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From logicist to probabilist cognitive science
Nick Chater
Department of Psychology
University College [email protected]
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
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 prop calculus O(A) deontic logic □A modal F(j) 1st order
John_must_singMust(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 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
Uncertainty: a logical invasion? 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
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)
0
20
40
60
80
100
MP DA AC MT
Inference
Pro
port
ion
End
orse
d (%
)
Data
Model
Low P(p), High P(q)
0
20
40
60
80
100
MP DA AC MT
Inference
Pro
port
ion
End
orse
d (%
)
Data
Model
High P(p), Low P(q)
0
20
40
60
80
100
MP DA AC MT
Inference
Pro
port
ion
End
orse
d (%
)
Data
Model
High P(p), High P(q)
0
20
40
60
80
100
MP DA AC MT
Inference
Pro
port
ion
End
orse
d (%
)
Data
Model
Negations implicitly vary probabilities (e.g., if Pr(Q)=.1; Pr(not-Q=.9)
If p then q
0
20
40
60
80
100
MP DA AC MT
Inference
Pro
port
ion
End
orse
d (%
)
Data
Model
If p then not-q
0
20
40
60
80
100
MP DA AC MT
Inference
Pro
port
ion
End
orse
d (%
)
Data
Model
If not-p then q
0
20
40
60
80
100
MP DA AC MT
Inference
Pro
port
ion
End
orse
d (%
)
Data
Model
If not-p then not-q
0
20
40
60
80
100
MP DA AC MT
Inference
Pro
port
ion
End
orse
d (%
)
Data
Model
Wason’s Selection task “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?P ¬P Q ¬ 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
Fits Observed preferences p > q > ¬q > ¬p
P ¬P
Q ¬ Q
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 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
Is logic dispensible?
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) = ??
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 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
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,…
So perhaps the fusion of logic and probability may be crucial
And note, too, that cognition can learn both from being told (i.e., logic?); and experience (probability?)
