Conditionals. Daniel Bonevac. February 12, 2013

Transcription

1 Neighborhood February 12, 2013

2 Neighborhood are sentences formed, in English, with the particle if. Some are indicative; some are subjunctive. They are not equivalent, as this pair seems to show: 1. If Oswald didn t shoot Kennedy, someone else did. 2. If Oswald hadn t shot Kennedy, someone else would have.

3 Neighborhood Crows The logic of conditionals is notoriously controversial. Callimachus said that debates among the Stoics were so intense that Even the crows on the rooftops caw about which conditionals are true, and they are no less intense in contemporary logic.

4 Neighborhood Stoic Logic The Stoics developed four theories of the conditional, two of which remain competitive today:

5 Neighborhood Philo The material conditional (Philo): a conditional is true if and only if it does not have a true antecedent and a false consequent. Thus, If A, then B is equivalent to not (A and not B). This approach to the conditional is standard in classical first-order logic, and is still popular as an account of indicative conditionals.

6 Neighborhood Material Conditional There are three primary arguments for the material conditional: A strong version of the deduction theorem (If A, then B follows from given information if and only if that information, together with A, implies B) implies that the conditional is material; If A, then B appears to be equivalent to Either not A or B, which is equivalent to not (A and not B); first-order logic requires a material conditional in its analysis of universally quantified sentences.

7 Neighborhood Chrysippus The strict conditional (Chrysippus): a conditional is true if and only if its antecedent implies its consequent. This view, as developed by C. I. Lewis, holds that a conditional is true only when the corresponding material conditional is necessary. It remains a popular view of conditionals in logic and mathematics. It enjoys a revival in dynamic semantics, where changes in context allow for flexibility.

8 Neighborhood Strict Conditional The chief argument for the strict conditional is its ability to avoid the paradoxes of material implication by preventing If A, then B from following from not A or from B.

9 Neighborhood Rules These two accounts agree in their rules for exploiting conditional information: If A then B; A; therefore B (modus ponens) If A then B; not B; therefore not-a (modus tollens)

10 Neighborhood Admissibility They agree on a general strategy from establishing conditionals: To show If A, then B, assume A and derive B. They differ, however, about what information may be admitted in the course of that derivation. The material conditional allows any information to be used; the strict conditional allows only necessary information. On many conceptions of necessity, this implies that strict conditionals, if true, are necessarily true.

11 Neighborhood Counterfactuals The most popular account of counterfactual conditionals differs from both of the above: The variably strict conditional (R. Stalnaker, D. Lewis): If A, then it would be the case that B is true at a possible world w if and only if B is true in all the A-worlds closest to w. Counterfactuals are not material conditionals, for their antecedents are typically false, but they are not thereby true. They are not strict conditionals, for they do not satisfy inference patterns such as transitivity, contraposition, and strengthening of the antecedent.

12 Neighborhood Other Logics Four other conceptions of the conditional are subjects of active research:

13 Neighborhood Probabilistic The probabilistic conditional (E. Adams, R. Jeffrey, B. Ellis): This account replaces truth with probability: The probability of a conditional If A then B is the conditional probability of B on the condition that A. D. Lewis showed that this fails if there are three possible but pairwise incompatible sentences (such as p and q, p and not q, and q and not p). Its advocates, however, have proposed revisions treating embedded conditionals differently or replacing truth with assertibility.

14 Neighborhood Conditional Assertion The conditional assertion (D. Edgington): do not have truth values, for they are not assertions. Instead, they assert the consequent conditional on the truth of the antecedent. If A then B asserts that B, if A is true, and asserts nothing otherwise. This view was recognized but generally rejected (for example, by Paul of Venice) in medieval times because of the difficulty it has in explaining embedded conditionals.

15 Neighborhood Defeasible The defeasible conditional (N. Asher, M. Morreau): If A, then B is true at a possible world w if and only if B is true in all the A-worlds that are normal from the perspective of w. This view denies the deductive validity of modus ponens, since a world may be abnormal by its own lights. But it develops a concept of defeasible validity, which modus ponens does satisfy.

