By Itzhak Gilboa, Larry Samuelson, David Schmeidler

The e-book describes formal versions of reasoning which are aimed toward taking pictures the best way that fiscal brokers, and determination makers usually take into consideration their atmosphere and make predictions in response to their earlier event. the point of interest is on analogies (case-based reasoning) and common theories (rule-based reasoning), and at the interplay among them, in addition to among them and Bayesian reasoning. A unified method permits one to check the dynamics of inductive reasoning by way of the mode of reasoning that's used to generate predictions.

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**Extra info for Analogies and Theories: Formal Models of Reasoning**

**Sample text**

Again, from the uniqueness part of Lemma 1 there are unique γ , δ > 0 such that (uy − ux ) = γ (vy − vx ) and (ug − uy ) = δ(vg − vy ). Summing up these two with (ux − ug ) = α(vx − vg ), we get 0 = α(vx − vg ) + γ (vy − vx ) + δ(vg − vy ) = αvx + γ (vy − vx ) − δvy . Thus (α − γ )vx + (γ − δ)vy = 0 . Since vx = vg = 0, vy = vg = 0, and vx = λvy if 0 = λ ∈ R, we get α = γ = δ. Plugging α = γ into (uy − ux ) = γ (vy − vx ) proves that uy = αvy + β. This completes the proof of Theorem 2. We now turn to complete the proof of Step 1.

To justify the term, we note the following. Observation: ∼ is an equivalence relation. Note that equivalence of cases is a subjective notion: cases are equivalent if, in the eyes of the predictor, they affect likelihood rankings in the same way. Further, the notion of equivalence is also context-dependent: two cases c and d are equivalent as far as a specific prediction problem is concerned. We extend the definition of equivalence to memories as follows. Two memories M1 , M2 ∈ M are equivalent, denoted M1 ∼ M2 , if there is a bijection 21 Analogies and Theories f : M1 → M2 such that c ∼ f (c) for all c ∈ M1 .

Step 2 proceeds to deal with the case in which |T| is unrestricted, but X is finite. Lastly, Step 3 deals with the general case in which both |X| and |T| are unrestricted. In all three steps, memories in M are represented by vectors of non-negative integers, counting how many cases of each type appear in memory. Formally, for every T ⊂ T define JT = ZT+ = {I | I : T → Z+ } where Z+ stands for the nonnegative integers. I ∈ JT is interpreted as a counter vector, where I(t) counts how many cases of type t appear in the memory represented by I.