# Is there any formal axiomatized definition of Determinism in First Order Logic (FOL), or any other logical system for that matter?

Is there any formal axiomatized definition of Determinism written in First Order Logic (FOL), with a semantic interpretation maybe rooted in Set Theory, or based on any other logical system and semantic interpretation, for that matter?

• Why you think so ? Why do you expect that can be relevant ? Commented May 17, 2018 at 5:53
• See Determinism for a description (I will be surprised if we can find something like a reasonable definition of it). Commented May 17, 2018 at 6:31
• Since determinism applies to physical world it only makes sense to define what it means for a physical model to be deterministic, not for FOL or set theory. For classical models determinism states that system's behavior at any time is determined by its state at the initial time, i.e. Cauchy problems for its evolution equations have unique solutions. Commented May 17, 2018 at 19:32

One form of determinism would be a Turing machine.

Turing machines have a formal definition.

Turing machines cannot decide every language. For example the Halting problem is undecided. However, if you generalize the notion of a Turing machine to have an infinite number of states, you can get around this and recognize any language: Consider a deterministic countably infinite "automata": with one state for each possible input string (it would look like an infinite binary tree). You can configure the accept/reject states however you want to recognize any language you want. This doesn't have a corresponding Turing machine because it has an infinite number of states.

If you generalize the notion of a "language" to be any set with higher cardinality, you could "construct" (basically "hard code") a corresponding "automata" that "decides" elements of that set. You would have a start state; each element of the set would transition to an accept state; other elements not in the set would transition to a reject state. This basically says determinism captures the notion of a boolean function.

Therefore anything you can define logically has a corresponding deterministic "automata" that decides it.

Therefore determinism is equivalent to logic.

• I wonder why you would need an infinite number of states? Is there a reference that might help explain that? Again, welcome to this SE! Commented Oct 15, 2018 at 0:40
• @FrankHubeny The infinite number of states is needed to recognize any language. Without it, some languages are not decided. I made that more clear. There no references; I thought of this myself. However I explained the "proof." It's trivial. Commented Oct 15, 2018 at 1:49

The question you are asking, is determinism not a metaphysical belief system?

The answer is it is a metaphysical belief. The definition of determinism in summary says,

Determinism is the philosophical theory that all events, including moral choices, are completely determined by previously existing causes. (Wikipedia)

As soon as you use the term "all events" if there is just one event that is not deterministic, the model fails. And this would require infinite knowledge and the ability to see everything in the past and the future. Such beliefs are outside the scope of science and are therefore metaphysical.

Ofcourse believers would love an absolute proof of their belief system, to conquer the non-believers, but that is the nature of faith.

• Do you know if the wikipedia definition you are quoting in bold can be formalized in FOL or other logical systems? Say, what would the "determinism axiom" look like?
– xwb
Commented May 17, 2018 at 18:19