Talk:Axiom S5

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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)

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 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

Contents

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)