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I have seen a short clip recently, in which someone used a proof by contradiction that a being cannot be omniscient (including past, present and future) and have free will at the same time. There are other arguments that involve omnipotence as well. I wonder if it is rational to use logic against god without a proper axiomatization of the relevant concepts.

There is a trivial first order theory in which an omnicient, omnipotent god with free will can exist. Have a constant symbol god, three unary predicates is_omniscient, is_omnipotent and has_free_will, as well as the axioms is_omnipotent(god), is_omniscient(god), and has_free_will(god). This theory is satisfiable. However, as it does not prove or disprove god, it should be declared as inadequate by theists and atheists.

It seems to me that an adequate theory would have to be very complicated. A universe of discource of such theory should not only include humans and gods, but all of physics and probably more.

One could argue that if there were a theistic logician, an atheistic logician could ask them to provide such formal theory of which they think that a god can exist.

However, Father John does not care about formal logic, he just has faith. In my opinion this shifts the responsibility of providing theories onto the atheistic logician. The latter cannot just pick some theory and prove gods nonexistence in it, but has to show that there isn't any adequate theory in which the existence of god is satisfiable.

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It seems the time has come to talk of cabbages and kings ... 😁 ... I mean theists and logicians. I never understood Wittgenstein, but supposing I did, I'd say that we need to unpack the words we use in the antipodal language games that are, allegedly, at play here.

True that omniscience is logically incompatible with free will, but what is free will and what is omniscience?

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Yes. The omnipotence - free will paradox has simply never been refuted. If one knows that X will happen, then nothing could happen otherwise.

Now of course, one can change the god hypothesis or propose that he works in mysterious ways to save this contradiction. But one can make ad hoc formulations about any concept in natural language to avoid contradiction.

For example, one can even say, “well John is a married bachelor but the way that he is married is different from the way we understand it so he remains a bachelor and married at the same time. Human beings are just no capable of understanding. He is beyond it.” This would be ludicrous.

God was proposed as timeless because he wasn’t found in time. He was proposed as invisible and immaterial because he wasn’t found. He was proposed to be able to do miracles at unverifiable times because they didn’t happen.

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Is a naïve logical argument against god rational?

It could be, but you would have to define the word "God" and every other word you use in your argument, which is certainly not a trivial effort and contradicts the idea of a naïve argument.

If I define God as something which is both here and not here, then we can infer logically that this God doesn't exist, or that if it exists at all, then its existence cannot make sense to us.

Logical arguments against the existence of God, for example the argument that Omnipotence is simply a logically inconsistent notion, do not prove that God doesn't exist, but show that if God exists, then its existence doesn't make any logical sense to us. Then it remains for each of us to decide whether such a God can exist or not. It is probably for the better that we can believe whatever we like without being corseted by logic. In effect, we can choose to just ignore logic. However, if we don't, then no omniscient being can possibly make sense.

In essence, we have to choose between an omniscient god and logic.

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You cannot prove or disprove the existence of imagined entities whose purported properties put them beyond the reach of our tools of analysis. If I tell you that the Universe was created by Spefflebaum, an ineffable being existing outside of time and space, you cannot prove that my claim is untrue, because it has an unverifiable nature. If I ascribe certain properties to Spefflebaum, the best you can do is to look for inconsistencies between them. If I say Spefflebaum is omniscient and omnipotent, you might say those two properties are inconsistent, but I can always define them in a way that eliminates the inconsistency, or I can tell you that the inconsistency you posit is a conclusion reached through the erroneous assumption that Spefflebaum is subject to human logic; either way, you are running a fool's errand by trying to apply logic to a matter that is inherently alogical.

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No, but that can be fixed.

The axioms of Hilbert spaces serve as a foundation for quantum mechanics. From there, the KS theorem can be proven, and it is an immediate consequence that omniscient beings are impossible.

The video you watched may have considered the axioms of Hilbert spaces to be obvious, already derived, or time-tested by decades of quantum-mechanics experiments. The video may also have been irrational, in the sense that it did not justify its axioms prior to deduction. However, since you did not share the video, we are left to steelman their position.

It seems to me that an adequate theory would have to be very complicated.

No, it only takes three to six axioms, depending on phrasing. The complicated part is category theory; while the KS theorem is a statement of linear algebra, the formalization of topological spaces requires a fair amount of dagger-category theory.