16 Neighborhood Relevant The relevant conditional (A. Anderson, N. Belnap, R. Routley, R. Meyer): If A, then B is true at a world w if and only if, for every pair < x, y > of test worlds relevant to w, if A is true at x, then B is true at y. Inspired by the Chrysippan thought that a conditional is true just when its antecedent implies its consequent, but seeking to deny implication in cases in which formulas are irrelevant to one another, this conditional satisfies modus ponens and counts a conditional valid only if antecedent and consequent share some propositional component.

17 Neighborhood Strict C. I. Lewis developed modal logic in large part to replace the material conditional of classical logic. The paradoxes of material implication are only the most obvious drawback of the material conditional. Lewis defines A B (A B), and proposes the strict conditional as an analysis of the natural language conditional.

18 Neighborhood Other Puzzling Inferences A A B B A B (A B) (B A) (A B) ( A B) ((A B) C) ((A C) (B C))

19 Neighborhood Paradoxes of Strict Implication It, however, is subject to the paradoxes of strict implication: A = A B B = A B

20 Neighborhood Other Inferences More seriously, it validates the following: Strengthening of the Antecedent. A B = (A C) B Transitivity. A B, B C = A C Contraposition. A B = B A

21 Neighborhood Counterfactuals But it seems that none is valid in general. Consider the counterfactuals: If Kennedy hadn t been assassinated, he would have been reelected. So, if Kennedy hadn t been assassinated, and, in fact, had never been born at all, he would have been reelected.

22 Neighborhood Counterfactuals If Kennedy hadn t been assassinated, he would have been reelected. If Kennedy had never been born, he wouldn t have been assassinated. So, if Kennedy had never been born, he would have been reelected.

23 Neighborhood Counterfactuals If Kennedy hadn t been assassinated, he would not have been reelected in So, if Kennedy had been reelected in 1964, he would have been assassinated.

24 Neighborhood Indicatives This might seem to distinguish subjunctive from indicative conditionals. But actually we can devise similar indicatives casting doubt on these inferences: If you strike the match, it will light. So, if you strike the match and there s no oxygen present, it will light. If Tweety is a bird, Tweety can fly. If Tweety is a penguin, Tweety is a bird. So, if Tweety is a penguin, Tweety can fly. If we don t operate, he won t survive. So, if he survives, we ll operate.

25 Neighborhood In Between The strict conditional looks at the entire set of possible worlds; A B holds iff every A-world is a B-world. The material conditional looks only at the world of evaluation. It seems reasonable to think that we need something in between. If we ask, What if Kennedy hadn t been assassinated?, we re asking about worlds other than the actual world but not about every world, or even every world in which Kennedy is assassinated. We presumably neglect worlds in which he is not shot in Dallas, but in which aliens take over the earth, or Buffy learns that he is a vampire and stakes him.

26 Neighborhood Well-Behaved Worlds? We might think that the relevant worlds are those that are in some sense well-behaved. It will not do, however, to isolate a set of well-behaved worlds, and limit our evaluations to those, for the above inferences would still be valid. Which set of worlds we choose depends on the antecedent under consideration. It seems plausible to think that it depends only on the antecedent and the world of evaluation, since we can meaningfully ask What if? questions, but that is a substantive hypothesis.

27 Neighborhood Selection Functions Let f w [A] be the set of worlds relevant to evaluating conditionals with A as antecedent in w. We might think of this as the set of closest A-worlds (Lewis), as the A-worlds most similar to the world of evaluation (also Lewis), or as the set of A-normal worlds (Asher and Morreau).

