I had a sneaking suspicion my last post would pose an irresistible temptation
I need to split my responses to you into a few groups, because I think there is a general level of discussion to be had here and an examination of the specifics as well.
In the first place, I would not characterize my argument as "proof" that God does not exist. The proof - such as it is - rests on some very particular premises. I would not ordinarily trot out an argument like this without those premises having been fairly well established and agreed upon before hand. The key premises:
1) We are dealing with at least an informally rigorous use of standard logic.
2) The subject (in this case, the God of organized religion, e.g., Christianity) sits entirely within the universe of our consideration, that is, the domain of objects upon which we have agreed to exercise the logical rigor stipulated in (1).
and finally:
3) We are working with definition of God that is sufficiently specific to yield an interesting discussion. In this case, the "transcendent" definition that God is the aggregation (and more!) of all that exists.
In your final paragraph above, you make the point that these three premises - particularly (2) - would not find agreement among many believers, and I think that's clear. Many religious folk would immediately balk at the idea that any rigorous logical "definition" of God is possible, at least to the human mind. In the context of this discussion, at that point we'd pretty much throw in the towel and say that here religion and logic part ways - God is simply not in the domain. However, you may note that was not the route Dr. Legend chose to take.
This is also my overall position on this topic - certain aspects of religious belief simply must be shielded from examination by logic; otherwise we end up with a doubly bad combination of logical inconsistencies and a puncturing of the religious experience. So really, the answer is the two are compatible only up to a certain point - on the whole and in their entirety, they are not.
On the second point - the vagueness of Dr. Legend's source - I think it's worded carelessly, but the concept here is certainly not novel, nor is it radically bad. Precursors of the notion of a transcendent divinity can certainly be found in Plato (the mini-narrative of Diotima from
The Symposium comes to mind), and some of the high points of the
Confessions clearly appropriate that Platonic notion of transcendence and reorganize it in a scriptural context (the man was a rhetorical genius). If anything, the good doctor's source is undermined by his resentment at the plebeian interpretation of his religion and fails to convey the sublime beauty of his vision. Who would have thought that the grace of humility and shedding of ego would become elusive in the presence of overweening pride and pretensions of superiority? Oh right,
that same guy would.
So those are my opening remarks. I do still owe you a detailed answer as to my appropriation of Russell's paradox, how I use the notion of existence, the zero analogy, and particulars regarding logical entailment.
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So here is how I have been thinking about concept of existence in propositional logic:
In my own simple-minded apprehension of mathematics (as it was taught to me), the notion of existence is really
the atomic primitive descriptor. For instance KP or ZF axioms for the most part consist of rules that govern how one can construct or derive new sets from existing sets, and then include at least one axiom that asserts the existence of a set (I like KP because it's clear how one assumes the existence of the empty set and then builds everything up from that). From these foundations the entirety of standard mathematics is then constructed.
However, none of the axioms explicitly define what this existence descriptor really means. It's not even formally defined, much less given any interpretive meaning. One might speculate that the intent is to form a representational analog to objective existence, but the approach is entirely formal, and proceeds from such abstract rudiments that it's just as clearly intended to stand on its own as the basis of a representational construct - e.g., mathematics.
It could be argued that some meaning is inherited via the semantics of predicate logic, but in this case I'm not using predicates, I'm just dealing in propositions. And even in predicate logic, existence as a descriptor is still not assigned meaning. Much as in set theory, it's a primitive state that is assumed to have propositional requirements. Much care is taken to ensure the universal and existential quantifiers operate to conform to the principle of non-contradiction, and to transfer existence safely (by inference) from one collection of objects to another, but existence itself is not assigned any meaning beyond its formal position in the semantics of predicate logic.
So in practice what we see is that existence is a primitive, undefined property that our entire axiomatic system is expressly designed to transfer safely from a small (defined or axiomatically stipulated) collection of primitives to other derived objects. As such it has no intrinsic meaning - it only serves to transfer a status from one collection of objects to others, to enable the construction of a contingent model. (If A exists, then so must B, and so on.) The governing rule is that existence must be propositional and non-contradictory: if we derive a state of affairs where an object both exists and does not exist, we have a paradox and have fallen into error. I am inclined to think every object of consideration must also have an existential status, but I'm not sure about that.
Now I know that in some philosophical contexts, there is an assumption of existence that accompanies an assertion. So, for instance, if I say consider all X, there may be an implicit assumption that there exists at least one X. In mathematical logic this is generally not the case, which gives rise to vacuously true implications.
We could clarify things by explicitly stating what the universe of consideration is. But even if we explicitly state that we want to consider a universe only of things that can be said to exist, it's clear we don't restrict ourselves only to contemplation of objects that exist in the practice of mathematical reasoning. Even though existence appears to be one of our central characteristics, the one characteristic of objects we most care - almost obsessively - about, some of our most cherished proofs spend a considerable amount of effort contemplating, defining, manipulating, and triumphantly discarding objects that we joyously conclude do not exist. Cantor's diagonal proof and the proof that the diagonal length of the unit square is not rational are two proofs that come easily to mind. And these are not obscure proofs, but rather some of the most celebrated, canonical proofs in the foundations of set theory and analysis.
So the "domain of contemplation" must quite prominently feature objects our contemplation itself informs us cannot exist, unless we want to toss proof by contradiction out the window. Sometimes it may even be the case that the proof would be impossible without contemplation of the non-existent. Now that would put us in a truly absurd situation: I could prove to you that this thing does not exist if only I were allowed to contemplate it.
From a philosophical perspective, I think the existence property forms a nice bridge between the conceptual realm and contexts of application. In the most stupidly obvious way I can, I pay tribute to Descartes in as much as, whatever idea pops into my head, can certainly be said to exist
as an idea I just had. Whether that existence can be properly transferred to another context is the more relevant question, and this is where the semantics of representational logic come into play.
In the context of the current discussion, if I have a working, purportedly well-formed definition of God that states God contains "all that exists" and on top of that claims to be consistent with propositional logic, then the notion of "all that exists" must be well defined, as must be the notion of containment. Even if, as in the examples above, I intend to show that "all that exists" cannot itself exist, I should be allowed to contemplate it, and even formulate it as a logical proposition. Despite the possibility there might be some philosophical objection to doing so, given that the body of working mathematicians appear to be comfortable with the contemplation of objects they intend to show do not exist, I think that's a high enough bar.