Talk:Axiom S5

Proposed disproof by showing the axiom implies every statement of any axiomatic system to be true

It seems to me that the S5 axiom is contradictory. My proof:

1. Consider an arbitrary axiomatic system.
2. Consider an arbitrary statement p.
3. The following must be true for any p: Either p is provably true, or p is not provably true. (This follows from bivalence.) (Note that "not provably true" does not necessarily mean "false", but that is beside the point.)
4. Therefore, it is possible that p is provably true.
5. Any statement that is provably true is necessarily true, since it follows from the axioms.
6. Therefore, p is possibly necessarily true.
7. Therefore, by axiom S5, p is true.

This works for any statement. Therefore, every statement imaginable will be true. Since the negation of p ("not p") is also a statement, every p is also false. This is a contradiction. But since the argument works for any axiomatic system, then every system is contradictory, under the assumption of axiom S5. Therefore, S5 itself is contradictory.

Is my reasoning here correct, or can anyone find a flaw in it? SpectrumDT 10:38, 8 February 2006 (UTC)

S5 is not part of the Gödel-Loeb provable modal logic, right? the problem starts with the axiom M, you should rework your argument to extend the modal logic article in what provability respects. — Preceding unsigned comment added by 189.178.233.172 (talk) 09:13, 4 November 2012 (UTC)

There is an equivocation on the sense of the modal vocabulary "possible" (step 4) and "necessarily" (step 5). The only sense of possibility in which an arbitrary statement p can be known to be possible is the subjective epistemic sense, on which a statement is possible as long as I don't know it to be false. However, the sense of necessity that follows from provability is different - a provable statement isn't necessary in a subjective epistemic sense, but just in an alethic sense, or possibly an objective epistemic sense like that of provability logic. Axiom S5 doesn't hold for provability logic, but it does for alethic modality, on which your step 4 fails. Just because I don't know something has to be true, doesn't mean that it's not necessary. Easwaran 03:34, 23 February 2006 (UTC)

What happens if the statement in step 4 is in the possible in the alethic sense? Certainly it is _both_ epistemically and alethically possible, for instance, for a large number of overload entities to exist (perfect islands, invisible pink unicorns, orbiting asteroids...). Since 5 is already in the alethic sense, the argument would hold only for specific instances of statements, but still not instances we'd like to accept. Madumlao (talk) 09:01, 2 December 2008 (UTC)

What is "alethic logic"? Is it the same as modal logic? SpectrumDT 21:39, 24 February 2006 (UTC)

And what if we edit our argument this way.

• If possibly necessarily p, then there is a possible world w0 at which p necessarily holds,
• Then, it is true at w0 that p is a broadly logically necessary truth, something whose negation would in a broadly logical sense be self-contradictory,
• If possibly necessarily not p, then there is a possible world w1 at which not p necessarily holds,
• Then, it is true at w1 that not p is a broadly logically necessary truth, something whose negation would in a broadly logical sense be self-contradictory,
• But if something is self-contradictory at some possible world, then it is self-contradictory at all worlds.

P and "not p" cannot be self-contradictory at the same time, but if something whose negation would in a broadly logical sense be self-contradictory then both p and "not p" would be self-contradictory in every world.

What I mean is that the negation of "not p" is "not not p" which is equal to p. And if negation of "not p" is in a broadly logical sense self-cntradictory in every world then "not p" (negation of p) is true for every world and can't be self-contradictory for w0 either. 83.27.13.45 16:26, 15 January 2007 (UTC)MaybeNextTime

4 does not follow from 3. For example, "2+2=5" is either provably true or not provably true (3). But it is not true that it's possible that "2+2=5" is provably true (4). 91.107.151.20 (talk) 00:03, 24 December 2008 (UTC)

Semantics

Perhaps this is just my lack of education in the field of modal logic speaking, but would it be worthwhile to give an explanation of why the second part of axiom S5 makes sense from the point of view of translating the statement into English? As far as I can tell, it seems to mean this:

If a world exists in which statement p is a necessary truth for all worlds, then statement p must be a necessary truth for all worlds.

Is this correct and, if so, would it be reasonable to include it in the article? Natsirtguy (talk) 09:59, 12 December 2007 (UTC)

Move?

It seems to me that this axioms is usually called just "axiom 5", not "axiom S5". The name S5 is for the modal logic K+T+5. Tizio 12:18, 10 August 2007 (UTC)

It's confusing if we call it just "5" as opposed to "S5" since lots of other logical systems include a 5th axiom. Jordan 22:59, 31 July 2008 (UTC) —Preceding unsigned comment added by Jordanotto (talk • contribs)

The name axiom 5 is well-established in the literature of modal logic and is used without confusion, usually being referred to as (5) and where there are conventions regarding how other axioms are referred to (see User:Nahaj's excellent resource on axiomatisations of modal logics). It is deeply confusing to call this axiom S5, since it is used in other familiar systems of modal logic: the SEP article on modal logic mentions K5, K45, K5B, D5, and D45. I propose that this article be moved to Axiom 5, though I can see a case for it to be deleted. If conflicts with other articles called axiom 5 crop up in the future, the article can be renamed to Axiom 5 (modal logic). I'll move it if there is no objection in the next 36 hours. — Charles Stewart (talk) 09:19, 18 March 2009 (UTC)

I agree to move or even remove, this article

Read the article in SEP, it is very clear, correct, and revised by an editor.

S5 is a modal logic, containing the axiom (5).

There is no need to have an entry for each modal axiom in wikipedia.

That is just an over-fragmentation of the modal logic article.

