Editing Philosophical logic (section)

=== Alethic modal ===
[[Alethic modal logic]] has been very influential in logic and philosophy. It provides a logical formalism to express what is ''possibly'' or ''necessarily true''.<ref name="Cambridge"/><ref name="Britannica"/><ref name="Garson">{{cite web |last1=Garson |first1=James |title=Modal Logic |url=https://plato.stanford.edu/entries/logic-modal/ |website=The Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |access-date=21 November 2021 |date=2021}}</ref><ref name="Benthem">{{cite web |last1=Benthem |first1=Johan van |title=Modal Logic: Contemporary View |url=https://iep.utm.edu/modal-lo/ |website=Internet Encyclopedia of Philosophy |access-date=4 December 2021}}</ref><ref>{{cite web |title=modal logic |url=https://www.britannica.com/topic/modal-logic |website=www.britannica.com |access-date=4 December 2021 |language=en}}</ref><ref name="Burgess3">{{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=3. Modal Logic}}</ref><ref name="MacMillanNonClassical"/> It constitutes an extension of first-order logic, which by itself is only able to express what is ''true simpliciter''. This extension happens by introducing two new symbols: {{nowrap|"<math>\Diamond</math>"}} for possibility and {{nowrap|"<math>\Box</math>"}} for necessity. These symbols are used to modify propositions. For example, if {{nowrap|"<math>W(s)</math>"}} stands for the proposition "Socrates is wise", then {{nowrap|"<math>\Diamond W(s)</math>"}} expresses the proposition "it is possible that Socrates is wise". In order to integrate these symbols into the logical formalism, various axioms are added to the existing axioms of first-order logic.<ref name="Garson"/><ref name="Benthem"/><ref name="Burgess3"/> They govern the logical behavior of these symbols by determining how the validity of an inference depends on the fact that these symbols are found in it. They usually include the idea that if a proposition is necessary then its negation is impossible, i.e. that {{nowrap|"<math>\Box A</math>"}} is equivalent to {{nowrap|"<math>\lnot \Diamond \lnot A</math>"}}. Another such principle is that if something is necessary, then it must also be possible. This means that {{nowrap|"<math>\Diamond A</math>"}} follows from {{nowrap|"<math>\Box A</math>"}}.<ref name="Garson"/><ref name="Benthem"/><ref name="Burgess3"/> There is disagreement about exactly which axioms govern modal logic. The different forms of modal logic are often presented as a nested hierarchy of systems in which the most fundamental systems, like ''system K'', include only the most fundamental axioms while other systems, like the popular ''[[S5 (modal logic)|system S5]]'', build on top of it by including additional axioms.<ref name="Garson"/><ref name="Benthem"/><ref name="Burgess3"/> In this sense, system K is an extension of first-order logic while system S5 is an extension of system K. Important discussions within philosophical logic concern the question of which system of modal logic is correct.<ref name="Garson"/><ref name="Benthem"/><ref name="Burgess3"/> It is usually advantageous to have the strongest system possible in order to be able to draw many different inferences. But this brings with it the problem that some of these additional inferences may contradict basic modal intuitions in specific cases. This usually motivates the choice of a more basic system of axioms.<ref name="Garson"/><ref name="Benthem"/><ref name="Burgess3"/>

Possible worlds semantics is a very influential formal semantics in modal logic that brings with it system S5.<ref name="Garson"/><ref name="Benthem"/><ref name="Burgess3"/> A formal semantics of a language characterizes the conditions under which the sentences of this language are true or false. Formal semantics play a central role in the [[Philosophy of logic#Conceptions based on syntax or semantics|model-theoretic conception of validity]].<ref name="Hintikka"/><ref name="McKeon"/> They are able to provide clear criteria for when an inference is valid or not: an inference is valid if and only if it is truth-preserving, i.e. if whenever its premises are true then its conclusion is also true.<ref name="Britannica"/><ref name="McKeon"/><ref name="Gómez-Torrente">{{cite web |last1=Gómez-Torrente |first1=Mario |title=Logical Truth |url=https://plato.stanford.edu/entries/logical-truth/ |website=The Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |access-date=22 November 2021 |date=2019}}</ref> Whether they are true or false is specified by the formal semantics. Possible worlds semantics specifies the truth conditions of sentences expressed in modal logic in terms of possible worlds.<ref name="Garson"/><ref name="Benthem"/><ref name="Burgess3"/> A possible world is a complete and consistent way how things could have been.<ref>{{cite web |last1=Menzel |first1=Christopher |title=Possible Worlds |url=https://plato.stanford.edu/entries/possible-worlds/ |website=The Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |access-date=25 November 2021 |date=2021}}</ref><ref>{{cite web |last1=Parent |first1=Ted |title=Modal Metaphysics |url=https://iep.utm.edu/mod-meta/ |website=Internet Encyclopedia of Philosophy |access-date=9 April 2021}}</ref> On this view, a sentence modified by the <math>\Diamond</math>-operator is true if it is true in at least one possible world while a sentence modified by the <math>\Box</math>-operator is true if it is true in all possible worlds.<ref name="Garson"/><ref name="Benthem"/><ref name="Burgess3"/> So the sentence {{nowrap|"<math>\Diamond W(s)</math>"}} (it is possible that Socrates is wise) is true since there is at least one world where Socrates is wise. But {{nowrap|"<math>\Box W(s)</math>"}} (it is necessary that Socrates is wise) is false since Socrates is not wise in every possible world. Possible world semantics has been criticized as a formal semantics of modal logic since it seems to be circular.<ref name="Oxford"/> The reason for this is that possible worlds are themselves defined in modal terms, i.e. as ways how things ''could'' have been. In this way, it itself uses modal expressions to determine the truth of sentences containing modal expressions.<ref name="Oxford"/>