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Philosophical logic
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== Deviant logics == === Intuitionistic === [[Intuitionistic logic]] is a more restricted version of classical logic.<ref name="Moschovakis">{{cite web |last1=Moschovakis |first1=Joan |title=Intuitionistic Logic: 1. Rejection of Tertium Non Datur |url=https://plato.stanford.edu/entries/logic-intuitionistic/#RejTerNonDat |website=The Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |access-date=11 December 2021 |date=2021}}</ref><ref name="Burgess6">{{cite book |last1=Burgess |first1=John P. |title=Philosophical Logic |date=2009 |publisher=Princeton, NJ, USA: Princeton University Press |url=https://philpapers.org/rec/BURPL-3 |chapter=6. Intuitionistic logic}}</ref><ref name="MacMillanNonClassical"/> It is more restricted in the sense that certain rules of inference used in classical logic do not constitute valid inferences in it. This concerns specifically the [[law of excluded middle]] and the [[double negation elimination]].<ref name="Moschovakis"/><ref name="Burgess6"/><ref name="MacMillanNonClassical"/> The law of excluded middle states that for every sentence, either it or its negation are true. Expressed formally: <math>A \lor \lnot A</math>. The law of double negation elimination states that if a sentence is not not true, then it is true, i.e. {{nowrap|"<math>\lnot \lnot A \to A</math>"}}.<ref name="Moschovakis"/><ref name="MacMillanNonClassical"/> Due to these restrictions, many proofs are more complicated and some proofs otherwise accepted become impossible.<ref name="Burgess6"/> These modifications of classical logic are motivated by the idea that truth depends on verification through a [[Formal proof|proof]]. This has been interpreted in the sense that "true" means "verifiable".<ref name="Burgess6"/><ref name="MacMillanNonClassical"/> It was originally only applied to the area of mathematics but has since then been used in other areas as well.<ref name="Moschovakis"/> On this interpretation, the law of excluded middle would involve the assumption that every mathematical problem has a solution in the form of a proof. In this sense, the intuitionistic rejection of the law of excluded middle is motivated by the rejection of this assumption.<ref name="Moschovakis"/><ref name="MacMillanNonClassical"/> This position can also be expressed by stating that there are no unexperienced or verification-transcendent truths.<ref name="Burgess6"/> In this sense, intuitionistic logic is motivated by a form of metaphysical idealism. Applied to mathematics, it states that mathematical objects exist only to the extent that they are constructed in the mind.<ref name="Burgess6"/> === Free === [[Free logic]] rejects some of the existential presuppositions found in classical logic.<ref name="Nolt1"/><ref name="Morscher">{{cite book |last1=Morscher |first1=Edgar |last2=Simons |first2=Peter |title=New Essays in Free Logic: In Honour of Karel Lambert |date=2001 |publisher=Springer Netherlands |isbn=978-94-015-9761-6 |pages=1โ34 |url=https://link.springer.com/chapter/10.1007/978-94-015-9761-6_1 |language=en |chapter=Free Logic: A Fifty-Year Past and an Open Future|doi=10.1007/978-94-015-9761-6_1 }}</ref><ref name="Lambert">{{cite book |last1=Lambert |first1=Karel |title=The Blackwell Guide to Philosophical Logic |date=2017 |publisher=John Wiley & Sons, Ltd |isbn=978-1-4051-6480-1 |pages=258โ279 |url=https://onlinelibrary.wiley.com/doi/10.1002/9781405164801.ch12 |language=en |chapter=Free Logics|doi=10.1002/9781405164801.ch12 }}</ref> In classical logic, every singular term has to denote an object in the domain of quantification.<ref name="Nolt1">{{cite web |last1=Nolt |first1=John |title=Free Logic: 1. The Basics |url=https://plato.stanford.edu/entries/logic-free/#1 |website=The Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |access-date=11 December 2021 |date=2021}}</ref> This is usually understood as an ontological commitment to the existence of the named entity. But many names are used in everyday discourse that do not refer to existing entities, like "Santa Claus" or "Pegasus". This threatens to preclude such areas of discourse from a strict logical treatment. Free logic avoids these problems by allowing formulas with non-denoting singular terms.<ref name="Morscher"/> This applies to [[proper names]] as well as [[definite descriptions]], and functional expressions.