Previously, we saw that the supervenience formula for “A supervenes on B” is:
¬◊(ΔA · ¬ΔB) (1)
Here are some examples to illustrate how we apply it. We begin with the famous example from philosophy of mind. We then consider a selection of disparate areas of inquiry where we could awkwardly use the language of supervenience. We want to show that talk about supervenience is just to talk about one aspect of the logic of change.
1. Philosophy of Mind
As originally stated by Davidson [1], the mental supervenes on the physical just in case it is impossible for there to be a change in our mental states without there also being a change in the physical state of our brains.
To say that the mental supervenes on the physical can be understood like this:
Let A be the collection of my mental states
Let B be the collection of my brain states
¬◊(ΔA · ¬ΔB)
Because the supervenience formula is logically equivalent to the strict implication ΔA ⇁ ΔB, someone who believes that the mental supervenes on the physical believes that a change in my mental states strictly implies a change in my brain states. So, if there is no change in my brain sates, there is no change in my mental sates, as expressed by the following valid inference:
ΔA ⇁ ΔB
¬ΔB
∴ ¬ΔA
Typically, such a philosopher assumes the negation of the converse supervenience claim, i.e., that it is possible that there is a change in my brain states without there being a change in my mental states. So, if we add this to the general claim that the mental supervenes on the physical, we are saying:
(ΔA ⇁ ΔB) · ¬ (ΔB ⇁ ΔA) (2)
When stating that the mental supervenes upon the physical, it would be useful to know whether one means (1) or (2).
Denying that the mental supervenes upon the physical is simply to assert:
◊(ΔA · ¬ΔB) ⇔ ¬ (ΔA ⇁ ΔB)
2. Elementary Mathematics
The rich language for talking about change in elementary mathematics could awkwardly be translated into talk about supervenience. Here's one simple example.
Let y=f(x) be a function. The value of y supervenes on the value of x, that is:
¬◊(Δy · ¬Δx)
Notice that in general it is not true that the value of x supervenes on the value of y. In other words,
◊(Δx · ¬Δy)
However, if we restrict our attention to one-to-one monotonic functions, then the value of y supervenes on the value of x and conversely:
(Δy ⇁ Δx) · (Δx ⇁ Δy) ⇔ Δx ⇌ Δy
3. Elementary Physics and Beyond
Because we can use supervenience to talk about functional relationships between two variables in elementary mathematics, we can carry over that talk to physics and any other area that models relationships among variables as mathematical functions. For example, in the kinetic theory of gases one could say that the temperature of a gas T supervenes on the mean kinetic energy of the molecues K:
¬◊(ΔT ·¬ΔK)
Since the functional relationship between temperature and mean kinetic energy is one-to-one and monotonic, the mean kinetic energy also supervenes on the temperature, i.e.,
ΔT ⇌ ΔK
This same type of co-supervenient relationship obtains between any two variables in a functional relationships in physics and beyond. For example, in physics, E = hν, where E is the energy of a quantum of light and ν is its frequency. Energy supervenes on frequency and conversely, though no one would say it quite that way. To cite an example from elementary economics, if one spends 10% of one's income on vacations annually, the amount one spends on vacations V supervenes on one's income I and conversely.
There are other interesting relationships among changes in physics that are not, strictly speaking, about how two quantites are functionally related. For example, let I be an inertial frame. And let V be the relative velocity of two inertially moving objects. The choice of inertial frame does not supervene on V, because it is possible to change the inertial frame and not change the relative velocity V:
◊(ΔI ·¬ΔV)
This is a special case of a failure of supervenience that always occurs when there is gauge freedom in the mathematical representation of a physical system (or “surplus structure”). For example, in general relativity, if S is a solution to the Einstein Field equations, so is d*S, where is a diffeomorphism. However, the spacetimes S and d*S are observationally indistinguishable. If we let O be the set of all observables defined on a spacetime, and we regard S and d*S as distinct spacetimes, then the spacetime does not supervene on the observable quantities since:
◊(ΔS ·¬ΔO)
4. Statistics & Quantum Mechnaics
Consider a fair coin that is tossed a large number of times. We adopt the relative frequency interpretation of probability. The probability of the coin coming up heads P(H)is equal to the probability of the coin coming up tails P(T). We have:
P(H) = P(T) = 50%
Suppose now that we let σ denote a particular sequence of heads and tails as they occurred in a series of many, many trials. The sequence of heads and tails does not supervene on the probability of getting heads because:
◊(Δσ ·¬ΔP(H))
Now consider a standard EPR-Bohm type configuration in elementary quantum mechanics. We measure spin in the Z-direction on either side of the experiment. On the left side, we get a random sequence of spin-up (+) and spin-down (-) measurement outcomes. As with the coin toss, the probability of getting spin-up is equal to the probability of getting spin-down:
P(+) = P(-) = 50%
Now let σL and σR be the sequences of spin measurement outcomes on the left side and right side of the experiment respectively, each of which we can represent as a random sequence + - + + + - - + +... If we focus on just one side of the experiment, we have the same situation as we had with the coin. The sequence of + and - measurement outcomes does not supervene on the probabilities P(+) or P(-).
However, the puzzling thing is that the sequence σLof + and - on the left supervenes on the sequence σR and conversely. The infamous strict anti-correlations of the EPR setup can be expressed like this:
ΔσL ⇌ ΔσR
Supervenience and Correlation
In general, for any two variables x and y, if x and y are strictly correlated or anti-correlated, x supervenes on y and conversely.