Earlier quoted context omitted.
There are types without values. That the type is itself a value does not mean that it has values itself. False is a type with no values. Consciousness is not a value. Replace type with concept and value with thing, and we get a rebuttal using the same language. You can also switch the terms, depending on how 'a thing' is interpreted. If you are more familiar with set theory, replace type with proposition, value with…
Nothing is fully defined formally. The most fundamental aspects of reality, and mathematics, are still not founded completely on formal definitions, without resorting to circular reasoning, and attempts to prove things assuming the things that are being proved. Existence isn't well defined. Yet Descartes' famous informal claim "I think, therefore I am" is still a compelling statement that each of us can use to verify…
My claim is that while tools to determine if consciousness exists do not presently exist, it is possible that they are discovered, and applied, and that the question of the existence of consciousness is open. Note that for me existence is a fundamentally formal notion. I cannot defend this except on grounds of its anecdotal usefulness because it is akin to an axiom or ground truth.
The notion of consciousness I consider is with respect to its 'actual' existence[1]. Suppose I imagine a circle, and then a square, but believe I imagined at once a round square. That I can consider the notion of 'a round square' or may have had a direct perception of it does not require me to accept its existence. Likewise, I do not need to accept the existence of a unifying singular notion of what might be perceived as consciousnesses. This also does not necessitate I reject it. If consciousness cannot find an agreed upon definition, assertions regarding it lack semantic meaning and cannot be 'reasonably' (waves hands) assigned truth values. If an exact definition is determined with no logical contradictions, I defer to the physical sciences for determining its plausibility.
The notion of conscious hearts was more credible in the past. The idea was that before the role of the heart was understood, a large population could have been convinced that 'the heart is the seat of consciousness', and thus a large population believing they were experiencing something a particular way may have been followed by the falsification of the existence of such a thing. At the time, it was very plausible that it was considered impossible to ascertain whether or not there were thinking hearts. I concede that the notion is not presently credible; for my purposes it is sufficient that it was ever credible.
If you regard existence proofs as being extra-logical, it may not be the case we can find agreement since we will be effectively talking in different languages. Personally, I am very skeptical of the ability of the human mind to find truth informally, which is why I lean on formal notions heavily.
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1: My favorite essay on this distinction is Quine's (admittedly polemic) "On What There Is". The distinction I consider is slightly different from his in that I am also expressing distrust of direct self-perceptions of the mind's functions.