<ref name="Nolt1"/><ref name="Lambert"/> Quantifiers, on the other hand, are treated in the usual way as ranging over the domain. This allows for expressions like {{nowrap|"<math>\lnot \exists x (x = santa)</math>"}} (Santa Claus does not exist) to be true even though they are self-contradictory in classical logic.<ref name="Nolt1"/> It also brings with it the consequence that certain valid forms of inference found in classical logic are not valid in free logic. For example, one may infer from {{nowrap|"<math>Beard(santa)</math>"}} (Santa Claus has a beard) that {{nowrap|"<math>\exists x (Beard(x))</math>"}} (something has a beard) in classical logic but not in free logic.<ref name="Nolt1"/> In free logic, often an existence-predicate is used to indicate whether a singular term denotes an object in the domain or not. But the usage of existence-predicates is controversial. They are often opposed, based on the idea that existence is required if any predicates should apply to the object at all. In this sense, existence cannot itself be a predicate.<ref name="Britannica">{{cite web |title=Philosophy of logic |url=https://www.britannica.com/topic/philosophy-of-logic |website=www.britannica.com |access-date=21 November 2021 |language=en}}</ref><ref>{{cite journal |last1=Moltmann |first1=Friederike |title=Existence Predicates |journal=Synthese |date=2020 |volume=197 |issue=1 |pages=311โ335 |doi=10.1007/s11229-018-1847-z |s2cid=255065180 |url=https://philpapers.org/rec/MOLEP}}</ref><ref>{{cite journal |last1=Muskens |first1=Reinhard |title=Existence Predicate |journal=The Encyclopedia of Language and Linguistics |date=1993 |pages=1191 |url=https://philpapers.org/rec/MUSEP |publisher=Oxford: Pergamon}}</ref> [[Karel Lambert]], who coined the term "free logic", has suggested that free logic can be understood as a generalization of classical predicate logic just as predicate logic is a generalization of Aristotelian logic. On this view, classical predicate logic introduces predicates with an empty extension while free logic introduces singular terms of non-existing things.<ref name="Nolt1"/> An important problem for free logic consists in how to determine the truth value of expressions containing empty singular terms, i.e. of formulating a [[semantics of logic|formal semantics]] for free logic.<ref name="Nolt3">{{cite web |last1=Nolt |first1=John |title=Free Logic: 3. Semantics |url=https://plato.stanford.edu/entries/logic-free/#3 |website=The Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |access-date=11 December 2021 |date=2021}}</ref> Formal semantics of classical logic can define the truth of their expressions in terms of their denotation. But this option cannot be applied to all expressions in free logic since not all of them have a denotation.<ref name="Nolt3"/> Three general approaches to this issue are often discussed in the literature: ''negative semantics'', ''positive semantics'', and ''neutral semantics''.<ref name="Lambert"/> ''Negative semantics'' hold that all atomic formulas containing empty terms are false. On this view, the expression {{nowrap|"<math>Beard(santa)</math>"}} is false.<ref name="Nolt3"/><ref name="Lambert"/> ''Positive semantics'' allows that at least some expressions with empty terms are true. This usually includes identity statements, like {{nowrap|"<math>santa = santa</math>"}}. Some versions introduce a second, outer domain for non-existing objects, which is then used to determine the corresponding truth values.<ref name="Nolt3"/><ref name="Lambert"/> ''Neutral semantics'', on the other hand, hold that atomic formulas containing empty terms are neither true nor false.<ref name="Nolt3"/><ref name="Lambert"/> This is often understood as a [[three-valued logic]], i.e. that a third truth value besides true and false is introduced for these cases.<ref>{{cite web |last1=Rami |first1=Dolf |title=Non-Standard Neutral Free Logic, Empty Names and Negative Existentials |url=https://philpapers.org/archive/RAMNNF.pdf}}</ref> === Many-valued === [[Many-valued logic]]s are logics that allow for more than two truth values.<ref name="Gottwald">{{cite web |last1=Gottwald |first1=Siegfried |title=Many-Valued Logic |url=https://plato.stanford.edu/entries/logic-manyvalued/ |website=The Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |access-date=11 December 2021 |date=2020}}</ref><ref name="MacMillanNonClassical"/><ref name="Malinowski">{{cite book |last1=Malinowski |first1=Grzegorz |title=A Companion to Philosophical Logic |date=2006 |publisher=John Wiley & Sons, Ltd |isbn=978-0-470-99675-1 |pages=545โ561 |url=https://onlinelibrary.wiley.com/doi/10.1002/9780470996751.ch35 |language=en |chapter=Many-Valued Logic|doi=10.1002/9780470996751.ch35 }}</ref> They reject one of the core assumptions of classical logic: the principle of the bivalence of truth. The most simple versions of many-valued logics are three-valued logics: they contain a third truth value. In [[Stephen Cole Kleene]]'s three-valued logic, for example, this third truth value is "undefined".