Hermes
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« Reply #15 on: March 16, 2012, 16:18:11 PM » 

"All humans are mortal" or "the earth is flat/spherical" are not held to be true because the converse would result in a collapse of logic or arithmetic, but based on sensory observations. They do therefore not qualify as axioms in compliance with Mefiante's requirements.



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st0nes


« Reply #16 on: March 16, 2012, 16:24:35 PM » 

"All humans are mortal" or "the earth is flat/spherical" are not held to be true because the converse would result in a collapse of logic or arithmetic, but based on sensory observations. They do therefore not qualify as axioms in compliance with Mefiante's requirements.
No, you are right, but that is just a trivial example quoted in almost all logic textbooks of valid logic whose conclusion is true if the premises are.



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Rigil Kent
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« Reply #17 on: March 17, 2012, 06:22:02 AM » 

... due to lack of belief in average human intelligence. I strongly believe in average human intelligence. An axiom must appeal to our intuitive, yet precise, understanding of it's components (the words making up the axiom.) Unless we can flawlessly conceptualize a point and a straight line, we cannot hope to understand that it is always possible to join two of the former with one of the latter. Because of this crucial link between the truth of an axiom, and our full understanding of it's components, an interesting, almost subconscious "reversal" of the logical process takes place. It appears that the axiom also reinforces our understanding of it's components in a backwards direction! For example, if I have problems logically grasping the axiomatic statement " concentric circles have a common center", then I know that it is time to reconsider my understanding of center, circle, and perhaps even common. And perhaps, in a similar though slightly more spooky way, a theorem reinforces the understanding of it's axioms too. Rigil


« Last Edit: March 17, 2012, 07:18:23 AM by Rigil Kent »

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Hermes


« Reply #18 on: August 13, 2012, 16:25:26 PM » 

If these laws of logic are subverted, it is not possible to construct a consistent arithmetic, so these axioms bootstrap themselves as universal truths as well because their falsity leads inevitably to absurdities that we cannot experience. 'Luthon64 Can one deduce that the antithesis of a paradox must be an axiom? If one examines a paradoxical statement such as "Truth does not exist," one finds that if the statement were true, at least one truth would exist, namely the statement itself. If the statement were false, truth would also exist. Therefore "Truth exists" must be an axiom. One could apply the same argument to the fact that the concept of "almighty" is paradoxical and therefore the nonexistence of an almighty god must be axiomatic.



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Mefiante
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« Reply #19 on: August 13, 2012, 17:09:01 PM » 

A very interesting question, but no, I don’t think that’s true as a universal rule. Selfreferential propositions, e.g. “This sentence is false”, are always tricky and normally require the use of (a hierarchy of) metalanguages for their resolution. In other words, they seem paradoxical when considered on a naïve level but are sensibly apprehendable in the correct framework and so it’s neither obvious what the correct antithesis would be, nor how any such possible antithesis would constitute an axiom. In the case of the given example, you might take “This sentence is true” or perhaps “Every sentence except this one is false” as the antithesis. The result is something that is tautological, incoherent and/or unfruitful. If we consider a paradox that does not rest on selfreference, e.g. Zeno’s AchillesandtheTortoise paradox, we note again that it’s only a paradox because there are perceptional assumptions that conspire to mislead our thinking. Zeno’s paradoxes are rigorously resolvable within the correct framework (specifically, infinite convergent series and calculus). The antithesis would be something perhaps like “Achilles will overtake Tortoise at some point”, which again is a truism (if it is true that Achilles can run faster than Tortoise). I can’t think of any way to reformulate an antithesis to this paradox so that it results in something that could rightly be labelled an axiom. 'Luthon64



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cyghost
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« Reply #20 on: August 14, 2012, 07:46:20 AM » 

You have a beautiful mind.



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Hermes


« Reply #21 on: August 14, 2012, 10:53:43 AM » 

Both "antithesis" and "paradox" are problematic here.
Let's try something related: If the falsity of a statement introduces selfcontradiction, the statement must be an axiomatic truth.



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Mefiante
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« Reply #22 on: August 14, 2012, 19:21:56 PM » 

You have a beautiful mind. If that was directed at moi, I’m truly touched that you should not only think so, but also say so. Merci beaucoup, patron. Both "antithesis" and "paradox" are problematic here. Yes, and this was a substantial part of what I meant to convey. If the falsity of a statement introduces selfcontradiction, the statement must be an axiomatic truth. 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 selfcontradictory (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 reemerges, so in this case your contention is valid. However, the system also has sentences that are socalled ‘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. 'Luthon64



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cyghost
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« Reply #23 on: August 15, 2012, 07:44:01 AM » 

If that was directed at moi, I’m truly touched that you should not only think so, but also say so. Merci beaucoup, patron. It was, been thinking it all along, it was just time to say it out loud dammit.



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Hermes


« Reply #25 on: 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 selfcontradictory (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 reemerges, so in this case your contention is valid.
However, the system also has sentences that are socalled ‘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.
'Luthon64 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.


« Last Edit: August 15, 2012, 12:54:32 PM by Hermes »

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Mefiante
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« Reply #26 on: 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 selfcontradictory 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 selfcontradictory 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 selfcontradictory 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 selfcontradiction 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. 'Luthon64



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Hermes


« Reply #27 on: 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.



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Mefiante
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« Reply #28 on: 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! 'Luthon64



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Hermes


« Reply #29 on: 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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