Formalization of the freedom of the will: This section
precisifies some of the things that are said in
Freedom of the Will.
The argument is that the freedom
of the Will, consciousness and
choice are interrelated. Here follows
such an argument.
One is conscious of the present in the
present, of the environment as rendered by one's senses, and of one's
thoughts that attempt to represent oneself and one's environment
selectively. Why is this so  why do humans have consciousness of some
if so much in one's body and environment run their courses unconsciously
and automatically, as it were?
Because it helps the body survive and
because it enables it here and now to choose whatever seems the
best chance for survival or for gratification of one's desires or for
the satisfaction of one's needs.
Consciousness is a selective
representation of parts and states of oneself and of one's environment,
namely such as one expects to help gratify one's desires, satisfy one's
needs, and serve one's interests  where most of these expectations are
either innate or acquired by education.
Now presupposing for the moment some material from the logic of
propositional attitudes, we can formalize relevant points here as
follows. I give the translations with the formulas.
T1. 0 <= (p(q)_{.t>0}) <= 1.
That is: Future events q of any kind
whatsoever have have any possible probability. This would follow from an
axiom like 0 <= p(q) <= 1, where tempolly unqualified statements are
supposed to hold for any time. The above differs from this only
instantiating this axiom to t>0 i.e. times later than now. The axiom is
part of standard probability theory
Therefore:
T2. 0 <= (p(a,q)_{.t>0}) <=
1.
That is: Future events q of any kind
whatsoever have have any possible personal probability. This follows as
a matter of logic from T1: If it holds for all probabilities, then it
also for a's probabilities, where a is an arbitrary person.
Next, we need an axiom:
A1. ~( 0 < (p(q)_{.t<=0}) <
1).
That is: Present and past events 
those before now  of any kind have an extreme probability. Put
otherwise: Present and past are determinate. This is common sense, at
least as regards the past. I think most will also agree that the present
is determinate if perhaps unknown.
T3. p(q_{i}._{t=0}) =1 V p(~q_{i}._{t=0})=1
Again simply a question of logic: What
happens now is so. This corresponds to (q_{i} V ~q_{i}) without temporal
qualification. It follows from A1.
D1. e(a,q)_{t=0} = p(a,qaCq )_{t=0}*v(a,q)_{t=0}
Person a's expectation for q is
a's probability for qaCq times a's value for q, all temporally for now
or later. By the way, it should be the case that e(a,q._{t<0})=1
V e(a,q._{t<0})=0, also if there is not enough information to
say what is the case.
D2. w(a,q_{i}._{t>=0}) = e(a,q_{i}._{t>=0}) if v(a,q_{i}._{t>=0})>0 else 0
a's weight for q_{i} is a's expectation if
a's value for q_{i} is greater than 0 and else 0, all temporally for now or
later.
D3. p(aCq_{i}._{t>=0}) = w(a,q_{i}._{t>=0}) : ∑j w(a,q_{j}._{t>=0})
The probability that a tries to cause
q_{i} is a's proportional weight for q_{i}, all temporally for now or later.
Note that this implies by D2 that a will not try to cause possible
events for which a's values are equal or less than 0, since their
probability equals 0.
A2. p( p(aCq_{i}._{t=0}) = 1) = p(aCq_{i}._{t>=0})
The probability that the probability
that a does try to cause q_{i} is true is the probability that a tries to
cause q_{i}.
A3. s rel aCq_{i}._{t=0} > (sea_{.t=0})
Any s is relevant to a's trying to
cause any q_{i} only if s is part of a.
Note that A3 expresses an important
feature of free will: Whatever is relevant to a's acts (that are free)
is part of a  there is nothing relevant to a's acts that is not part of
a.
And note also that this may be backed up by considerations of
locality, as are normal in physics: The only thing capable of effecting
something must be in direct contact with it, and thus in the same
locality.
A4. (Eq_{i})(Ea)(p(a,qaCq_{i}._{t>=0}) >
p(a,q~aCq_{i}._{t>=0}) &
p(qaCq_{i}._{t>=0}) > p(q~aCq_{i}._{t>=0}) ).
There are events q_{i} and
persons a such that a's probability for q_{i} if a tries to
cause q_{i} exceeds a's probability for q_{i} if a does
not try to cause q_{i} and indeed the real probability of q_{i}
given that a tries to cause q_{i} exceeds the real probability
of q_{i} given that a does not try to cause q_{i}.
This is the reason for having free
will, or the reason why it helps survive: At least sometimes when
someone believes himself to make a difference to the chances of
something happening this really is so.
There are several possible variants
here to consider. But the present form is neat and nice in stipulating
what amounts to a knowledge condition: One's personal probabilities for
one's effectiveness are at least qualitatively true, sometimes. And it
is also minimalistic.
Note that A4. can be written as
T4. (Eq_{i})(Ea)(aK (q_{i}
prel aCq_{i})._{t>=0})
And we have
T5. p(aCq_{i} ) = p(aCq_{i.t>=0}) if t>=0
= 1 if aCq_{i.t=0}
= 0 if ~aCq_{i.t=0}
This follows from earlier assumptions.
There are things to be clarified,
straightened out and added, but let's consider two general questions
A. What level of animal may have this?
B. Can computers have it?
To start with (B): Obviously, it all
depends, then, on whether computers can assign probabilities and values;
can represent themselves at least partially; and can act independently
from their environment only bases on some of their own internal states
and unconstrained except by nonextreme probabilities.
It's a bit difficult to say, for in one
sense the answer is obviously yes, and in another sense it much depends
on what one means by "assigns" and, especially by "values". The last
problem may be considered by noting computers have no natural needs,
though obviously ends may be programmed into them, including
selfmaintenance and "be kind to humans, even if they are boringly
stupid and disgustingly mean".
Now consider (A): In principle, it
would seem, whatever is alife and moves has some selfinterest in having
free choice, if it is in it's interest or feels well to keep living, and
escape the maws of whatever preys on one and the pains of starvation. So
something as simple as phototropism may involve some amount of choice.
However, as I noted before, both cases
(A) and (B) might be covered in principle by a shortcircuit that
produces the specious present, consciousness, and the need to choose by
making up one's mind afresh, given the present.
This also need only involve a small
part of one's brain or processor: The point of principal importance is,
after all, merely that the past is dissociated from the present, thereby
making some present, however small, simple and partial.
