Logic with a Probability Semantics by Theodore Hailperin

By Theodore Hailperin

The current research is an extension of the subject brought in Dr. Hailperin's Sentential chance Logic, the place the standard true-false semantics for common sense is changed with one dependent extra on chance, and the place values starting from zero to one are topic to likelihood axioms. in addition, because the notice "sentential" within the identify of that paintings shows, the language there into account used to be restricted to sentences made from atomic (not internal logical elements) sentences, by way of use of sentential connectives ("no," "and," "or," etc.) yet now not together with quantifiers ("for all," "there is").

An preliminary advent provides an summary of the booklet. In bankruptcy one, Halperin provides a precis of effects from his past booklet, a few of which extends into this paintings. It additionally incorporates a novel therapy of the matter of mixing facts: how does one mix goods of curiosity for a conclusion-each of which individually impart a likelihood for the conclusion-so as to have a likelihood for the realization in keeping with taking either one of the 2 goods of curiosity as facts?

Chapter enlarges the chance common sense from the 1st bankruptcy in respects: the language now comprises quantifiers ("for all," and "there is") whose variables diversity over atomic sentences, now not entities as with typical quantifier common sense. (Hence its designation: ontological impartial logic.) a collection of axioms for this common sense is gifted. a brand new sentential notion—the suppositional—in essence as a result of Thomas Bayes, is adjoined to this common sense that later turns into the root for making a conditional chance logic.

Chapter 3 opens with a suite of 4 postulates for likelihood on ontologically impartial quantifier language. Many houses are derived and a basic theorem is proved, specifically, for any likelihood version (assignment of likelihood values to all atomic sentences of the language) there'll be a distinct extension of the likelihood values to all closed sentences of the language.

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The diagrams of Figure 1 show that it is possible for P (z | xy) to have the extreme values 0 and 1, as well as any value in between. y x P (xyz) =0 P (xy) z z x y P (xyz) = 1. P (xy) Figure 1. Could the additional premises P (z | x) = p and P (z | y) = q sufficiently 15 Contained in the memoir Boole 1857, reprinted as Essay XVI in Boole 1952, of which the relevant pages are 355–367. On p. 356 of this book in the display labeled (1) there is a typographical error. The first numerator should be ‘Prob xz’.

Should one prefer to say “suppositional probability” or is it too late to change? 6. 2 III above) the basic semantic components are statements of the form P (φ) ∈ α, where P is an arbitrary probability function (becoming specific with a choice of probability model), and α is a subset of [0, 1]. Such subsets need not be explicitly given but may only be described—for example, the set {x ∈ [0, 1] | F(x, P (ψ1 ), . . 2 III. This enlargement of the probability semantic language beyond the simple V (φ) = 1 of verity logic brings in the need for sharper distinction between syntax and semantics, a distinction barely noticable in verity logic.

Functions are as follows. Each pk (k = 1, . . , m), Q’s primary functions, will have the same number of argument places as the predicate Pk and the values of each of these functions will range (respectively) over the m different residue classes of natural numbers modulo m (thus keeping the ranges of the pk from overlapping). This is achieved by setting: pk (x1 , . . , xmk ) = mN (mk ) (x1 , . . , xmk ) + k − 1, where N (n) (x1 , . . , xn ) is a primitive recursive function mapping n-tuples of natural numbers (= numerals, for us) to the natural numbers (numerals).

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