Logical omniscience : from epistemic logic to rational choice Mikaël Cozic [email protected]

(Philosophy, Paris IV-Sorbonne and IHPST (CNRS-Paris I))

M.Cozic, ECCE1, Logical omniscience – p. 1/31

introduction ¥ logical omniscience = closure under logical consequence ,→ no deductive ignorance : if someone knows chess rules, he or she knows if White have a winning strategy ; if someone knows Peano axioms, he or she know all theorems of number theory... ¥ huge idealization : "It is not only mathematicians who need to worry about their failure to know all the consequences of their knowledge. Any context where an agent engages in reasoning is a context that is distorted by the assumption of deductive omniscience, since reasoning (at least deductive reasoning) is an actitivity that deductively omniscient agents have no use for." (Stalnaker)

M.Cozic, ECCE1, Logical omniscience – p. 2/31

introduction, cont. Plan of the talk :

¥ epistemic logic : Hintikka, Stalnaker, Fagin, Halpern... ¥ Question : how extend to probability and decision making ? ¥ Plan of the talk ¨ Section 1 : logical omniscience and epistemic logic ¨ Section 2 : the probabilistic case ¨ Section 3 : the decision-theoretic case

M.Cozic, ECCE1, Logical omniscience – p. 3/31

sec. 1, epistemic propositional language Definition : The set of formulas of an epistemic propositional language L B( At), Form(L B( At)), is the subset of L B( At) such that (i) if p ∈ At, p ∈ Form(L B( At)), (ii) if φ ∈ Form(L B( At)), then ¬φ ∈ Form(L B( At)), (iii) if φ, ψ ∈ Form(L B( At)), then ( ϕ ∧ ψ), ( ϕ ∨ ψ), ( ϕ → ψ) ∈ Form(L B( At)), and (iv) if ϕ ∈ Form(L B( At)), then Bϕ ∈ Form(L B( At)). (v) only strings of symbols generated by (i)-(iv) in a finite number of steps are in Form(L B( At)). M.Cozic, ECCE1, Logical omniscience – p. 4/31

sec. 1, kripke structures Definition : let L B( At) an epistemic propositional language ; a Kripke structure for L B( At) is a 3-tuple M = (S, π, R) where (i) S is a state space, (ii) π : At × S → {0, 1} is a valuation (iii) R ⊆ S × S is an accessibility relation

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sec. 1, kripke structures, cont. ¯ called the satisfaction relation, extends π to every Definition : π, formula of the langage according to the following conditions : (i) π¯ (s, p) = π (s, p) if p ∈ At (ii) π¯ (s, φ ∧ ψ) = 1 iff π¯ (s, φ) = 1 and π¯ (s, ψ) = 1 (iii) π¯ (s, φ ∨ ψ) = 1 iff π¯ (s, φ) = 1 or π¯ (s, ψ) = 1 (iv) π¯ (s, ¬φ) = 1 iff π¯ (s, φ) = 0 (v) π¯ (s, Bφ) = 1 iff ∀s0 s.t. sRs0 , π¯ (s, φ) = 1 (possible-state analysis of belief ) M.Cozic, ECCE1, Logical omniscience – p. 6/31

sec.1, kripke structures, example p, q

p

s1

s2

s3

s4

q

M.Cozic, ECCE1, Logical omniscience – p. 7/31

sec. 1, logical omniscience Definitions : (i) the proposition expressed by φ, or informational content of φ is [[φ]]M = {s : π¯ (φ, s) = 1} (ii) φ M-implies ψ if [[φ]] M ⊆ [[ψ]]M (iii) φ and ψ are M-equivalent if [[φ]] M = [[ψ]]M

Proposition: for all Kripke structure M, - Deductive monotony : if φ M-implies ψ, then Bφ M-implies Bψ - Intensionality : if φ and ψ are M-equivalent, then Bφ and Bψ are M-equivalent M.Cozic, ECCE1, Logical omniscience – p. 8/31

sec.1, neighborhood structures Definition : a neighborhood structure is a 3-tuple M = (S, π, V ) where (i) S is a state space, (ii) π : At × S → {0, 1} is a valuation, (iii) V : S → ℘(℘(S)), called the agent’s neighborhood system, associates to every state a set of propositions. Doxastic satisfaction condition : π¯ ( Bφ, s) = 1 iff [[φ]] M ∈ V (s)

