958120744By examining various philosophies involving knowledge, one can develop a “hierarchy ofknowability”. Notably, at the very top of this hierarchy would lie a priori knowledge andbelow it would lie the gamut of a posteriori knowledge. Two major philosophies that regardthemselves with this are Descartes, and Kant. Descartes defines “I think therefore I am” asthe basis for all a priori knowle dge, while Kant defines two classes of knowledge that relateto their knowability. []This paper is concerned not with the hierarchy itself, but only with where logic, in partic-ular, fits in this hierarchy. At first glance, Descartes would appe ar to label logic a posterioriknowledge, as would Kant, though Ayer’s objection to Kant labels logic a priori knowledge. []It is [also] important to separate out two aspects of logic: the language, and the underly-ing meaning. It is easy to argue against the language of logic as a priori knowledge, because itdoes not arise necessarily. Indeed, the language of logic is arbitrary and could be replaced byany other isomorphic language. It just so happens that the language of logic that we use wasdeveloped to conveniently represent what was inductively discovered about the world. Thisprecisely parallels the development and nature of arithmetic, as I will dis cuss later. However,the underlying concept expressed by the language of logic is where logic truly lies, and thisis what must be analyzed with regards to logic’s nature as a priori or a posteriori knowledge.Descartes stated in his Meditations that “I think therefore I am” is the only truly know-1able fact. Thus, given a hierarchy of “knowability”, Cartesians would place existence andnothing else at the very top of this hierarchy. This would be the only thing that is truly know-able a priori, while everything else, including logic, would merely be a posteriori knowledge,and further down the hierarchy.Thus, logic must fit in with a posteriori knowledge according to Desc artes, and must besomething garnered from experience and developed through induction. It is not necessarythat the nature of logic be as it is. However, if a posteriori knowledge can only be gained bymeans of induction, then the rules of logic themselves would have to be arrived at inductively.Logic states that induction of this form is not valid. Thus, logic itself would say that it isfalse, and clearly this is a contradiction. However, this contradiction is rather meaningless,because the very concept of contradiction, as well as the logic used to conclude it, are definedby the rules of logic.Taking another look at Descartes’ statement, “I think therefore I am”, reveals that thestatement itself is in a logical form. This could be argued by saying the logical statement of itis not necessary for the concept itself, but is only a convenient means by which to present theunderlying idea. However, as I pointed out before, it is necessary to separate the languageof logic from the concept of logic. Because the accepted language of logic is used to presentthe idea, then there exists a logical representation of the underlying idea, and, even if thelanguage were changed, this underlying idea would still be the same, and if the languagewere changed enough to make it no longer a statement of logic, then the underlying conceptmay no longer be within the realm of logic, but the idea would be fundamentally different.2Thus, by the fact that the underlying idea lies within the realm of logical representation, itbecomes effectively inseparable from logic. If it were restated such that it were no longerwithin the realm of logic, it would not be the same idea. Therefore, logic necessarily e xists.Now, suppose that Descartes’ statement were only to imply the existence of itself withinthe logical realm (that is, it doesn’t imply the existence of all of logic, but only the subset oflogic that is the statement itself). This would mean that logic is not fundamental, and thatthe remainder of it could be developed in a synthetic manner. However, logic is based onthe fact that all logical statements are equivalent (everything can be resolved into Aristotle’sLaw of Identity, A = A). Thus, every logical statement is equivalent to Descartes’ statement,and thus equally true, and equally knowable. Therefore, logic not only exists, but is complete.Kant, on the other hand, took a different approach, and formalized a concept of analyticaland synthetic judgments, in which analytical judgments were those that could be known apriori, and synthetic judgments were those that could be known a posteriori. Thus, Kantianthought would place analytical judgments at the very top of the hierarchy of “knowability”,and synthetic judgments would descend from there.Interestingly, Kant defines analytic judgments as “adding nothing through the predicateto the concept of the subject, but merely breaking it up into those constituent concepts thathave all along been thought in it”. Or, as Ayer puts it, “analytic judgments are devoid offactual content”. This has the interesting effect of making the only things that are at thevery top of our hierarchy of knowability contain no factual information. []3[iamhere]We looked at the world and saw that it was conservative. Thus, we defined addition,the fundamental operator of conservation (and the operator from which all other arithmeticoperators can be defined).What is only knowable a posteriori is that the laws of conservation hold. If the laws ofphysics were different, 1 + 1 would still be 2, but it may be of less interest to those seekingan a posteriori understanding of their universe.[Ayer away!][symbolic thing and Peano thing and arithmetic thing]Another interesting question that arises from this is the following: Is the statement “Ithink therefore I am”, simply a consequence of the nature of implication (and thus a Kan-tian analytical statement)? Or is it just another tautology; a way to “say in a round-aboutfashion ’A = A’ ”, as Poincar´e put it? [perhaps move with equivalence argument]There is a fundamental flaw in any reasoning regarding the universality of logic, and anysuch reasoning will suffer from a fallacy similar to that of absolute relativism. Namely, ifone believes that logic is not fundamental, but is, instead, a synthetic development, thenthey have just undermined any mechanism by which they could argue their point. On theother hand, if one believes that logic is