2. To believe that logic is not necessarily universal is to believe that A is not necessarily not non-A, that positive characteristics do not necessarily exclude their opposites.
3. If positive characteristics do not necessarily exclude their opposites, then no positive characteristic can ever be brought forward to exclude the reality of any opposing characteristic, or no proposition can ever be used to exclude any opposing proposition. For example, “there is a book on the shelf” could not be used to exclude “there is not a book on the shelf.”
4. If no proposition can be used to exclude any opposing proposition, and no characteristic implies the absence of an opposing characteristic, then all propositions are meaningless. For the meaning of a proposition is always bound up with the assumption that true propositions exclude false propositions that oppose them. For example, there is no meaning to the statement that “evolution really happened” if we do not assume that it excludes “evolution didn't really happen” (keeping the same definitions of words in both sentences, of course). If the former proposition's truth does not necessarily imply the latter proposition's falsehood, then there is no meaning in the former proposition.
5. If all propositions are meaningless, then knowledge is impossible. For knowledge also depends on the assumption that positive characteristics exclude their opposites and that propositions exclude opposing propositions. Using the same example, if “evolution really happened” does not necessarily exclude “evolution didn't really happen,” then there is no actual content to the knowledge claim that “evolution really happened.”
6. Related to #5, if positive characteristics and propositions do not exclude their opposites, then it will be impossible to prove anything. For no matter how much evidence one gathers to support any proposition, if that proposition does not inherently and necessarily exclude its opposite, then one cannot argue from the truth of one proposition to the non-truth of an opposing proposition, and therefore one cannot establish the truth of any meaningful propositions or knowledge claims. No matter how much evidence one gathers to prove that “evolution really happened,” if this claim does not necessarily exclude the claim that “evolution didn't really happen,” then the first claim has no meaning and thus cannot be an element of knowledge, and therefore cannot be proven. It cannot even be more likely, for any evaluation of a truth claim will depend on that truth claim having some positive meaning.
7. If propositions and positive characteristics do not necessarily exclude their opposites, then even the proposition that “propositions and positive characteristics do not necessarily exclude their opposites” does not necessarily exclude the contrary proposition, “propositions and positive characteristics DO necessarily exclude their opposites,” and therefore the former proposition has no meaning. So the very claim that “propositions and positive characteristics do not necessarily exclude their opposites,” or that “logic is not necessarily universal,” is self-contradictory and therefore meaningless, because to make such a claim is to assume that the claim has meaning while the claim itself implies that it does not have meaning.
8. No one can avoid making propositional truth claims or believing various knowledge claims. Therefore, if propositions and positive characteristics do not necessarily exclude their opposites, then all people, so far as they are conscious, are always engaging in self-contradictory behavior, believing meaningless and self-contradictory truth claims and asserting meaningless and self-contradictory propositions. Including me, right now, and including any responses to this. And also not including these things. And both. And neither.
9. Since all people who are conscious always continue to believe truth claims and make propositional statements AS IF they have meaning and knowledge can be had, AS IF positive characteristics and propositions necessarily exclude their opposites—that is, AS IF logic is necessarily universally valid—therefore it is clear that no one really believes that logic is not necessarily universally valid, and all people see that it is, however much they confuse themselves into thinking that they don't see this. And that makes sense, because it is self-evident that propositions and positive characteristics necessarily exclude their opposites—that is, that logic is necessarily universally valid. It is as self-evident, for example, that redness excludes non-redness and being excludes non-being as it is that something exists, that I (whoever you are, apply it to yourself) exist, etc.
10. Since logic is necessarily universally valid, the central claim of rationalist epistemology is true, which is that logical, deductive arguments can be legitimately used to establish true knowledge about the real world. If logic is necessarily universally valid, no self-contradictory or illogical proposition (such as, “the past is infinite”) could possibly be true, and any proposition that is required by logic must be true.