Previously, I offered a terse definition of determinism. But I've given the topic more thought, and there are better ways to understand determinism. Here is a brief understanding in which I drop the notion of a causal law.
Definition 1 (state of affairs). A state of affairs is a non-empty set of well-formed formulas (wffs) in our first-order modal logic language L. Let S be a state of affairs, and let L1 be the set of all wffs in L; then:
(S ≠ ∅) ∧ (S ⊆ L1)
The propositional content of each φ ∈ S is determined by the interpretation of L. Moreover, we say that S obtains iff each wff (or proposition) in S is true:
(∀x ∈ S)(x)
Remarks on Definition 1. States of affairs are sometimes understood as abstract objects whose internal contents are usually denoted by sentences with the being gerundive, such as “Grass’ being green” and “John’s being engaged to Sally”. However, such an understanding coats the concept with a cloud of mystery; hence, to have well-defined states, we use the simpler definition in terms of sets of wffs (or propositions), WLOG.
Definition 2 (temporal split). Let S be a state of affairs. A temporal split on S is a double S` = <S1, S2> such that:
| 1. S1 ∪ S2 = S. |
| 2. S1 ≠ S2. |
| 3. ∅ ∉ {S1, S2}. |
| 4. S1 is conceptually, or descriptively, temporally prior S2. |
Example 1 (temporal split). Let t and t` be times such that t < t`, and let S be the following state of affairs:
S = {“1 + 1 = 2”, “Jones is alive at t”, “Jones is dead at t`”}
Then, the only possible temporal split on S, S` = <S1, S2>, is such that:
S1 = {“1 + 1 = 2”, “Jones is alive at t”}
S2 = {“1 + 1 = 2”, “Jones is dead at t`”}
Definition 3 (implies). Let S0 and S1 both be states of affairs. We say that S0 implies S1 iff S1 obtains in every possible world in which S0 obtains. More precisely, let M = <W, R, V> be a Kripke model, where W is the set of all possible worlds, R is a binary accessibility relation on W, and V is a function that assigns to each φ ∈ S0 ∪ S1 a set V(φ) ⊆ W. Then, S0 implies S1 iff:
(∀x ∈ W){(∀y ∈ S0)[x ∈ V(y)] ∧ (∀z ∈ S1)[x ∈ V(z)]}
Definition 4 (determinism). Let w be the actual world and, thus, in our model M, w ∈ W. Let Sw be the state of affairs that completely describes w, that is:
(∀x ∈ Sw)[w ∈ V(x)] ∧ ¬(∃x){(x ∉ Sw) ∧ [w ∈ V(x)]}
Moreover, let:
Ixy: x implies y
Then, determinism is the thesis that, for every temporal split on Sw, <x, y>, x implies y. Let A be the set of all possible temporal splits on Sw; then:
(∀x ∈ A)(Ix0x1)