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Normally i do not like to utilize multiple posts while making topics, but in this case it seems prudent. In this post i would like to address several issues which have been raised, but were not directly related to the main topic. So far there are two that come to mind. This post will be as concise as i can make it, so please tell me if anything needs clarification.

The Law of Non-Contradiction

This law is interesting, as it is in fact a universal negative. However, denying it is typically viewed as impossible, as denying it requires one to use it in the denial, a clear example of circular logic.

This can be easily misunderstood. The Law of Non-Contradiction is not true in the slightest. Just as it is impossible to deny it, it is also impossible to prove it. Consequently, it is indemonstrable. It is neither verifiable nor falsifiable. It has no more inherent value than the statement, "An invisible pink unicorn!"

The Law of Non-Contradiction is useful, and is in fact an unspoken aspect of a great deal of things. However, it is important to understand while it is useful, this law is not "true."

Is Anything Provable?

Another issue which has been raised claims rejecting the original contention requires one accept nothing is provable. To a certain extent the conclusion is true. In the strictest of terms, nothing is provable. Indeed, it is already apparent the very structure used to "prove" anything is unprovable.

This should not be surprising, and there should be no disagreement on this point. However, it is worth noting the conclusion is not based upon the position i stated. The inability to prove anything does not stem from the inability to prove a universal negative. It is in fact the exact opposite.

By a literal interpretation, a universal negative cannot be proven, nor can a universal positive. However, it is possible to "prove" a universal affirmative. The statement, "Dogs exist" can be "proven." All it takes is to point to a dog. The statement, "Dogs do not exist" cannot be "proven," as it would be impossible to search all spots for dogs.

The difference here lies in the difference between prove and "prove." The first is literal, and impossible. Things like the Law of Non-Contradiction are first level statements. They are what high level "proofs" are based. All higher level proofs, such as, "Dogs exist," require the first level proofs be accepted (as axiomatic beliefs).

As any proof builds upon another proof, it adds another level. The things which are considered "universal positives" and "universal negatives" are second level proofs. At this level, only universal positives are provable. While this leaves a large amount of unprovable claims, it is still a substantial increase over the amount of provability on the first level.

On the third level, it becomes possible to prove some negatives. At the same time, there will still be unprovable claims. The pattern holds for each successive level, bounded only by the logical constraints created by language and human thought.

Then your "local negative" is just a "particular negative" in new clothes, if defined that way. You're not inventing anything new. Your definition is hence redundant. The quibble over the definition of a universal negative is a false controversy. As revealed below.

The distinction between a universal negative and "a different negative" is a matter of quantification, not being "unbounded by any systems." (You've defined "systems," but not "unbounded," BTW.)



Fallacy: Shifting the burden of proof. If you "negate it," ("The debate circles around the contention, which i negate") then you go beyond the fallback position (which is mere skepticism) and positively assert the opposite. If you assert the truth of something, then you assume a burden of proof. Your position is the same as that of a strong atheist saying, "I deny God exists. Prove me wrong." Of course, you must shift the burden of proof in doing so, or fall into self-contradiction.



This proves that there is no real quibble over the definition of a universal negative. If you accept that "No reptiles have fur" is an example of a universal negative proposition, then you agree that the way one expresses universal negative propositions in Aristotleian logic is by the form "No S are P." Which has nothing to do with your gratuitous "local" or "unbounded by systems" mumbo-jumbo. When a person says, for example, "The non-existence of unicorns cannot be established because you can't prove the universal non-existence of a thing," in terms of Aristotleian logic, they're saying that "No E-type ["No S are P"] propositions are provable," which is itself an E-type proposition, incidentally. In other logics, you can express a universal negative by means of symbols which translated into English, mean "For all of X, if X is decribed by S, X must not be described as P."

It is true that the syllogism requires the truth of the universal negative in a premise for the universal negative in the conclusion to be true, but there are other ways of proving universal negatives than the four syllogisms I gave you. For example. Modus ponendo tollens. Reductio ad absurdum. Self-contradiction: "No bachelors are married" is true by definition. You can only reject these other methods by rejecting deductive logic itself.



