Conditionals, Suppositions and Uncertainty Lecture 2

1. Conditionals, Suppositions and UncertaintyLecture 2 Dorothy Edgington Paris June 2019

2. Reasons for a suppositional theory of conditionals • 1. Referring expressions in consequents which don’t refer are no problem. • 2. Easier extension to other kinds of speech act. • 3. The main reason: it is the only theory which handles well the fact that our conditional judgements are typically uncertain.

3. The role of referring expressions in the consequents of conditionals • If there’s an ant in my shoe, it’s not very large. • If you do that, the consequences will be dire. • If you press the switch, your action will cause the light to go on.

4. Uniform extension to other speech acts • If the patient is still alive in the morning, change the dressing. • Make it the case that if the patient is still alive in the morning you change the dressing.

5. Conditional uncertainty • ‘The probability that two … events will both happen is … the probability of the first [multiplied by] the probability of the second on the supposition that the first happens’ (Bayes, 1763)

6. Example • The probability of heads is ½. The probability of not-heads is ½. The probability of tails is ½. The probability of ‘Heads and Tails’ is 0. the probability of ‘Heads and not-tails’ is ½. The probability of ‘Heads on 1st toss and heads on 2ndtoss’ is ¼.

7. Ramsey • ‘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; … they are fixing their degrees of belief in q given p.’ (1929, in 1931, p. 247) • ‘Many sentences express cognitive attitudes without being propositions; … . This is even true of the ordinary hypothetical.’ (1929, 147-8)

8. If she tosses, it will land heads H H 25% 25% 50% CP: 50% Truth-function: 75% Lewis-style: 25% Strict: 0% (e.g. Lycan, Gillies) Toss T Don’t toss

9. If offered, will decline 9% H Accept Offered Not offered Decline 1% CP: 10% Truth-function: 91% Lewis-style: 1% Strict: 0% 90%

10. Two questions • 1. You are certain that ¬(A&¬B) but not certain that ¬A. Should you be certain that if A, B? • t-f: yes; n-t-f: no; supp: yes • 2. You think it unlikely that A. Might you still think it unlikely that if A, B? • t-f: no; n-t-f: yes; supp: yes. • So we see that supp differs from both, and gets the right answer to both questions!

11. The truth-functional conditional (i) (ii) (iii) A B AB ¬AB A¬B T T T T F T F F T T F T T T T F F T F T

12. The non-truth-functional conditional (iv) (v) (vi) A B AB ¬AB A¬B T T T T/F F T F F T/F T F T T/F T T/F F F T/F F T/F

13. Adams on validity • For a valid argument without conditionals, it is demonstrable that the improbability (1 – probability) of the conclusion cannot exceed the sum of the improbabilities of the premises. • (Because: ¬C entails ¬A1 v … v ¬An. And the maximum probability for a disjunction is the sum of the probabilities of the disjuncts.)

14. Adams-validity continued • So if you have a valid argument with two premises, each 90% likely, the conclusion must be at least 80% likely. • If each premise is 99% likely, the conclusion must be at least 98% likely. • Two premises each 50% likely compatible with conclusion 0%. (e.g.: A or B; ¬A; so B) • A huge number of premises each very likely compatible with conclusion 0 or very low. (Paradox of Lottery and Paradox of Preface)

15. Extension to conditionals Adams proposed that we extend this very valuable feature of valid arguments to arguments with conditionals assessed by conditional probability. Call the improbability of ‘If A, B’ 1 – p(BA). The valid arguments are such that the improbability of the conclusion cannot exceed the sum of the improbabilities of the premises. And a logic for conditionals emerges. (It coincides with Stalnaker’s over their common domain.)

16. A difference Say ‘It’s possible that A’ when the probability of A is non-zero. For valid arguments without conditionals, with two or more logically independent premises, we can have all premises possible and conclusion 0. E.g. A, B, so A&B A or B, ¬A, so B. Now consider modus ponens: Poss(A); Poss(If A, B) so Poss(B). Another sign that conditionals differ from propositions.

17. Some invalid inferences • B; so if A, B • I can consistently be close to certain that Sue is lecturing right now, while thinking it highly unlikely that if she had a heart attack on her way to work, she is lecturing just now. • ¬A; so if A, B • I can be close to certain that the Republicans won’t win, while thinking it highly unlikely that if they win they will double income tax.

18. And more • Strengthening of the antecedent: If A, B; so if C&A, B • It’s very likely that if you strike the match, it will light; but very unlikely that if you dip it in water and strike it, it will light. • Transitivity: If A, B; if B, C; so if A, C. • Very likely, if Jones is elected, Brown will resign immediately afterwards; very likely, if Brown dies before the election, Jones will be elected; but not: if Brown dies before the election, Brown will resign immediately after the election!

19. Comments • We do have restricted transitivity: if A, B; if A&B, then C; so if A, C. • Also worthy of note that Conditional Proof fails. ‘¬(A&B); A; so ¬B’ is valid. But as we saw, ‘¬(A&B); so if A, ¬B’ is not. • We still have a good logic—the same as Stalnaker’s over their common domain.

20. The bête noire Construing conditionals as conditional probabilities, we have no established theory of how to evaluate, in terms of truth conditions, or in terms of probabilities, conjunctions or disjunctions of conditionals, conditionals embedded in other conditionals, and so on. We don’t even know how to construe negations of conditionals.

21. “What about compound sentences that have … conditionals as constituents? We think we know how the truth conditions for compound sentences of various kinds are determined by the truth conditions of constituent subsentences, but this knowledge would be useless if any of those sentences lacked truth conditions. Either we need new semantic rules for many familiar connectives and operators when applied to indicative conditionals … or else we need to explain away all seeming examples of compound sentences with conditional constituents.” (Lewis, ‘Probabilities of Conditionals and Conditional Probabilities’ 1976, reprinted in Jackson (ed.) Conditionals p. 85.

22. Examples • ‘It’s more likely than not that if the gardener did it he used a spade and if the cook did it he used a knife’ • ‘If she’ll be angry if she gets a B, she’ll be furious if she gets a C’

23. Adams’ remark • There’s a smooth theory of embeddings in the truth-functional operators if and only if conditionals have truth conditions. For if there is a smooth theory, conditionals or their negations can occur along with atomic sentences in Carnapian state descriptions. We can have an exclusive and exhaustive set of these, whose probabilities must sum to 1. The probability of a conditional will be the sum of the probs. of the state descriptions in which it occurs un-negated, i.e. the probability of their truth; and their content is the set of ‘worlds’ in which they are true.

24. We’ll look at some attempts to solve this problem in my fourth lecture