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  • Thanks for the pointer to Albert's sock drawers. I do not support the conclusion, though. You take the inability to know all nine drawers' contents as evidence against omniscience. But hypothetical Jim Bell said it himself: "there simply aren't any such facts"! This is as if I said that god is not omniscient because he does not know my brother, when I do not have a brother. If anything, it shows that people have different notions of omniscience which would produce different formal theories.
    – Damian
    Commented Aug 15, 2023 at 19:12
  • A quantum universe is classically simulatable (albeit slowly). No theorem starting from known physics can rule out the possibility that the universe is running on a classical computer with a lot of RAM, with an operator who can pause it and inspect its entire working state at any time. You can only get a contradiction if you assume that your omniscient being is a quantum object subject to the same rules that we are.
    – benrg
    Commented Aug 15, 2023 at 19:37
  • @Damian: Sure, but you'd have to explain how a Stern-Gerlach apparatus functions; clearly, it separates silver ions into those which are "spin up" and "spin down" at the moment of measurement, despite the KS theorem proving that those ions do not have a definite spin prior to measurement. If the silver ions were measured by any observer prior to separation, then we would observe a deflection; more generally, we can detect certain kinds of tampering with quantum information.
    – Corbin
    Commented Aug 15, 2023 at 19:50
  • @benrg: You are espousing support for local hidden-variable theories, which are ruled out by Bell tests. In addition to cosmic Bell tests, we have e.g. polarized sunglasses, which behave according to quantum optics. The ability to examine the state of a simulation does not grant the ability to make undetectable quantum measurements within the simulation. For an introduction to the topic, see this video.
    – Corbin
    Commented Aug 15, 2023 at 19:53
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    Even we lowly human beings have pseudorandom number generators that are indistinguishable from random by any known test (CSPRNGs). The state of the PRNG could be inspectable. I'm not saying this scenario is very plausible, but it is a counterexample. You linked Scott Aaronson's entire "Quantum Computing Since Democritus" course, and I'm not sure where the relevant argument is, but if it's a proof that you need true randomness then locality is one of the assumptions.
    – benrg
    Commented Aug 15, 2023 at 22:05
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This is a theological claim about the nature of a hypothetical existing god, not a claim about the existence of gods.

A 2018 Pew Study showed only a quarter of American theists assert a belief that God determines what happens to them all the time.

Speaking for myself, I don't know how anybody can read the Bible, Torah, or Koran and conclude that any of the characters in it know everything about the future, occasional out-of-context superlatives notwithstanding. And I don't believe anybody who purports to have a consistent belief that God does, if they go around making reasonable choices in their own life and asking God to change his mind every time they pray. I'm sure most of that 27% honestly believe that God knows everything about the future when they're thinking about how much about the future God knows. But when they're thinking about what to make for dinner, they believe the future is uncertain. When they're thinking about whether God will heal their friend's sickness, they believe that God can change his mind.

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As an atheist: god is supposed to be almighty. So human logic might prove that god doesn’t exist, but being almighty, he/she/it/they would just ignore our human logic and exist totally undisturbed by it. And being almighty, an existing god could make anyone find a totally convincing proof that he/she/it/they doesn’t exist.

On the other hand I haven’t seen any evidence of god’s existence.

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Not only are the axioms to which we apply logic variable enough for someone in a closed system of belief to be warranted, on their own terms, in thinking that there is a God, but logic itself is variable enough to open the door to personally justified inferences to the existence of such a God. As Onora O'Neill put it (channeling Immanuel Kant), we are permitted to hope that it is possible for there to be a God, though we might not be obligated to claim to know that there actually is a God:

... any adequate account of what we may hope will have to incorporate some account of anything that we must hope. There might, however, be many distinct answers to the question “What may I hope?” that had in common only those aspects of hope that are required. It may, for example, be the case that various quite distinct hopes for human destiny incorporate a convincing account of what we must hope. ... Even if the abstract claims of deism and the tenets of traditional Christian faith provide two specific answers to the question “What may I hope?” Kant’s arguments may not show that either forms part of every answer to the narrower question “What must I hope?”

Perhaps there are conclusive public justifications for some axioms and inference rules, such that we might publicly disavow the admissibility of belief in God as such; but you note that you are asking about, and focusing on, more private cases. By analogy, one might adopt a set of idiosyncratic axioms in set theory, and/or a peculiar logic (maybe a system with 19 truth values, one connective (TONK), etc.), and go on to arrive at peculiar conclusions. Those inferences will be relatively justified to some extent; of course, if it came time to apply them to the judgments and in turn actions of other people, other considerations will come into play. But if it's just oneself and one's own opinions that are at stake, then yes, it does seem acceptable to hope for, or maybe even believe in, the possibility or actuality of God's existence.


Moreover, lest one overzealously claims to have come up with a perfect, consensus definition of divine attributes, let them read through the lack of consensus (LOC) regarding the following:

  1. LOC in definitions of "omniscience."
  2. LOC re: "omnipotence."
  3. LOC re: "perfect goodness."
  4. LOC re: "omnipresence."
  5. LOC re: "divine eternity."
  6. LOC re: concepts of God or "ultimates" in general.

