Logic is a natural and useful intellectual/scientific tool for testing the consistency of our understanding of concepts and ideas, claims and assertions, hypotheses and theories (both scientific and colloquial), casual day-to-day decision-making, etc., and is an essential part of mathematics, science, engineering, philosophy, law, cooking, art (music, writing, painting, etc.), plus many other professional disciplines.

Symbolic keymap

¬ = is not (negation)
B = belief/believing (psychological)
D = deity/deities (a deity is a goddess or a god)
K = knowledge/knowing (epistemological)
Note:  Parenthesis clarify logical grouping.

Hereunder, the symbolic keymap (above) clarifies the meanings of select vocabulary that is used in public discourse, linguistic research, philosophical debate, epistemological exploration, etc.

Occasionally, the subject of epistemology arises, but this is a non-sequitur that adds unnecessary confusion since atheism, anti-theism, and theism address a psychological characteristic without venturing into epistemological aspects.  The following entries included here resolve this confusion:

In summary, while atheism (a.k.a., "not theism") is a psychological state of "not believing in deities," it differs from agnosticism (a.k.a., "not gnosticism") which is the epistemological subject of "not having knowledge about deities."

Propositional logic

In propositional logic, the casual question "Do you believe in any deities?" (which may be expressed as "I believe in one or more deities") seeks to yield a response of acceptence or non-acceptence since the objective of propositions is to confirm only the validity of a given proposition.  (When using truth tables to document responses to propositions, "true" commonly indicates acceptance while "false" indicates non-acceptance without implying some other position, including an opposite position {even if a dichotomy is central to the proposition}.)

Although a theist will "accept" the proposition, we can't discern from this response whether the theist is also a monotheist or a polytheist since confirming the proposition doesn't address the number deities, so at best we can only deduce that they are a theist (of which monotheists and polytheists are the two possible subsets).

Although an atheist will "not accept" the proposition, we can't discern from this response whether the atheist is also an anti-theist because their answer doesn't address anti-theism, so at best we can only deduce that they are an atheist (of which anti-theists are a possible subset).

Although an anti-theist will "accept" the proposition, it comes as no surprise that both theists (because they believe in one or more deities), and atheists who don't also take the anti-theistic position, will "not accept" the proposition.

Although a gnostic will "accept" the proposition, in contrast an agnostic will "not accept" it.

Excluded Middle, the principle of

The Principle of the Excluded Middle is a well-known logical law that asserts that a logical statement can only be "true" or "false."  While this law is useful, the law itself is not exempt from causing false dichotomies when applied where more than two options are logically valid possibilities (as is the case with traffic lights), and so in the spirit of striving for objectivity it is important that practitioners of logic be aware of the risk of bifurcated thought that can arise from unnecessarily or illogically excluding additional options and/or positions (as can occur in classical logic where a "false" response to a disjunctive syllogism is incorrectly interpreted as an affirmative denial that infers an opposing position).

As a straight-forward example, a false dichotomy arising from the incorrect application of the "excluded middle" can be demonstrated with a basic system of signal lights on a motor-vehicle.  While these signal lights are best known for communicating the driver's intention to perform a left-hand or right-hand turn or lane change, limiting the options to "left" and "right" signalling is a false dichotomy because it unnecessarily excludes the default of "not signalling."  Additionally, a well-known fourth option of "hazard lights," which utilizes both sets of signal lights simultaneously, is used to indicate a temporary full stop due to a mechanical breakdown, etc.

Along this line of thought is that "not believing that deities exists" (atheism) is distinctly different from "believeing that deities do not exist" (anti-theism) and its opposite of "believing that one or more deities do exist" (theism).  While the former does not carry a burden of proof since it is not an assertion, a belief, or a claim; the remaining two are belief-positions that do carry burdens of proof insofar as they qualify as claims or assertions, and/or are extended as expectations that others should adopt the same belief-positions.

In summary, applying the Principle of the Excluded Middle to choices that aren't naturally limited to "true" and "false" is illogical, even though some debaters have been known to utilize this principle in an attempt to unnecessarily impose a burden of proof onto adversaries where either a valid alternative or no particular position was adopted.  Therefore, the attitude that an individual only has the option of "believing in deities" (theism) or "believing that deities don't exist" (anti-theism) is illogical and qualifies as a bifurcation fallacy because it excludes the valid null hypothesis (default) of "not believing in any deities" (atheism).

Further reading

• atheistic values - logic
• Quotes of interesting perspectives from interesting people