<ref name="Gottwald"/><ref name="Malinowski"/> According to [[Nuel Belnap]]'s four-valued logic, there are four possible truth values: "true", "false", "neither true nor false", and "both true and false". This can be interpreted, for example, as indicating the information one has concerning whether a state obtains: information that it does obtain, information that it does not obtain, no information, and conflicting information.<ref name="Gottwald"/> One of the most extreme forms of many-valued logic is fuzzy logic. It allows truth to arise in any degree between 0 and 1.<ref name="Cintula">{{cite web |last1=Cintula |first1=Petr |last2=Fermรผller |first2=Christian G. |last3=Noguera |first3=Carles |title=Fuzzy Logic |url=https://plato.stanford.edu/entries/logic-fuzzy/ |website=The Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |access-date=11 December 2021 |date=2021}}</ref><ref name="Gottwald"/><ref name="MacMillanNonClassical"/> 0 corresponds to completely false, 1 corresponds to completely true, and the values in between correspond to truth in some degree, e.g. as a little true or very true.<ref name="Cintula"/><ref name="Gottwald"/> It is often used to deal with vague expressions in natural language. For example, saying that "Petr is young" fits better (i.e. is "more true") if "Petr" refers to a three-year-old than if it refers to a 23-year-old.<ref name="Cintula"/> Many-valued logics with a finite number of truth-values can define their logical connectives using truth tables, just like classical logic. The difference is that these truth tables are more complex since more possible inputs and outputs have to be considered.<ref name="Gottwald"/><ref name="Malinowski"/> In Kleene's three-valued logic, for example, the inputs "true" and "undefined" for the conjunction-operator {{nowrap|"<math>\land</math>"}} result in the output "undefined". The inputs "false" and "undefined", on the other hand, result in "false".<ref>{{cite journal |last1=Malinowski |first1=Grzegorz |title=KLEENE LOGIC AND INFERENCE |journal=Bulletin of the Section of Logic |date=2014 |volume=43 |issue=1/2 |pages=3โ52}}</ref><ref name="Malinowski"/> === Paraconsistent === [[Paraconsistent logic]]s are logical systems that can deal with contradictions without leading to all-out absurdity.<ref name="StanfordParaconsistent">{{cite web |last1=Priest |first1=Graham |last2=Tanaka |first2=Koji |last3=Weber |first3=Zach |title=Paraconsistent Logic |url=https://plato.stanford.edu/entries/logic-paraconsistent/ |website=The Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |access-date=14 December 2021 |date=2018}}</ref><ref name="MacMillanNonClassical"/><ref name="Zach"/> They achieve this by avoiding the [[principle of explosion]] found in classical logic. According to the principle of explosion, anything follows from a contradiction. This is the case because of two rules of inference, which are valid in classical logic: [[disjunction introduction]] and [[disjunctive syllogism]].<ref name="StanfordParaconsistent"/><ref name="MacMillanNonClassical"/><ref name="Zach"/> According to the disjunction introduction, any proposition can be introduced in the form of a disjunction when paired with a true proposition.<ref name="StanfordDisjunction">{{cite web |last1=Aloni |first1=Maria|author-link=Maria Aloni |title=Disjunction |url=https://plato.stanford.edu/entries/disjunction/#DisjClasLogi |website=The Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |access-date=14 December 2021 |date=2016}}</ref> So since it is true that "the sun is bigger than the moon", it is possible to infer that "the sun is bigger than the moon or Spain is controlled by space-rabbits". According to the disjunctive [[syllogism]], one can infer that one of these disjuncts is true if the other is false.<ref name="StanfordDisjunction"/> So if the logical system also contains the negation of this proposition, i.e. that "the sun is not bigger than the moon", then it is possible to infer any proposition from this system, like the proposition that "Spain is controlled by space-rabbits". Paraconsistent logics avoid this by using different rules of inference that make inferences in accordance with the principle of explosion invalid.