Free Will and Determinism, part 6

Introduction: http://www.themindisaterriblething.com/2011/08/free-will-and-determinism.html
Part 2: http://www.themindisaterriblething.com/2011/08/free-will-and-determinism-part-2.html
Part 3: http://www.themindisaterriblething.com/2011/08/free-will-and-determinism-part-3.html
Part 4: http://www.themindisaterriblething.com/2011/08/free-will-and-determinism-part-4.html
Part 5: http://www.themindisaterriblething.com/2011/08/free-will-and-determinism-part-5.html
Part 6: http://www.themindisaterriblething.com/2011/09/free-will-and-determinism-part-6.html
Part 7: http://www.themindisaterriblething.com/2011/09/free-will-and-determinism-part-7.html

Having established a logical basis with which to discuss Free Will and foreknowledge, we now must define those terms using the language of that logical model. We have previously identified two primary interpretations of Free Will: the ability to do (choose) otherwise, and to be the ultimate source of one's choices. We will first tailor our definition toward the first, and then deal with the second later.

As a note and caveat, we will talk of Free Will as, primarily, the ability to do otherwise. This is easiest for visualization purposes. We implicitly acknowledge the objection to defining Free Will discussed in part 2, and note that, to avoid this objection, we can simply substitute "choose" wherever we use "do."

Furthermore, we need to define Free Will that addresses the issues brought up in part 3, by using the tree/bridge example and applying it to our logical model.

To recap: One comatabilist refutation has been that, we can have Free Will, even if the present and future are determined by the past, by noting that the past could have been otherwise, therefore the present could have been otherwise and, therefore, we could have done otherwise, thus meeting the criteria for Free Will.

This refutation was criticized by constructing a scenario that meets this interpretation but, nevertheless, results in the conclusion that we did not have a Free Choice available to us:

There is a storm which knocks down a tree blocking a bridge.

Across the bridge is an injured person, but I cannot reach him.

Since it could have been that the storm did not knock down the tree, it also could have been that the bridge was free and I went across to help him.

Therefore, I had Free Will in not helping the injured person

This is nonsensical. In the case where the tree fell, no choice was even presented for us to choose, freely or otherwise. Displayed visually:

We only have a choice to make in the green circle, and that is the only circumstance in which we can reasonable say we have Free Will. To suggest that we have Free Will in the red circle would suggest that, we chose not to help the person and, by most moral codes, are morally responsible for that person suffering.

This representation can be easily translated to our logical system. Each circle above represents a state, and the text in there represents the propositions true at that state.

Propositions:

w = "a storm happens"
f = "a tree falls"
~f = "a tree does not fall"
h = "I help the person"
~h = "I do not help the person"

Sets:
AP = {w, f, ~f, h, ~h}
s0 = {w}
s1 = {f}
s2 = {~f}
s3 = {h}
s4 = {~h}
R = {(s0, s1), (s0, s2), (s1, s4), (s2, s3), (s2, s4)}

If we replace the circles with the labels for their respective states, we get:

Note that, in tracing a path from one state to the next, each step can be found in R. That is, this is a visual representation of all possible paths. With this representation we can formulate a definition of Free Will:

Free Will occurs when we are presented with a choice such that, for all outcomes of that choice, we can transition to a state where the outcome of that choice has occur.

More formally, we can define a choice between A and B as C(A,B) and our definition as: If C(A,B) is true in some state si and we have Free Will with regards to that choice, then either: A is true in some state sj, such that (si,sj) is in R (a valid transition); or B is true in some state sk, such that (si,sk) is in R (a valid transition).

Another representation:

Thus, even if we consider the red circle to contain a choice between A (Helping) and B (Not Helping) the choice isn't free because, while there is a valid transition between that state and B there is no valid transition between that state and A. Thus, no Free Will.

Next: Determinism