Personal probability: A
person's belief about the probability of something. The
term 'personal probability' (like 'real probability') is used in a
somewhat special sense in this Dictionary. The reason to say this is
that there is a related senses of the term  see:
subjective probability

that is not intended. What is intended is that any person may have a
personal probability: A personal belief what about what the probability
of a statement is, expressed by a real number between 0 and 1.
PP  A system of personal probability
What follows is a system of personal probability that consists mostly
of definitions and conventions, and rules of reasoning with these.
The concept and terminology to be explained and rendered precise by
assumptions and definitions is this:
"p(α,Q)=x" =d "the probability of Q for person α equals x"
regardless of how α arrived at this estimate of his
probability for Q.
This may be done by
assumptions like the following  where it should be noted that in what
follows all assumptions for α are supposed to hold at one and the same
time, which is left out in the current treatment so as not to have a
cluttered notation or a more complicated basic set of assumptions and
definitions than is necessary.
A1. (a)(P)(p(a,P) e R &
0
≤
p(a,P)
≤
1)
A1 says that all personal
probabilities are real numbers between 0 and 1 inclusive.
The reasons for real numbers are mathematical, and A1 is mostly
conventional.
A2. (EP) ((Ex)(p(α,P)=x
)
A3. (EP)(EQ)
((Ex)(p(α,P)=x) & (Ey)(p(α,Q)=y) &
(Ez)(p(α,P&Q)=z))
A4. (EP)(EQ) ((Ex)(p(α,P)=x) & (Ey)(p(α,Q)=y) &
~(Ez)(p(α,P&Q)=z))
A2 says that for every person α there are propositions with some personal
probability  and so it may also be the case that there are for α
propositions for which α does not have a
personal probability.
A3 says that for every person α there are pairs
of propositions such that both propositions have some probability and
so does their conjunction
for α.
A4 says that for
every person α there are
pairs of propositions such that both propositions have some
probability for α but their conjunction does not.
Note that, accordingly, it is not assumed here what is more commonly assumed,
namely that all conceivable facts or statements have some probability and value for
everyone. According to the above, there may be statements, or conjunctions
or disjunctions thereof, which have no probability or value for any
specific person α. And indeed α may not have thought about these if he
knows them, or may have made no relevant judgments, or α simply may not know
of them.
This differs from nonpersonal probability and it seems factually
adequate, in that real persons don't have probabilities for
all conceivable statements, nor even for
all statements they have thought of.
Next, one can define probability for conditionals and denials,
supposing α does have the requisite probabilities occuring on the
right hand side:
D1. p(α,QT)
=def p(α,Q&T) : p(α,T)
D2. p(α,~Q)
=def 1p(α,Q)
D3. p(α,~QT) =def 1p(α,QT)
This defines conditional probability, and the probability
of denials for both unconditional probabilities and conditional
probabilities. Using D1D3 we can get something much like standard probability
