One important question is whether the will is free. Here is a general theory of willing, choosing or deciding, here formulated in terms of willing, that has the merit of being fairly clear and of answering some important questions, such as: how can one be free to will as one pleases and determined by one's beliefs and desires to will the way one does; how to explain what one would have done, if so-and-so; and in general how to explain how one acts using one's desires and beliefs.

First, there is an

• Assumption: Human beings have a faculty of willing, that is active only in the present; that is based on beliefs humans have that relate their experience of the present to their conception of their interests; and that initiates bodily and mental acts and states.

This faculty makes judgements, and these judgements are of three fundamental kinds: one wills to do, to believe or to desire, and the general formula comes to this:

• One believes that at this moment it is in one's interests to do q, to believe r or to desire s, and one does q, believes r or desires s if one now wills to do so, for the given reason.

Three important points to note here are that

(1) There is supposed to be a capacity of willing, given by a supposed faculty of the will, that determines what one does, and that is active only here and now.
(2) One wills not only some of one's bodily behavior but also some of one's mental behavior, notably one's beliefs and desires.
(3) One wills all the time, and as long as one survives the majority of one's willfull acts have been successful.

To explain how this can be used is most easily done - if it is to be done fairly precisely, using some logical tools, that involve the logics of propostions, probability and propositional attitudes, and also explain these to some extent.

Formula-wise, one can write what is involved as follows, with the following notation:

"αWs" = "α wills that s"
"αCs" = "α tries to cause that s"
"αBs" = "α believes that s"
"αDs" = "α desires that s"
" q |-s " = "q entails s"

At this point a system of personal probablity is assumed that provides assumptions to reason with the expression

p(α,q)=x  =def the probability of q for α equals y

This system is now extended with assumptions to reason with the expression for willing that start with two assumptions about the values of person α, that is written thus:

v(α,q)=x  =def the value of q for α equals x

To begin with, there is an axiom to the effect that every situation α can think of has some value which is given by some natural number:

AV1. (En) ( v(α,q)=n & neN)

Next, there is this simple axiom for α's values, that can be seen as an assumption of minimum consistency for valuations:

AV2. v(α,q) > 0 |- v(α,~q) < 0

This makes intuitive sense, and allows also that α's value of ~q needs not at all be the same as α's value of q taken negatively. All that is necessary to value consistently is that one takes the values of the denials of what one values positively as negative values.

Using the concept of value we introduce the concepts of weight w and proportional weight pw that simplify reasoning with values:

D1. w(α,q) = z    =def  v(α,q) = z & z ≥ 0 else z=0
D2. pw(α,q_i) = z  =def  w(α,q_i) : ∑_j w(α,q_j)

Next, there person is α's expectation for q_i, and his related proportional expectation for q_i which conforms to the standard notion for expectation, except that it is framed in terms of α's probabilities and α's values

D3. e(α,q_i) = z   =def  p(α,q_i|αCq_i)*w(α,q_i) = z
D4. pe(α,q_i) = z  =def  e(α,q_i) : ∑_j e(α,q_j)

In words, α's expectation for q is α's probability for q_i|αCq_i times a's weight for q_i. Note this always is 0 whever α's value for is q_i negative or zero: weights are positive values or zero, and expectations are defined using weights.

Also, we shall need a condition called well-foundedness, that can be defined in many ways, but that minimally incorporates something to the following effect:

D5. wf(α,q)  =def (Et)( p(α,t)>1/2 & p(α,q|t)=1 )

That is, q is well-founded for α if α believes there is some t that α believes in about which α believes that t logically implies q. One can introduce degrees of well-foundedness by considering α's probabilities for t, q and q|t, but this will not be done here, as it is fairly obvious given the rest, and as it will not be required here.

Now we have the tools to formulate axioms for willing.

First, willing (in the present) is effective

AW1. αWαCq |- αCq
AW2. αWαBr |- αBr
AW3. αWαDs |- αDs

If α wills to do q (to believe r, to desire s), then as matter of fact thereby α does q (believes s, desires r), and also irrespective of what was the case before now: By willing one initiates something new, also if this is nothing but a restart of something one did, believed or desired before, and had given up meanwhile, for some reason, cogent or not.