Historically, it was not always the case that theologians worked primarily from pre-theoretical conglomerates of lower-order predicates, to define their God, but the appeal was more deeply made to formal predicates of a superlative character:

... the concept of “infinite being” has a privileged role in Scotus’s natural theology. As a first approximation, we can say that divine infinity is for Scotus what divine simplicity is for Aquinas. It’s the central divine-attribute generator [emphasis added]. But there are some important differences between the role of simplicity in Aquinas and the role of infinity in Scotus. The most important, I think, is that in Aquinas simplicity acts as an ontological spoilsport for theological semantics. Simplicity is in some sense the key thing about God, metaphysically speaking, but it seriously complicates our language about God. God is supposed to be a subsistent simple, but because our language is all derived from creatures, which are all either subsistent but complex or simple but non-subsistent, we don’t have any way to apply our language straightforwardly to God. The divine nature systematically resists being captured in language.

For Scotus, though, infinity is not only what’s ontologically central about God; it’s the key component of our best available concept of God and a guarantor of the success of theological language. That is, our best ontology, far from fighting with our theological semantics, both supports and is supported by our theological semantics.

We can even see this with respect to Aquinas' model of a triune deity, where the model is built from a theory of relations (polyadic predicates) modulo an essentialism (and essence/existence distinction) inherited from Aristotle. (C.f. the difference between (A) a philosophy of mathematics that takes at least the natural numbers for granted and (B) one which seeks to ground 0 and ℕ in something else (e.g. Frege's peculiar introduction of the empty class as the class of items with contradictory properties, or Cantor's transmutation of ℕ into the ordinal ω and the cardinal ℵ0).)


Addendum: Gödel's God

I should like to add that Kurt Gödel's ontological argument was read off the notion of an ens realissimum (c.f. Olivier Esser's positive set theory), which I will account for by quoting Kant:

This conception of a sum-total of reality is the conception of a thing in itself, regarded as completely determined; and the conception of an ens realissimum is the conception of an individual being, inasmuch as it is determined by that predicate of all possible contradictory predicates, which indicates and belongs to being. It is, therefore, a transcendental ideal which forms the basis of the complete determination of everything that exists, and is the highest material condition of its possibility—a condition on which must rest the cogitation of all objects with respect to their content. Nay, more, this ideal is the only proper ideal of which the human mind is capable; because in this case alone a general conception of a thing is completely determined by and through itself, and cognized as the representation of an individuum.

Kant goes on to explain that this primal unity is a cognitive structure, something carved into the architectonic of reason in seeking to simplify and condense its elementary premises as much as possible. So it is not, "just like that," the mark of an actual God. Nevertheless, even per Gödel's reasoning later, we can see that the definition of an ens realissimum is intuitively noncontradictory by itself, either because it is defined in terms of quintessential consistency in abstracto/from the outside, or because the definition is too generalized and simplistic to license enough substantial inferences to infer a substantial contradiction from. Modulo the incompleteness theorems, we would say, "The theory of an ens realissimum does not allow us to derive enough of arithmetic for the incompleteness problem to confront the theory; the theory is so weak that its consistency would be provable, and then is too simple to be proven contradictory." Rather than just coherent definitions of God being epistemically possible, then, it seems as if we know of a coherent definition of God that is already actual, if perhaps trivial (in the negative limit).

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  • I downvoted because I think that it's a dodge to handwave axioms. It turns out that we don't need to axiomatize a deity in order to disprove them; we merely need to formalize the claimed properties of deities.
    – Corbin
    Commented Aug 15, 2023 at 14:47
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    @Corbin reasons can be given for axioms, e.g. we can appeal to perception/intuition as non-inferential grounding. And some definitions of words like "omnipotence" are contradictory, but other definitions aren't. But then if I define my use of the word "God" to refer to Kant's ens realissimum, for example, I have a coherent enough definition in play (waiving the problem of unrestricted quantification in abstracto, at least), and then morally the issue is more one of may-vs.-must: I may believe in/hope for the ens realissimum, but I don't have to. Commented Aug 15, 2023 at 15:01
  • Now if we were talking about public reason more than private opinion, not only would the desire for overlapping consensus come into play as a theoretical principle, but it would occur as a practical social issue: does my personal definition of the word "God" find purchase with other definitions, so as to mediate discourse using the word? However, the OP seems to be focused on the private question, not the one of public reason, and I'm not narcissistic enough to command people to accept my definitions "just like that." Commented Aug 15, 2023 at 15:04
  • Also, the axioms for Chu spaces are not compatible with omnipotence, using the interpretation of Chu spaces as enumerations of points with states: there are contradictory states, so that an omnipotent being cannot bring about contradictory states for a single point. This makes your first paragraph slightly silly: yes, there's logics which prove that God exists, but also logics which prove that God cannot possibly exist.
    – Corbin
    Commented Aug 15, 2023 at 15:12
  • @Corbin isn't the OP question about whether one is privately justified in accepting this-or-that system of axioms/inference rules in the absence of a universally compelling public proof to the contrary of whichever system? I am not going to pretend to have solved the problem of the regress of reasons out of nowhere, and I don't see that it matters whether my answer is "silly" (a subjective description, after all) if it's adequate to the question being asked in the OP. Perhaps the question is not a strong one, but I'm not making that judgment either. Commented Aug 15, 2023 at 15:19

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