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3 AM Philosophy

Atheism, Theism and Agnosticism: Square of Opposition and Semantic Collapse (Short version)

Argument: Atheism, Theism and Agnosticism: Square of Opposition and Semantic Collapse

 

Reviewed by mathematician Josh Garver.

 Note: This is a work in progress. Fair and honest critiquing is appreciated!

Dr. Graham Oppy notes this in “A Companion to Atheism and Philosophy | | Introduction”:

“Some wish to distinguish different kinds of atheism: some distinguish between ‘strong’ – ‘hard’, ‘positive’ – atheism and ‘weak’ – ‘soft’, ‘negative’ – atheism. But, given that atheists can differ in all of the ways discussed in the preceding two paragraphs, and in many other ways as well, it is very hard to believe that any useful purpose could be served by stipulation of a context‐independent distinction between strong atheism and weak atheism. In particular, it seems to me to be a mistake to use a distinction between strong atheism and weak atheism to subsume agnosticism under atheism: strong atheists reject the claim that there are gods, while weak atheists refrain from accepting the claim that there are gods. For, if we accept that there is this distinction between strong atheism and weak atheism, we should surely accept that there is a similar distinction between strong theism and weak theism: strong theists reject the claim that there are no gods, while weak theists merely refrain from accepting the claim that there are no gods. And then we shall have it that agnostics are both weak atheists and weak theists.”

Abstract: Defining subalternations with the same term will result in sematic collapse. Allowing for Bsg ->~Bs~g with both Bsg and ~Bs~g labled as “theism”, Bs~g->~Bsg with both Bs~g and ~Bsg as “atheism”, and agnostic donated as ~Bs~g ^ ~Bsg will effectually result in the possibility of semantic condition of someone being concurrently an atheist, theist and agnostic.

Given:
g is the proposition that at least God/god exists or “God/god(s) exists”
B is the epistemic disposition of “Belief” that affirms a proposition is True.
s is Subject (indexical)

Atheism = Bs~g
Theism = Bsg
Agnosticism= ~Bsg ^ ~Bs~g
“Weak atheism” = ~Bsg
“Weak theism” = ~Bs~g
Bs~g -> ~Bsg is a subalternation
Bsg -> ~Bs~g is a subalternation
Bsg v Bs~g is a contrariety
~Bs~g v ~Bsg is a subcontrariety
Bsg ^ ~Bsg is a contradiction
Bs~g ^ ~Bs~g is a contradiction
Subalternation = by entailment or implication, not based upon truth values. Contrariety = one of two propositions that both can be False, but cannot both be True.
Subcontrariety = one of two propositions that can both be True, but cannot both be False.
Contradiction= one of them is True if and only if the other is False. Can not both be True, nor can both be False.

More specifically, Smessaert H., Demey L. (2014) defines these Aristotelian relations as:

φ and ψ are contradictory iff S ⊨ ~(φ ∧ ψ) and S ⊨ ~(~φ ∧ ~ψ),
φ and ψ are contrary iff S ⊨ ~(φ ∧ ψ) and S ⊨ ~(~φ ∧ ~ψ),
φ and ψ are subcontrary iff S ⊭ ~(φ ∧ ψ) and S ⊨ ~(~φ ∧ ~ψ)
φ and ψ are in subalternation iff S ⊨ φ → ψ and S ⊭ ψ → φ.

By using this schema we can show that any semantic labeling of subalternations as the same term will result in semantic collapse:

Argument:

Given φ and ψ are in subalternation iff S ⊨ φ → ψ and S ⊭ ψ → φ, then any form of  φ → ψ, where S ⊭ ψ → φ, by S holding to ψ ^ ~φ will result in semantic collapse.

Let φ be Bs~g, and ψ be ~Bsg:

φ->ψ
Bs~g->~Bsg
~φ =~Bs~g


Then:
If ~Bsg and ~Bs~g, then ~Bsg ^ ~Bs~g. (conjunction introduction)

Semantic instantiation: Weak atheism and weak theism, then agnosticism. If then we allow “weak atheism” to be atheism and “weak theism” to be theism then: atheism, theism and agnosticism.

Example:

Theism = Bsg

Bsg->~Bs~g or if you believe God exists, you do not believe God does not exist. You can not be ~Bsg as that would be a contradiction.
You can not be Bs~g as contrariety only one can be True.
You are either ~Bs~g or ~Bsg as subcontrariety as both can not be False.
Since you can’t be ~Bsg as that is a contradiction, then you must be ~Bs~g which is the subalternation Bsg->~Bs~g.

We can label these as follows on the square of opposition (Agnostic being the conjunction of the subcontrarities ~Bs~g and ~Bsg):

If atheists label “weak atheism” (~Bsg) as atheism, instead of the normative Bs~g, theist can rename the subcontrariety of “weak theism” (~Bs~g) as theism, and by failing to allow them to do so you’re guilty of special pleading. (See WASP argument: https://greatdebatecommunity.com/2020/02/27/if-bp-is-held-as-atheism-then-bp-can-be-held-as-theism-else-you-are-guilty-of-special-pleading/)

Conclusion: By defining atheism in the weak case we are forced to accept that it results in a semantic collapse where if person is ~Bsg, without being B~g, then they are ~Bsg, ~Bs~g, and ~Bsg ^ ~Bs~g; or atheist, theist and agnostic at the same time.

 


References:

Demey, Lorenz (2018). A Hexagon of Opposition for the Theism/Atheism Debate. Philosophia, (), –. doi:10.1007/s11406-018-9978-5

Smessaert H., Demey L. (2014) Logical and Geometrical Complementarities between Aristotelian Diagrams. In: Dwyer T., Purchase H., Delaney A. (eds) Diagrammatic Representation and Inference. Diagrams 2014. Lecture Notes in Computer Science, vol 8578. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44043-8_26

Burgess-Jackson, K. (2017). Rethinking the presumption of atheism. International Journal for Philosophy of Religion, 84(1), 93–111.doi:10.1007/s11153-017-9637-y

Smessaert H., Demey L. (2014) Logical and Geometrical Complementarities between Aristotelian Diagrams. In: Dwyer T., Purchase H., Delaney A. (eds) Diagrammatic Representation and Inference. Diagrams 2014. Lecture Notes in Computer Science, vol 8578. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44043-8_26

Oppy, Graham (2019). A Companion to Atheism and Philosophy || Introduction. , 10.1002/9781119119302(), 1–11. doi:10.1002/9781119119302.ch0

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