Editing Modal logic (section)

==Modal logics in philosophy==

===Alethic logic===
{{main|Subjunctive possibility}}
Modalities of necessity and possibility are called ''alethic'' modalities. They are also sometimes called ''special'' modalities, from the [[Latin]] ''species''. Modal logic was first developed to deal with these concepts, and only afterward was extended to others. For this reason, or perhaps for their familiarity and simplicity, necessity and possibility are often casually treated as ''the'' subject matter of modal logic. Moreover, it is easier to make sense of relativizing necessity, e.g. to legal, physical, [[nomological]], [[epistemic]], and so on, than it is to make sense of relativizing other notions.

In [[classical modal logic]], a proposition is said to be
*'''possible''' if it is ''not necessarily false'' (regardless of whether it is actually true or actually false);
*'''necessary''' if it is ''not possibly false'' (i.e. true and necessarily true);
*'''contingent''' if it is ''not necessarily false'' and ''not necessarily true'' (i.e. possible but not necessarily true);
*'''impossible''' if it is ''not possibly true'' (i.e. false and necessarily false).

In classical modal logic, therefore, the notion of either possibility or necessity may be taken to be basic, where these other notions are defined in terms of it in the manner of [[De Morgan duality]]. [[Intuitionistic modal logic]] treats possibility and necessity as not perfectly symmetric.

For example, suppose that while walking to the convenience store we pass Friedrich's house, and observe that the lights are off. On the way back, we observe that they have been turned on.
* "Somebody or something turned the lights on" is ''necessary''.
* "Friedrich turned the lights on", "Friedrich's roommate Max turned the lights on" and "A burglar named Adolf broke into Friedrich's house and turned the lights on" are ''contingent''.
* All of the above statements are ''possible''.
* It is ''impossible'' that [[Socrates]] (who has been dead for over two thousand years) turned the lights on.
(Of course, this analogy does not apply alethic modality in a ''truly'' rigorous fashion; for it to do so, it would have to axiomatically make such statements as "human beings cannot rise from the dead", "Socrates was a human being and not an immortal vampire", and "we did not take hallucinogenic drugs which caused us to falsely believe the lights were on", ''ad infinitum''. Absolute certainty of truth or falsehood exists only in the sense of logically constructed abstract concepts such as "it is impossible to draw a triangle with four sides" and "all bachelors are unmarried".)

For those having difficulty with the concept of something being possible but not true, the meaning of these terms may be made more comprehensible by thinking of multiple "possible worlds" (in the sense of [[Gottfried Wilhelm Leibniz|Leibniz]]) or "alternate universes"; something "necessary" is true in all possible worlds, something "possible" is true in at least one possible world. 

====Physical possibility====
Something is physically, or nomically, possible if it is permitted by the [[physical law|laws of physics]].{{citation needed|date=January 2016}} For example, current theory is thought to allow for there to be an [[atom]] with an [[atomic number]] of 126,<ref>{{cite news|title=Press release: Superheavy Element 114 Confirmed: A Stepping Stone to the Island of Stability|url=http://newscenter.lbl.gov/2009/09/24/114-confirmed/|work=Lawrence Berkeley National Laboratory|date=24 September 2009}}</ref> even if there are no such atoms in existence. In contrast, while it is logically possible to accelerate beyond the [[speed of light]],<ref name="Feinberg67">{{cite journal |last=Feinberg |first=G. |year=1967 |title=Possibility of Faster-Than-Light Particles |journal=[[Physical Review]] |volume=159 |issue=5 |pages=1089–1105 |bibcode=1967PhRv..159.1089F |doi=10.1103/PhysRev.159.1089}} See also Feinberg's later paper: Phys. Rev. D 17, 1651 (1978)</ref> modern science stipulates that it is not physically possible for material particles or information.<ref>{{cite journal | last = Einstein | first = Albert | author-link = Albert Einstein | title = Zur Elektrodynamik bewegter Körper | journal = Annalen der Physik | volume = 17 | pages = 891–921 | date = 1905-06-30|bibcode = 1905AnP...322..891E |doi = 10.1002/andp.19053221004 | issue = 10 | url = http://sedici.unlp.edu.ar/handle/10915/2786 | doi-access = free }}</ref> Physical possibility does not coincide with empirical possibility.<ref>{{cite journal | last = Gyenis | first = Balazs | author-link = Balazs Gyenis | title = Physical, Empirical, and Conditional Inductive Possibility | journal = Philosophy of Physics | volume = 3 | pages = 1-22 | date = 2025-03-03|doi = 10.31389/pop.148 | issue = 1 | url = https://philosophyofphysics.lse.ac.uk/articles/10.31389/pop.148 | doi-access = free | url-access = subscription }}</ref>

