Editing Philosophical logic (section)

=== Temporal ===
[[Temporal logic]], or tense logic, uses logical mechanisms to express temporal relations.<ref name="StanfordTemporal">{{cite web |last1=Goranko |first1=Valentin |last2=Rumberg |first2=Antje |title=Temporal Logic |url=https://plato.stanford.edu/entries/logic-temporal/ |website=The Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |access-date=15 December 2021 |date=2021}}</ref><ref name="MacMillanNonClassical"/><ref name="MacMillanModal"/><ref name="Burgess2"/> In its most simple form, it contains one operator to express that something happened at one time and another to express that something is happening all the time. These two operators behave in the same way as the operators for possibility and necessity in alethic modal logic. Since the difference between past and future is of central importance to human affairs, these operators are often modified to take this difference into account. [[Arthur Prior]]'s tense logic, for example, realizes this idea using four such operators: <math>P</math> (it was the case that...), <math>F</math> (it will be the case that...), <math>H</math> (it has always been the case that...), and <math>G</math> (it will always be the case that...).<ref name="StanfordTemporal"/><ref name="MacMillanNonClassical"/><ref name="MacMillanModal"/><ref name="Burgess2"/> So to express that it will always be rainy in London one could use {{nowrap|"<math>G(Rainy(london))</math>"}}. Various axioms are used to govern which inferences are valid depending on the operators appearing in them. According to them, for example, one can deduce {{nowrap|"<math>F(Rainy(london))</math>"}} (it will be rainy in London at some time) from {{nowrap|"<math>G(Rainy(london))</math>"}}. In more complicated forms of temporal logic, also [[binary operators]] linking two propositions are defined, for example, to express that something happens until something else happens.<ref name="StanfordTemporal"/>

Temporal modal logic can be translated into classical first-order logic by treating time in the form of a singular term and increasing the arity of one's predicates by one.<ref name="Burgess2"/> For example, the tense-logic-sentence {{nowrap|"<math>dark \land P(light) \land F(light)</math>"}} (it is dark, it was light, and it will be light again) can be translated into pure first-order logic as {{nowrap|"<math>dark(t_1) \land \exists t_0(t_0 < t_1 \land light(t_0)) \land \exists t_2(t_1 < t_2 \land light(t_2))</math>"}}.<ref name="Goranko">{{cite web |last1=Goranko |first1=Valentin |last2=Rumberg |first2=Antje |title=Temporal Logic |url=https://plato.stanford.edu/entries/logic-temporal/ |website=The Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |access-date=13 December 2021 |date=2021}}</ref> While similar approaches are often seen in physics, logicians usually prefer an autonomous treatment of time in terms of operators. This is also closer to natural languages, which mostly use grammar, e.g. by [[Grammatical conjugation|conjugating]] verbs, to express the pastness or futurity of events.<ref name="Burgess2">{{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=2. Temporal Logic}}</ref>