As we saw in Part 1, the supervenience formula works to capture the meaning of supervenience in philosophy of mind. We began to see how it can be applied to kinds of changes in mathematics, elementary physics, and quantum mechanics. To illustrate the variety of ways in which the supervenience formula can be used, we now consider cases from disparate areas of inquiry: Computing and AI, learning and foolishness, public policy, and organizational change.
The main point of these examples is to show that what we really need is to develop a logic of change. Supervenience then turns out to be just one type of logical relationship among pairs of changes. No wonder that the concept of supervenience finds such wide application in philosophy; for unless one is an extreme monist, one's philosophy has to deal with change.
5. Computing and AI
In a standard algorithm in which randomness does not figure, the output O supervenes on the input I. It is impossible for there to be a change in the output and there not be a change in the input:
¬◊(ΔO · ¬ΔI) (1)
The converse does not hold. For example, every time we enter 5 + 2 in a calculator and then press = we get 7. However, we can get 7 as an output after an indefinite number of operations. So, the input does not supervene on the output:
◊(ΔI · ¬ΔO) (2)
We can, as before, use the strict conditional to describe the relationship between change in output and change in input in an algorithm that does not include randomness. The output strictly implies the input, i.e., there is a change in the output only if there is a change in the input, which we write ΔO ⇁ ΔI, but not conversely.
Even a passing acquaintance with generative AI shows us that in those algorithms, i.e., neural nets that include randomness, it is possible for there to be a change in the output and there not be a change in the input. In these kinds of algorithms, the output does not supervene on the input:
◊(ΔO · ¬ΔI) (3)
In generative AI that creates text output, it's not so clear whether it is possible to have a change in the input but no change in the output. The main reason for this is that even with the same input it is extremely improbable that one will get exactly the same output.
Foolishness and Learning
Frequently, people repeat the tired refrain that it is the definition of “insanity” (let's just call it foolishness) to continue doing the same thing and expect different results. It is foolish to believe, we could now say, that the results R do not supervene on what we are doing D:
◊(ΔR · ¬ΔD) (4)
6. Foolishness is failing to recognize that the results supervene on what we are doing.
However, the tired refrain is insensitive to the Drano effect. When the drain is slow and I apply clog remover, there is no noticeable change in the rate at which the water drains. However, if I repeat the application of the clog remover several times, the rate at which the water drains improves significantly. If we let R be the rate at which the water drains and D what we are doing, i.e., applying the same amount of drain remover, then:
◊(ΔR · ¬ΔD) (5)
Naturally, we do not believe in general that for any physical process change accumulates from nothing. The key is simply that we did not notice the “microscopic” changes taking place in R, which eventually accumulate to a noticeable change.
Now suppose I am trying to learn a skill, such as how to juggle a soccer ball or how to play a song on the piano. The old adage that “practice makes perfect” has recently been updated to “practice makes permanent”. To work toward perfection, common wisdom dictates, we don't just practice, i.e., repeat the same thing over and over. We engage in intentional and reflective practice making small changes in light of a careful analysis of what leads to improved outcomes. If O is our desired outcome and P is the set of practice behaviors, the “new” wisdom about practicing is that O supervenes on P:
¬◊(ΔO · ¬ΔP) (6)
This relationship between practice and outcomes is not new at all, of course. It is, for example, the centerpiece of Dewey's view of learning as much as it is of Popper's “trial and modification” view of scientific knowledge. Arguably, it goes much further back in philosophy at least to the time of Aristotle and his conception of a moral virtuous circle the ascent through which leads to a better character. We'll discuss this separately later.
6. Public Policy
Suppose someone wanted to defend the claim that we cannot change the rate of gun violence G without changing gun legislation L. They would be arguing that gun violence supervenes on gun legislation. Using the supervenience formula, we could write:
¬◊(ΔG · ¬ΔL) (7)
Interpreting “only if” as indicating strict implication, such a person is claiming that there is a change in rates of gun violence only if there is a change in legislation. This supports the inference that if there is no change in gun legislation, there will be no change in the rate of gun violence.
A proponent of changes to gun legislation will recognize, nevertheless, even if we change gun legislation, it may happen that there is no change in gun violence: a change in gun legislation does not supervene on changes in gun violence, i.e.,
◊(ΔL · ¬ΔG) (8)
While rates of gun violence supervene on gun legislation, the converse may only hold ceteris paribus. Exactly what counts as all other things being equal in this case is specific to this case. However, this is just a special case of types of changes in which a supervenience relationship holds in one direction but the converse holds only ceteris paribus
7. Organizational Change
A business wishes to change how it delivers a service S to its clients. Employees are convinced that such a change will be accompanied by a change (specifically an increase) to their workload W. The firm claims that it is possible to change how the service is delivered and not change work load. The employees claim that a change in service supervenes on their work load:
¬◊(ΔS · ¬ΔW) (9)
Management denies this claim and maintains that the change in service does not supervene on working conditions:
◊(ΔS · ¬ΔW) (9)
Supervenience Simplified
In six of the seven examples we have explored talk of supervenience seems philosophically heavy-handed. Yet, it can be a compact way of expressing the relationship between how changes in one thing are related to changes in another thing. For us, the supervenience formula is particularly helpful because it clearly articulates the relationship among the changes in pairs of things without importing associations with the word supervenience, for example with the concept of emergence, which would in many contexts be out of place.
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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.
- Details
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/>.
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