Conditionals as Random Variables?

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Conditionals as Random Variables?

Richard Bradley

Department of Philosophy, Logic and Scientific Method

London School of Economics

[email protected]

PROGICGroningen, September 09

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The Ramsey-Adams Hypothesis

• General Idea: Rational belief for conditionals goes by conditional belief for their consequents on the assumption that their antecedent is true.

“If two people are arguing ‘If p will q?’ and are both in doubt as to p, they are adding p hypothetically to their stock of knowledge and arguing on that basis about q …” (Ramsey, 1929, p.155)

• Adams Thesis: The probability of an (indicative) conditional is the conditional probability of its consequent given its antecedent:

(AT) )|()( ABpBAp

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The Truth Conditional Orthodoxy O1. Semantics: The semantic content of a sentence, A, is

given by its truth conditions.

a) Bivalence: vw(A) {0,1}.

b) Boolean: vw(AB) = vw(A).vw(B); vw(¬A) = 1 - vw(A)

O2. Pragmatics: The probability of a sentence is its probability of truth:

][

)()().()(Aww

w wpwpAvAp

O3. Logic: A B iff [A] [B]

O4. Explanation: The Priority of Semantics

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Routes to Reconciliation

• Abandon O4:– Give an independent pragmatic explanation for AT and PC (Lewis,

Jackson, Douven)

• Abandon O1(a) and O4:– Allow the truth-value of a conditional to depend on pragmatic factors

(McGee)– Treat conditionals as random variables (Jeffrey & Stalnaker)

• Abandon O1(a) and (b):– 3-valued logics (Milne, Calabrese, McDermott)– Proposition-valued functions (Bradley)

• Abandon both O1 and O2. – Non-factualism (Edgington, Gibbard).

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Jeffrey & Stalnaker (1994)

• The semantic contents of sentences are random variables taking values in the interval [0,1] at each world. In particular:

vw(AB) = 1 if w A and B, 0 if w A and ¬Bp(B|A) if w ¬A

• The values (at a world) of compounds are Boolean functions of the values of their constituents (O1b.)

• The probability of a sentence is its expected semantic value:

w

w wpAvAp )().()(

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• It follows that:

(AT) Adams’ Thesis:

p(A B) = 1.p(AB) + 0.p(A¬B) + p(B|A).p(¬A)

= p(B|A)

(IN) Independence: If AC is a logical falsehood, then:

p(C(A B)) = 1.p(ABC) + p(B|A).p(¬AC)

= p(C).p(A B)

• Two questions:

1. How do we interpret the semantic values and the corresponding probabilities?

2. How do we explain AT?

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McGee (1989)

• McGee adopts a modified version of a ‘Stalnaker’ semantics’ by which a simple conditional sentence A B is true at w iff B is true at the ‘nearest’ world in which A is true.

• But ..“purely semantic considerations are … only able to tell us which world is the actual world. Beyond this, to try to say which of the many selection functions that originate at the actual world is the actual selection function, we rely on pragmatic considerations in the form of personal probabilities”

• These pragmatic considerations suffice to determine (only) an expected truth value for a conditional at a world:

i

i

f

iAwfw fqBvBAv )().()( ),(

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• The weight q on a selection function f is obtained in turn from the probabilities of state description sentences - conjunctions of the form (A f(A))(B f(B))…., - where these are determined by a probability calculus containing both AT and IN.

• In summary:– The semantic contents of conditionals are not propositions, but

random variables mapping states (qua selection functions) to propositions.

– The contents of complex sentences (including conditionals) are Boolean functions of the contents of their constituents.

– Adams’ Thesis is explained by non-semantic constraints on rational belief.

