Maarten Maartensz:    Philosophical Dictionary | Filosofisch Woordenboek                      

 W - Will - Freedom of - Formalized


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 


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(qi.t=0) =1 V p(~qi.t=0)=1

Again simply a question of logic: What happens now is so. This corresponds to (qi V ~qi) without temporal qualification. It follows from A1.

D1. e(a,q)t=0 = p(a,q|aCq )t=0*v(a,q)t=0

Person a's expectation for q is a's probability for q|aCq 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,qi.t>=0) = e(a,qi.t>=0) if v(a,qi.t>=0)>0 else 0

a's weight for qi is a's expectation if a's value for qi is greater than 0 and else 0, all temporally for now or later.

D3. p(aCqi.t>=0) = w(a,qi.t>=0) : ∑j w(a,qj.t>=0)

The probability that a tries to cause qi is a's proportional weight for qi, 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(aCqi.t=0) = 1) = p(aCqi.t>=0)

The probability that the probability that a does try to cause qi is true is the probability that a tries to cause qi.

A3. s rel aCqi.t=0  --> (sea.t=0)

Any s is relevant to a's trying to cause any qi 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. (Eqi)(Ea)(p(a,q|aCqi.t>=0) > p(a,q|~aCqi.t>=0) &
                  p(q|aCqi.t>=0) > p(q|~aCqi.t>=0) ).

There are events qi and persons a such that a's probability for qi if a tries to cause qi exceeds a's probability for qi if a does not try to cause qi and indeed the real probability of qi given that a tries to cause qi exceeds the real probability of qi given that a does not try to cause qi.

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. (Eqi)(Ea)(aK (qi prel aCqi).t>=0)

And we have

T5. p(aCqi ) = p(aCqi.t>=0) if t>=0
                 = 1 if   aCqi.t=0
                 = 0 if ~aCqi.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 non-extreme 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 self-maintenance 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 self-interest 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 short-circuit 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.



See also: Freedom of the will, Logic, Willing



 Last edited: 12 December 2011.   Top