What is *causal determinism*? An initial answer is: nobody really knows! Much of the literature that discusses causal determinism avoids a deep, precise definition of the term, and leaves us with a vague notion that is no clearer than a drunk man’s vision. But if you squeeze the determinist hard enough, he might drip out a definition along these lines:

The physical world is determined iff, given a certain way things are at a time t, the way things are after t is fixed and could not have been otherwise.

But a mere pinch of reflection reveals that the above definition is too vague. To see this, let us try understand what it means for one state of affairs to determine another.

**Definition 1 (state of affairs)**. If S is a state of affairs, then S is a non-empty set of propositions that describe part, or the entirety, of reality. Let Px = “x is a proposition that describes reality”, then:

(S ≠ ∅) ∧ [(∀x∈S)(Px)]

We say that S obtains iff each proposition in S is true. Let Ox = “x obtains”, and let V be the set of all states of affairs (i.e., V = { x : x is a state of affairs }), then:

∀x∈V[Ox ≡ (∀y∈x)(y)]

*Remarks*. Now, what does it mean for one state of affairs to causally determine another? According to the definition above, it would mean that, for any two states of affairs, x and y, if x obtains, and if, necessarily, x’s obtaining implies that y obtains, then x causally determines y. But this is incorrect because, for one thing, it entails that all necessary states of affairs are determined.

**Lemma 1**. Let Dxy = “x causally determines y”, let Ox = “x obtains”, and let our domain of discourse (D) be V, that is, D = { x : x is a state of affairs }; then:

¬(∀x,y){[Ox ∧ □(Ox → Oy)] ≡ Dxy}

*Proof 1.* Let a = {“Grass is green”} and b = {“2 + 2 = 4”}. Moreover, let us assume the non-controversial view that b is necessary (i.e., b obtains in all possible worlds), and that a does not determine b (for how can grass’ being green determine that 2 + 2 = 4?); then:

1. Oa | Premise |

2. □Ob | Premise |

3. ¬Dab | Premise |

4. (∀x,y){[Ox ∧ □(Ox → Oy)] ≡ Dxy} | Assumption |

5. □Ob → (□Oa → □Ob) | 2, →-addition |

6. □Ob → □(Oa → Ob) | 5, Distribution |

7. □(Oa → Ob) | 2, 6, Modus ponens |

8. Oa ∧ □(Oa → Ob) | 1, 7, Conjunction |

9. [Oa ∧ □(Oa → Ob)] ≡ Dab | 4, 8, Universal instantiation |

10. {[Oa ∧ □(Oa → Ob)] → Dab} ∧ {Dab → [Oa ∧ □(Oa → Ob)]} | 9, Definition of ≡ |

11. [Oa ∧ □(Oa → Ob)] → Dab | 10, ∧-elimination |

12. Dab | 8, 11, Modus ponens |

13. ⊥ | 3, 12 |

14. ¬(∀x,y){[Ox ∧ □(Ox → Oy)] ≡ Dxy} | QED |

*Proof 2*. Let a = {“It is raining in Paris at t”} (for some time t) and b = {“It is cloudy in Paris at t”}. Both a and b are contingent (not necessary), yet, necessarily, a implies b: □(a → b). Moreover, if a occurs, then a does not causally determine b to occur; our intuition screams how obvious this is. But if we assume that [Oa ∧ □(Oa → Ob)] ≡ Dab, then a does causally determine b. We thus reach a contradiction, so our assumption must be false.

*Remarks*. Lemma 1 highlights the grand separation of material entailment from causality. Thus, we need to introduce the notion of causality, even if we leave it undefined, at least for now. So, let Cxy = “x causes y”; then, for any two distinct states of affairs, x and y, if x causes y, then x obtains and y obtains:

(∀x,y)[Cxy → Ox ∧ Oy ∧ (x ≠ y)]

We might now understand “causally determines” as follows:

(∀x,y){[Cxy ∧ □(Ox → Oy)] ≡ Dxy}

But certain issues poison this understanding. First, *causality* remains undefined. What does it mean, exactly, for *x to cause y*? Despite our best efforts, a satisfactory account of causation keeps jumping out of our reach, laughing and waggling its tongue at us like a mischievous schoolboy.^{1} So, we are trying to understand a baffling concept – determinism – by building on an even more mystifying concept – causation.

Second, causal determinism is usually (or perhaps virtually always) restricted to the physical world – the rough idea is that *physical* states of affairs causally determine other *physical* states of affairs. But the above understanding, (∀x,y){[Cxy ∧ □(Ox → Oy)] ≡ Dxy}, does not consider this restriction.

Third, our intuitive concept of causation cries out that there is some causal law, so to speak, that governs causality in the universe. But the above understanding, once again, does not consider this.

What can we do to solve these issues? I do not know. We might remain unable to solve them all, I’m afraid; but perhaps we can make some progress. So, let us introduce the notion of a causal law.

**Definition 2 (causal law)**. If A and B are non-empty sets of states of affairs:

(A ∩ B ⊃ V) ∧ (∅∉{A, B})

then the *causal law* between A and B is the function f: A → B such that, for all x∈A, and for all y∈B, f(x) = y iff the obtaining of x implies both that y obtains and that x causes y:

(∀x∈A)(∀y∈B){[f(x) = y] ≡ [Ox → (Oy ∧ Cxy)]}

If the graph of f, R_{f} = { <x, y> : f(x) = y }, is empty (i.e., R_{f} = ∅), then we say that a causal law between A and B *does not* exist; else a causal law between A and B *exists*.

*Remarks*. Although causation remains undefined and we must rely on our intuition about it, at least a causal law is now on our radar. Moreover, a causal law offers flexibility in defining causal determinism, as we will see next.

**Definition 3 (causally determines)**. For any two distinct states of affairs, x and y, x *causally determines *y iff (1) both x and y obtain, and (2) necessarily, there exists a causal law, f: A → B, between two sets of states of affairs, A and B, such that x∈A and y∈B:

(∀x,y){[Ox ∧ Oy ∧ □(f(x) = y)] ≡ Dxy}

*Remarks*. A seemingly but artificial paradox of Definition 3 is that a state of affairs, in which a person has free will, can be determined. We will not provide a proof for that here, but it is easy to see that no logical contradiction exists in a person’s being causally determined (as per Definition 3) to have free will.

**Definition 4 (causal determinism)**. Causal determinism is the thesis that, for any physical state of affairs (i.e., a state of affairs that only describes physical reality) x, if x obtains, then there exists a distinct physical state of affairs y such that y causally determines x. Let our domain be the set of all physical states of affairs:

D = { x : x is a state of affairs that only describes physical reality }

then:

(∀x)[Ox → (∃y)(Dyx)]

*Remarks*. Perhaps this is the best we can do in defining causal determinism? I am unsure. However, does this definition of causal determinism have pimples and warts and other hidden ailments? I think it does; but that is a discussion for another time.

See, for example, https://iep.utm.edu/causation/↩︎