====Metaphysical possibility====
{{Main|Modal metaphysics}}
[[Philosophers]]{{who|date=April 2012}} debate if objects have properties independent of those dictated by scientific laws. For example, it might be metaphysically necessary, as some who advocate [[physicalism]] have thought, that all thinking beings have bodies<ref>{{cite web|last1=Stoljar|first1=Daniel|title=Physicalism|url=http://plato.stanford.edu/entries/physicalism/|website=The Stanford Encyclopedia of Philosophy|access-date=16 December 2014}}</ref> and can experience the passage of [[time]]. [[Saul Kripke]] has argued that every person necessarily has the parents they do have: anyone with different parents would not be the same person.<ref>Saul Kripke ''Naming and Necessity'' Harvard University Press, 1980, p. 113.</ref>

[[Metaphysical possibility]] has been thought to be more restricting than bare logical possibility<ref>{{cite book|last1=Thomson|first1=Judith and Alex Byrne|title=Content and Modality : Themes from the Philosophy of Robert Stalnaker|date=2006|publisher=[[Oxford University Press]]|location=Oxford|page=107|url=https://books.google.com/books?id=JXeOkXnCwb8C|access-date=16 December 2014|isbn=9780191515736}}</ref> (i.e., fewer things are metaphysically possible than are logically possible). However, its exact relation (if any) to logical possibility or to physical possibility is a matter of dispute. Philosophers{{who|date=April 2012}} also disagree over whether metaphysical truths are necessary merely "by definition", or whether they reflect some underlying deep facts about the world, or something else entirely.

===Epistemic logic===
{{Main|Epistemic logic}}

'''Epistemic modalities''' (from the Greek ''episteme'', knowledge), deal with the ''certainty'' of sentences. The □ operator is translated as "x is certain that…", and the ◇ operator is translated as "For all x knows, it may be true that…" In ordinary speech both metaphysical and epistemic modalities are often expressed in similar words; the following contrasts may help:

A person, Jones, might reasonably say ''both'': (1) "No, it is ''not'' possible that [[Bigfoot]] exists; I am quite certain of that"; ''and'', (2) "Sure, it's ''possible'' that Bigfoots could exist". What Jones means by (1) is that, given all the available information, there is no question remaining as to whether Bigfoot exists. This is an epistemic claim. By (2) he makes the ''metaphysical'' claim that it is ''possible for'' Bigfoot to exist, ''even though he does not'': there is no physical or biological reason that large, featherless, bipedal creatures with thick hair could not exist in the forests of North America (regardless of whether or not they do). Similarly, "it is possible for the person reading this sentence to be fourteen feet tall and named Chad" is ''metaphysically'' true (such a person would not somehow be prevented from doing so on account of their height and name), but not ''alethically'' true unless you match that description, and not ''epistemically'' true if it is known that fourteen-foot-tall human beings have never existed.

From the other direction, Jones might say, (3) "It is ''possible'' that [[Goldbach's conjecture]] is true; but also ''possible'' that it is false", and ''also'' (4) "if it ''is'' true, then it is necessarily true, and not possibly false". Here Jones means that it is ''epistemically possible'' that it is true or false, for all he knows (Goldbach's conjecture has not been proven either true or false), but if there ''is'' a proof (heretofore undiscovered), then it would show that it is not ''logically'' possible for Goldbach's conjecture to be false—there could be no set of numbers that violated it. Logical possibility is a form of ''alethic'' possibility; (4) makes a claim about whether it is possible (i.e., logically speaking) that a mathematical truth to have been false, but (3) only makes a claim about whether it is possible, for all Jones knows, (i.e., speaking of certitude) that the mathematical claim is specifically either true or false, and so again Jones does not contradict himself. It is worthwhile to observe that Jones is not necessarily correct: It is possible (epistemically) that [[Goldbach's conjecture#History|Goldbach's conjecture]] is both true and unprovable.

