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Is this statement always true (= a tautology)? Nothingness here does not mean absolutely nothing, instead it means: nothing but logical laws. I'd formalize it as ∅ |= ◻ ◇ F (F stands for any wff) or maybe ∅ |= ◇ F and as far as my modal logic knowledge goes it is a valid semantic consequence. Am I right?

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    Does "anything can happen from nothing" mean "if nothing is assumed (other than logical consistency) then anything logically possible can happen"? That is, indeed, a tautology. But your axiom cannot apply to "any" wff, F= A∧⌐A is a wff. It should only apply to (logically) satisfiable wff.
    – Conifold
    Oct 12 at 9:36
  • Not very clear... Do you mean that F is a formula whatever? What do you think about p ∧ ¬p? Oct 12 at 9:36
  • If instead F is a "logical law" (a formula always true), like e.g. p ∨ ¬p, then YES: ⊨ ◻ F Oct 12 at 9:39
  • I see. If F is a (well-formed) formula which is satisfiable, i.e. F could be true at least in one world/instance, then ∅ |= ◻ ◇ F is a valid sem. consequence (tautology as a statement), even ∅ |= ◇ F. But if F is just a wff, i.e. also possibly p & ~p, then ∅ |= ◻ ◇ F as well as ∅ |= ◇ F are not valid sem. consequences. Right?
    – Pippen
    Oct 12 at 10:21
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    Which worlds are possible depends on how you set up your modal logic and semantics. One typically adds modality to a background first order theory, and all theorems of the theory are then incorporated as necessarily true by the necessitation axiom. That theory can be just pure predicate calculus, or it can include PA, ZF, ZFC, or whatever one considers "standard mathematics". As far as logical possibility is concerned, there are, of course, possible worlds where ZFC, or even PA, are false, but in typical works people usually incorporate "standard mathematics".
    – Conifold
    Oct 12 at 11:26
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Following from your comments, I think your question is not so much about whether it is logically necessary that something exists, but about whether probability theory can be used to demonstrate that it is highly probable.

To start with the question as you have asked it, logic itself cannot tell you whether something exists or not. In the standard way that first-order predicate logic is set up, we assume that something exists. This is implicit in the way the rule of universal instantiation works, and corresponds within model theory to the assumption that the domain is always non-empty. If we wish to reason about potentially empty domains, we have to use a non-standard logic, viz, one of the free logics. Even then, the logic itself does not tell us what exists and what doesn't.

Using modal logic does not change anything. A thing might possibly exist, in the sense that it exists in some possible world. But, the logic itself does not assure us that if a thing is possible then it is actual. If we wanted to be able to reason in that way, we would have to make use of some assumptions to that effect and incorporate them into the logic. Either way, they are still assumptions.

You say in a comment, "...probability theory ... does not belong to logic". Whether probability theory should be regarded as part of logic is not the issue here. The important point is that Carrier is abusing probability theory itself. He says that since there could be any number of universes from zero to infinity, and each of these possibilities is equally probable, the probability of there being no universe is (approximately) zero. This is nonsense. One cannot have a proper uniform probability distribution over an unbounded range. Sometimes, in Bayesian theory, one can proceed from an improper prior to a proper posterior, but that is only feasible when you can update on known conditional probabilities.

In the case of the existence of the universe (or universes), there is no way to get a handle on such things. We don't know anything about the probability distribution of possible universes, or whether we, or anyone, would be around to observe them if they existed. It is precisely this that makes it impossible to assess things like the probability of the universe existing, or the probability of God existing, or whether the fine tuning argument tells us anything about either.

As a corollary, note that Carrier's argument, if it were sound, would also prove the existence of gods. Since there might be no gods, or one god, or two, or three, or any number up to infinity, then if we assume that all of these possibilities are equally likely, by Carrier's reasoning, it would be overwhelmingly probable that some gods exist. Not a conclusion that he would be happy to endorse!

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  • Thanks! What would be your take on my criticism of Carrier that he uses probability theory as if it consists only of logical axioms, but in fact probability theory uses only non-logical axioms. So beyond your points above one could argue that the very tool he uses (probability theory) itself is only possible and could be wrong in our world and whatnot other worlds, so he uses a tool that any theist could attack as speculative. I suggested to him to define nothingness as nothing but only logic and probability theory. Then his argument could work because he just wants to refute nihil ex nihilo…
    – Pippen
    Oct 13 at 3:08
  • i.e. he just wants to show that is it highly probable that something comes out of nothing.
    – Pippen
    Oct 13 at 3:09
  • It doesn't matter whether you think of probability theory as being part of logic or not. There are probability logics that combine the two. The main point is that within probability, you cannot (except with a few exceptions) use improper probability distributions, or just help yourself to the assumption that any specified range of possibilities is equiprobable. I'm not sure how probability theory could be wrong, though I suppose there could be alternatives. In any event, logic and probability do not tell you what exists, so they do not address the issue of whether something comes from nothing.
    – Bumble
    Oct 13 at 11:35
  • Carrier just wants to show that if you assume nothingness (= only logical laws) then 1) it follows necessarily that anything (satisfiable) can happen and 2) if you then assume that any thing has an equal chance of happening (which makes sense) then you can show that P(something) is very likely. This is all he wants to show because he wants to refute "nothing can come from nothing" as very unlikely. So it is crucial to determine if he can just apply probability theory in his argument since he just assumes logic and nothing more. I think he cannot, I think he cheats based on his premises.
    – Pippen
    Oct 13 at 13:35
  • I understand what he is trying to do, but it is entirely spurious. You cannot use an improper distribution like that, and there is nothing plausible about assuming that a given range of possibilities is equiprobable without some justification. Neither logic nor probability provides any reason to suppose that something exists.
    – Bumble
    Oct 13 at 14:37

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