Absolute certainty and universal truth.

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Hermes (August 15, 2012, 12:29:05 PM):
I think this may be true for some statements within the framework of an axiomatic formal system, but not so generally. If we take an axiom of a formal system and simply negate it, we obtain something (technically, it’s called a ‘sentence’ of the system) that is self-contradictory (i.e. incoherent and thus false) in that system because it directly contradicts one of its axioms. If we now negate this false sentence, the original axiom re-emerges, so in this case your contention is valid.

However, the system also has sentences that are so-called ‘theorems’ (i.e., sentences that are provably true in that system) which consist of (some of) the system’s axioms put in a particular logical arrangement. The negation of such a theorem, while clearly false (because it contradicts what is valid in that system), is not itself an axiom because the theorem itself (which is the negation of its own negation) requires more than one axiom of the system for its proof. In this case, the contention fails.

The short symbolic version is ~(~P) = P where P can be either an axiom or a theorem.

I think it is obvious that a selfcontradiction must contain an untruth. What I am testing, is if its negation may lead us to a special category of truth. Surely the negation of a theorem will create a contradiction with the system, but I would like to see an illustration where the negation of a theorem creates a selfcontradiction.
Mefiante (August 16, 2012, 18:49:35 PM):
Take the following general sentence of propositional logic:

S = P → ~P (In words: “If P is true then P is false.”)

This is clearly an archetype of a self-contradictory statement.

The truth of a logical implication S = P → Q can be negated in just one way, namely when you observe a case where P entails ~Q instead of Q, and so directly contradicts S. That is, there is a complementary instance of S, its negation ~S = S' = P → ~Q.

If we replace Q with ~P, we have our original self-contradictory sentence S, and its negation becomes S' = P → ~(~P) = P → P, which is always true. (Technically, it’s called a tautology, which is indeed a special category of truth, viz. something trivially true.)

However, S' is not an axiom unless P itself is an axiom, in which case S' is merely an awkward restatement of P.

So, once again, the negation of a self-contradictory proposition is not necessarily an axiom.

By turning the situation around and starting with the tautology S = P → P where P could be an axiom or a theorem, we can now negate S in the opposite direction as before, and so arrive at the situation where the negation of a theorem creates the self-contradiction S = P → ~P.

If the above still doesn’t satisfy your requirements then I don’t understand properly what you are driving at, in which case perhaps you need to delve into some formal logic and the arcane field of axiomatic formal systems to crystallise your meaning.

Hermes (August 17, 2012, 12:58:45 PM):
This answers my questions most satisfactorily. Wikipedia describes a tautology in logic as follows:
[A] tautology is a formula whose negation is unsatisfiable. Unsatisfiable statements, both through negation and affirmation, are known formally as contradictions.

It follows that the negation of a selfcontradiction must lead to an irrefutable truth. Intuitively it seems selfevident, but it would not suffice to rely on intuition when it comes to proving irrefutability. That the negation of a selfcontradiction leads to a tautology rather than (necessarily) an axiom, is of secondary importance. This is a useful tool, because selfcontradictions are much easier to spot and point out than tautologies.

Mefiante, thank you for your (widely admired) guidance.
Mefiante (August 17, 2012, 13:25:46 PM):
Always glad to be of service where I can in the pursuit of understanding.

And thanks for the compliments — both of them! ;) :-*

Hermes (August 17, 2012, 15:14:22 PM):
However, S' is not an axiom unless P itself is an axiom, in which case S' is merely an awkward restatement of P.
Oops, now I have doubts about this. I don't think that the characteristics of S' should be a function of P's veracity at all. Even if P is an axiom, it should not make S' an axiom. S' appears to me to be an "if...then" statement in which P is both the condition and the fulfillment.


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