Editing Philosophical logic (section)

==== 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"/>