Epistemic possibilities also bear on the actual world in a way that metaphysical possibilities do not. Metaphysical possibilities bear on ways the world ''might have been,'' but epistemic possibilities bear on the way the world ''may be'' (for all we know). Suppose, for example, that I want to know whether or not to take an umbrella before I leave. If you tell me "it is ''possible that'' it is raining outside" – in the sense of epistemic possibility – then that would weigh on whether or not I take the umbrella. But if you just tell me that "it is ''possible for'' it to rain outside" – in the sense of ''metaphysical possibility'' – then I am no better off for this bit of modal enlightenment.

Some features of epistemic modal logic are in debate. For example, if ''x'' knows that ''p'', does ''x'' know that it knows that ''p''? That is to say, should □''P'' → □□''P'' be an axiom in these systems? While the answer to this question is unclear,<ref>cf. [[Blindsight]] and [[Subliminal perception]] for negative empirical evidence</ref> there is at least one axiom that is generally included in epistemic modal logic, because it is minimally true of all normal modal logics (see [[#Axiomatic systems|the section on axiomatic systems]]):
* '''K''', ''Distribution Axiom'': <math> \Box (p \to q) \to (\Box p \to \Box q)</math>.

It has been questioned whether the epistemic and alethic modalities should be considered distinct from each other. The criticism states that there is no real difference between "the truth in the world" (alethic) and "the truth in an individual's mind" (epistemic).<ref>{{cite book| last=Eschenroeder |first=Erin |author2=Sarah Mills |author3=Thao Nguyen |title=The Expression of Modality|editor=William Frawley|publisher=Mouton de Gruyter| date=2006-09-30 |series=The Expression of Cognitive Categories|pages=8–9|url=https://books.google.com/books?id=72URszHq2SEC&pg=PT18| isbn=978-3-11-018436-5 | access-date=2010-01-03}}</ref> An investigation has not found a single language in which alethic and epistemic modalities are formally distinguished, as by the means of a [[grammatical mood]].<ref>{{cite book|last=Nuyts|first=Jan|title=Epistemic Modality, Language, and Conceptualization: A Cognitive-pragmatic Perspective|publisher=John Benjamins Publishing Co|date=November 2000|series=Human Cognitive Processing|page=28|isbn=978-90-272-2357-9}}</ref>

===Temporal logic===
{{Main|Temporal logic}}

Temporal logic is an approach to the semantics of expressions with [[Grammatical tense|tense]], that is, expressions with qualifications of when. Some expressions, such as '2 + 2 = 4', are true at all times, while tensed expressions such as 'John is happy' are only true sometimes.

In temporal logic, tense constructions are treated in terms of modalities, where a standard method for formalizing talk of time is to use ''two'' pairs of operators, one for the past and one for the future (P will just mean 'it is presently the case that P'). For example:

:'''F'''''P'' : It will sometimes be the case that ''P''
:'''G'''''P'' : It will always be the case that ''P''
:'''P'''''P'' : It was sometime the case that ''P''
:'''H'''''P'' : It has always been the case that ''P''

There are then at least three modal logics that we can develop. For example, we can stipulate that,

:<math> \Diamond P </math> = ''P'' is the case at some time ''t''
:<math> \Box P </math> = ''P'' is the case at every time ''t''

Or we can trade these operators to deal only with the future (or past). For example,

:<math> \Diamond_1 P </math> = '''F'''''P''
:<math> \Box_1 P </math> = '''G'''''P''

or,

:<math> \Diamond_2 P</math> = ''P'' and/or '''F'''''P''
:<math> \Box_2 P </math> = ''P'' and '''G'''''P''

The operators '''F''' and '''G''' may seem initially foreign, but they create [[normal modal logic|normal modal systems]]. '''F'''''P'' is the same as ¬'''G'''¬''P''. We can combine the above operators to form complex statements. For example, '''P'''''P'' → □'''P'''''P'' says (effectively), ''Everything that is past and true is necessary''.

It seems reasonable to say that possibly it will rain tomorrow, and possibly it will not; on the other hand, since we cannot change the past, if it is true that it rained yesterday, it cannot be true that it may not have rained yesterday. It seems the past is "fixed", or necessary, in a way the future is not. This is sometimes referred to as [[accidental necessity]]. But if the past is "fixed", and everything that is in the future will eventually be in the past, then it seems plausible to say that future events are necessary too.

Similarly, the [[problem of future contingents]] considers the semantics of assertions about the future: is either of the propositions 'There will be a sea battle tomorrow', or 'There will not be a sea battle tomorrow' now true? Considering this thesis led [[Aristotle]] to reject the [[principle of bivalence]] for assertions concerning the future.

Additional binary operators are also relevant to temporal logics (see [[Linear temporal logic]]).