28 Neighborhood We can then characterize truth conditions for conditionals. A > B is true at w iff B is true at all the selected A-worlds: ν w (A > B) = 1 f w [A] [B]

29 Neighborhood Accessibility We can develop the same idea in terms of accessibility relations. Let a model I =< W, {R A : A F}, ν > be a triple consisting of a set of worlds, a set of accessibility relations for each formula, and an assignment function. We can define the accessibility relations in terms of the selection function, or vice versa; f w [A] = {w : wr A w }. We can thus express the truth condition as: ν w (A > B) = 1 w (wr A w ν w (B) = 1)

30 Neighborhood Modal Logic Notice the similarity between the truth condition expressed in this form and the truth condition for necessity. We have in effect defined a necessity operator relative to formulas: A > B A B.

31 Neighborhood Tableau Rules Tableau rules for > allow us to go from A > B, i and ir A j to B, j, and from (A > B), i to ir A j and B, j, where j is new. The logic that results, C, is very weak; among the few valid schemata are Weakening of the Consequent: A > B = A > (B C). A paradox of strict implication also holds; B = A > B. Even idempotence (A > A) and modus ponens (A > B, A = B) fail in C.

32 Neighborhood C and C+ C+ adds conditions to C that make idempotence and modus ponens valid. Nothing in C requires that the worlds relevant to evaluating conditionals with A as antecedent are A-worlds. Adding a rule allowing us to go from ir A j to A, j makes idempotence valid. That corresponds to a semantic constraint of facticity: f w [A] [A].

33 Neighborhood Closest A-worlds This is implicit in the terminology that Lewis introduces; the closest A-worlds are indeed A-worlds.

34 Neighborhood Centering We may validate modus ponens by splitting a branch into two: the left containing A, i, the right containing A, i and ir A i. That corresponds to a semantic constraint of Weak Centering: w [A] w f w [A]. The world of evaluation, if it is an A-world, is among the closest A-worlds to itself.

35 Neighborhood C+ Are there reasons for declining to adopt these conditions? Facticity is appealing, but conditional obligation would not accord with it. It may also be useful to deny it to devise a concept of an anomaly. There are some powerful reasons to reject modus ponens, which we will examine further in the context of nonmonotonic logic.

36 Neighborhood Spheres We might rank worlds according to their degree of closeness to w, forming a sequence of spheres S w 0 Sw 1... W. We can then define the selection function in terms of the system of spheres: f w [A] = if [A] = ; f w [A] = S w [A] for i the smallest S w overlapping [A]. i This validates the facticity and weak centering constraints.

37 Neighborhood Constraints It also validates three other constraints: [A] f w [A] f w [A] [B] f w [B] [A] f w [A] = f w [B] f w [A] [B] f w [A B] f w [A] The first of these is obvious: if there are A-worlds, there are closest A-worlds. The second says that if all the closest A-worlds are B-worlds, and all the closest B-worlds are A-worlds, then the closest A-worlds are exactly the closest B-worlds. That corresponds to the schema: (A > B) (B > A) = (A > C) (B > C)

38 Neighborhood Counterfactual Antecedents Say that A and B are tantamount to each other in w iff A > B and B > A are both true in w. This says that formulas that are tantamount to each other have the same counterfactual consequences. Since logically equivalent formulas are tantamount to each other, they too have the same counterfactual consequences; [A] = [B] f w [A] = f B (w). This justifies our thinking of the selection function as defined not by formulas but by propositions, that is, sets of worlds: A > B is true at w iff f [A] (w) [B].

39 Neighborhood Weak Transitivity This makes valid a series of weak transitivity inferences: A > B, B > A, B > C = A > C (A B), A > C = B > C A > B, (B > A), B > C = A > C

40 Neighborhood Strong Centering Lewis and Stalnaker extend this system in different ways. Lewis maintains that, if w is an A-world, it is the unique closest A-world to itself. The sphere system implies that w is, in those circumstances, among the closest A-worlds. The constraint of Strong Centering asserts that it is the only one; no world can be as close to w as w is to itself. w [A] w f w [A] w = w (or equivalently w [A] f w [A] = {w})

41 Neighborhood Strong Centering This validates the inference from a conjunction to a corresponding conditional: A B = A > B.

