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From logicist to probabilist cognitive science Nick Chater Department of Psychology University College London [email protected] Overview Logic and probability A competitive perspective How much deduction is there? Evidence from human reasoning A cooperative perspective Compare and contrast Logic as a theory of representation Probability as a theory of uncertain inference Probability over logically rich representations The probabilistic mind Two traditions 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 Neither Boole nor Bayes distinguished between normative and descriptive principles: Error or insight? Logic and the consistency of beliefs 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 Different depths of representation John must sing A O(A) □A F(j) prop calculus deontic logic modal 1st order John_must_sing Must(John_sings) Necessarily(John_sings) Must_sing(John) Probability and the consistency of subjective degrees of belief 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) Subjective degrees of belief consistent if they have a (probabilistic) model So difference of emphasis Believe it (or not) Calculus of certain inference Degrees of belief* Calculus of uncertain inference *Tonight’s Evening Session – Chater, Griffiths, Tenenbaum, Subjective probability and Bayesian foundations Overview Logic and probability A competitive perspective How much deduction is there? Evidence from human reasoning A cooperative perspective Compare and contrast Logic as a theory of representation Probability as a theory of uncertain inference Probability over logically rich representations The probabilistic mind Uncertainty: a logical invasion? Philosophy of science T implies D Popper D is false T is false Statistics (!) Fisher (Sampling theory) AI Non-monotonic logic If A then B, and A, and no reason to the contrary, infer B Why the invasion won’t work 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) A probabilistic counter-attack? Everyday inference is defeasible There is no deduction! So cognitive science should focus on probability Conditionals: probability encroachng on logic? Inference Additional premise Candidate conclusion Logical validity Probabilistic comparison MP: Modus Ponens P Q Y Pr(Q|P) Pr(Q) DA: Denial of the Antecedent Not-P Not-Q N Pr(not-Q|not-P) Pr(not-Q) AC: Affirming the Consequent Q P N Pr(P|Q) Pr(P) MT: Modus Tollens Not-Q Not-P Y Pr(not-P|not-Q) Pr(not-P) • Probabilistic predictions are graded • Depend on Pr(P) and Pr(Q) • Fit with data on argument endorsements… Varying probabilities in conditional inference (Oaksford, Chater & Grainger, 2000) Low P(p), Low P(q) Low P(p), High P(q) 100 Proportion Endorsed (%) Proportion Endorsed (%) 100 80 60 40 20 Data Model 0 MP DA AC 80 60 40 20 Data Model 0 MT MP Inference AC MT Inference High P(p), Low P(q) High P(p), High P(q) 100 Proportion Endorsed (%) 100 Proportion Endorsed (%) DA 80 60 40 20 Data Model 0 MP DA AC Inference 80 60 40 20 Data Model 0 MT MP DA AC Inference MT Negations implicitly vary probabilities (e.g., if Pr(Q)=.1; Pr(not-Q=.9) If p then q If p then not-q 100 Proportion Endorsed (%) Proportion Endorsed (%) 100 80 60 40 20 Data Model 0 MP DA AC 80 60 40 20 Data Model 0 MT MP Inference AC MT Inference If not-p then q If not-p then not-q 100 Proportion Endorsed (%) 100 Proportion Endorsed (%) DA 80 60 40 20 Data Model 0 MP DA AC Inference 80 60 40 20 Data Model 0 MT MP DA AC Inference MT Wason’s Selection task Each card has P/¬P on one side, Q/ ¬Q on the other Test If P then Q Which cards to turn? P ¬P Q ¬Q “logical” Popperian view: aim for falsification only: turn P, ¬Q But people tend to ‘seek confirmation’ choosing P, Q Bayesian view: assess expected amount of information from each card (cf Lindley 1956) And expected amount of information (Shannon) depends crucially on Pr(P), Pr(Q) normally most things don’t happen, i.e., assume rarity P ¬P Q ¬Q Fits Observed preferences p > q > ¬q > ¬p 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) Overview Logic and probability A competitive perspective How much deduction is there? Evidence from human reasoning A cooperative perspective Compare and contrast Logic as a theory of representation Probability as a theory of uncertain inference Probability over logically rich representations The probabilistic mind Is logic dispensible? Just a special case of probability? Quantification (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 The bane of confirmation theory Fa, Fb, Fc Pr(x.Fx) = ?? Modality x.Fx Pr(□Fx) = ?? 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 Reconciliation: Logic as representation; Probability for belief updating 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 Representation is crucial, not just for natural language 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… Cf. e.g., Tenenbaum; Kemp; Goodman; Russell; Milch and more at this summer school… Overview Logic and probability A competitive perspective How much deduction is there? Evidence from human reasoning A cooperative perspective Compare and contrast Logic as a theory of representation Probability as a theory of uncertain inference Probability over logically rich representations The probabilistic mind Two Probabilistic Minds Probability as a theory of internal processes of neural/cognitive calculation Probability as a meta-language for description of behaviour 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 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) The paradox of human probabilistic reasoning 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 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 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 A speculation Maybe the key is having the right representations Not just heavy-duty numerical calculations Qualitative structure of probabilistic reasoning Including predication, quantification, causality, modality,… 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