Negation of existential quantifier ... and indeterminism.

Question

I wrote these propositions, for personnal purpose:

(1) {E, F, G} ∈ Σ

(2) ∀E ∃F [(E ↷ F) ∧ ∀G ((E ↷ G) → (F = G))]

E, F and G describe 3 states of the system Σ, ↷ is a peculiar function of this system.

I need to write the negation of proposition (2). This is my suggestion:

(3) ∃E ∀F ¬(E ↷ F) ∨ ∀E ∃F ∃G [((E ↷ F) ∧ (E ↷ G) ∧ ¬(F = G))]

My question are:

a) Are these well-formed propositions?

b) Are there better formulations?

c) Is negation (3) correct?

Thank you.

Answer 1

Thank you for your comments Mauro.

Maybe I should add some context, despite it's more about analytical philosophy than about mathematics. To respect the forum's rules, I don't intend to discuss the philosophical aspects here.

My reflexion comes from the reading of neuroscience books, where the authors often can not help but write some (often) flimsy and botched pages about “free will”. Most of the time, they don't even try to give a thorough definition of what “free will” means for them. They often contrast free will against determinism (and they don't define determinism either). However, the negation of determinism is not free will, it's indeterminism. So I try to investigate these notions in stricter frames, among which logic. - I'm not a logician and my basic knowledge of logic dates back from my long gone college years. So this is a rather heroic, or quixotic try. I hope you won't mind.

First, I decided to ground my reflexion in a simple example: a basic system, which consists of 10 states that follow each other. In a deterministic setting, when one state E (from the French “état”) occurs, at time t0, one and only one state F follows it, at time t1.

This is what I try to formulate in proposition (2):

∀E ∃F [(E ↷ F) ∧ ∀G ((E ↷ G) → (F = G))]

The function “↷” means here “is followed by”.

If I'm correct, the negation of this proposition is (3):

∃E ∀F ¬(E ↷ F) ∨ ∀E ∃F ∃G [((E ↷ F) ∧ (E ↷ G) ∧ ¬(F = G))]

(Adapted from Brian M. Scott's answer to this thread: Negation of Uniqueness Quantifier)

This leads to unexpected conclusions to me:

In this indeterministic system,

a) There should exist at least one state that is followed by no other state.

Or

b) Each state is followed by at least 2 simultaneous, or superposed, states.

So far, I don't know how to interpret a).

I'm surprised by b), which seems to forsee the superposed states of quantum mechanics. Btw, this is different from the intuitive definition I would have given from the indeterministic system, i.e.: when E occurs, any state may follow.

I suppose that these are naive question about the status of the logical propositions… but your thoughts are most welcome.

Answer 2

Indeed, the logical conclusion is obvious, but I didn't expect it before I began to use my rudimentary logic.

This was my reasoning:

Lets take the mental states, M1, M2, M3… and the strictly corresponding physical states P1, P2, P3... Here, you may reason about the mental or the physical states as well. Lets use the (less scary) physical states.

Intuitively, I would have said that, in a deterministic world, P1 ↷ P2, meaning P1 is followed by P2 and nothing else. In an indeterministic world, I would have said that P1 ↷ P2 OR P1 ↷ P3 OR P1 ↷ Px. However, the proposition (3) says P1 ↷ P2 AND P1 ↷ P3, which seems to imply that, in an indeterministic world defined that way, one would get several states at the same time, rather than any single possible state (with its relative frequency as the measure of its probability).