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Conceptual Puzzles and Theoretical Elegance: Exploring Cognition Beyond Numbers

This conceptual exploration delves into the intricate realm of cognition, beyond mere numbers and algorithms. It emphasizes the significance of multiple representations in studying gradience and the interplay of probability distributions. Theoretical frameworks like MaxEnt and Harmony Theory are discussed, alongside catastrophes like Counting Catastrophe and Epenthesis. Various cognitive concepts, rankings, and biases are analyzed to understand the complexity of induction and generalization processes.

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Conceptual Puzzles and Theoretical Elegance: Exploring Cognition Beyond Numbers

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  1. Conceptual Puzzles & Theoretical Elegance (principledness? naturalness?)

  2. Need to add cognition — not just numbers, experiments, and algorithms • Multiple representations are tapped differentially by different tasks/conditions (Boersma, 3:19 pm) • Especially important in study of gradience • Probability distribution family* • *but see distribution-free methods (PAC learning) • Rankings: Natural for (partial) orders? (MaxEnt: 1) • Outputs: Natural for induction/generalization? (MaxEnt maximizes entropy) • Bias:Bayesian inference

  3. Constraint interaction through numbers • The Counting Catastrophe • WSP [σH main stress]; M-PARSE • Numerical/HG typology: *σH σH ⋯σH iff #σH’s > n∀n, • OT typology: *σH σH ⋯ σH iff #σH’s > n for n {1,  } • Variation through Losers • Relative Harmony predicts relative frequency • Harmony Theory/MaxEnt and HG; also Pater’s relativized H in HG (yesterday) • The Epenthesis Catastrophe • MAX≫ DEP/batak/  bataka  but /batak/  batakatatata ≻ /batak/ bata  • Fate of harmonically bounded forms?

  4. Variation through winners of multiple rankings • Number of rankings producing an output predicts relative frequency • All other approaches (?) • Bogus Rankings Catastrophe • A, B conflict: Proportion of A-obeying output (A ≫ B) • {A; B}: 1/2 • X bogus; no (relevant) violations • {X ≫ A; B}: 1/3; {A ≫ X; B} : 2/3; {A ≫ X, X ≫ B}: 1 • Ordinal, not just quantitative, effects

  5. Analogical Combinatorial Catastrophe • Theorem.Lexical Similarity Theory cannot explain Basic Combinatorial Generalization. • Lex = {ta, ki}  ti/ka ≻ pa/tu • Smolensky, P. 2006. On theoretical facts and empirical abstractions. In Wondering at the natural fecundity of things: Essays in honor of Alan Prince, eds. E. Bakovic, J. Ito, and J. McCarthy. Linguistics Research Center [http://repositories.cdlib.org/lrc/prince/13] & ROA. • Role of puzzles, thought experiments & generality of proposed solutions

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