Two logical arguments against the use of “Agnostic Atheist” multi-axial diagrams

I wanted to give some quick notes on the use of these types of epistemological image which people can use when these types of images come up, so they can quickly point out a few glaring issues with such a multi-axial approach of having a x-axis of knowledge, and a y-axis of belief. Never mind of course the fact that I can’t seem to find a single solitary publication that contains these diagrams, or any accredited college that has it in its course curriculum. These diagrams seemingly were just created by someone who really didn’t think critically about the logic thoroughly or very well at all, when they created it. People who continue to use them seem to fail to think logically as well and hopefully, once you see the obvious issues with it, you’ll stop using it and recognize them as being epistemologically untenable.


K = Knowledge
B = Belief
p= proposition
a = agent
~ = Not (or negation)
^ = And (conjunction)
-> = implies (or by implication)

Quadrant I – Gnostic Atheist
Quadrant II – Agnostic Atheist
Quadrant III – Agnostic Theist
Quadrant IV – Gnostic Theist


1. Quadrant I vs Quadrant IV

Both use “gnostic” as “claim to know”, the Gnostic Atheist claims to know that no god exists, while the Gnostic Theist claims to believe a God exists, and also claims to know that god exists. This is represented as Bap and Ka~p respectfully.

However, Quadrant IV has theism as “Believes God exists” and “Gnostic Atheist” in quadrant I has “Does not believe any God exists”. Are you already starting to see the problem? Knowledge is a subset of belief, and as such the logical relationship would be Kap -> Bap -> ~Ba~p (agent that knows p, also believes p, and does not believe ~p). Or for the negation of p, you have Ka~p -> Ba~p -> ~Bap (agent that knows ~p, also believes ~p, and does not believe p)

So in Quad IV you have Bap ^ Kap (agent Believes p, and agent Knows p), while in quad I you have ~Ba~p ^ Kap (agent does not believe p, and agent knows p). This can only make logical sense if the agent ALSO believes ~p. Since Ka~p -> Ba~p! That is the only way Quad I could make epistemic sense, is if “atheist” has a positive epistemic status, since “to know p” is clearly a positive epistemic status. You can’t know p, unless you believe p! So you can’t know ~p , if you don’t believe ~p. If you disagree, please explain how an agent can have a positive epistemic status of knowledge, but fail to have a positive epistemic status of belief.


Additionally Quad I should be: Ka~p ^ Ba~p or “Believes God does not exist” + “Claims to know god does not exist”. However, This is redundant, since Ka~p -> Ba~p, this just reduces to Ka~p. (Think about it in the same way that Ba~p -> ~Bap or Bap -> ~Ba~p. If the agent believes p is false, then the agent does not believe p is true, and can just say “believes false”, as the rest (does not believe p) is superfluous as understood by implication.

Conclusion: Quad I should just be Kap, “agent knows ~p” or “Claims to know God does not exist”, and that “atheism” only makes sense if understood as a positive epistemic status.

2. Quadrant II vs Quadrant III

Quadrant III of “Agnostic Theist” is given as “Believes a god exists” + “Doesn’t claim to know this belief is true”. One problem here obviously, is that this is superfluous as “theism” is generally understood as “believes a god exists, but doesn’t claim to know this belief is true”, as theism is a doxastic position, with a positive epistemic status that the proposition is true. So this term “agnostic theist” is superfluous. Logically it would be: Bap ^ ~Kap, which just reduces to Bap, and that is normally considered to be just known as theism. It is superfluous to add “does not know p” here.

Quadrant II of “Agnostic Atheist” is given as “Does not believe any god exists” + “Doesn’t claim to know that no god exists”. Logically this is: ~Bap ^ ~Ka~p. This suffers from the same problem as quadrant III. Atheism is generally understood as a doxastic position, not a knowledge one. Regardless if atheism is held as Ba~p or ~Bap, neither imply any claim about knowledge. So it is either ambiguous, or superfluous, to have Ba~p ^ ~Ka~p, as that just reduces to Ba~p. If you hold atheism as ~Bap, then you have ~Bap ^ ~Ka~p, which if you don’t believe ~p, then you again clearly have no knowledge claim, and it reduces to just ~Bap. In the later case however, it oddly becomes the only quadrant in which you can have atheism as ~Bap. Quad I, III and IV only make sense if there is some sense of positive epistemic status. Quad II and quad III have theism as a positive epistemic status. Quad I has a positive epistemic status as it is a knowledge claim that as it has been explained implies a belief with a positive epistemic status.

So why is quad II the odd man out, being that it is a non-position? In addition, quad II is the only quadrant that doesn’t have a positive epistemic status, so the symmetry here is obviously broken. For it to keep a symmetrical relationship, quad II should be Ba~p ^ ~Ka~p (Agent believes p, but doesn’t know p). If however it is kept as ~Bap ^ ~Ka~p this becomes indistinguishable from agnostic being ~Bap ^ ~Ba~p as neither make any knowledge claims. Making “Agnostic Atheist” in that case collapses to just “agnostic” if there is no positive epistemic status. So in one case “Agnostic Atheist” reduces to “atheist” if atheist has a positive epistemic status, or to agnostic if it does not, so we can conclude that “Agnostic Atheist” in quadrant II is ambiguous.

Summary for this type of chart to even begin to make sense:


Quadrant I – Gnostic Atheist = ~Bap ^ Ka~p
Quadrant II – Agnostic Atheist = ~Bap ^ ~Ka~p
Quadrant III – Agnostic Theist  = Bap ^ ~Kap
Quadrant IV – Gnostic Theist  = Bap ^ Kap

Updated to at least make some sense:

Quadrant I – Gnostic Atheist = Ba~p ^ Ka~p (since Ka~p -> Ba~p)
Quadrant II – Agnostic Atheist = Ba~p ^ ~Ka~p
Quadrant III – Agnostic Theist  = Bap ^ ~Kap
Quadrant IV – Gnostic Theist  = Bap ^ Kap

Then of course, we can see that quad II just reduces to Ba~p, and quad III reduces to Bap…meaning that adding “agnostic” is superfluous.

To make further sense (reduced):

Quadrant I – Gnostic Atheist = Ba~p ^ Ka~p = Ka~p
Quadrant II – Atheist = Ba~p
Quadrant III -Theist  = Bap
Quadrant IV – Gnostic Theist  = Bap ^ Kap = Kap

We then can notice that the entire -y axis can go away as non-needed, which effectively removes the need for the entire axis, and we can use a single axial diagram to show “gnostic” to “agnostic” relationship if a person is dead set on using those labels. Using one of Dr. Malpass’ diagram (which doesn’t use “a” for agent):

Kp area would be “Gnosic Theist”.
K~p area would be “Gnostic Atheist”
Bp would be TheistB~p area would be atheist
~Bp ^ ~B~p area would be agnostic

 Conclusion: This single axial schema would work fine with out ambiguity, nor being superfluous. This also demonstrates that the phrase “agnostic atheist” is either ambiguous or superfluous, and that multi-axial diagrams are epistemologically untenable.

If a multi-axial approach is used I have shown conclusively that in order to avoid ambiguity, and retain symmetry, it logically reduces to:



Author: Steve McRae