Get Analogies and Theories: Formal Models of Reasoning PDF

By Itzhak Gilboa, Larry Samuelson, David Schmeidler

ISBN-10: 0198738021

ISBN-13: 9780198738022

The e-book describes formal versions of reasoning which are aimed toward shooting the best way that monetary brokers, and choice makers regularly take into consideration their surroundings and make predictions according to their previous 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 technique 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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In particular, we would like to thank Bruno Jullien, Klaus Nehring, and Ariel Rubinstein. 30 Inductive Inference: An Axiomatic Approach In conclusion, there are classes of counterexamples to our axioms that result from under-specification of cases, of eventualities, or of memories. There are others that are more fundamental. Among these, two seem to deserve special attention. First, there are situations where second-order induction is involved, and the similarity function itself is learned. Indeed, our model deals with accumulated evidence but does not capture the emergence of new insights.

For every ordered list (x, y, z, w) ∈ X, choose M ∈ M such that x M y M z M w. Such an M exists by A4. Let M0 be the union of all sets M so obtained. , M0 ∈ M. Let T0 be the set of types (equivalence classes) of cases in M0 . Choose g ∈ X. Apply Theorem 2 to obtain T a representation of { I }I∈JT by vT0 : X × T0 and (3) for all I ∈ JT0 ≡ Z+0 , such 0 that vT0 (g, ·) = 0. For every finite T ⊂ T such that T0 ⊂ T, apply Theorem 2 again to obtain a representation of { I }I∈JT by vT : X × T and (3) for all I ∈ JT ≡ ZT+ , such that vT (g, ·) = 0 and such that vT extends vT0 .

Choose an arbitrary act, say, g in X. Define vg = 0, and for any other alternative, x, define vx = vxg , where the vxg -s are from Lemma 3. 41 Analogies and Theories Given I ∈ QT + and x, y ∈ X we have: x I y ⇔ vxy · I ≥ 0 ⇔ (vxg + vgy ) · I ≥ 0 ⇔ (vxg − vyg ) · I ≥ 0 ⇔ vx · I − vy · I ≥ 0 ⇔ vx · I ≥ vy · I The first implication follows from Lemma 1(i), the second from the Jacobi identity of Lemma 3, the third from Lemma 1(vi), and the fourth from the definition of the vx -s. Hence, (3) of the theorem has been proved.

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

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