Monday 9 July 2018

Material implication and conditional probability

A simple argument shows that in general the ratio formula for conditional probability cannot be the probability of the material conditional. But there is still controversy over both.

Despite the undoubted success of probability theory in providing tools for inference, statistical analysis and decision making, there remain concerns about its foundations. A major concern is with the status of conditional probability and its relationship with logical implication (indicative conditional). In propositional logic material implication provide the formal concept. Although this is often glossed over in standard texts it is taken seriously by E. W. Adams. However his solution giving primacy to conditional probability is also open to criticism.  These points are of practical importance as status of inference and its foundations in logic, probability and set theory are fundamental to the development of Artificial Intelligence

Adams’ thesis is that the assertability of the indicative conditional $A \to B$ is given by the conditional probability of $B$, given $A$. For example, he writes: “Take a conditional which is highly assertible, say, ‘If unemployment drops sharply, the unions will be pleased’; it will invariably be one whose consequent is highly probable given the antecedent. And, indeed, the probability that the unions will be pleased given unemployment drops sharply is very high”

The default standard foundation of probability theory is the axiomisation of A.N. Kolmogorov. This takes as one of its primitives a function denoting the probability of a set and these sets are called random events. An event is something that happens or has the potential to happen.

In Kolmogorov's theory probability space $\left( \Omega, \Sigma,\mu \right)$ consists of a set $\Omega$ (called the sample space), a $\sigma$-algebra $\Sigma$ of subsets of $\Omega$ (i.e., a set of subsets of $\Omega$ containing $\Omega$ and closed under complementation and countable union, but not necessarily consisting of all subsets of $\Omega$) whose elements are called measurable sets, and a probability measure $\mu:\Sigma \rightarrow \lbrack 0,1\rbrack$ satisfying the following properties:
P1. ${\mu}\left(X \right){\geq 0}$ for all $X \in \Sigma$
P2. ${\mu}\left( \Omega \right){= 1}$
P3. ${\mu}\left( {\bigcup}_{i = 1}^{\infty}{\ }X_{i} \right){= \ }\sum_{i = 1}^{\infty}{\mu(}X_{i}{)}$, if the $X_{i}$'s are pairwise disjoint members of $\Sigma$.
P4. $\mu(A | B) = \frac{\mu(A \cap B) }{\mu(B)}$
Postulate P4 provides an analysis of conditional probability. It is more often referred to as the definition. However conditional probability was current as a concept prior to the axiomisation. In the sense of
The probability of $A$ given $B$,
The probability of "if $B$ then $A$"
The probability that $B$ implies $A$.
 In the usage prior to the formalisation of probability $A$ and $B$ are not sets but usually statements or propositions. So a relationship between propositions and sets is needed.

In propositional logic, material implication is a rule of replacement that allows for a conditional statement to be replaced by a disjunction in which the antecedent is negated. The rule states that $P$ implies $Q$ is logically equivalent to not-$P$ (in symbols $\neg P$) or $Q$.
$$ P \to Q \Leftrightarrow \neg P\lor Q$$
where $\Leftrightarrow$ denotes logical equivalence.

There is a straight forward mapping between the sets and connectives in the set based axiomisation and the the propositions and connectives in propositional logic. The correspondence of connectives is:
  • $\cup$ corresponds to $\lor $
  • $\cap$ corresponds to $\land$
  • $\Omega$ corresponds to $\mathbf{t}$ (the single extension of all tautologies)
  • $\emptyset$ corresponds to $\mathbf{f}$ (the single extension of all falsehoods)
  • The set complement ($\bar{A}$ for any $A \in \Sigma$) corresponds to $\neg$ (negation). 
This would mean
  • $\bar{A} \cup B$ corresponds to $P \to Q$, where proposition $P$ pertains to the event represented by $A$ and $Q$ pertains to the event represented by $B$.
So, what is the relationship between $\mu(A | B)$ and $\mu(\bar{A} \cup B)$? A simple analysis shows that they are only equal in a very special case. Consider the partition of $\Omega$ shown in the diagram below.

 From this it follows:
$$ \mu(B|A) = \frac{c}{a+c}$$
$$ \mu(\bar{A} \cup B) = b+c+d = 1-a$$
Therefore, in this case,
$$ \mu(B|A) = P(\bar{A} \cup B) \Rightarrow P(A)=1$$.

So the two terms are only equal when $A$ is the certain event. In general, the ratio formula for conditional probability cannot be the probability of the material conditional. In general, the relationship is
$$\mu(\bar{A} \cup B) = 1 - \mu(A)(1- \mu(B|A))$$
This is equivalent to the result stated by E. W. Adams in "The Logic of Conditionals: An Application of Probability to Deductive Logic" page 3.

The morphism between propositional logic and set theory is used extensively in interpreting theories of probability. It preserves structure but does not extend to implication and it does not entail that meaning or ontological status is preserved. It is from the direction of metaphysical analysis of the ontological status of conditionals, both logical and probabilistic, that progress may be made. In a recently published book,  What Tends to Be, Rani Lill Anjum and Stephen Mumford provide a synthesis of this analysis.

In probability theory alternative axiom systems may be the answer and candidates exist from Renyi and Popper. However, the ontological analysis may indicate that the eventual practical answers lie in something more akin to physics rather than logic or mathematics. Future posts will engage critically with this work.

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