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sec.1, neighborhood structures, example p, q

p

s1

s2

s3

s4

V(s1) = {{s1 , s2 }}

q

M.Cozic, ECCE1, Logical omniscience – p. 10/31

sec.1, awareness structures Definition : an awareness structure is a 4-tuple (S, π, R, A) where (i) S is a state space, (ii) π : At × S → {0, 1} is a valuation, (iii) R ⊆ S × S is an accessibility relation, (iv) A : S → Form(L B( At)) is a function which maps every state in a set of formulas ("awareness set"). Doxastic satisfaction condition : π¯ ( Bφ, s) = 1 iff ∀s0 s.t. sRs0 , s0 ∈ [[φ]]M and φ ∈ A(s) M.Cozic, ECCE1, Logical omniscience – p. 11/31

sec.1, awareness structures, example p, q

p

s1

s2

s3

s4

A(s1) = {p}

q

M.Cozic, ECCE1, Logical omniscience – p. 12/31

sec.1, non-standard structures ¯ R) where Definition : a non-standard structure is a 4-tuple M = (S, S 0 , π, (i) S is a space of standard states, (ii) S0 is a space of non-standard states, (iii) R ⊆ S ∪ S0 × S ∪ S0 is an accessibility relation, (iv) π : Form(L B( At)) × S → {0, 1} is a satisfaction relation standard on S Subjective informational content : [[φ]]∗M = {s ∈ S∗ : π (φ, s) = 1} M.Cozic, ECCE1, Logical omniscience – p. 13/31

sec.1, non-standard structures, example p, q

p

s1

s2

s3

s4

π(p ∨ q, s5) = 0 s5

q

M.Cozic, ECCE1, Logical omniscience – p. 14/31

sec.1, respective powers

AWARENESS STRUCTURES

NON-STANDARD STRUCTURES

NEIGHBORHOOD STRUCTURES

KRIPKE STRUCTURES

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sec. 2, probabilistic structures Definition : let L( At) a propositional language ; a probabilistic structure for L( At) is a 3-tuple M = (S, π, P) where (i) S is a state space, (ii) π is a valuation, (iii) P is a probability distribution on S. An agent believes to degree r a formula ϕ ∈ Form(L( At)), PB( ϕ) = r, if P([[ ϕ]]M ) = r

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sec.2, probabilistic omniscience Proposition : for all probabilistic structure M, - deductive monotony : if φ M-implies ψ, then PB(φ) ≥ PB(ψ). - intensionality : if φ and ψ are M-equivalent, then PB(φ) = PB(ψ). Case r= 1 : - if φ M-implies ψ, if it is certain that φ, it is certain that ψ - if φ and ψ are M-equivalent, φ is certain iff ψ is certain.

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sec.2, probabilistic non-standard structures Definition : let L( At) a propositional language ; a non-standard probabilistic structure for L( At) is a 4-tuple M = (S, S 0 , π, P) where (i) S is a standard state space, (ii) S0 is a non-standard space, (iii) π : Form( L( At)) × S ∪ S 0 → {0, 1} is a satisfaction relation which is standard on S, (iv) P is a probability distribution on S ∗ = S ∪ S0 .

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sec.2, example p, q 1/8 π(p ∨ q, s5) = 0

p 1/8

s1

s2

s3

s4

q 1/8

1/8

1/2 s5

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sec.2, special topic, deductive information and learning ¥ What does it mean to learn that φ implies ψ ? According to the possible-state analysis of belief : exclude states where φ is true but ψ false ; hence, to learn the event I (φ, ψ) = S∗ − ([[φ]]∗M − [[ψ]]∗M )

¥ Compatibility between conditionalization on I and deductive monotony : Proposition : if I (φ, ψ) is learned according to Bayes rule, then deductive monotony is regained, ie PB I (φ) ≤ PB I (ψ).

