An algebraic synthesis of the foundations of logic and probability

Abstract

This paper develops a unified algebraic theory of logic and probability based upon an initial Boolean algebra (logic) of propositions together with its extensionally associated Boolean algebra of models (probabilistic events) that satisfy the axioms of the Boolean algebra of propositions. A proposition may be true in some models and false in others, but each model must unambiguously assign “ true” or “ false” to each proposition. Thus the examples (models) in which different propositions are true may be all, some, or none according as those propositions are necessary, possible, or impossible. A probability measure P on the collection of models (events) induces a probability measure on the propositions. The crucial “if-then” relation is defined so that P(if p then q) equals $\text{P(q¦p)}$, the conditional probability of q given p. This (conditional) probability is less than the probability of the material conditional, “q or not p”, unless P(p)=1 or $\text{P(q¦p)=1}$. At these two extremes the two propositional constructions coalesce. Thus an additional relation, $\text{(q¦p)}$, is needed in logic and probability, namely “q given p”. This is accomplished in the algebraic logic generated by the various sum ideals (entailments) of various propositions that may serve as the explicit condition of a conditional proposition $\text{(q¦p)}$. The corresponding relation, $\text{(B¦A)}$, in the model realm defines the notion of a conditional event. In this context of algebraic logic and model theory, a new algebraic structure called the conditional closure $\text{L}\text{L}$ of a Boolean logic L is defined, consisting of the ordered pairs $\text{(q¦p)}$ of all propositions of L excluding those for which the condition p is equivalent ( = ) to 0, that is, impossible. The conditional closure is not altogether a Boolean algebra, not altogether distributive for instance, but it has many subalgebras that are Boolean, including L itself. This paper contains partially assumed formulas for operating on the conditional propositions, $\text{(q¦p)}$, using “and”, “or”, “not”, and, of course, “if-then”. This allows complex conditional expressions to be reduced to simple conditionals of Boolean propositions, which have a conditional probability. An example calculation from the dice table illustrates how to simplify a compound conditional proposition and compute its probability. The structure of $\text{L}\text{L}$ is explored, including the finite conditional closures of finite Boolean logics. Formulas are given for the probabilistic difference between familiar conditional propositions that are equivalent in the two-valued logic (where all propositions are either necessary or impossible) but not equivalent in the conditional closure of a logic with possible propositions. The probability that a proposition p is a theorem (true in all models) is contrasted from the probability of the models in which the proposition is true, the latter being P(p) while the former is P(p =1). This leads to a natural connection with fuzzy set theory, since “p =1” can also be expressed as “p ϵ (1)”, where (1) is the sum ideal (filter) of necessary propositions given the Boolean axioms. This paper ends with a representation of each conditional proposition $\text{(q¦p)}$ as the P-measurable two-valued truth function of q restricted to those models that satisfy p.