Reflections on Logic and Probability in the Context of Conditionals

摘要

Various controversies surrounding the meaning of conditionals like "A given B" or "if B then A" are discussed including that they can non-trivially carry the standard conditional probability, are truth functional but have three rather than two truth values, are logically and probabilistically non-monotonic, and can be combined with operations that extend the standard Boolean operations. A new theory of deduction with uncertain conditionals extends the familiar equations that define deduction between Boolean propositions. Several different deductive relations arise leading to different sets of implications. New methods to determine these implications for one or more conditionals are described. An absent-minded coffee drinker example containing two subjunctive (counter-factual) conditionals is solved. The use of information entropy to cut through complexity, and the question of the confidence to be attached to a maximum entropy probability distribution, are discussed including the results of E. Jaynes concerning the concentration of distributions at maximum entropy.