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 Maarten Maartensz:    Philosophical Dictionary | Filosofisch Woordenboek                      

 P - Probability - personal

 

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 non-personal 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,Q|T)      =def  p(α,Q&T) : p(α,T)
D2.     p(α,~Q)       =def  1-p(α,Q)
D3.     p(α,~Q|T)    =def  1-p(α,Q|T)

This defines conditional probability, and the probability of denials for both unconditional probabilities and conditional probabilities. Using D1-D3 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 t1 and t2           : t1 =def t1 |- t1=t2
R2.  For propositions p1 and p2: p1 =def p2 |- p1 IFF p2

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,Q|T)+p(α,~Q|T)=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 in-α 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 ind-α Q =def   (EkeK)(p(α,Q|T&k)=p(α,Q|~T&k) & p(α,k)≥)
D6. T irr-α Q  =def ~(EkeK)(p(α,Q|T&k)p(α,Q|~T&k) & p(α,k)≥)

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 in-α except that in case of ind-α 0 < p(α,T) < 1 and the added condition must have at least probability i.e. be minimally credible. (We need the case of 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 irr-α Q and p(α,k)=1 then T ind-α Q. And one may also write the more convenient "α-independent" and "α-irrelevant". 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,Q|T)=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,Q|T) 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,Q|T)*p(α,T) p(α,Q|~T)*p(α,~T) | p(α,QT)
~Q  c  d | p(α,~Q&T) p(α,~Q&~T) | p(α,~Q|T)*p(α,T) p(α,~Q|~T)*p(α,~T) | p(α,~QT)

  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,Q|T)*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,Q|T), 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,QT) in the fundamental table:

D9. p(α,QT) =def  p(Q|T)*p(T)+p(Q|~T)*p(~T)

That is: p(α,QT) is the probability of Q for α calculated by reference to T. In case of p(α,QK) α has calculated Q by reference to (presumed) knowledge K; in case of p(α,QE) α has calculated Q by reference to (presumed) empirical proceedings E. These distinctions will be useful below, and imply that e.g. p(α,QE) and p(α,QT) 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,Q|T)=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,Q|T)=1 &
                              ~(ES)(Sepr(α) & p(Q|S)=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,Q|T)=1    |-  p(α,Q)=1
D14.  abduction(α,T,Q) = def  p(α,QK)=1 & be(α,T,Q)     |- T |-α Q
D15.  induction(α,T,Q)  = def  p(α,QE)=1 & p(α,T|Q)=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,QK)=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,QE)=1 --> p(α,T)=p(α,T|Q)*p(α,QE)+p(α,T|~Q)*p(α,~QE)
                                  =p(α,T|Q)

which comes by way of D1 and T3 and the fact that p(α,~QE)=0 if p(α,QE)=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,T|Q) we have p(α,QT) rather than p(α,QE).

Next, there is this further argument (see also The Problem of Induction, esp. Chapter V):

T7.   P irr-α Q --> P irr Q |-α T 

which is to say that if P and Q are α-irrelevant, they are also α-irrelevant for any T, where P irr Q |-α T iff p(α,P|Q&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 LPA-prefix "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 irr-α Q  =def αK [~(EkeK)(p(α,Q|T&k)p(α,Q|~T&k) & p(α,k)≥) ]

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,QE)=p(β,QE)=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 ind-α Q =def   (EkeK)(p(α,Q|T&k)=p(α,Q|~T&k) & p(α,k)=1)
D6'. T irr-α Q  =def ~(EkeK)(p(α,Q|T&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,Q|T&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)≥ " 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 wysiwyg-html-editor like Frontpage, that I use. And it may produce gargle in browsers that are not like MS IE 6.

Since this is a side-remark 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.

 


See also: Extended Propositional Logic, Proportional Probability


Literature:

Maartensz
 

 Original: Apr 6, 2005                                                Last edited: 12 December 2011.   Top