Versions of temporal logic can be used in [[computer science]] to model computer operations and prove theorems about them. In one version, ◇''P'' means "at a future time in the computation it is possible that the computer state will be such that P is true"; □''P'' means "at all future times in the computation P will be true". In another version, ◇''P'' means "at the immediate next state of the computation, ''P'' might be true"; □''P'' means "at the immediate next state of the computation, P will be true". These differ in the choice of [[Accessibility relation]]. (P always means "P is true at the current computer state".) These two examples involve nondeterministic or not-fully-understood computations; there are many other modal logics specialized to different types of program analysis. Each one naturally leads to slightly different axioms.

===Deontic logic===
{{Main|Deontic logic}}

Likewise talk of morality, or of [[obligation]] and [[norm (philosophy)|norms]] generally, seems to have a modal structure. The difference between "You must do this" and "You may do this" looks a lot like the difference between "This is necessary" and "This is possible". Such logics are called ''[[deontic logic|deontic]]'', from the Greek for "duty".

Deontic logics commonly lack the axiom '''T''' semantically corresponding to the reflexivity of the accessibility relation in [[Kripke semantics]]: in symbols, <math>\Box\phi\to\phi</math>. Interpreting □ as "it is obligatory that", '''T''' informally says that every obligation is true. For example, if it is obligatory not to kill others (i.e. killing is morally forbidden), then '''T''' implies that people actually do not kill others. The consequent is obviously false.

Instead, using [[Kripke semantics]], we say that though our own world does not realize all obligations, the worlds accessible to it do (i.e., '''T''' holds at these worlds). These worlds are called ''idealized'' worlds. ''P'' is obligatory with respect to our own world if at all idealized worlds accessible to our world, ''P'' holds. Though this was one of the first interpretations of the formal semantics, it has recently come under criticism.<ref>See, e.g., {{cite journal |first=Sven |last=Hansson |title=Ideal Worlds—Wishful Thinking in Deontic Logic |journal=Studia Logica |volume=82 |issue=3 |pages=329–336 |year=2006 |doi=10.1007/s11225-006-8100-3 |s2cid=40132498 }}</ref>

One other principle that is often (at least traditionally) accepted as a deontic principle is ''D'', <math>\Box\phi\to\Diamond\phi</math>, which corresponds to the seriality (or extendability or unboundedness) of the accessibility relation. It is an embodiment of the Kantian idea that "ought implies can". (Clearly the "can" can be interpreted in various senses, e.g. in a moral or alethic sense.)

====Intuitive problems with deontic logic====
When we try to formalize ethics with standard modal logic, we run into some problems. Suppose that we have a proposition ''K'': you have stolen some money, and another, ''Q'': you have stolen a small amount of money. Now suppose we want to express the thought that "if you have stolen some money, it ought to be a small amount of money". There are two likely candidates,
: (1) <math>(K \to \Box Q)</math>
: (2) <math>\Box (K \to Q)</math>

But (1) and ''K'' together entail □''Q'', which says that it ought to be the case that you have stolen a small amount of money. This surely is not right, because you ought not to have stolen anything at all. And (2) does not work either: If the right representation of "if you have stolen some money it ought to be a small amount" is (2), then the right representation of (3) "if you have stolen some money then it ought to be a large amount" is <math>\Box (K \to (K \land \lnot Q))</math>. Now suppose (as seems reasonable) that you ought not to steal anything, or <math>\Box \lnot K</math>. But then we can deduce <math>\Box (K \to (K \land \lnot Q))</math> via <math>\Box (\lnot K) \to \Box (K \to K \land \lnot K)</math> and <math>\Box (K \land \lnot K \to (K \land \lnot Q)) </math> (the [[contrapositive]] of <math>Q \to K</math>); so sentence (3) follows from our hypothesis (of course the same logic shows sentence (2)). But that cannot be right, and is not right when we use natural language. Telling someone they should not steal certainly does not imply that they should steal large amounts of money if they do engage in theft.<ref>Ted Sider's ''Logic for Philosophy'', unknown page. http://tedsider.org/books/lfp.html</ref>

=== Doxastic logic ===
{{Main|Doxastic logic}}

''Doxastic logic'' concerns the logic of belief (of some set of agents). The term doxastic is derived from the [[ancient Greek]] ''doxa'' which means "belief". Typically, a doxastic logic uses □, often written "B", to mean "It is believed that", or when relativized to a particular agent s, "It is believed by s that".