Also, one may will one's actions, beliefs and desires, and any deliberate action is so because it involves a judgment that one willed to do so.

Second, the probability that α wills α's trying to do q is given by α's proportional expectation for q, provided this is well-founded for α and likewise for α's beliefs and desires.

AW4. p(αWαCq)=x IFF wf(α,pe(α,q)=x)
AW5. p(αWαBr)=y IFF wf(α,p(α,r)=y)
AW6. p(αWαDs)=z IFF wf(α,pw(α,s)=z)

Note these allow for the possibilities that α may be willing to believe something that α  does not believe is certain or most probable, and that α may be willing to desire something that α does not believe is the most desirable.

Both conform to the known facts about people, and to the need to come to beliefs to act on, and the need to come to desires to guide one's actions by. In order to put this in notation and make clear what is involved, the following assumptions may be added:

D6. αBr IFF p(α,r) > 1/2
D7. αDs IFF v(α,s) > 0

In effect, these relate α's believing and desiring s to α's probabilities and values. Note that both the belief and the desire may be faint or strong, and that they  have strength in proportion to their size.

It is also of some importance to define wishful thinking:

D8. wishful(α,q) =def p(αWαBq|αDq) > p(αWαBq|~αDq)

That is, α engages in wisful thinking concerning q if α's probabilty of believing q depends probabilistically on α's desire for q.

This is very common: it is very human to engage to some extent in wishful thinking about most things one desires, and this is indeed more probable to the extent one desires it stronger.

Third, willing is free for α to the extent that it is independent of everything outside α:

D9.     free(α, αWq) =def (s)(s rel aWq --> αBs V αDs)
AW7.  (α)(Eq)(free(α, αWq)

That is, α is free to will q, or α wills q freely if and only if everything that is relevant to α's willing q is among α's beliefs or desires. Here the sense of "relevant" is probabilistic: s is relevant to q if the probability of q is different if s is true from what the probability of q is if s is not true.

The axiom simply claims that everybody is free to will some things in the defined sense.

Now, how does the above explain the general questions this lemma started with? Here are the answers.

To start with how to explain how one acts using one's desires and beliefs.

If the above is true, one acts on one's own desires and beliefs in the manner explained, namely with what one takes to be well-founded beliefs, values and desires, and how these currently work out as proportions, according to D1-D4.

It follows then that what one really does conforms to AW1-AW3, that describe how one starts to have an action, belief or desire, and why, while the probabilities that one starts these actions, beliefs and desires are given by AW4-AW6.

Accordingly, these probabilities are proportional to the beliefs and desires one has at the moment.

Second, how to explain what one would have done, if so-and-so.

In general, one cannot do better than provide the probabilities given by AW4-AW6. But within that limitation, one can explain what one would have done if so-and-so, namely by finding what one's corresponding well-founded proportions would have been, and by comparing them with the proportions if not so-and-so.

Thus, if the well-founded proportional expectation for doing q if so-and-so is much smaller than the well-founded proportional expectation for doing ~q if not so-and-so, then to that extent one would have more probably done ~q than q if not so-and-so.

Third, how can one be free to will as one pleases and determined by one's beliefs and desires to will the way one does.

In fact, this is due to AW1-AW7 collectively.

How what one eventually wills does depend on the probabilities and values one has at that moment has been explained by AW1-AW7, including what are the probabilities of what one will do.

And this is the best one can do for such acts one does that are free, for these do not depend on anything outside one by AW7, while this does not determine more than a probability between 0 and 1 exclusive normally, which also guarantees that when willing what one does one is not constrained to do it, but can will something else.

Hence the freedom of the will, as construed here, is due to the combination of its depending on the values and probabilities one has, without being made certain by these.

Thus one is called to decide upon many things one may but need not do, that one does decide with nothing but one's values and probabilities.

In almost all cases one consciously does something, one could have done otherwise, whether perversely, by intentionally doing something else than what seems best, or whether by whim, by intentionally trying an alternative that is not the best to see how it works out, or whether by the present facts, that made one intentionally take another alternative than what seemed best until the present arrived.