[The following is a response to the contention that atheism violates a basic tenet of logic made by @ToAtheists and @memotoworld on Twitter. The former first shrouded himself in some mysterious claims and refused to engage in a debate with me, so it took me a while to figure out that both the latter and he meant the following: Atheism, they purport, is the claim that no (kind of) god exists. It is wide-spread knowledge that the non-existence of an entity cannot be proved, but only the existence of one. Not so common are the (possible) reasons for this, which are part of my refutation.]
Now, there are several aspects to this contention far more complicated, however, than it appears to presume, being at least:
(1) the logical aspect as opposed to the empirical aspect;
(2) the linguistic aspect in relation to the psychological aspect;
(3) the logical aspect in relation to the linguistic aspect; and
(4) the logical aspect in relation to the psychological aspect.
First, that one cannot prove a negative proposition such as ‘There is no X’ or ‘X does not exist’, is rather an empirical problem than a logical one. From a pure logical point of view, which is a pure theoretical point, if you could search any given point of spacetime for X in every possible aspect, and X were nowhere to be found, you would, by definition, have proven beyond a shadow of a doubt that X does not exist, after all. This ideal situation entails that X has no means to occupy a place concealed from your attention, that is to say, X cannot, by any means, hide from you. As this theoretical framework is not given, however, it is impossible to prove a negative proposition of said kind.
Second, there is a linguistic difficulty resulting from the fact that you can combine both a positive propositional attitude with a negative proposition [pPA nP], and a negative propositional attitude with a positive proposition [nPA pP] so as to, at least on the face of it, express the same psychological fact. As to atheism, there are these possibilities:
(nPA pP) I do not believe [that any kind of god exists].
(pPA nP) I believe [that no kind of god exists].
This begs the question, then, whether (nPA pP) and (pPA np) are simply linguistic rephrasings of the same psychological facts, or, rather, render two different psychological facts. In the latter case, a follow-up question would be whether, under any possible circumstances, (nPA pP) and (pPA np) psychologically entail each other.
Also, this connects to the question whether there be a distinction to be made between conscious and subconscious beliefs, that is to say, those beliefs you are aware of and those you are not aware of. For instance, you certainly do not believe that ordinary chocolate you can buy at the supermarket is poisoned. But now that I have mentioned this, I have brought it to your awareness. Does this mean that you, in virtue of my mentioning, came from not believing that ordinary chocolate you can buy at the supermarket is poisoned to believing that ordinary chocolate you can buy at the supermarket is not poisoned? To be sure, you are now in a different mental state, yet did your former mental state entail your present mental state? This is not evident at all.
Third, how do language and logic connect to and influence each other? Nautral languages, as opposed to technical ones, were not constructed to fit (methodo)logical needs. Instead, they evolved from the need of (more) detailed communication. Natural languages often obscure, even distort, logical connections. Therefore, arguments can be presented in several forms, both valid and invalid, or correct and incorrect, without implying that one invalid or incorrect presentation renders the entire argument invalid or incorrect in each of its possible forms of presentation.
Fourth, how, if at all, do psychological facts and mental states bear on logical aspects? Evidently, mental states do bear on our capability to think logically. If we are swept by jealousy, numbness, hatred, love, or countless other affects or emotions, we are prone to cast logic aside and act impulsively. Beliefs connected to strong emotions tend to betray logic by foregone conclusions.
On the other hand, logical reasoning on its own does not guarantee factually true conclusions, but only theoretically true (deduction) or correct (induction, abduction) ones. Thus, the premises
P1 All cats can fly
P2 All dogs are cats
lead to the conclusion
C All dogs can fly.
Logically, this is perfectly valid, but, at the same time, empirically false in our actual world.
In the face of these considerations, it is not evident at all that atheism, as originally contended, violates a basic tenant of logic, especially not the one specified, and is therefore an untenable position; quod erat demonstrandum.
[In case you still have not figured it out, my use of ‘tenant’ in place of ‘tenet’ is a philosophical joke referring to Plato’s theory of forms, but also, and more basically, hinting at the original contention’s unsoundness.]