False. It requires the acceptance of definitions, not axioms. The definitions in Euclid come before the axioms (the postulates), and the specific controversy here has no relation with the postulates. If we are in agreement with what a circle, a square, and a side is, the conclusion that circles lack four sides follows. If you reject that on the grounds that the term "circle" can mean anything, then your argument is one of linguistic nihilism and thus doesn't affect the point, since that doesn't affect the actual concept of a circle, only the terminology. Moreover that kind of nihilism destroys your own position as well. It also reminds me of the position of Roscelin of Compiègne that concepts are just "flatus vocis." You're not in very good company...

In any case, axioms are supposed to be self-evidently true.

You also commit the fallacy of redefinition. You agreed that the proposition, "No reptiles have fur" is an example of a universal negative proposition.



False. The non-quadrilateral nature of the circle can be known deductively from the known properties of the square and the circles, and hence, certainly. In the same manner, the infinitude of the primes and the irrational nature of the square root of 2 can also be proven. Again, if you want a more rigourous proof of why circles don't have four sides, I can provide it for you.



False. Not circular logic. Stealing the concept. You didn't even get the right name of the kind of fallacy is involved in denying it. When your argument attempts to deny a concept, but that concept must be assumed to be true for your argument against it to be correct in the first place, you've self-refuted yourself by committing the fallacy of the stolen concept.



False. The Law of Non-Contradiction is not some kind of arbitrary a**-pull. The truth of the LNC is defendable via retortion, much like the Socratic elenchus, and it puts the denier of the LNC into a state of aporia. "Proving" it is not necessary, since the LNC stands above any proof (anyone who thinks you need to "prove" logic doesn't understand logic in the first place), and denying it is impossible since you have to count on the LNC being true when you are in the process of arguing that it is false. To contradict it is to affirm it. If the LNC is false, then there are two truths at the same time: you can say that a proposition is true and that it is also false (doublethink). But to say "The LNC is false" asserts a unity of truth, which contradicts what would be the case if the LNC were actually false. And when you say "The LNC is true and it is false," then you are in a situation of radical self-contradictory incoherence, since the LNC denies that a proposition can both be true and false at the same time, or that two antithetical propositions can both be true at the same time. So:

1. "The LNC is false." Self-refuting.
2. "The LNC is both true and false." Incoherent.

Since there is only one more option, "The LNC is true," the conclusion can only be that the LNC is incorrigible, which means that the LNC can also be regarded as a self-evidently true axiom. We can not only consider the LNC to be true, or even "true without error," we can consider it to be "of the essence of truth."

For you to state that the LNC can neither be proven or denied, and also to claim that it is "not true in the slightest" is to commit the fallacy of arguing from ignorance. It is also self-contradiction, since to say it is "not true in the slightest" means you do deny it, which you admit is not possible.

Moreover, you believe in the LNC anyway. Your position implies "Universal negatives are either provable or they are not provable, both positions cannot be true at the same time," which positively asserts the truth of the LNC. So why are you arguing against something you undoubtedly believe anyway? You've refuted yourself out of your own mouth, so why should I waste any more time with you?



Also false. There are limitations on doubt itself, and it could not be otherwise. Doubt is a form of consciousness, so it is impossible to doubt the existence of consciousness, as Descartes and others have pointed out. The reality of consciousness can therefore be taken as certain, as well as the reality of existence. Given consciousness and existence, the axioms of logic can be a priori deduced by any sentient mind. Once those axioms are made explicit, you have the means to prove things deductively. Deductively obtained knowledge is certain knowledge, as long as you don't stray from true premises, your conclusions will also be proven ones in the strict sense of the word.

On the other hand, if what you are saying is true, it is not proven in the strict sense either, and can therefore be doubted anyway. But I doubt that you actually want people to go around doubting what you are saying about provability.



If all reasoning were inductive reasoning, it would be true that universal negatives could never be proven. But there is deductive reasoning, too. So your argument commits the fallacy of composition: of taking a part of reasoning for the whole of reasoning. It also commits the fallacy of the stolen concept since your argument is in the structure of a deductive argument, and you're arguing against the validity of deductive reasoning. It is also self-contradictory, since we would have at least one universal negative whose truth could never be in doubt: "No universal negative can be proven inductively" which implies there is a non-inductive realm of reasoning, something the original argument implictly denied exists.



True. But the first level statements are neither arbitrary nor corrigible, which is what you seem to believe. So it is not true that all our reasoning lies on feet of sand.