theory, for those propositions that α does have probabilities for.
To do this we need two rules for definitions, where "=def" may
be read as "is definable by":
R1. For terms t_{1} and t_{2
}: t_{1} =def t_{1}  t_{1}=t_{2}
R2. For propositions p_{1} and p_{2}: p_{1}
=def p_{2}  p_{1} IFF p_{2}
With these we can replace definitions by statements of equality or
equivalence.
Here are some fundamental theorems, with sketches of proofs:
T1.
p(α,T)+p(α,~T)=1
This follows from D2 and R1, as does the following from D3 by R1
T2.
p(α,QT)+p(α,~QT)=1
Next we have from T2, D1 and R1
T3. p(α,T)
= p(α,Q&T)+p(α,~Q&T)
Since the concepts of
probabilistic
independence and
irrelevance are important
here are relevant
definitions. First, we have a special conjunction to cover the case that
may arise by A4:
D4.
p(α,Q.T)
=def p(α,T)*p(α,Q)
Thus, if
~(Ez)(p(α,T&Q)=z) then a has an alternative
probabilistic conjunction for T and Q if
(Ex)(p(α,T)=x) and
(Ey)(p(α,Q)=y).
And now have the tools to
define and contrast
independence and
irrelevance. The general definition
of independence in PT
comes to this
D5A. T inα
Q =def
(Ez)(p(α,T&Q)=z) &
p(α,T&Q)=p(α,T)*p(α,Q))
but we will use a slightly
more specific and slightly
more general pair of
definitions that
distinguish the two by
using background knowledge
K:
D5. T indα Q =def (EkeK)(p(α,QT&k)=p(α,Q~T&k) &
p(α,k)≥½)
D6. T irrα Q =def ~(EkeK)(p(α,QT&k)≠p(α,Q~T&k)
& p(α,k)≥½)
Thus T is independent
of Q for α iff α has a
minimally credible belief
k on which the probability
of Q given T and k is the
same as the probability of
Q given ~T and k, for α.
This is like inα except
that in case of indα 0 <
p(α,T) < 1 and the added
condition must have at
least probability ½ i.e.
be minimally credible. (We
need the case of ½ for
fair coins and the like).
And T is irrelevant
of Q for α iff α has no
minimally credible belief
k on which the probability
of Q given T and k differs
from the probability of Q
given ~T and k, for α.
(*)
It follows that if T irrα
Q and p(α,k)=1 then T
indα Q. And one may also
write the more convenient
"αindependent" and
"αirrelevant". And of
course the presumptions
that matter are those of
α.
Next, one can introduce definitions for entailment and
valid formula using what was assumed and defined:
D7. T α Q =def p(α,QT)=1 V p(α,T)=0
D8. α Q =def p(α,Q)=1
This defines valid implication a.k.a. entailment and
valid formula for α in terms of the
personal probability of α. Note that in fact only 1 and 0 are used here.
When restricted to 1 and 0, what we have assumed so far is also
sufficient to surrect standard
propositional logic.
This is here taken for granted.
An important point here involving D7 and D8 that concerns
probabilities is this theorem:
T4. T α Q  p(α,T) ≤ p(α,Q)
since p(α,T&~Q)=0 if T α Q.
This is interesting and
useful in itself, and from
T4 we have the important theorem
T5. T α Q  p(α,T) = p(α,Q)
that says all logical equivalents for α have the same probability for
α.
Now if α has a conditional probability p(α,QT) and α
has a conditional probability for p(α,Q~T) then with these plus
p(α,T) all entries for
α can be
calculated and listed in a
fundamental table. Indeed here is such a fundamental table in three
forms:

