Willing: The decision that one
wills to do, to believe or to desire something; or also the capacity
or faculty of this, also called will.
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 soandso; 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.
Formulawise, 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 wellfoundedness,
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,qt)=1 )
That is, q is wellfounded 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 wellfoundedness by
considering α's probabilities for t, q and qt, 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 wellfounded
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(αWαBqαDq) > 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
wellfounded beliefs, values and desires, and how these currently
work out as proportions, according to D1D4.
It follows then that what one really does conforms to AW1AW3,
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 AW4AW6.
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 soandso.
In general, one cannot do better than provide the probabilities
given by AW4AW6. But within that limitation, one can explain what
one would have done if soandso, namely by finding what one's
corresponding wellfounded proportions would have been, and by
comparing them with the proportions if not soandso.
Thus, if the wellfounded proportional expectation for doing q if
soandso is much smaller than the wellfounded proportional
expectation for doing ~q if not soandso, then to that extent one
would have more probably done ~q than q if not soandso.
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 AW1AW7 collectively.
How what one eventually wills does depend on the probabilities
and values one has at that moment has been explained by AW1AW7,
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.
