Mathematics has a formal language for describing change. Philosophers, if confined to natural languages, describe logical relationships among changes using words. "Supervenience" describes a specific type of relationship between how one object's change depends on another's.
Discussions of supervenience, as represented for instance in the excellent article on the Stanford Encyclopedia of Philosophy [1], typically talk about two sets of properties A and B of two objects. We'll work at a slightly more general level by focusing on the state of two systems. Later, we can ask whether the state of a system at a given time is fully determined by its properties at that time.
We focus first on writing the supervenience relation using modal logic.
The Supervenience Formula
Let A and B be two states of two systems SA and SBwhose changes of state may be related. Supervenience is typically described by the slogan “there cannot be an A-difference without a B-difference”
To simplify:
Let A be the state of system SA
Let B be the state of system SB
Let Δ represent “A change in...”
Let □ and ◊ be the usual modal operators
¬◊(ΔA · ¬ΔB)
The formula reads:
It is not the case that it is possible for there both to be a change in the state of system A of SA and there not be a change in the state of the system B of SB.
More colloquially, our supervenience formula simply says:
It is impossible for there to be a change in A and there not be a change in B.
The Value of the Supervenience Formula
The main value of writing our supervenience formula surfaces when we use familiar logical equivalences in propositional logic and modal logic to re-write it as:
□(ΔA → ΔB)
This formula just says that a change in A strictly implies a change in B. Using ⇁ for strict implication, we have:
ΔA ⇁ ΔB
Using strict implication is convenient because we know from modal logic (S5) that strict implication supports two common inferences:
ΔA ⇁ ΔB
ΔA
∴ Δ B
ΔA ⇁ ΔB
¬ΔB
∴ ¬ΔA
If we agree to treat all conditionals expressed in natural language as strict conditionals, we can also say:
A supervenes on B means that there's a change in A only if there's a change in B.
Finally, writing our supervenience formula also helps us identify clearly what it means to deny a superveneience claim. To say that A does not supervene on B is simply to say that it is possible for there to be a change in A but no change in B:
◊(ΔA · ¬ΔB)
References
[1] McLaughlin, Brian and Karen Bennett, "Supervenience", The Stanford Encyclopedia of Philosophy (Summer 2021 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/sum2021/entries/supervenience/>.