Editing Philosophical logic (section)

=== Deontic ===
[[Deontic logic]] extends classical logic to the field of [[ethics]].<ref name="McNamara">{{cite web |last1=McNamara |first1=Paul |last2=Van De Putte |first2=Frederik |title=Deontic Logic |url=https://plato.stanford.edu/entries/logic-deontic/ |website=The Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |access-date=14 December 2021 |date=2021}}</ref><ref name="MacMillanNonClassical"/><ref name="MacMillanModal">{{cite book |last1=Borchert |first1=Donald |title=Macmillan Encyclopedia of Philosophy, 2nd Edition |date=2006 |publisher=Macmillan |url=https://philpapers.org/rec/BORMEO |chapter=Modal Logic}}</ref> Of central importance in ethics are the concepts of [[obligation]] and [[Permission (philosophy)|permission]], i.e. which actions the agent has to do or is allowed to do. Deontic logic usually expresses these ideas with the operators <math>O</math> and <math>P</math>.<ref name="McNamara"/><ref name="MacMillanNonClassical"/><ref name="MacMillanModal"/><ref name="Garson"/> So if {{nowrap|"<math>J(r)</math>"}} stands for the proposition "Ramirez goes jogging", then {{nowrap|"<math>O J(r)</math>"}} means that Ramirez has the obligation to go jogging and {{nowrap|"<math>P J(r)</math>"}} means that Ramirez has the permission to go jogging.

Deontic logic is closely related to alethic modal logic in that the axioms governing the logical behavior of their operators are identical. This means that obligation and permission behave in regards to valid inference just like necessity and possibility do.<ref name="McNamara"/><ref name="MacMillanNonClassical"/><ref name="MacMillanModal"/><ref name="Garson"/> For this reason, sometimes even the same symbols are used as operators.<ref>{{cite journal |last1=HANSON |first1=WILLIAM H. |title=Semantics for Deontic Logic |journal=Logique et Analyse |date=1965 |volume=8 |issue=31 |pages=177–190 |jstor=44083686 |url=https://www.jstor.org/stable/44083686 |issn=0024-5836}}</ref> Just as in alethic modal logic, there is a discussion in philosophical logic concerning which is the right system of axioms for expressing the common intuitions governing deontic inferences.<ref name="McNamara"/><ref name="MacMillanNonClassical"/><ref name="MacMillanModal"/> But the arguments and counterexamples here are slightly different since the meanings of these operators differ. For example, a common intuition in ethics is that if the agent has the obligation to do something then they automatically also have the permission to do it. This can be expressed formally through the axiom schema {{nowrap|"<math>O A \to P A</math>"}}.<ref name="McNamara"/><ref name="MacMillanNonClassical"/><ref name="MacMillanModal"/> Another question of interest to philosophical logic concerns the relation between alethic modal logic and deontic logic. An often discussed principle in this respect is that [[ought implies can]]. This means that the agent can only have the obligation to do something if it is possible for the agent to do it.<ref>{{cite web |title=Ought implies can |url=https://www.britannica.com/topic/ought-implies-can |website=Encyclopedia Britannica |access-date=8 September 2021 |language=en}}</ref><ref>{{cite journal |last1=Chituc |first1=Vladimir |last2=Henne |first2=Paul |last3=Sinnott-Armstrong |first3=Walter |last4=Brigard |first4=Felipe De |title=Blame, Not Ability, Impacts Moral "Ought" Judgments for Impossible Actions: Toward an Empirical Refutation of "Ought" Implies "Can" |journal=Cognition |date=2016 |volume=150 |pages=20–25 |doi=10.1016/j.cognition.2016.01.013 |pmid=26848732 |s2cid=32730640 |url=https://philpapers.org/rec/CHIBNA}}</ref> Expressed formally: {{nowrap|"<math>O A \to \Diamond A</math>"}}.<ref name="McNamara"/>