Talk:Axiom S5
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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)
- 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)
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- 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