I added the advice to remove the article, but I fixed it before, to make it more easy to reuse the content in the S5 entry. — Preceding unsigned comment added by 189.178.233.172 (talk) 07:58, 4 November 2012 (UTC)

the "less intuitively understood" portion is fallacious

i dunno who wrote this, but they do not know anything about logic. Axiom 5 (or S5) is an axiom stating that if something is necessarily possible, than it is possible. It says nothing about something being necessary if it is possibly necessary. An example of this is that if string theory were true (which is a possibility), it would be necessary for matter to be composed of strings. Since it is possibly necessary for matter to be composed of strings, this incorrect reasoning would mean that matter is necessarily composed of strings (i.e. matter is composed of strings, merely because it is possibly necessary). I'm going to write this into the article and replace the incorrect section eventually, but if anyone else can do so before I get around to it, please do.

oops, forgot to sign. Wing gundam 03:58, 2 September 2007 (UTC)

• Starting your comment with an insult is poor form... particularly when you then get your facts wrong. Your statment says S5 is "if something is necessarily possible, than it is possible" That is not axiom 5, that axiom (\Box\Diamond p \to \Diamond p) is an instance of axiom T. Axiom 5 is either ($\Box p\to \Diamond \Box p$) or its dual ($\Box \Diamond p\to \Diamond p)$) [Completely equivalent forms in systems with duality such as KT5(S5)). Given your premise is false, it would be hard for the rest of the argument to follow. However, what was on the page was wrong (and opinionated...) I've fixed it, Hughes and Cresswell's "Introduction to Modal Logic" and Brian Challas' "Introduction to Modal Logic" cover this... as does J.Zemans "Modal Logic, the Lewis Systems" for those that want to go into more detail. Nahaj —Preceding signed but undated comment was added at 20:57, 17 September 2007 (UTC)

Nahaj: I erased your reference because because strict implication is not mentioned here and the notation is confusing in your page.

There are symbols in LaTeX and unicode for strict implication, here they are some variants:

⥼ name: lfisht \strictfi

⥽ name: rfisht \strictif

⥾ name: ufisht \rotateleft{\strictif} (not used in logic)

I hope that this is usefulfor you. — Preceding unsigned comment added by 189.178.233.172 (talk) 08:55, 4 November 2012 (UTC)

• the "not intuitive" bit in the article is meant to highlight that fact that people who reject S5 often do so because of the difficult in conceptualizing the proposition -- if possibly necessarily P, then necessarily P. Jordan 23:08, 31 July 2008 (UTC) —Preceding unsigned comment added by Jordanotto (talk • contribs)

You are missing the point, I hope that at this time you got it.

It depends on the meaning that you want 'necessary' and 'possible' have in your logic, to introduce or not this axiom. By adding axioms to K (or M) you tune the logic restricting the models that fit the logic, giving to modal operators the desired meaning, certainly it is not an easy task to do.

For that reason it makes no sense to have a separate article for each axiom, this part, and the S5 modal logic article, should be part of the main article modal logic, where it makes sense.

Possible Necessity implies Necessity

The Stanford Encyclopedia of Philosophy's article states that 00...[]p is equivalent to []p. Would you say that the author does not "know anything about logic", Wing gundam? —Preceding unsigned comment added by 69.255.189.114 (talk) 15:45, 16 September 2007 (UTC)

I agree with you. — Preceding unsigned comment added by 189.178.233.172 (talk) 08:09, 4 November 2012 (UTC)

Even if the SEP article were not well written by an expert in logic, you can work with the logic, as one should do when studying maths. Apply the inference rules and axioms to any formula of the form OOOO[]p and you will get []p. Use []p=~<>~p to see why it works for <> too.

If you are not convinced that it works for every formula of such form.

Try to prove it by induction!!!

Thank's God, discussions in formal disciplines end with a formal proof. Just do it!

error in header

Part one of s5 in the header is definitely true, and does not correspond to the diagram. It says necesary possibly=>possibly. For this reason, I think the header is flawed. 68.144.80.168 (talk) 12:35, 24 June 2008 (UTC)

merge this page

The title is wrong should be axiom (5) not axiom S5, S5 is a modal logic that contains the axiom (5). See any of the cited bibliography/references in the life of the page. Even if correctly renamed it would be redundant because it should be part of S5 (modal logic), which in turn should be a section in modal logic which needs to be extended. This is needed to understand why there are many variants of modal logic. Wikipedia is not a dictionary, entries should not be over-fragmented in very short articles.

Do not take this as a vandal act, read the bibliography and the previous discussion you will agree with me that this page needs to be merged first in S5 (modal logic)) then with modal logic among other pages which may be separated. It is unfair that an automatic procedure makes this judgement based just in the number of changes in the page in the same day. I did it, because I want to save the partial changes, to protect against a power failure that disconnects me from the world.

I won't do this procedure of merging because I have never do that, and I do not want to introduce inconsistencies.

This page is 5 years old, and have never been fixed. Depending on the response to my petition I will help to improve the entry, but I had bad experiences in the past from inexperienced wikipedians destroying my additions, because they have no more knowledge than their class notes of some high-school course.

I am not signing, because you can act according with the arguments, not from who is the author. — Preceding unsigned comment added by 189.178.233.172 (talk) 10:49, 4 November 2012 (UTC)

Explanation for Non-Mathematicians?

You do understand that this is a very complicated problem explained in the worst possible language, that of experts that have studied the related field for years and acquired the jargon to work with? This article is missing any attempt to explain its scope and content in an understandable way for normal users that want to research the term... --37.24.226.11 (talk) 19:36, 1 July 2013 (UTC)