Editing Philosophical logic (section)

=== 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"/>