From logicist to probabilist cognitive science

1. Nick Chater Department of Psychology University College London n.chater@ucl.ac.uk From logicist to probabilist cognitive science

2. Overview • Logic and probability • Compare and contrast • A competitive perspective • How much deduction is there? • Evidence from human reasoning • A cooperative perspective • Logic as a theory of representation • Probability as a theory of uncertain inference • Probability over logically rich representations • The probabilistic mind

3. George Boole (1815-1864) “Laws of thought” Logic (and probability) Vision of mechanizable calculus for thought Boolean algebra developed and applied to computer circuits by Shannon Symbolic AI Thomas Bayes (1702-1761) “Bayes theorem” for calculating inverse probability Making probability applicable to perception and learning Machine learning Two traditions Neither Boole nor Bayes distinguished between normative and descriptive principles: Error or insight?

4. When is a set of beliefs consistent? If  = {A, B, ¬C} is inconsistent, then A, B deductively imply C Beliefs consistent if they have a model A variety of logics provide rules for avoiding inconsistency Focus on internal structure of beliefs Logic and the consistencyof beliefs

5. John must sing A prop calculus O(A) deontic logic □A modal F(j) 1st order Different depths of representation John_must_sing Must(John_sings) Necessarily(John_sings) Must_sing(John)

6. Probability When is a set of subjective degrees of belief consistent? Defined over formulae of prop. calculus P, P&Q, ¬Q Pr(P) = .5 Pr(Q|P) = .5 Pr(P&Q) = .3 Inconsistency To avoid this, must follow laws of probability (including Bayes theorem) Probability and the consistencyof subjective degrees of belief Subjective degrees of belief consistent if they have a (probabilistic) model

7. Believe it (or not) Calculus of certain inference Degrees of belief* Calculus of uncertain inference So difference of emphasis • *Tonight’s Evening Session – Chater, Griffiths, Tenenbaum, Subjective probability and Bayesian foundations

8. Overview • Logic and probability • Compare and contrast • A competitive perspective • How much deduction is there? • Evidence from human reasoning • A cooperative perspective • Logic as a theory of representation • Probability as a theory of uncertain inference • Probability over logically rich representations • The probabilistic mind

9. Philosophy of science Popper Statistics (!) Fisher (Sampling theory) AI Non-monotonic logic T implies D D is false T is false If A then B, and A, and no reason to the contrary, infer B Uncertainty: a logical invasion?

10. Methods for certain reasoning fail, because they can only reject Or remain agnostic, not favouring one option or another No mechanism for gaining confidence in a hypothesis (though Popper’s corroboration of theories) Why the invasion won’t work

11. A probabilistic counter-attack? • Everyday inference is defeasible • There is no deduction! • So cognitive science should focus on probability

12. Conditionals: probability encroachng on logic? • Probabilistic predictions are graded • Depend on Pr(P) and Pr(Q) • Fit with data on argument endorsements…

13. Varying probabilities in conditional inference (Oaksford, Chater & Grainger, 2000)

14. Negations implicitly vary probabilities (e.g., if Pr(Q)=.1; Pr(not-Q=.9)

15. “logical” Popperian view: aim for falsification only: turn P, ¬Q But people tend to ‘seek confirmation’ choosing P, Q Each card has P/¬P on one side, Q/ ¬Q on the other Test If P then Q Which cards to turn? Wason’s Selection task P ¬P Q ¬ Q

16. And expected amount of information (Shannon) depends crucially on Pr(P), Pr(Q) normally most things don’t happen, i.e., assume rarity Bayesian view: assess expected amount of information from each card (cf Lindley 1956)

17. P ¬P ¬ Q Q Fits Observed preferences p > q > ¬q > ¬p

18. And also if priors Pr(P), Pr(Q) are experimentally are manipulated… A fits with science---where we attempt to confirm hypotheses (and reject them if we fail)

19. Overview • Logic and probability • Compare and contrast • A competitive perspective • How much deduction is there? • Evidence from human reasoning • A cooperative perspective • Logic as a theory of representation • Probability as a theory of uncertain inference • Probability over logically rich representations • The probabilistic mind

20. Just a special case of probability? (when Probs are 0 and 1) Not yet! Probability doesn’t easily handle: Objects, Predicates, Relations though see BLOG, Russell, Milch. Morning session Monday 16 July Quantification The bane of confirmation theory Fa, Fb, Fc  Pr(x.Fx) = ?? Modality x.Fx  Pr(□Fx) = ?? Is logic dispensible?

21. Why logic is not dispensible: An example • John must sing or dance • ? (John must sing) OR (John must dance) • ? If ¬(John sings) then (John must dance) • ?There is something that John must do • P □P(j) (second order logic, and modals) • From an apparently innocuous sentence to the far reaches of logical analysis

22. Logic What is the meaning of a representation Especially, in virtue of its structure Probability How should my beliefs be updated Aim: probabilistic models over complex structure, including logical languages Reconciliation: Logic as representation; Probability for belief updating

23. Diseases cause symptom in the same person only People can transmit diseases (but it’s the same disease) Effects cannot precede causes Can try to capture by “brute force” in, e.g., a Bayesian network But no representation of “person” Two people having the same disease Etc… Representation is crucial, not just for natural language Cf. e.g., Tenenbaum; Kemp; Goodman; Russell; Milch and more at this summer school…

24. Overview • Logic and probability • Compare and contrast • A competitive perspective • How much deduction is there? • Evidence from human reasoning • A cooperative perspective • Logic as a theory of representation • Probability as a theory of uncertain inference • Probability over logically rich representations • The probabilistic mind

25. Two Probabilistic Minds • Probability as a theory of internal processes of neural/cognitive calculation • Probability as a meta-language for description of behaviour

26. Probability as description is a push-over • The brain deals effectively with an probabilistic world • Probability theory elucidates the challenges the brain faces…and hence a lot about how the brain behaves Cf. Vladimir Kramnik vs. Deep Fritz

27. But this does not imply probabilistic calculation • Indeed, tractability considerations imply that the brain must be using some approximations • (e.g., general assumption in this workshop) • But are they so extreme, as not be recognizably probabilistic at all? • (e.g., Simon; Kahneman & Tversky, Gigerenzer, Judgment and Decision literature – cf Busemeyer, Wed, 25 July)

28. Good Parsing and classifying complex real world objects Learning the causal powers of the everyday world Commonsense reasoning, resolving conflicting constraints, over a vast knowledge-base Bad Binary classification of simple artificial categories; Associative learning Multiple disease problems Explicit probabilistic and ‘logical’ reasoning The paradox of human probabilistic reasoning

29. The puzzle • Where strong, human probabilistic reasoning far outstrip any Bayesian machine we can build • Spectacular parsing, image interpretation, motor control • Where weak, it is hopelessly feeble • e.g., hundreds of trials for simple discriminations; daft reasoning fallacies

30. Resolving the paradox? • Interface solution: • Some problems don’t allow interface with the brain’s computational powers? • 2-factor solution: • Perhaps there are two aspects to probabilistic reasoning • the brain is good at one; • But as theorists, we only really understand the other

31. Maybe the key is having the right representations Not just heavy-duty numerical calculations Qualitative structure of probabilistic reasoning Including predication, quantification, causality, modality,… A speculation And note, too, that cognition can learn both from being told (i.e., logic?); and experience (probability?) So perhaps the fusion of logic and probability may be crucial