Reasoning about causal belief Kaibo Xie Institute for Logic, - - PowerPoint PPT Presentation

Reasoning about causal belief

Kaibo Xie

Institute for Logic, Language and Computation

July 27, 2018

Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 1 / 20

Causal Logic in Halpern (2016)

Signature

a signature is defined as a triple (U, V, R) where U is the set of exogenous variables and V is the set of endogenous variables, and R is a function that indicates the range of possible values of each causal variables.

Causal Model

Given such a signature S, a causal model is a pair (S, F) where F associates with every endogenous variable X a function denoted FX which characterize the value of X given the value of all the other variables in U ∪ V.

Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 2 / 20

An Example

U1 A P B U2

A and B stand for two assassins. P stands for whether the president is

• killed. U1 and U2 represent external factors that determine whether

assassin A or B will shoot the president. Exogenous variables: U = {U1, U2}; Endogenous variables:V = {A, B, P}. Structural equations: A = U1, B = U2, P = A ∨ B

Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 3 / 20

What if we prevent A from shooting

Suppose in the real world, A receives order to shoot the president and B does not, in this case the president is killed (U1 = A = 1, U2 = B = 0, P = 1). What will happen if we prevent A from shooting? Is the president alive in this case?

Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 4 / 20

What if we prevent A from shooting

Suppose in the real world, A receives order to shoot the president and B does not, in this case the president is killed (U1 = A = 1, U2 = B = 0, P = 1). What will happen if we prevent A from shooting? Is the president alive in this case?

Intervention

1 Set the value of A to 0: replace F with FA=0 where FA=0 is the

result of replacing the equation for A in F by A = 0 (by turning FA into constant functions whose output is 0) and leaving the remaining equations untouched.

2 Check: whether in all possible solutions to the structural equations

• btained after setting A to 0 (namely FA=0), P = 0 holds whenever

U1 = 1, U2 = 0

Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 4 / 20

Halpern’s Causal Language

(Basic) causal formula

A basic causal formula is in the form of [Y1 = y1, ..., Yk = yk]φ where Y1, ..., Yk are distinct causal variables, and φ is a boolean combination

• f formulas in the form of X = x.

A causal formula is a boolean combination of basic causal formulas.

Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 5 / 20

Halpern’s Causal Language

(Basic) causal formula

A basic causal formula is in the form of [Y1 = y1, ..., Yk = yk]φ where Y1, ..., Yk are distinct causal variables, and φ is a boolean combination

• f formulas in the form of X = x.

A causal formula is a boolean combination of basic causal formulas. For instance, [Y1 = y1, ..., Yk = yk]X = x means: in all possible solutions to the structural equations obtained after setting Yi to yi, i = 1, ..., k, The random variable X has value x

Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 5 / 20

Reasoning about causal belief

1 If Y = y had been the case, X = x would been the case 2 It is believed that setting the value of Y to y results in X having the

value x

3 After revising my belief state with Z = z, it is believed that setting

the value of Y to y results in X having the value x

Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 6 / 20

Epistemic operators

Knowledge: Kφ (the agent knows φ) Belief: Belφ (The agent believes φ) Conditional Belief: Belψφ (The agent believes φ after revising its belief with ψ)

Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 7 / 20

Epistemic operators

Knowledge: Kφ (the agent knows φ) Belief: Belφ (The agent believes φ) Conditional Belief: Belψφ (The agent believes φ after revising its belief with ψ) For instance ∧x∈R(X) ∨y∈R(Y ) K[X = x](Y = y)

Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 7 / 20

Epistemic model (basic)

An epistemic model is a tuple (W , , Π)

1 W is a set of possible worlds 2 is a plausibility ordering over W 3 Π is an information partition over W : for each w ∈ W , Π(w) tells us

which possible worlds are indistinguishable for the agent

Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 8 / 20

Epistemic Model

¯ PQ

w1

PQ

w2

¯ P ¯ Q

w3

P ¯ Q

w4 w1 w2 w3 w4 Π(w1) = Π(w2) = {w1, w2}; Π(w3) = Π(w4) = {w3, w4}

Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 9 / 20

Causal Epistemic Model

A causal epistemic model is a tuple S, F, Π, ≤

1 S is a tuple (U, V, R) 2 F is a set of structural equations, for each X ∈ V, FX is a function

from (×Z∈UR(Z) × (×Y ∈V−{X}R(Y )) to R(X). F has no causal loops.

3 Π is an information partition over W where W = ×X∈U∪VR(X). 4 ⊂ W × W is a total pre-order on W satisfying certain constraints.

is known as the plausibility ordering.

Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 10 / 20

Combine the languages

Let S = (U, V, R), the language for S, write L(S), is defined as follows: X = x (if X ∈ U ∪ V and x ∈ R(X) |φ ∧ ψ|¬φ|[X1 = x1, ..., Xi = xi]φ (if X1 = x1, ..., Xi = xi is a sequence of distinct atomic sentences with X1, ..., Xi ∈ V and φ is a formula without intervention

• perators|Belφ ∈ L(S)|Belψφ|Kφ

Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 11 / 20

Semantics

Let S = (U, V, R) be a signature and M = S, F, Π, is a causal epistemic model.

Boolean Cases

Let w be a possible world in W in the form of (y1, ..., yn). M, (y1, ..., yn) | = Xi = xi (1 i n) if and only if xi = yi. The boolean combinations are defined in the usual way.

Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 12 / 20

Epistemic Operators (Baltag and Smets (2006))

Belief

M, w | = Belφ if and only if φ holds on the most plausible worlds in Π(w) Believing φ means φ is true at the most plausible worlds

Conditional Belief

M, w | = Belψφ if and only if for any s ∈ Min({t ∈ W |M, t | = ψ} ∩ Π(w)), M, s | = φ. It means that the agent has the conditional belief “given ψ, then φ” if and only if φ holds on the most plausible ψ worlds. Believing φ conditional on ψ means φ is true at the most plausible ψ-worlds.

Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 13 / 20

Find the counterpart of Halpern’s intervention operator

We need to define the output of setting the value of X to x in ( u, v), write fX=x(( u, v)) = ( a, b)

The output of setting the value of X to x

FX=x is defined as the result of replacing the equation for X1, ..., Xn in F by X1 = x1, ...Xn = xn (namely FX1, ..., FXn become constant functions whose output are x1, ..., xn and leaving the remaining equations untouched). Let V = V1, ..., Vn, v = v1, ..., vn, v′ = v′

1, ..., v′ n,

b = b1, ..., bn. Define ( a, b) as: a = u; for any 1 ≤ i ≤ n, bi = v′

i if X Vi, otherwise bi = vi.

Y Z means “Y affects Z” as an abbreviation for the formula ∨

X⊂V, x∈×X∈VR(X),y∈R(Y ), u∈×U∈UR(U),z=z′∈R(Z)([

X = x, Y = y]Z = z′ ∧ [ X = x, Y = y]Z = z)

Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 14 / 20

The intervention operator

The truth condition of the a sentence with intervention operator

Given a signature S = (U, V, R) and a causal epistemic model M = S, F, Π, ≤. Let w be a possible world in W in the form of (y1, ..., yn). M, w | = [ X = x]φ if and only if M, f

X= x(w) |

= φ where f

X= x is defined as

before.

Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 15 / 20

Plausibility ordering is not arbitrary

Belief should consistent with agents’ causal information.

Constraint on the plausibility ordering

For any w1, w2 ∈ W if w1 ≺ w2 and w2 ⊀ w1 then w1 < w2 (where w1 ≺ w2 is defined as there is X ∈ V such that w1 complies FX and w2 does not.)

Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 16 / 20

Applications of the combination in the philosophy of language

The distinction between indicative conditionals and subjunctive conditionals Bel[ X = x]φ does not imply Bel

X= xφ

Example from Adams (1970)

“If Oswald had not killed Kennedy, then someone else would have” does not imply “If Oswald did not kill Kennedy, then someone else did”

Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 17 / 20

Subjunctive conditional

Kratzer’s King Ludwig Example

King Ludwig of Bavaria likes to spend his weekends in Leoni Castle. Whenever the king is in the castle, the lights will be on and the royal flag will be up. A traveler watches the castle from a distance and sees that the lights are on. The flag, however, is not up. He says If the flag had been up, the king would have been in the castle.

U K F L

Nobody will agree that the king would be brought into the castle by hoisting the flag. [L = 1]K = 0 is rejected in this model under the setting.

Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 18 / 20

Condition of Acceptance in Two Readings

Ontic Reading

A counterfactual “if X = x had been the case then Y = y would have been the case” is accepted in the ontic reading (on the actual world) iff Bel[ X = x] Y = y is true at the actual world.

Epistemic Reading

A counterfactual “if X = x had been the case then Y = y would have been the case” is accepted in the epistemic reading (at the actual world) iff [ X = x]Bel( Y = y) is true at the actual world.

Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 19 / 20

Thank you!

Kaibo Xie (Institute for Logic, Language and Computation) Reasoning about causal belief July 27, 2018 20 / 20