Editing Modal logic (section)

===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>