T 
~T 
 
T 
~T 
 
T 
~T 
 


Q 
a

b 
 
p(α,Q&T) 
p(α,Q&~T) 
 
p(α,QT)*p(α,T) 
p(α,Q~T)*p(α,~T) 
 
p(α,Q^{T}) 
~Q 
c 
d 
 
p(α,~Q&T) 
p(α,~Q&~T) 
 
p(α,~QT)*p(α,T) 
p(α,~Q~T)*p(α,~T) 
 
p(α,~Q^{T}) 


p(α,T) 
p(α,~T) 
 
p(α,T) 
p(α,~T) 
 
p(α,T) 
p(α,~T) 
 
1 
In the table a=p(α,Q&T)=p(α,QT)*p(α,T); b=p(α,Q&~T)=p(α,Q~T)*p(α,~T) etc. and one may note that all four values
a,b,c and d can be calculated having only p(α,QT), p(α,Q~T) and
p(α,T) (for this can be proved once one has surrected probability
theory with the above).
Also it should be noted what is the sense of p(α,Q^{T}) in
the fundamental table:
D9. p(α,Q^{T}) =def p(QT)*p(T)+p(Q~T)*p(~T)
That is: p(α,Q^{T}) is the probability of Q for α calculated
by reference to T. In case of p(α,Q^{K}) α has calculated Q by
reference to (presumed) knowledge K; in case of p(α,Q^{E})
α has calculated Q by reference to (presumed) empirical proceedings E.
These distinctions will be useful below, and imply that e.g. p(α,Q^{E})
and p(α,Q^{T}) need not be the same at all.
Now, since we can calculate all of the probabilities that make up a
fundamental table from three of them, it is convenient to define a
complete distribution of a for Q and T thus:
D10. cd(α,Q,T,h,i,j) =def p(α,QT)=h & p(α,Q~T)=i
and
p(α,T)=j
Sofar, what we have set up is a set of assumptions and definitions that explain how
one can reason with personal probabilities about facts and hypotheses of
any kind.
Now we can at this point also give reasonable definitions of the
three basic modes of inference, namely
deduction,
abduction and
induction,
supposing that
α has a complete
distribution for T and Q.
To do so for abduction a definition of currently best
explanation for
α of Q is required, which also makes it convenient to define the probabilities of
α
as the set of propositions for which
α does have
a probability at the time:
D11. pr(α) =def {Q:
(Ex)(p(α,Q)=x}
D12. be(α,T,Q) =def Qepr(α) & Tepr(α) & p(α,QT)=1 &
~(ES)(Sepr(α) & p(QS)=1 & p(α,S)≥p(α,T))
D12 simply says that for the time being α does not have any more
probable deductive explanation for Q
than T.
Here are definitions of
the
three basic modes of inference, that in PP all are valid modes of
inference:
D13. deduction(α,T,Q) =
def p(α,T)=1
& p(α,QT)=1  p(α,Q)=1
D14. abduction(α,T,Q) = def p(α,Q^{K})=1
& be(α,T,Q)  T α Q
D15. induction(α,T,Q) = def p(α,Q^{E})=1
& p(α,TQ)=x  p(α,T)=x
That deduction is
valid follows from D1 and D7 using R1
and R2.
The abductive inference is  trivially  deductively valid in that
p(α,Q^{K})=1.
This would be useless if it were not for the premiss
be(α,T,Q), which gives
α's
real current reason for
T α Q,
which may change with more information.
The validity of induction follows from this argument
T6. p(α,Q^{E})=1 > p(α,T)=p(α,TQ)*p(α,Q^{E})+p(α,T~Q)*p(α,~Q^{E})
=p(α,TQ)
which comes by way of D1 and T3 and the fact that
p(α,~Q^{E})=0 if
p(α,Q^{E})=1 by
D2 and R1.
Therefore induction is also a valid inference, and allows us to upgrade or
downgrade the probability of a theory depending on the empirical facts according
to the rules of probability. And note that in p(α,TQ) we have p(α,Q^{T})
rather than p(α,Q^{E}).
Next, there is this further argument (see also
The Problem of Induction,
esp. Chapter V):
T7. P irrα Q > P irr Q α T
which is to say that if P and Q are αirrelevant, they are also αirrelevant
for any T, where P irr Q α T iff p(α,PQ&T)=p(α,P~Q&T). In other words, if P
and Q are irrelevant for α, then P and Q are also irrelevant given T for α.
For T7 follows from D5 (putting first "P" for "T" and then "T" for "k" in the
result).
In words: If α doesn't know (α believes) that there is any ground for P and
Q to be relevant, the consequence most follow for any particular T, all for α.
To the argument that this is ab ignorantiam there are two replies:
First, one can add the LPAprefix
"αK" i.e. α knows of this irrelevance (due to α's not knowing any ground for
them to be relevant). That is indeed a strengthening, but more to the point is
this:
Second, it is about personal probabilities, and so about personal
distributions of belief, ignorance and probabilities. The issue is not what one
knows or not, but whether one is consistent about what one knows.
And indeed, what α knows that α does not know is α's firmest
and most certain knowledge also, in many ways.
