The principle of empirical verifiability by David Hume is flawed.

Let me begin by apologizing to the readers. I often get carried away and forget that I am debating for other people's benefit, and they might not have any expertise in the areas being debated. With that being said, I will try to fully explain any technical philosophical or complex mathematical terms that I use, and provide links for additional information. First, it is essential to understand the difference between an analytic and synthetic claim. "Analytic" sentences, such as "Ophthalmologists are doctors," are those whose truth seems to be knowable by knowing the meanings of the constituent words alone, unlike the more usual "synthetic" ones, such as "Ophthalmologists are ill-humored," whose truth is knowable by both knowing the meaning of the words and something about the world (. http://plato.stanford.edu...). Therefore, my opponent is correct to assert that "all triangles have 3 sides" is an analytic claim because "3 sided figure" is contained within the definition of triangle. However, my opponent stops too soon, which will be evident once one understands what a synthetic a priori claim entails. Synthetic a priori reasoning is presented by Immanuel Kant in response to Hume in "A Critique of Pure Reason. " Kant claims that there are certain types of knowledge that one knows a priori, or before experience as my opponent defined it. This type of knowledge is related to the lenses that we see the world through. This is known as the Copernican Revolution in philosophy. Kant claims that spatial, temporal, and causal knowledge are built into our innate lenses. For example, I might not know that there is a computer on my desk at any given time, but I do know that if a computer does exist in my room, it will have a spatial relationship with my desk. Therefore, Kant claims that there is certain a priori knowledge that a human knows from birth. However, this does not cover all kinds of synthetic a priori knowledge that one can obtain, and does not address my objections to Hume's theory. . http://plato.stanford.edu... The second kind of knowledge that one can obtain that is synthetic and a priori was determined by Kant in response to two of Hume's assertions, both of which my opponent pointed out. The first is the problem of inductive reasoning, and the second is the assertion in question, i. e. the principle of empirical verifiability. My opponent made a common error by asserting that mathematical concepts are learned, and, thus, must be known a posterori. Rather, math is built upon axiomatic (an assertion that we cannot prove, but that seems obvious, i. e. a=a, a+b=b+a, etc. ) foundations. From these axiomatic foundations, one can deductively prove a number of amazing conclusions that are clearly synthetic (. http://thesamovar.net...). However, the question on point is whether or not one can also consider them a priori, and this is where Hume's own assertions come back to haunt him. Hume claims that all knowledge is either analytic if it is a priori knowledge, and a posteriori (after experience) if it is synthetic. Hume also claims that only analytic knowledge can be known to be true through deductive means (see opponent's explanation about the problem of induction). However, certain claims in math seem to be both synthetic and are known to be true before one experiences them every time, since one uses deductive methods to determine their truth. This is known as synthetic a priori knowledge. For example, a triangle contains 180 degrees, but the idea that a triangle contains 180 degrees is not contained within the definition of a triangle. One could come by this knowledge, admittedly, by measuring the angles of many triangles, however, this is not the point. The point is that once one has determined that 3 sided figures contain 180 degrees, one knows that all 3 sided shapes in the future will also contain 180 degrees. Hence, one has knowledge about an entity before one experiences it. Since one can obtain knowledge about all 3 sided figures in the future before one experiences them, then one has obtained knowledge through non-empirical means. Thus, Hume's theory is flawed. I chose the example of Gabriel's Horn in the above section to point out a non-contentious synthetic a priori claim. One obtains the knowledge of Gabriel's Horn through deductive processes and reason, and Gabriel's Horn clearly cannot exist in the physical universe as we know, nor can we put together any pieces of our experiences to produce Gabriel's horn, and once we determine the result of the deduction, we know that the processes will produce the same results into the future (. http://en.wikipedia.org...). Thus, Hume's theory is flawed based on all synthetic a priori knowledge that can be gained through mathematics. I will now move on to the second prong of my argumentation, which ought to have been secondary since, as my opponent points out, math is the most obvious objection to Hume's theory. The second prong of my argument dealt with the appearance of the logical positivists in the early 20th century. The logical positivists are essentially empiricists like Hume, and wished to show that no claims are synthetic and a priori. A. J. Ayer, who I mentioned earlier, was among this group. Ayer proposed that "a sentence is factually significant to any given person, if, and only if, he knows how to verify the proposition which it purports to express—that is, if he knows what observations would lead him, under certain conditions, to accept the proposition as being true, or reject it as being false" ("Language, Truth and Logic" p 35). Ayer goes on to emphasize that a statement need only be verifiable "in principle" to be meaningful. Clearly there are parallels between Ayer and Hume, but Ayer's theory is more complete, as I attempted to point out in previous rounds. First, claims such as Einstein's theory of relativity are meaningless under Hume's theory until they are empirically verified. However, since Einstein's theory clearly proposed a way to verify it's truthfulness, and because it seemed intelligent and logical, one ought to include it among sensible sentences. The same can be said about nearly all existential claims on a more basic level (see rabbit example from round 1). Since these scientific theories and existential claims are clearly synthetic and not analytic as my opponent claims, and since Ayer's theory is better able to embrace these seemingly meaningful statements, Ayer's theory is clearly superior, and, hence, Hume's theory is flawed. The final point of my argumentation deals with the concepts such as 'truth,' 'justice,' 'love,' 'God' etc, however, I would like to retract this argument, even if it is to my detriment. My opponent is correct to point out that these claims do have analytic a priori truths, i. e. definitional truths, but Hume has no problem embracing these contingencies.