## Sunday, February 14, 2016

### Necessary and Sufficient Causes

One of the common formulations of determinism is (quoting from Wikipedia)

Determinism is the philosophical position that for every event, including human interactions, there exist conditions that could cause no other event.
A world that satisfies this definition of determinism can evolve in only a single way over time. This definition is stonger than merely saying that the event's causes are necessary, sufficient, or even necessary and sufficient. To see why that is so, I will demonstrate a series of model worlds each of which is an intentionally flawed Interpretation of "Determinism". Each model will be a very simple imaginary "world"; it is not intended to reflect the real world, but is rather just an abstract thing that helps us understand the implications of the definition. Through this series of models, we'll see that the definition of determinism above is stronger than even saying that every cause necessary and sufficient for its effect.

For these models we have the following concepts:
• Variable: Some fact or condition about the state of the world. In our example will use a letter to designate each abstract fact about the world.
• State: The set of variables designating the state of the world at some time. In our example we will use a set of letters to designate the state of the world, but it will always happen to be a single variable at any given time.
• Cause: Some variable or variables that lead to an Event occurring (resulting in the specific Effect of that Event)
• Event: Something that can occur from one moment to the next and has a specific Effect
• Effect: A change in the state of the world
• Initial Condition: The state of the world at the beginning of some time of interest

### Every cause being necessary does not imply determinism.

First lets us examine necessary causes. According to Wikipedia
If x is a necessary cause of y, then the presence of y necessarily implies the presence of x. The presence of x, however, does not imply that y will occur.
A good example of a necessary cause in nature is particle decay. A particle that can decay is a necessary cause of the particle's decay, because the particle's decay necessarily implies that there was a particle there that could decay. The presence of a particle that can decay, however, does not imply that the particle's decay will occur. While the presence of a particle that can decay is necessary in order for the decay to occur, it is not sufficient for the decay to occur. The laws of physics (as best we understand them today) do not say that there is any sufficient condition for the particle to decay.

One might imagine that the claim that there is no sufficient condition for particle decay is merely a limitation of our knowledge, rather than a fundamental feature of nature. Perhaps there are hidden variables that cause the decay to occur at some particular time, but we simply do not know how to (or even cannot) measure these variables? That is not the case. It has been shown that the assumption that there are local (i.e. consistent with relativity) hidden variables underlying quantum indeterminacy contradicts observations of entanglement in experiments like the double-slit experiment and those demonstrating Bell's inequality. In any case, the point of our discussion here is to understand the implications of the definitions of causality, not to determine which of them are satisfied by nature.

Our first model simplifies the behavior of the KS meson, which can decay in more than one way. For simplicity we shall pretend that it can decay in precisely two ways. At any given moment it might not decay, or it might decay in one of these two ways. We describe our simplified model of the world as follows:
Initial condition: K (meaning the KS meson is present)
Possible events:
E1: precondition: K is present
change in state: no change in state (K does not decay)
E2: precondition: K is present
change in state: remove K from the state, and add P (decay to particle P)
E3: precondition: K is present
change in state: remove K from the state, and add Q (decay to particle Q)
There are more than one ways in which this world can evolve. We can have an unending series of occurrences of E1 (meaning the particle never decays). We can either have zero or more occurrences of E1 (meaning nothing happens for a time), followed by E2. Finally, we can have zero or more occurrences of E1, followed by E3. These latter two possibilities result in a different final state. This model world is thus a counterexample to the proposition that "every event has a necessary cause" implies that the world is deterministic.

### Every cause being necessary and sufficient does not imply determinism.

Let us examine sufficient causes. According to Wikipedia
If x is a sufficient cause of y, then the presence of x necessarily implies the presence of y. However, another cause z may alternatively cause y. Thus the presence of y does not imply the presence of x.
We can imagine an example of sufficient cause being the burning of a book. It would be sufficient for the book to be thrown into an incinerator for the book to burn. However, that is not necessary, as there are other sufficient conditions for it to burn (for example, being thrown on a bonfire). Being a sufficient cause is neither stronger nor weaker than being a necessary cause. It is worth asking: what if every cause is both necessary and sufficient for its effect? Would that be enough to imply that the world is deterministic? We can demonstrate that this is not so by another model world, as follows:
Initial condition: [A,B] (two different facts about the world are true: A and B)
Possible events:
E1: precondition: A is present
change in state: remove A from the state, remove Z from the state if it is present, and add X
E2: precondition: B is present
change in state: remove B from the state, and add Y and Z
In this example, the cause A is both necessary and sufficient for its effects (it always leads to E1 occurring). The cause B is both necessary and sufficient for its effects too. There are two ways in which this world can evolve. We can have an event produced by the first rule, followed by an event produced by the second rule. In that case the final state of the world is [X,Y,Z]. Or we can have an event produced by the second rule, followed by an event produced by the first rule. In that case the final state of the world is [X,Y]. As we can see, this world is not deterministic. This model world is thus a counterexample to the proposition that "every event has a necessary and sufficient cause" implies that the world is deterministic.

