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 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 [ & ROA.
Role of puzzles, thought experiments & generality of proposed solutions