Reflections on Logic & Probability in the Context of Conditionals

摘要

This paper discusses various controversies surrounding the meaning and use of such conditionals as 'A given B' or 'If B then A' including that such Boolean fractions (1) can non-trivially carry the standard conditional probability, (2) are truth functional but with three rather than two truth values, (3) are logically and probabilistically non-monotonic, (4) can be combined with operations that extend the standard Boolean operations, and (5) allow definitions that extend Boolean deduction but do not serve as deductions themselves thereby avoiding the so-called paradoxes identified by E. Adams. A new theory of deduction with uncertain conditionals is defined in terms of the new operations on conditionals by extending the familiar equations that define deduction between Boolean propositions. This leads to several plausible forms of deduction between conditionals. These different deductive relations on conditionals give rise to different sets of implications. Methods to determine the implications of one or more conditionals with respect to the various different deductive relations are described. Three examples of deduction with uncertain conditionals are extensively examined and solved. An example about an absent-minded coffee drinker contains two so-called subjunctive or counterfactual conditionals, which pose no additional difficulty. The issue of practical computation with conditionals is addressed and the use of information entropy to cut through complexity is discussed and illustrated. Lastly there is the question of how much confidence can be attached to a probability distribution having maximum entropy. In this regard the results of E. Jaynes concerning the concentration of distributions at maximum entropy are described along with two other theoretical approaches to this problem.