Analogies and Theories: Formal Models of Reasoning by Itzhak Gilboa, Larry Samuelson, David Schmeidler

By Itzhak Gilboa, Larry Samuelson, David Schmeidler

The publication describes formal versions of reasoning which are geared toward taking pictures the way in which that financial brokers, and choice makers typically take into consideration their atmosphere and make predictions in accordance with their prior event. the focal point is on analogies (case-based reasoning) and basic theories (rule-based reasoning), and at the interplay among them, in addition to among them and Bayesian reasoning. A unified process permits one to review the dynamics of inductive reasoning by way of the mode of reasoning that's used to generate predictions.

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For example, A4 will be violated if a row in the matrix dominates another row. Since A4 is not necessary for a representation by a matrix v via (2) (even if it respects case equivalence), one may wonder whether it can be dropped. The answer is given by the following. Proposition: Axioms A1, A2, and A3 do not imply the existence of a matrix v that satisfies (2). Some remarks on cardinality are in order. Axiom A4 can only hold if the set of types, T = C/ ∼, is large enough relatively to X. For instance, if there are two distinct eventualities, the diversity axiom requires that there be at least two different types of cases.

Observe that by definition and A1∗ : Axy ⊂Bxy , Bxy ∩Ayx = ∅, and Bxy ∪ Ayx = T Q+ . 1 For every distinct x, y ∈ X there is a vector vxy ∈ RT such that, T | vxy · I ≥ 0}; (i) Bxy = {I ∈ Q+ T | vxy · I > 0}; (ii) Axy = {I ∈ Q+ T | vxy · I ≤ 0}; (iii) Byx = {I ∈ Q+ T | vxy · I < 0}; (iv) Ayx = {I ∈ Q+ (v) Neither vxy ≤ 0 nor vxy ≥ 0; (vi) −vxy = vyx . Moreover, the vector vxy satisfying (i)-(iv), is unique up to multiplication by a positive number. The lemma states that we can associate with every pair of distinct eventualities x, y ∈ X a separating hyperplane defined by vxy · ξ = 0 (ξ ∈ RT ), such that x I y iff I is in the half space defined by vxy · I ≥ 0.

Similarly, statements such as “The weather always surprises” or “History repeats itself” are about sequences of cases, and are therefore likely to generate violations of the combination axiom. Second-order induction An important class of examples in which we should expect the combination axiom to be violated, for descriptive and normative purposes alike, involves learning of the similarity function. For instance, assume that one database contains but one case, in which Mary chose restaurant x over y.

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