic. In Stanford Encyclopedia of Philosophy [Web]. Stanford: the Metaphysics Research Lab. Retrieved October 28, 2006, from http://plato.stanford.edu/entries/logic-classical/</ref><ref name="haack">Haack, Susan, (1996). ''Deviant Logic, Fuzzy Logic: Beyond the Formalism''. Chicago: the University of Chicago Press.</ref>

	The intended semantics of classical logic is bivalent. With ~~The~~ advent of algebraic logic it became apparent however that classical propositional calculus admits oTher semantics. In Boolean valued semantics (for classical propositional logic), ~~The~~ truth values are ~~The~~ elements of an arbitrary Boolean algebra; "true" corresponds to ~~The~~ maximal element of ~~The~~ algebra, and "false" corresponds to ~~The~~ minimal element. Intermediate elements of ~~The~~ algebra correspond to truth values oTher than "true" and "false". ~~The~~ principle of bivalence holds only when ~~The~~ Boolean algebra is taken to be ~~The~~ two-element algebra, which has no intermediate elements.		The intended semantics of classical logic is bivalent. With the advent of algebraic logic it became apparent however that classical propositional calculus admits oTher semantics. In Boolean valued semantics (for classical propositional logic), the truth values are the elements of an arbitrary Boolean algebra; "true" corresponds to the maximal element of the algebra, and "false" corresponds to the minimal element. Intermediate elements of the algebra correspond to truth values oTher than "true" and "false". the principle of bivalence holds only when the Boolean algebra is taken to be the two-element algebra, which has no intermediate elements.

	== Examples of classical logics ==		== Examples of classical logics ==
	* [[Aristotle]]'s [[Organon]] introduces his [[Theory]] of syllogisms, which is a logic with a restricted form of judgments: assertions take one of four forms, ''All Ps are Q'', ''Some Ps are Q'', ''No Ps are Q'', and ''Some Ps are not Q''. These judgments find Themselves if two pairs of two dual operators, and each operator is ~~The~~ negation of anoTher, relationships that Aristotle summarised with his ''square of oppositions''. Aristotle explicitly formulated ~~The~~ law of ~~The~~ excluded middle and law of non-contradiction in justifying his system, although These laws cannot be expressed as judgments within ~~The~~ syllogistic framework.		* [[Aristotle]]'s [[Organon]] introduces his [[Theory]] of syllogisms, which is a logic with a restricted form of judgments: assertions take one of four forms, ''All Ps are Q'', ''Some Ps are Q'', ''No Ps are Q'', and ''Some Ps are not Q''. These judgments find Themselves if two pairs of two dual operators, and each operator is the negation of anoTher, relationships that Aristotle summarised with his ''square of oppositions''. Aristotle explicitly formulated the law of the excluded middle and law of non-contradiction in justifying his system, although These laws cannot be expressed as judgments within the syllogistic framework.
	* George Boole's algebraic reformulation of logic, his system of [[Boolean logic]];		* George Boole's algebraic reformulation of logic, his system of [[Boolean logic]];
	* ~~The~~ first-order logic found in Gottlob Frege's [[Begriffsschrift]].		* the first-order logic found in Gottlob Frege's [[Begriffsschrift]].

	== Non-classical logics ==		== Non-classical logics ==
	* Computability logic is a semantically constructed formal [[Theory]] of computability, as opposed to classical logic, which is a formal [[Theory]] of truth; integrates and extends classical, linear and intuitionistic logics.		* Computability logic is a semantically constructed formal [[Theory]] of computability, as opposed to classical logic, which is a formal [[Theory]] of truth; integrates and extends classical, linear and intuitionistic logics.
	* Many-valued logic, including fuzzy logic, which rejects ~~The~~ law of ~~The~~ excluded middle and allows as a truth value any real number between 0 and 1.		* Many-valued logic, including fuzzy logic, which rejects the law of the excluded middle and allows as a truth value any real number between 0 and 1.
	* [[Intuitionistic logic]] rejects ~~The~~ law of ~~The~~ excluded middle, double negative elimination, and ~~The~~ De Morgan's laws;		* [[Intuitionistic logic]] rejects the law of the excluded middle, double negative elimination, and the De Morgan's laws;
	* [[Linear logic]] rejects idempotency of [[Logical consequence\|entailment]] as well;		* [[Linear logic]] rejects idempotency of [[Logical consequence\|entailment]] as well;
	* [[Modal logic]] extends classical logic with non-truth-functional ("modal") operators.		* [[Modal logic]] extends classical logic with non-truth-functional ("modal") operators.
	* Paraconsistent logic (e.g., [[dialeTheism]] and [[relevance logic]]) rejects ~~The~~ law of noncontradiction;		* Paraconsistent logic (e.g., [[dialeTheism]] and [[relevance logic]]) rejects the law of noncontradiction;
	* [[Relevance logic]], [[linear logic]], and [[non-monotonic logic]] reject monotonicity of entailment;		* [[Relevance logic]], [[linear logic]], and [[non-monotonic logic]] reject monotonicity of entailment;

	In ''Deviant Logic, Fuzzy Logic: Beyond ~~The~~ Formalism'', Susan Haack divided non-classical logics into deviant, quasi-deviant, and extended logics.<ref name="haack" />		In ''Deviant Logic, Fuzzy Logic: Beyond the Formalism'', Susan Haack divided non-classical logics into deviant, quasi-deviant, and extended logics.<ref name="haack" />

	== FurTher reading ==		== FurTher reading ==