Conditionals as Random Variables?

1. PROGIC Groningen, September 09 Conditionals as Random Variables? Richard Bradley Department of Philosophy, Logic and Scientific Method London School of Economics r.bradley@lse.ac.uk

2. 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)

3. The Truth Conditional Orthodoxy O1. Semantics: The semantic content of a sentence, A, is given by its truth conditions. • Bivalence: vw(A)  {0,1}. • Boolean: vw(AB) = vw(A).vw(B); vw(¬A) = 1 - vw(A) O2. Pragmatics: The probability of a sentence is its probability of truth: • O3.Logic: AB iff [A]  [B] • O4. Explanation: The Priority of Semantics

4. 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).

5. Jeffrey & Stalnaker (1994) • The semantic contents of sentences are random variables taking values in the interval [0,1] at each world. In particular: vw(AB) = 1 if w  A and B, 0 if w  A and ¬B p(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:

6. 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: • How do we interpret the semantic values and the corresponding probabilities? • How do we explain AT?

7. 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:

8. 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.

9. Fair Coin (Uncertain Selection) w1 = (H) Lands heads (T) Toss coin w2 = Lands tails (¬T) Don’t toss w3 f(w3, T) = w1, q(f) = 0.5 f*(w3, T) = w2, q(f*) = 0.5 p(T H) = p(w1) + p(w3).q(f) = 0.5

10. Biased Coin (Uncertain Worlds) (T) Toss coin w1 = (H) Lands heads w2 (¬T) Don’t toss (Bh) Biased heads (T) Toss coin w3 = Lands tails (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) (¬T) Don’t toss w4

11. Biased Coin (Uncertain Worlds) w1 = (H) Lands heads (T) Toss coin (Bh) Biased heads w2 (¬T) (T) Toss coin W3 = Lands tails (Bt) Biased tails (¬T) w4 Independence fails: p(T H|¬TBt) = 0 ≠ p(T H) = p(Bh) > 0 Hence so too must O1b:vw(T H & ¬TBt) ≠ vw(T H).vw(¬TBt)

12. B ¬B w1 A w2 ¬A w3 w4

14. A

15. A B

16. Probability p(Wi) is probability of Wi. q(WAi) is probability of Wi on the hypothesis that A. Pr(Wi,WAi) is their joint probability. A-worlds ¬A-worlds

17. Restricted Independence: Pr(WAi|A) = Pr(WAi) A-worlds ¬A-worlds Note that AT holds: Pr(A B) = Pr(W1) + Pr(W1|A).Pr(¬A) = Pr(W1|A) = Pr(B|A)

18. Probability is expectation of expected truth! A-worlds ¬A-worlds Note that (unrestricted) Independence can fail.

19. Independence (Biased Coin) Pr(T H|¬TBh) = 1 But: Pr(T H) = Pr(W1) + Pr(W2) < 1

20. It’s Elementary! • It is very probable that it wasn’t the cook. • It is very probable that if it wasn't the butler, then it was the cook. So: • 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!

21. 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.