Agnosticism, Atheism, Theism and LOGIC 101:

Agnosticism, atheism, theism and LOGIC 101:
p V ~p
p is a proposition
“V” means OR
~p is the negation of the proposition
Since p V ~ p is always true, this is called a tautology…It also means it is the case p is true or it is not the case p is true. Which is the same as saying p is either true or false. (This is know as the Law of Excluded Middle…that a proposition can only be T or F.)
Given the proposition of theism:
p=”at least one God exists” (includes existed)
p V ~p
It is either the case that p is T or p is F. Or in other words it is the case that “at least one God exists” or it is NOT the case that “at least one God exists”. This is an ontological dichotomy (God(s) exists or God(s) does not exist).
The theist affirms p to be true. That they believe it is the case that “at least one God exists”. The atheist (philosophical atheism) affirms the negation that p is false that it is not the case that “at least one God exists”.
This is logically shown as:
Bp = Believes p is true (Theist)
B~p= Believes p is false (Atheist)
Bp V B~p however is not an ontological (or even epistemic) dichotomy. One does NOT have to believe either that p is true or p is false.
The agnostic does not hold to either theism nor atheism and does not believe p to be true, nor believes p is false. They logically hold to ~Bp (I do not believe p) and ~B~p (I do not believe not p).
The agnostic suspends judgment on the proposition…but because they do not affirm theism, they do not believe in Gods, however they are not asserting that they do not exist. This is often the stance of “weak” atheist (~Bp) who while not believing God(s) exists, do not affirm the negation of theism or that God(s) does not exist (doesn’t explicitly state they also hold to ~B~p, however a logical proof can show that all who hold to ~Bp that do not hold to B~p must then hold to ~Bp & ~B~p (proof at bottom)). 
Agnostics are not theist, nor are they atheist, and are ostensibly logically the same thing as a “weak atheist” or “weak theist”. (see proof below)

(This is inevitably when someone says “But if you don’t believe in God you’re an atheist! “. Well, sorry…that is just NOT true. That is how many atheist think of atheism as they are told that by other atheists…but logically, as I have shown, you do not have to be neither theist nor atheist. Not believing in God is a necessary, but not sufficient condition and does not by entailment nor by necessity mean that if you do not believe in Gods you are an atheist. Logic > misinformation you have been told. What they are merely doing is taking A V ~A and saying “Theist or “NOT theist” and relabeling “not theist” as “atheist”. This is like making a disjunction (p V q) and saying if not p then q. I can do that with anything. p = theist, q=car. p V q, theist or car, If you are not a theist you are a car. See?)

What atheism and theism actually are called contradictories. Meaning that if there is a God and theism is true, then atheism must be false…and vice versa, such that if there isn’t a God then theism is false and atheism is true. Neither can both true, and neither can be both false. One must be the the case, and the other must not be the case. This is slightly different than contraries such as atheistic religions or theistic religions where if one is true the other must be false, both can not be true, but both can be false.



Proof: If p=”at least one God exists”

p1) A lack of belief for p logically is ~Bp
P2) A lack of belief for ~p logically is ~B~p
p3) A lack of belief atheist holds to ~Bp
p4) Holding to ~Bp without holding to B~p must entail holding to ~B~p.
p5) A lack of belief atheist who holds to ~Bp (p3) but does not hold to B~p must then hold to ~Bp & ~B~p (p3-p4). (constructive elimination)
p6) Agnostic holds to ~Bp & ~B~p
C) Agnostic logically is the same as a lack of belief atheist as both hold to ~Bp & ~B~p.

(But they are epistemically different, much like “Christian” and “Jew” are logically the same (both are Bp), but epistemically different.)


Author: Steve McRae

10 thoughts on “Agnosticism, Atheism, Theism and LOGIC 101:

  1. You said that agnostics do not believe that any gods exist nor believe that no gods exist. As agnostics in particular do not bekieve in any gods agnostics are atheists.

  2. Atheists are right to cite the null hypothesis. Because theists disagree as to which god is the true god and they offer no evidence for their god other than their own belief there is no reason to believe that any of theIr gods are real. To say I don’t believe in any gods does not carry any burden of proof. I could simply be waiting for at least one god to step forward with evidence of its existence.

  3. The discussion here clarifies my own issue between atheism and agnosticism. I really like the introduction of the operator B, which reminds me greatly of the existential (∃) and universal (∀) quantifiers. I’m curious if the operator B is used regularly, because I think it would make the atheist/agnostic discussion much clearer.

    I did have some trouble parsing the proof at the end, but I can express it point in a different way. Feel free to correct me if I am misunderstanding the logic or definitions being used.

    I think the issue can be stated succintly in a question of commutativity. Does the operator B commute with the negation ¬? That is, are B¬p and ¬Bp the same? To explain, consider the following calculation.

    ≡ ¬Bp ∧ T
    ≡ ¬Bp ∧ (B¬p ∨ ¬B¬p)
    ≡ (¬Bp ∧ B¬p) ∨ (¬Bp ∧ ¬B¬p).

    If the predicate B¬p is true, then the subject believes ¬p, meaning the subject believes there is no god. This would be an atheist by definition. Thus, (¬Bp ∧ B¬p) defines someone who does not believe that there is a god, and believes that there are no gods. This would be the “lack of belief atheist”, as I understand.

    On the other hand, if B¬p is false, then the subject does not believe ¬p, meaning the subject does not believe there is no god. Thus, (¬Bp ∧ ¬B¬p) defines someone who does not believe that there is a god, and does not believe that there are no gods. This would be the “agnostic”, as I understand. Equivalently, applying De Morgan,
    (¬Bp ∧ ¬B¬p) ≡ ¬(Bp ∨ B¬p),
    meaning an agnostic is someone who rejects either Bp or B¬p.

    Here is where a serious temptation arises. Someone will want to apply the Litotes (double-negative) law: ¬¬p ≡ p. However, please note that in “¬B¬p”, the operator B separates the two ¬’s.

    If B and ¬ commute, then one has
    (¬Bp ∧ ¬B¬p) ≡ (¬Bp ∧ B¬¬p) ≡ (¬Bp ∧ Bp) ≡ F
    by Litotes. In this case, holding “does not believe there is a god” and “does not believe there are no gods” would be impossible, i.e. “agnosticism is impossible”. As this conclusion is false, I would conclude that B and ¬ do not commute.

    Am I mistaken? Please correct me if I have made an error here.

  4. “B” is just a predication for the epistemic or doxastic disposition of “Believes”
    B¬p and ¬Bp are not commutative.
    B¬p entails ¬Bp , but ¬Bp does not entail B¬p ( B¬p ⊨ ¬Bp |¬Bp ⊭ B¬p)

    If B¬p is T then yes that means S (Subject) believes that p is T. Dr. Burgess-Jackson uses the form Bs~g and Bsg where subject believes ~g or g. Same logical notation, just adding in a specific agent and “g” stands for the proposition of god exists.

    Agnostic would be as you said ¬Bp ∧ ¬B¬p which holds no belief either way.

    Your conclusion is correct for the reasons you gave. They do not commute.

    1. Ah! Thank you for the detail. I figured there was an implication one way or the other between ¬Bp and B¬p, but I knew it couldn’t be bidirectional. Dr. Burgess-Jackson’s use of the operator sounds much more in-line with the logic that I use regularly. I’ll have to check it out.

      The algebraist side of me is quite amused that so much confusion and argument can be essentially described by a non-commuting square.

      More precisely, the one-way implication would be interpreted as a singular 2-cell in a 2-category. Thus, the confusion could be interpreted as mistaking the singular 2-cell as an invertible 2-cell.

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