Adams Conditionals and Non-Monotonic Probabilities

1. It is very probable that it wasn't the cook. 2. It is very probable that if it wasn't the butler, then it was the cook. Hence: 3. It is very improbable that if it wasn't the butler, then it was the gardener. Now: 4. It is certain, given that it wasn't the cook, that if it wasn't the butler, then it was the gardener. 5. It is impossible, given that it was the cook, that if it wasn't the butler, then it was the gardener. Hence: 6. It is very probable that if it wasn't the butler, then it was the gardener. A Triviality Theorem Let Ω be a set of sentences closed under the sentential operations of conjunction and negation and Π be the set of all probability measures on Ω. Let → denote a two place operation on sentences and assume that for some A and X in Ω, the sentence A→X is in Ω. Let P be a probability variable ranging over the members of Π. Adams' Theses 1. Ordinary inference is governed by considerations of probabilistic validity: roughly, the high probability of the premises ensures the high probability of the conclusion. 2. The probability of a conditional is the conditional probability of its antecedent given its consequent. over them. b. A two-place numerical function, p. c. One or more operation symbols, with the (unextended) operation of concatenation occurring in every system. d. The domain of interpretation is left unspecified Axiom System M A1. 0 ≤ p(x, z) (lower bound) A2. p(x, z) ≤ p(y, y) (upper bound) A3. xz(p(x, z) ≠ 0) (non-triviality) M1. p(xy, z) ≤ p(x, z) (monotonicity) M2. p(xy, z) = p(x, yz).p(y, z) (product rule) In M it is derivable that: 1. Idempotence: p(xx, z) = p(x, z) 2. Commutativity: p(xy, z) = p(yx, z) 3. Associativity: p(x(yz), w) = p((xy)z, w). Logic Probabilistic Implication: x  y = Df z(p(x, z) ≤ p(y, z)) Probabilistic Equivalence: x ~ y = Df x  y and y  x  is a partial order and ~ an equivalence relation. However, to identify probabilistically equivalent terms, replacement in p's second argument must be assured: A5. x ~ y  p(z, x) = p(z, y) By addition of A5 to M we obtain system M +. The models of M + are reducible to lower semi-lattices. Introduction of a join operation via: J. …