Aug 31, 2011

Free Will and Determinism, part 5

Part 2:
Part 3:
Part 4:
Part 5:
Part 6:
Part 7:

Having provided support for the incompatiblist view, we have only shown the relationship between determinism and Free Will. This is only relevant in the event that one or the other is true (which makes the other false) or in examining world views that incorporate compatibilism as a fundamental component (which makes those world views false).

Causal determinism is generally dismissed as being false, at least to the degree necessary to support incompatabilist arguments. What other kinds of determinism are there?

Divine foreknowledge (or any sort of perfect foreknowledge, regardless of its nature) is a form of determinism, but we must demonstrate this.

In other forays into this topic, I have tried to use modal logic (previously discussed) to represent the concepts here, but have found it to be insufficient. I have found other forms of logic that work better. Rather than simply dive into them, I will build up to them from the ground level.

A proposition is any statement that can have a truth value assigned to it. "I have a five dollar bill in my pocket" is a proposition. It can be true or false. "Come here" is a command, not a proposition; it has no truth value.

An atomic proposition is a proposition that cannot be broken up into simpler propositions. "I have a five dollar bill and a ten dollar bill in my pocket" is not an atomic proposition as it can be broken up into "I have a five dollar bill in my pocket" and "I have a ten dollar bill in my pocket," each of which are atomic propositions.

For simplicity, we can represent propositions by using variables, rather than write them out each time we wish to refer to them. For example: p = "I have a five dollar bill in my pocket" and q = "I have a ten dollar bill in my pocket." Usually, we assign lables only to atomic propositions and represent more complex propositions by connecting atomic propositions with the appropriate logical connectives:

p AND q = "I have a five dollar bill in my pocket and I have a ten dollar bill in my pocket."

Sets are a mathematical/logic concept. A set is a collection of, well, anything you want. In logic, we generally talk about a set of propositions, objects, and other logical concepts. Here we will be talking about a set of propositions. Like atomic propositions, sets are referred to using variables:

A = {p, q}

I have defined the set A as containing the propositions p and q.

Moving forward, I will use the following propositions:
D = "I am dead."
A = "I am alive."

And the following set:
AP = {D, A}

AP stands for "atomic proposition" and is the set of all atomic propositions relevant to our discussion. In this case, D and A.

A state represents the universe at some point in time. More specifically, a state is a set of all atomic propositions true at the same time, at some arbitrary point in time. Any set that represents a state is a subset of AP. That is, a state set can only contain propositions that are also in AP. Essentially AP contains all of our building blocks, and states are different ways to combine those building blocks. Given the AP we defined above, possible states include:

s0 = {} (The Null, or empty, set)
s1 = {D}
s2 = {A}
s3 = {D, A}

The null set is of no use to us, and semantically, s3 is contradictory - I can't be alive and dead at the same time. If we limit ourselves to only the sets that make sense, and renumber, we get:

s0 = {D}
s1 = {A}

So far we have AP, the set of all atomic propositions that are relevant to this discussion, and the set of all (acceptable) states (s0, s1, s2...) that are composed of atomic propositions found within AP. Semantically, each state should represent some possible world, similar to modal logic. But we will add another component by relating the states to each other. The relation set, R, is a set of ordered pairs of states such as (s1, s2). Given our states above, the possible ordered pairs are:

(s0, s0)
(s0, s1)
(s1, s0)
(s1, s1)

The set R is a subset of this list; it can't contain anything not on this list. Since each of our states only have one atomic proposition, we can for illustrative purposes, replace the state with its corresponding proposition:

(D, D)
(D, A)
(A, D)
(A, A)

So what does R represent? In this type of logic, R represents an acceptable transition. That is, if we are currently in the state represented by the first item in the ordered pair found in R, then it is acceptable for us to transition to the state represented by the second element in that ordered pair. The above list is reprinted below, with the meaning of the transition:

(D, D) - "I am dead now, and I can continue being dead in the next moment"
(D, A) - "I am dead now, and I can be alive in the next moment"
(A, D) - "I am alive now, and I can be dead in the next moment"
(A, A) - "I am alive now, and I can continue being alive in the netx moment"

We discard the second ordered pair as being unacceptable, we get get our set R:

R = {(s0, s0), (s1, s0), (s1, s1)}

Here we begin to fix some of the problems previously discussed with modal logic. It's not enough that a state represent a possible world, but we want to know if we can get to a future state based upon a previous, or initial state. If I can't get to a future state from the state I'm currently in, then we can't say that I have Free Will regarding choices that exist in those inaccessible future states. This gets us closer to where we need to be, but we're not quite there yet.

To recap:

AP - the set of relevant atomic propositions

States - a set of propositions in AP that can be true at the same time.
R - the set of ordered pairs that represent valid state transitions

What does this give us that modal logic doesn't?

Modal logic only talks about a finite number of propositions and whether they are: true in the actual world, true in some possible world, true in all possible worlds.

There doesn't appear to be a way to easily:

- Talk about whether or not some number of propositions are true in the same possible world (unless that possible world is the actual world)
- Incoporate issues involving time and ordering of events
- Determine if a possible truth can become an actual truth. That is, to determine if the actual world we are in can, through a series of valid state transitions, end up in a possible world where a given proposition is true.

It is this latter one that is important to us, but we are not yet ready to address it in the context, first we have to describe determinism and Free Will in the logic we have modeled above.

More details on the logical concepts discussed above:

I have borrowed heavily from Computation Tree Logic (CTL), specifically the construction of Kripke structures. I have am not, by far, bringing the full force of that logical system to bear here, I am only bringing in the elements I think I necessary to illustrate the points I am trying to make. For one, I want to keep this as simple as possible and two, my research shows different ways of expressing this logic and I want to keep this as simple as possible.

CTL is related to, and is actually an evolution of, modal logic. Bare modal logic has no consideration of time. Adding a temporal element gives you temporal modal logic, which contains operators that denote tense, such as "past" and "future" in the same sense as "possible" and "necessary." Linear Time Logic, is a type of temporal modal logic. CTL is related to LTL, and adds a branching element, though the latter is not fully subsumed into the former.

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