Talk:Axiom S5

message: It seems to me that the S5 axiom is contradictory. My proof: 
# Consider an arbitrary axiomatic system. 
# Consider an arbitrary statement p. 
# 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.) 
# Therefore, it is possible that p is provably true. 
# Any statement that is provably true is necessarily true, since it follows from the axioms. 
# Therefore, p is possibly necessarily true. 
# 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)
replies: text:  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)

text: 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)
replies: text: 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)

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

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

message: 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)
thread_title: Semantics

message: ::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)
thread_title:  I agree to move or even remove, this article 

message: 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)
thread_title:  error in header 

message: 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)
thread_title:  merge this page 

message: 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)
thread_title:  Explanation for Non-Mathematicians?