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sec.2, special topic, additivity ¥ Basic idea : control additivity by means of logical connectives Definitions : (i) M is (logically) additive if, when φ and ψ are logically incompatible, PB(φ) + PB(ψ) = PB(φ ∨ ψ). (ii) M is ∨-standard if for every formulas φ, ψ, [[φ ∨ ψ]]∗M = [[φ]]∗M ∪ [[ψ]]∗M . Proposition If M is ∨-standard, then it is (logically) subaddditive.

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sec.2, special topic, additivity, cont. Definitions : (i) M satisfies the (logical) inclusion-exclusion rule if PB(φ ∨ ψ) = PB(φ) + PB(ψ) − PB(φ ∧ ψ) (ii) M satisfies (logical) submodularity (resp. supermodularity or convexity) if : PB(φ ∨ ψ) ≤ PB(φ) + PB(ψ) − PB(φ ∧ ψ) (resp. PB(φ ∨ ψ) ≥ PB(φ) + PB(ψ) − PB(φ ∧ ψ)) Proposition. Suppose that M is ∨-standard ; - if M is negatively ∧-standard, then submodularity holds. - if M is positively ∧-standard, then supermodularity holds. M.Cozic, ECCE1, Logical omniscience – p. 22/31

sec.3, model of decision making, primitives Primitives of decision making under set-theoretic uncertainty : (i) a set A of opportunities, (ii) a state space S, (iii) a set of accessibility K ⊆ S, (iv) a set of consequences C, (v) a consequence function C : A × S → C, (vi) an utility function u : C → R. M.Cozic, ECCE1, Logical omniscience – p. 23/31

sec.3, model of decision making, structure CHOICE MODEL DOXASTIC MODEL

AXIOLOGICAL MODEL

subset K of S

u:C

choice

R

criterion

choice of acts

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sec. 3, the dilemma ¥ Question : how to integrate the alternative doxastic model ? Option 1 (conservative option) : keep the argument’s type NS’s case : take as argument the smallest proposition in the neighborhood set

,→ makes logical ignorance innocuous Option 2 (heroic option) : build a new argument’s type NS’s case : the whole set of propositions as argument

,→ if possible, looses naturalness of choice criterion M.Cozic, ECCE1, Logical omniscience – p. 25/31

sec.3, the dilemma, example p, q

p

s1

s2

s3

s4

V(s1) = {{s1 , s2 }}

q

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sec. 3, non-standard model of choice, target situation ¥ Target : an agent knows in principle the consequence function, ie knows in principle what follows from each pair ( action, state), but is not able to infer from that the exact value of each argument. K

s1

? c1

s2

s4

s3

consequence of action 1

? c2

consequence of action 2 c3 c4

c5

c6

c7

c8

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sec. 3, non-standard model of choice, extended states Definitions : (i) an extended state is a pair w = (s, C w ) where s ∈ S and Cw : A → C (ii) an extended state w = (s, C w ) is standard if for all a ∈ A, Cw ( a) = C T ( a, s).

w5

w1

w2

s1

s2 Cw2

Cw1 s1

Cw5

s4s4

s3

Cw4

Cw3

w4

w3

extended K

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sec.3, applications ¥ set-theoretic uncertainty : consequences on choices according to the criterion example : non-standard maximin Sol MaxMinNS = arg maxa∈ A minw∈ R u(Cw ( a)) Remark : in set-theoretic uncertainty, for usual choice criterion, non-standard models are equivalent to consequence correspondences (Ghirardato 2001) : for any correspondence model, there exists a non-standard equivalent model and conversely

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applications, cont. ¥ expected utility See Lipman 1999 Classical case : (i) given S and C, (ii) if certain conditions on the preference relation ¹ on F = C S hold, (iii) then there exists a probability distribution P on S and an utility function u on C s.t. ¹ is SEU-representable. Non-standard case : (i) given S and C, (ii’) to which conditions on ¹ (iii’) does there exist a non-standard state space S ∗ ⊇ S, a probability distribution P∗ on S∗ and an utility function s.t. ¹ is SEU-representable ?

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conclusion ¥ Two main limitations : - a framework, not a theory - difficulty of choice criterion is omitted

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