Finally, let me reformulate the RHS of D6 as explicit knowledge of α:
D6A. T irrα Q =def αK [~(EkeK)(p(α,QT&k)≠p(α,Q~T&k)
& p(α,k)≥½) ]
where one can define
D16. αKq =def p(α,q)=1 & q
i.e. α knows q iff α's probability for q is 1 and indeed q is true.
With this all three basic modes of
inference are deductively valid,
though there remain minor
points to be clarified, and thus we have explained
human reasoning in some important ways and have
done so on the foundation of some rather obviously true assumptions and sensible definitions concerning personal probability.
It is well to note that while PP is  a sketch of  a system of personal
probability, it is not more relativistic than it needs to be, in that two
persons may very well have quite different personal probabilities for many
statements, but that as long as they agree on the experimental facts they will
agree on how this must influence whatever probabilities they had before the
experimental facts became known to them, if they agree to the assumptions for
PP.
And specifically this means that if for α and β
it is true
that p(α,Q^{E})=p(β,Q^{E})=1,
then they must both agree in which direction their personal probabilities for T
change, and also by what amounts if they had given their probabilities.
Note: The above is a sketch, and anyone who wants to use, instead of D1 up to
and including T3, a standard system for probability like Kolmogorov's axioms,
but relativized to a person's probabilities, is welcome to do that.
The important points in this sketch are
 That there are personal
probabilities.
 That most or all persons have no
personal probabilities for all propositions they consider.
 That it is conventient to have a
standard probabilistic conjunction that equals the product of the
probabilities of its factors.
 That a distinction between
independence and irrelevance is useful, that involves reference to
conjunctions a person has no probabilities for.
 That it is helpful to define logical
implication and validity in probabilistic terms, with probabilities
restricted to 1 and 0.
 That probabilities may be calculated
with reference to diverse assumptions.
 That the three basic modes of
reasoning are all definable in probabilistic terms, as deductively
valid inferences.
(*) It is noteworthy that there are obvious simpler
versions of D5 and D6
D5'. T indα Q =def (EkeK)(p(α,QT&k)=p(α,Q~T&k) &
p(α,k)=1)
D6'. T irrα Q =def ~(EkeK)(p(α,QT&k)≠p(α,Q~T&k)
& p(α,k)=1)
but D6' seems to admit fuzzing in case of a k' such that
p(α,QT&k)=p(α,Q~T&k) & p(α,k)=0.9), say: Intuitively, it is a bit odd
to say T is irrelevant of Q on the ground that though one knows of a
highly probable k' that makes them relevant, this k' is not certain.
This is the reason to adopt the originals with " p(α,k)≥½ " to be
understood as: " k is at least minimally credible for α ". With this
restored T is irrelevant of Q on the ground that α knows no minimally
credible k' that makes them relevant.
And of course anyone may think of anything not minimally credible to
make any T and Q relevant in the defined sense. One point of the more
involved concepts of independence and irrelevance is precisely to
exclude these cases.
A note on the fonts and eventual display, relating to mathematical symbols:
The text uses a combination of the Verdana font and Unicode for Greek letters
and mathematical symbols that are not in the Verdana font. It has turned out
that this may easily get mixed up in a wysiwyghtmleditor like Frontpage, that
I use. And it may produce gargle in browsers that are not like MS IE 6.
Since this is a sideremark anyway, but one that concerns the readability of
this site on the one hand, and the labor of maintaining it on the other hand:
It really is still a considerable practical problem that one cannot write
basic mathematics, with the usual symbols, in almost all fonts, at least, on a
computer screen and that one is forced to revert to tricks, such as inserting
bits of Unicode, or to using special small bitmaps or glyphs.
This is a problem that is not properly addressed on this site, and indeed I
prefer not to, for it involves a lot of boring work. What I do hope is that
standard fonts are extended to include standard mathematical and logical
notation, also including Greek letters.