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w1 = (H) Lands heads

w2 = Lands tails

(T) Toss coin

(¬T) Don’t tossw3

Fair Coin (Uncertain Selection)Fair Coin (Uncertain Selection)

f(w3, T) = w1, q(f) = 0.5f*(w3, T) = w2, q(f*) = 0.5p(T H) = p(w1) + p(w3).q(f) = 0.5

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w1 = (H) Lands heads

w2

(T) Toss coin

(¬T) Don’t toss w4

Biased Coin (Uncertain Worlds)Biased Coin (Uncertain Worlds)

w3 = Lands tails

(¬T) Don’t toss

(T) Toss coin

(Bh) Biased heads

(Bt) Biased tails

f(w2, T) = w1 f(w4, T) = w3 p(T H) = p(w1) + p(w2) = p(Bh)p(H|T) = p(w1)/[p(w1) + p(w3)] = p(Bh|T) = p(Bh)

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w1 = (H) Lands heads

w2

(T) Toss coin

(¬T)w4

Biased Coin (Uncertain Worlds)Biased Coin (Uncertain Worlds)

W3 = Lands tails

(¬T)

(T) Toss coin

(Bh) Biased heads

(Bt) Biased tails

Independence fails: p(T H|¬TBt) = 0 ≠ p(T H) = p(Bh) > 0Hence so too must O1b: vw(T H & ¬TBt) ≠ vw(T H).vw(¬TBt)

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A

¬A

B ¬B

w1

w4

w2

w3

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Counterfactual A-worlds

Worlds WA1 WA2

W1

W2

W3

W4

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Counterfactual A-worlds

Worlds WA1 WA2

W1

W2

W3

W4

A

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Counterfactual A-worlds

Worlds WA1 WA2

W1

W2

W3

W4

A B

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Counterfactual A-worlds

Worlds WA1 WA2

W1 p(W1) 0

W2 0 p(W2)

W3 ? ?

W4 ? ?

A-w

orld

s¬

A-w

orld

sProbability

p(Wi) is probability of Wi.

q(WAi) is probability of Wi on the hypothesis that A.

Pr(Wi,WAi) is their joint probability.

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Counterfactual A-worlds

Worlds WA1 WA2

W1 Pr(W1) 0

W2 0 Pr(W2)

W3Pr(W1|A).Pr(¬A) Pr(W2|A).Pr(¬A)

W4

A-w

orld

s¬

A-w

orld

sRestricted Independence: Pr(WAi|A) = Pr(WAi)

Note that AT holds: Pr(A B) = Pr(W1) + Pr(W1|A).Pr(¬A) = Pr(W1|A) = Pr(B|A)

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Counterfactual A-worlds

Worlds WA1 WA2

W1 1.p(W1) 0.p(W1)

W2 0.p(W2) 1.p(W2)

W3Pr(W1|A).p(¬A) Pr(W2|A).p(¬A)

W4

A-w

orld

s¬

A-w

orld

s

Note that (unrestricted) Independence can fail.

Probability is expectation of expected truth!

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Worlds in which coin is tossed

Worlds

Biased Heads

WT1 = T&H

Biased Tails

WT3 = T&¬H

W1 = T&Bh Pr(W1) 0

W2 = ¬T&Bh Pr(W2) 0

W3 = T&Bt 0 Pr(W3)

W4 = ¬T&Bt 0 Pr(W4)

Independence (Biased Coin)

Pr(T H|¬TBh) = 1But: Pr(T H) = Pr(W1) + Pr(W2) < 1

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It’s Elementary!1. It is very probable that it wasn’t the cook.

2. It is very probable that if it wasn't the butler, then it was the cook.

So:

3. It is very improbable that if it wasn't the butler, then it was the gardener.

Furthermore:

4. It is certain, given that it wasn't the cook, that if it wasn't the butler, then it was the gardener.

5. It is impossible, given that it was the cook, that if it wasn't the butler, then it was the gardener.

So since Pr(X) = Pr(X|Y).Pr(Y) + Pr(X|¬Y).Pr(¬Y), we can conclude:

6. It is very probable that if it wasn’t the butler, then it was the gardener.

This contradicts 3!

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Conclusion

• Indicative and counterfactual conditionals can be treated in the same way.

• Adams’ Thesis holds in virtue of the way in which degrees of belief interact with ‘similarity’ judgements.

• We need not abandon the priority of semantics.

• We do need to abandon the Boolean assumption O1(b) because Independence fails.