Editing Philosophical logic (section)

== Extended logics ==
=== 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"/>

=== 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"/>

=== 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>

=== Epistemic ===
[[Epistemic logic]] is a form of modal logic applied to the field of [[epistemology]].<ref name="StanfordEpistemic">{{cite web |last1=Rendsvig |first1=Rasmus |last2=Symons |first2=John |title=Epistemic Logic |url=https://plato.stanford.edu/entries/logic-epistemic/ |website=The Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |access-date=14 December 2021 |date=2021}}</ref><ref name="BritannicaEpistemic">{{cite web |title=applied logic - Epistemic logic Britannica |url=https://www.britannica.com/topic/applied-logic/Epistemic-logic |website=www.britannica.com |access-date=14 December 2021 |language=en}}</ref><ref name="MacMillanModal"/><ref name="Britannica"/> It aims to capture the logic of [[knowledge]] and [[belief]]. The modal operators expressing knowledge and belief are usually expressed through the symbols {{nowrap|"<math>K</math>"}} and {{nowrap|"<math>B</math>"}}. So if {{nowrap|"<math>W(s)</math>"}} stands for the proposition "Socrates is wise", then {{nowrap|"<math>K W(s)</math>"}} expresses the proposition "the agent knows that Socrates is wise" and {{nowrap|"<math>B W(s)</math>"}} expresses the proposition "the agent believes that Socrates is wise". Axioms governing these operators are then formulated to express various epistemic principles.<ref name="MacMillanModal"/><ref name="StanfordEpistemic"/><ref name="BritannicaEpistemic"/> For example, the axiom schema {{nowrap|"<math>K A \to A</math>"}} expresses that whenever something is known, then it is true. This reflects the idea that one can only know what is true, otherwise it is not knowledge but another mental state.<ref name="MacMillanModal"/><ref name="StanfordEpistemic"/><ref name="BritannicaEpistemic"/> Another epistemic intuition about knowledge concerns the fact that when the agent knows something, they also know that they know it. This can be expressed by the axiom schema {{nowrap|"<math>K A \to KK A</math>"}}.<ref name="MacMillanModal"/><ref name="StanfordEpistemic"/><ref name="BritannicaEpistemic"/> An additional principle linking knowledge and belief states that knowledge implies belief, i.e. {{nowrap|"<math>K A \to B A</math>"}}. Dynamic epistemic logic is a distinct form of epistemic logic that focuses on situations in which changes in belief and knowledge happen.<ref>{{cite web |last1=Baltag |first1=Alexandru |last2=Renne |first2=Bryan |title=Dynamic Epistemic Logic |url=https://plato.stanford.edu/entries/dynamic-epistemic/ |website=The Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |access-date=13 December 2021 |date=2016}}</ref>

=== Higher-order ===
[[Higher-order logics]] extend first-order logic by including new forms of [[Quantifier (logic)|quantification]].<ref name="Cambridge"/><ref name="Väänänen"/><ref name="Ketland">{{cite book |last1=Ketland |first1=Jeffrey |title=Encyclopedia of Philosophy |date=2005 |url=https://www.encyclopedia.com/humanities/encyclopedias-almanacs-transcripts-and-maps/second-order-logic |chapter=Second Order Logic}}</ref><ref name="Computing">{{cite book |title=A Dictionary of Computing |chapter=predicate calculus |url=https://www.encyclopedia.com/computing/dictionaries-thesauruses-pictures-and-press-releases/predicate-calculus}}</ref> In first-order logic, quantification is restricted to singular terms. It can be used to talk about whether a predicate has an extension at all or whether its extension includes the whole domain. This way, propositions like {{nowrap|"<math>\exists x (Apple(x) \land Sweet(x))</math>"}} (''there are some'' apples that are sweet) can be expressed. In higher-order logics, quantification is allowed not just over individual terms but also over predicates. This way, it is possible to express, for example, whether certain individuals share some or all of their predicates, as in {{nowrap|"<math>\exists Q (Q(mary) \land Q(john))</math>"}} (''there are some'' qualities that Mary and John share).<ref name="Cambridge"/><ref name="Väänänen"/><ref name="Ketland"/><ref name="Computing"/> Because of these changes, higher-order logics have more expressive power than first-order logic. This can be helpful for mathematics in various ways since different mathematical theories have a much simpler expression in higher-order logic than in first-order logic.<ref name="Cambridge"/> For example, [[Peano arithmetic]] and [[Zermelo-Fraenkel set theory]] need an infinite number of axioms to be expressed in first-order logic. But they can be expressed in second-order logic with only a few axioms.<ref name="Cambridge"/>

But despite this advantage, first-order logic is still much more widely used than higher-order logic. One reason for this is that higher-order logic is [[Completeness (logic)|incomplete]].<ref name="Cambridge"/> This means that, for theories formulated in higher-order logic, it is not possible to prove every true sentence pertaining to the theory in question.<ref name="Hintikka"/> Another disadvantage is connected to the additional ontological commitments of higher-order logics. It is often held that the usage of the existential quantifier brings with it an ontological commitment to the entities over which this quantifier ranges.<ref name="Britannica"/><ref name="Schaffer">{{cite journal |last1=Schaffer |first1=Jonathan |title=On What Grounds What |journal=Metametaphysics: New Essays on the Foundations of Ontology |date=2009 |pages=347–383 |url=https://philpapers.org/rec/SCHOWG |access-date=23 November 2021 |publisher=Oxford University Press}}</ref><ref name="Bricker">{{cite web |last1=Bricker |first1=Phillip |title=Ontological Commitment |url=https://plato.stanford.edu/entries/ontological-commitment/ |website=The Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |access-date=23 November 2021 |date=2016}}</ref><ref name="Quine">{{cite journal |last1=Quine |first1=Willard Van Orman |title=On What There Is |journal=Review of Metaphysics |date=1948 |volume=2 |issue=5 |pages=21–38 |url=https://philpapers.org/rec/QUIOWT-7}}</ref> In first-order logic, this concerns only individuals, which is usually seen as an unproblematic ontological commitment. In higher-order logic, quantification concerns also properties and relations.<ref name="Britannica"/><ref name="Väänänen"/><ref name="HaackLogics1"/> This is often interpreted as meaning that higher-order logic brings with it a form of [[Platonism]], i.e. the view that [[Universal (metaphysics)|universal]] properties and relations exist in addition to individuals.<ref name="Cambridge"/><ref name="Ketland"/>