You have provided no counter argument to my thesis statement (T), "Agnosticism is the most rationally acceptable default position," nor to the conclusion drawn from my main argument (C), "The subset of humans who value intellectual honesty must by default begin from an agnostic position," but rather you conclude (-A) "[the proposition] 'The Abrahamic Deity exists' cannot be true," and (-O) that, "Agnosticism, with respect to 'the existence of the Abrahamic Deity,' does not obtain."
Those are not are not refutations to either my thesis statement or the concluding proposition in my main argument. Those are obviously the negations of their own respective propositions: (A) "[the proposition] 'The Abrahamic Deity exists' is true," and (O) "Agnosticism, with respect to 'the existence of the Abrahamic Deity,' obtains." These are, again, obviously not the propositions that I argue for in the OP.
You need to conclude with the negating propositions presented by the OP, (-T), "Agnosticism is NOT the most rationally acceptable default position," or (-C), "[It is NOT the case that] The subset of humans who value intellectual honesty must by default begin from an agnostic position," in order to avoid committing a fallacy of relevance as I tried to point out before. An example of what a relevance fallacy looks like and how one may occur can be elucidated within the pattern of the following argument:
Interlocutor 1 asserts proposition (P), "Being agnostic with regards to the truth-value of a proposition, or that propositions negation on the grounds of insufficient information is warranted."
Interlocutor 2 argues against a superficially similar proposition (B) by asserting that propositions negation (-B), "Being agnostic with regards to the truth-value of the proposition (P), 'The Abrahamic God of monotheism exists,' or that propositions negation (-P), 'The Abrahamic God of monotheism DOES NOT exists,' on the grounds of insufficient information, whenever you feel as if there is sufficient information to warrant the propositions negation (-P), is NOT warranted,"
falsely, as if an argument against (B) were an argument against (P).
Lets isolate all the concluding propositions and negations presented by us both so far and include their respective counterparts (opposing propositions and negations):
(T) Agnosticism is the most rationally acceptable default position.
(-T) Agnosticism is NOT the most rationally acceptable default position.
(C) [it is the case that] The subset of humans who value intellectual honesty must by default begin from an agnostic position.
(-C) [it is NOT the case that] The subset of humans who value intellectual honesty must by default begin from an agnostic position.
(A) [the proposition] "The Abrahamic Deity exists" is true.
(-A) [the proposition] "The Abrahamic Deity exists" is NOT true.
(O) Agnosticism, with respect to "the existence of the Abrahamic Deity," obtains.
(-O) Agnosticism, with respect to "the existence of the Abrahamic Deity," DOES NOT obtain.
Now, the wording doesn't necessarily have to be identical to these example representations (e.g., (-A) could be written as, "[the proposition] 'The Abrahamic Deity exists' is FALSE," rather than, ". . . NOT true.") but they do however have to retain and not omit relevant context. In other words, we cannot exclude, exchange, or include extra context (e.g., "Agnosticism, with respect to 'the existence of the Abrahamic Deity'. . .") to the proposition in question if we want to provide a refutation of it. By changing the relevant context of a proposition, it is no longer the proposition in which our aim (hopefully) is to represent and subsequently refute. It becomes a type of straw man, essentially - a member of the family of fallacies known as 'relevance fallacies' - and I hope that I have clarified this issue for us, in that regard.
Next, I want to represent your entire (attempted counter) argument and offer my critique as well as raise your attention to any issues of relevant context that I find.
(ax. 1) If theism is cognitive, then its claims (e.g. "the Abrahamic Deity exists") are demonstrably true or not true — (excerpts) QED & Other Stigmata
I briefly mentioned a few things that immediately jumped out at me as being problematic with this conditional statement. First, it is not 'axiomatic' as implied by (ax. 1), because an axiom is "an unprovable statement generally held as a first principle which is granted as true because it is self-evident. The statement, "A whole is greater than it's parts," for example, is an axiom. The argument to substantiate this claim also serves to expose an error in reasoning, which is a good segue into my second issue.
Second, it is not self-evidently true that if a thing is "cognitive" (capable of being reduced to empirical factual knowledge) that the thing must be "demonstrably true or not true" (a demonstrable fact or quality which can be shown to be true or not true). For example, the number of birds in the sky at precisely this moment is a cognitive quantification (i.e., it is simple arithmetic), however it is not demonstrable. There surely is some number of birds in the sky at a given moment in time, but that number cannot be demonstrably shown to be true, and it would be irrational to say that because the total number of birds in the sky is something that can be known, yet isn't, and thus cannot be demonstrated or proved, that therefore there is no such number of birds in the sky at a given moment.
Third, notice all of the context added and omitted. I am arguing for a broad and universal applicability for agnosticism which includes all contexts and varies between subject to subject, agent to agent, and the information available to both. For example, it would be rational to be agnostic regarding the existence of the sun if you spent the entirety of your life underground without any information about it. When you include specific context by using terms such as 'theism,' 'Abrahamic Deity,' and the concept of 'existence' you take away from the broad universal scope and narrow it down to one that is specific and particular.
(ax. 2) If theism is noncognitive, then its claims (e.g. "the Abrahamic Deity exists") are mere poetry (i.e. figures of speech). [from ax. 1] — (excerpts) QED & Other Stigmata
First, besides the aforementioned added context, just because something is "noncognitive" (incapable of being reduced to empirical factual knowledge) doesn't mean that it is "mere poetry" (i.e., figures of speech). I understand that poetry is an expression of emotions and feelings that doesn't render empirical factual knowledge, but neither does motivational speeches. Motivational speeches are not poetry. There are also forms of cognitive poetry. Second, if you are not using these terms literally, then you are using them rhetorically (which is hilarious given the context of this critique).
(prop. 1) Any claim that "the Abrahamic Deity exists" entails that nature is "supernatural", or that its otherwise law-like regularities are arbitrarily, purposefully, changed (suspended). [from ax. 1] — (excerpts) QED & Other Stigmata
Not necessarily, it is to claim that there is a metaphysical realm which has extra-spatial and extra-temporal dimensions. Both this proposition and its negation are well beyond our capacity to suppose or even ask well formed questions of so to begin to generalize. This is essentially an argument from ignorance. Just because there is an absence of evidence does not mean there is evidence of absence. Aphorism aside, that is a legitimate refutation.
I think I'll stop here before this starts to resemble a textbook rather than a critique.