We can now see why the usual philosophical definition of determinism is stronger:
Determinism is the philosophical position that for every event, including human interactions, there exist conditions that could cause no other event.
It is worth asking whether or not any of these definitions might correspond to the real world we live in. As far as we know, time is continuous, rather than discrete. Since these models require discrete time, they don't fit the real world, though there are tantalizing theories (yet to be tested) that time might be quantized. In addition, the usual philosophical definition of determinism requires a fixed ordering of all events throughout the universe. Relativity teaches us that there is no objective order of the occurrence of events: event ordering varies from one (subjective) inertial reference frame to another. These two facts make it nearly impossible to reconcile any widely accepted modern theory of the physical world with philosophy's concept of "determinism". A more useful definition might be found in the physicist's use of the term "deterministic" to describe a system that is capable of having only one possible future evolution. The most widely held scientific theories that explain the results of quantum measurement are not deterministic in that sense.

### Some other definitions do not imply determinism

There are other definitions of determinism that one might find discussed on the web. It is sometimes hard to determine if they are stronger, weaker, or equivalent to the usual philosophical definition. For example, there is a model frequently discussed on http://breakingthefreewillillusion.com/. As far as I can tell the author's intended definition does not correspond to any of the definitions above. He has told me that the definition does not include sufficient causality, as that is logically inferred from his definition, and he has directed my attention to his proofs at http://breakingthefreewillillusion.com/must-lead-to-causality/ and http://breakingthefreewillillusion.com/otherwise-causal-contradiction/. Similarly, his definition does not include necessary causality: see http://breakingthefreewillillusion.com/necessary-sufficient-causality/. As I was unable to find a concise description of the definition he was using, I engaged with him over the course of a few weeks to draw up a definition to use for analysis. This is what we came up with:
• Event: A change in state from one moment to the next, described by its State Change.
• Causal event: An event that is caused by some set of Variables and whose Effect is derived entirely from those Variables.
• Cause: a set of pre-existing variables that derives (causes) the Effect in it's entirety (of a causal event). The variables of a cause includes all physical parts and physical laws that are described by formulas, which in the case of a causal event, force the output of the effect.
• Variables: The "parts" of a specific physical configuration (parts of the state). We shall indicate each variable by a letter like "A"; or a letter, and equals sign, and a number like "x=10".
• State Change: The ontological change in the state that results from an event. For State Change we need not include in the change mention of variables that are not changed, but the variables that do not change may play a role in the variables that do.
• Effect: A State Change for a Causal Event.
• State: The "whole" encompassing all variables in a specific physical configuration. We shall indicate a state by a list of variables between square brackets, for example "[x=10,A]"
• Scenario: Specific sequence of causes and effects that can be contrasted with other scenarios. We shall indicate a scenario notationally by a sequence of states, starting with the Initial State, indicating for reference the law of physics that describes the event between each pair of states.
• Initial State: The first state in a scenario.
• Laws of Physics (LoP): The constraining behavior of the universe, which is also inherent in all existing physical variables in the universe.
• Formulas for LoP: We shall represent the laws of physics using Equations, formulas, mathematics, or any sort of symbolic language.
• Determinism: "every event is causal"
A scenario starts with a given Initial State, and evolves through a sequence of steps by Events produced according to the world's Laws of Physics, each step producing a new State.

The scenario must continue until no further events are possible.

Our goal is to evaluate the following hypothesis:
Hypothesis: Given a particular Initial State, if a Scenario that begins with that Initial State satisfies Determinism, then every Scenario that begins with that Initial State is the same.
The author asked me to add this comment to this set of definitions:
"These definitions are one's that Trick [the author] and I, after much deliberation, have agreed to for the assessment I will make. It isn't the case that, for example, Trick's regular usage of the concept of causality requires an entirely physicalist account, but for the sake of our discussion we are both agreeing on a physical universe only. There are also some words that Trick normally wouldn't use, but for the sake of clarity we added them."
Now we can try the model world from our analysis of necessary and sufficient causes:
Initial state: [A,B]
The laws of physics permit the following possible events:
E1: precondition: A is present
change in state: remove A from the state, remove Z from the state if it is present, and add X
E2: precondition: B is present
change in state: remove B from the state, and add Y and Z
Scenario S1: [A,B] E1 [X,B] E2 [X,Y,Z]
Scenario S2: [A,B] E2 [A,Y,Z] E1 [X,Y]
To evaluate the hypothesis, we will analyze S1. There are two events. The first, E1, is a causal event with cause [A]. The second, E2, is a causal event with cause [B]. Thus this scenario satisfies the definition of Determinism. Similarly scenario S2 satisfies the definition of Determinism. However, we can see that scenarios S1 and S2 are different from each other. Moreover, they end with different world states. Thus this counterexample disproves the hypothesis. We can therefore conclude that this definition is weaker than the usual philosophical definition of Determinism, and the definition discussed at http://breakingthefreewillillusion.com/, if satisfied by a world, does not imply that the world's timeline "could not have been otherwise".

One might ask what causes one event to occur before the other in one scenario but vice-versa in the other scenario; what causes the ordering of events? In the ontology of this definition of determinism, an event (a change in state from one moment to the next) or an effect (a change in state for a causal event) can be caused, but the "order" of these changes is not something that can be caused. So the question does not make sense. The whole point of this example is to surface the fact that this definition of determinism does not require events to be ordered in any particular way. That is how it differs from the usual philosophical definition of determinism.

In his defense, the author has not evaluated my argument in detail, but has explained to me that I must be either
1. Asserting a variable with self-contradictory properties; or
2. Smuggling in an acausal event (an event without any cause)
though I do not know of any variable that is contradictory, or any event does not have a cause in this example.