<ref name="StanfordParaconsistent"/><ref name="MacMillanNonClassical"/><ref name="Zach"/> An important motivation for using paraconsistent logics is dialetheism, i.e. the belief that contradictions are not just introduced into theories due to mistakes but that reality itself is contradictory and contradictions within theories are needed to accurately reflect reality.<ref name="Zach"/><ref>{{cite book |last1=Haack |first1=Susan |title=Deviant Logic, Fuzzy Logic: Beyond the Formalism |date=1996 |publisher=Chicago and London: University of Chicago Press |url=https://philpapers.org/rec/HAADLF |chapter=Introduction}}</ref><ref name="StanfordParaconsistent"/><ref name="StanfordDialetheism">{{cite web |last1=Priest |first1=Graham |last2=Berto |first2=Francesco |last3=Weber |first3=Zach |title=Dialetheism |url=https://plato.stanford.edu/entries/dialetheism/ |website=The Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |access-date=14 December 2021 |date=2018}}</ref> Without paraconsistent logics, dialetheism would be hopeless since everything would be both true and false.<ref name="StanfordDialetheism"/> Paraconsistent logics make it possible to keep contradictions local, without ''exploding'' the whole system.<ref name="MacMillanNonClassical"/> But even with this adjustment, dialetheism is still highly contested.<ref name="Zach"/><ref name="StanfordDialetheism"/> Another motivation for paraconsistent logic is to provide a logic for discussions and group beliefs where the group as a whole may have inconsistent beliefs if its different members are in disagreement.<ref name="Zach">{{cite web |last1=Weber |first1=Zach |title=Paraconsistent Logic |url=https://iep.utm.edu/para-log/ |website=Internet Encyclopedia of Philosophy |access-date=12 December 2021}}</ref> ==== Relevance ==== [[Relevance logic]] is one type of paraconsistent logic. As such, it also avoids the principle of explosion even though this is usually not the main motivation behind relevance logic. Instead, it is usually formulated with the goal of avoiding certain unintuitive applications of the material conditional found in classical logic.<ref name="StanfordRelevance">{{cite web |last1=Mares |first1=Edwin |title=Relevance Logic |url=https://plato.stanford.edu/entries/logic-relevance/ |website=The Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |access-date=14 December 2021 |date=2020}}</ref><ref name="MacMillanNonClassical"/><ref name="MacMillanRelevance">{{cite book |last1=Borchert |first1=Donald |title=Macmillan Encyclopedia of Philosophy, 2nd Edition |date=2006 |publisher=Macmillan |url=https://philpapers.org/rec/BORMEO |chapter=RELEVANCE (RELEVANT) LOGICS}}</ref> Classical logic defines the material conditional in purely truth-functional terms, i.e. {{nowrap|"<math>p \to q</math>"}} is false if {{nowrap|"<math>p</math>"}} is true and {{nowrap|"<math>q</math>"}} is false, but otherwise true in every case. According to this formal definition, it does not matter whether {{nowrap|"<math>p</math>"}} and {{nowrap|"<math>q</math>"}} are relevant to each other in any way.<ref name="StanfordRelevance"/><ref name="MacMillanNonClassical"/><ref name="MacMillanRelevance"/> For example, the material conditional "if all lemons are red then there is a sandstorm inside the Sydney Opera House" is true even though the two propositions are not relevant to each other. The fact that this usage of material conditionals is highly unintuitive is also reflected in [[informal logic]], which categorizes such inferences as [[Informal fallacy#Fallacies of relevance|fallacies of relevance]]. Relevance logic tries to avoid these cases by requiring that for a true material conditional, its antecedent has to be relevant to the consequent.<ref name="StanfordRelevance"/><ref name="MacMillanNonClassical"/><ref name="MacMillanRelevance"/> A difficulty faced for this issue is that relevance usually belongs to the content of the propositions while logic only deals with formal aspects. This problem is partially addressed by the so-called ''variable sharing principle''. It states that antecedent and consequent have to share a propositional variable.<ref name="StanfordRelevance"/><ref name="MacMillanRelevance"/><ref name="MacMillanNonClassical"/> This would be the case, for example, in {{nowrap|"<math>(p \land q) \to q</math>"}} but not in {{nowrap|"<math>(p \land q) \to r</math>"}}. A closely related concern of relevance logic is that inferences should follow the same requirement of relevance, i.e. that it is a necessary requirement of valid inferences that their premises are relevant to their conclusion.<ref name="StanfordRelevance"/>
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