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 L_{1} be the set of all wffs in L; then:

(S ≠ ∅) ∧ (S ⊆ L_{1})

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` = <S_{1}, S_{2}> such that:

1. S_{1} ∪ S_{2} = S. |

2. S_{1} ≠ S_{2}. |

3. ∅ ∉ {S_{1}, S_{2}}. |

4. S_{1} is conceptually, or descriptively, temporally prior S_{2}. |

**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` = <S_{1}, S_{2}>, is such that:

S

_{1}= {“1 + 1 = 2”, “Jones is alive at t”}S

_{2}= {“1 + 1 = 2”, “Jones is dead at t`”}

**Definition 3 (implies).** Let S_{0} and S_{1} both be states of affairs. We say that S_{0} *implies* S_{1} iff S_{1} obtains in every possible world in which S_{0} 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 ∈ S_{0} ∪ S_{1} a set V(φ) ⊆ W. Then, S_{0} implies S_{1} iff:

(∀x ∈ W){(∀y ∈ S_{0})[x ∈ V(y)] ∧ (∀z ∈ S_{1})[x ∈ V(z)]}

**Definition 4 (determinism).** Let w be the actual world and, thus, in our model M, w ∈ W. Let S_{w} be the state of affairs that completely describes w, that is:

(∀x ∈ S_{w})[w ∈ V(x)] ∧ ¬(∃x){(x ∉ S_{w}) ∧ [w ∈ V(x)]}

Moreover, let:

Ixy: x implies y

Then, *determinism* is the thesis that, for every temporal split on S_{w}, <x, y>, x implies y. Let A be the set of all possible temporal splits on S_{w}; then:

(∀x ∈ A)(Ix_{0}x_{1})