Probability Logic Test

This Demonstration presents a simple test for probability logic. The value of compound propositions is evaluated according to probability laws.
A simple two-dimensional area is occupied by white or gray triangles, squares, and pentagons. A disk indicates that the shape of the element is not known; in this case a proposition of type Shape() has value 1/3. A gray-white figure means that the color of the figure is not known; in this case a proposition of type Color(x) has value 1/2.

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DETAILS

The procedure used should work on any propositional sentence. Using Mathematica's built-in function LogicalExpand eliminates all propositional connectives except ¬, ∧, and ∨. Negation is eliminated by rules, such as . Another application of LogicalExpand gives a disjunctive normal form without negations, that is a disjunction of conjunctions. There are special cases if the form reduces to a single conjunction, a single atomic statement, or True or False.
A conjunction is treated as a list of statements . In case the same figure appears with different shapes or colors, the probability of the conjunction is 0. (For instance: ) If no such case occurs, the set of statements is independent.
The shape and color of the same figure are independent of each other; the shapes of two different figures are independent, and so on. In these kinds of cases, and the basic formula for disjunction is .

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