Regarding the Claims! Claims! Claims! stream and the ternary system.
or ternarinception

Let’s employ some programming skills with ternary operators and a bit of logic and see where it gets us.

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definitions and notation
Before I get into the argument let me set up some tools. You can skip or glance over these, but they're there if you need them later.

We assume that it is only possible to hold one of three positions on any proposition

A = Accept
B = Reject
C = Abstain

Given any proposition P one can then make three derived propositions P' of the form:

PA ⇔ position on P is 'Accept'
PB ⇔ position on P is 'Reject'
PC ⇔ position on P is 'Abstain'

The ternary system must then be applicable to each of those statements but must retain logical consistency, so each statement must infer meaning about the other statements.

PA ⇔ PAA ∧ PBB ∧ PCB

In other words:
Accepting P is equivalent to Accepting PA AND Rejecting PB AND Rejecting PC

And the relation can be written as:

PAA ⇔ PBB ∧ PCB
Accepting PA is equivalent to Rejecting PB And Rejecting PC
In other words:
Accepting P is equivalent to neither Rejecting nor Abstaining from P

Solving this derived set of propositions gives complete information about the position on P.

This chain of derived statements should hold all the way down to the axioms if the system is to be consistent

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disagreeing with TheRumpus

has claimed that statements like not-accept are illegal because they lead to contradiction.

I disagree. Statements that are contradictions are illegal, but statements that are not contradictions are fine-just-fine (but might be incomplete).

The way in which this becomes crystal clear for me is to see it from an outsiders perspective trying to find out what some person's position is

So lets write the legal positions.

The set where I agree with TheRumpus

PAA ⇒ PBB ∧ PCB
position is known: the person Accepts P (e.g., Theism)

PBA ⇒ PAB ∧ PCB
position is known: the person Rejects P (e.g., Hard Atheism)

PCA ⇒ PAB ∧ PBB
position is known: the person Abstains on P (e.g., Agnosticism)

Since the position is known, any of these sufficiently answers the question:
“What position does the person hold on P?”

The set TheRumpus calls illegal

¬PAA ⇒ (PBA ∨ PCA)
position is unknown: the person either Rejects or Abstains on P (e.g. non-theism)
Along with PAA, sufficiently answers the question “Does the person Accept P?”
Technically the full form is: ¬PAA ⇒ (PBA ∨ PCA) ∨ (PBC ∧ PCC) but it can be reduced with no loss.

¬PBA ⇒ (PAA ∨ PCA)
position is unknown: the person either Accepts or Abstains on P
Along with PBA, sufficiently answers the question “Does the person Reject P?”
Technically the full form is: ¬PBA ⇒ (PAA ∨ PCA) ∨ (PAC ∧ PCC) but can be reduced with no loss.

¬PCA ⇒ (PAA ∨ PBA)
position is unknown: the person either Accepts or Rejects P
Along with PCA, sufficiently answers the question “Does the person Abstain on P?”
The long form would contain a contradiction in the second part of the statement so that part must be excluded (giving a consistent reduction)

And there are other non-contradictory values in the derivations such as
PAC ∧ PBC ∧ PCC which would reduce to PC but I would argue that it a possible answer in case the person doesn’t know of P (literally, doesn’t know of the proposition itself)

Contrast this with ternary values (T - true, F - false, U - undefined).
A negation of ternary values would work something like:

¬T=F
¬F=T
¬U=U

And applying that to the examples in which this ternary system is proposed, you could get the following exchange:
"Are you an Agnostic?" - "No!" - "Oh, so you're an Agnostic then."

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Conclusion

All of this violates the excluded middle rule but maps much better to how people actually think (since ignorance is a major component of cognition) and converse.

It could be interesting to derive the mathematical properties and rules of the system and construct truth tables (or rather the position tables) to see how the positions interact and sum up to see which form of ternary logic would fit best.
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