Probabilities as Shapes

By focusing on intrinsic properties of the objects under study---notions of identity (what the objects are), reference (what the concepts refer to), and extension (which objects satisfy a given condition)---traditional logical frameworks have a limited ability to express the many non-mathematical attributes of Logic and Probability. Once instantiated as mathematical objects within a formalism that emphasizes intrinsic properties of the objects rather than extrinsic meaning of the system as a whole, logical and probabilistic concepts become captured by their mathematical representations. Whatever lies beyond the math is lost.

Mathematical representations of logical concepts attain meaning not through what they {\em are} (i.e., frequencies, sizes, prices) but in how they {\em are used} (e.g., for prediction, inference, betting). Axiomatic systems developed within the orthodox quantitative frameworks of probability (see Chapter 4) are useful for applications in which a number (frequency, size, price, or other) adequately represents the primary object of interest.
But sets and numbers can only represent properties that sets and numbers themselves possess, and both logical propositions and probabilities have attributes beyond what these formalizations can encode.

A suitable logic for Intuition and Common Sense, as we seek here, cannot be so constrained that it distorts or eliminates fundamental patterns of intuitive reasoning. Inductive inferences, qualitative probabilistic inferences, context dependence, and generic subjective judgments are inherent to the Intuition yet incompatible with classical logic and orthodox probability theory. That which simply `makes sense' is the gold standard of Intuition, but wholly unacceptable in the traditional formal paradigm. The goal of this chapter is to initiate a theory that does justice to the Intuition and Common Sense, as a sound basis for discerning when something makes sense.

Note: This is a chapter in the author's forthcoming book Probability, Intuition, and Common Sense.