42 Neighborhood Stalnaker Stalnaker goes further. He assumes that the selection always yields a singleton, a unique closest world. His constraint: w f w [A] w f w [A] w = w (or equivalently w f w [A] f w [A] = {w }) This validates Conditional Excluded Middle: (A > B) (A > B).

43 Neighborhood Might Lewis defines might in terms of would, as follows: A B (A > B). Stalnaker plainly cannot do the same, since it would make might equivalent to would not in his system. So, he defines it differently: for Stalnaker, A B (A > B).

44 Neighborhood Ramsey Test We might follow the lead of Ramsey and Gentzen in taking testing as essential to conditionality. introduce antecedent-oriented situations, and then perform consequent-determined tests on them. Ramsey (1929, 155), for example, gives this account of what conditionals do: 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.

45 Neighborhood Antecedents Abstracting from the epistemic character of this understanding, we can talk of what is the case, and think of the world-relativity of the conditional as linked specifically to the antecedent. The selection function f builds this into the semantics of the conditional; the proposition representing the antecedent A relative to a world w is a function f w [A] of the antecedent proposition [A] itself and w. We can think of f w [A] as the set of test worlds relative to A and w. On a pointwise conception, the conditional asserts that each test world satisfies B: f w [A] [B].

46 Neighborhood Generalizing The conditional, on a more general conception, relates the test proposition f w [A] to the consequent proposition [B]. We might express this as R(f w [A], B) [B] h(f w [A]) or, by analogy with the power set version, f w [A] g[b].

47 Neighborhood Neighborhood Semantics f w [A], on the neighborhood conception, picks out the modal neighborhood relevant to the testing of A in w. The conditional A B asserts that that neighborhood has the property associated with B. The test conception thus naturally leads to the truth condition w [A B] f w [A] g[b]

48 Neighborhood Weakness That truth condition is nevertheless extremely weak so weak, in fact, that it threatens to make conditionals inferentially inert. We have so far said nothing, however, about the nature of or relations between the functions f and g. The inferential features of the conditional spring from the assumptions we make about those natures and relations.

49 Neighborhood Weaker Logics We might think about placing some constraints on f and g to give us a richer logic and characterize the behavior of conditionals more faithfully.

50 Neighborhood Conservativity The earliest treatments of conditionals (e.g., in Theophrastus) linked their analysis to syllogisms; they reduced If A then B to All A-cases are B-cases. We might find something about this attractive even if we reject the assimilation of conditionals to universal affirmatives. Suppose we were to assume that a conditional corresponds to a quantifier of some type over cases or worlds. Then the general form of a truth conditional analysis ought to be something like Q([A], [B]) or, perhaps, Q(f w [A], [B]). But that is just a notational variant of f w [A] g[b].

51 Neighborhood Conservativity One linguistic universal applying to quantifiers but not to relations in general, however, is Conservativity (C): X g(y) X g(x Y)

52 Neighborhood Living on an Antecedent This constraint validates an inference pattern indicates that a conditional lives on its antecedent: A B = A (A B)

53 Neighborhood Monotonicity One attractive constraint stems naturally from the inferential role of the conditional. Suppose we introduce a subderivation starting with the assumption that A. We are now in a neighborhood f w [A] relevant to testing what happens if A. We proceed to apply logical rules to derive other formulas within the subproof.

54 Neighborhood Monotonicity Ignore for the moment the complication that some of these might come from outside the subproof, and think solely of the applications of rules within the subproof itself. If such application is possible, then we are assuming that if the neighborhood has the property associated with B, and B entails C, the neighborhood has the property associated with C.

55 Neighborhood Monotonicity Assuming that we wish our system of inference to permit the application of logical rules in subderivations, we would naturally want to assume that the function g giving the extensions of the associated properties is monotonic. Monotonicity (M): X Y g(x) g(y)

56 Neighborhood Monotonicity This constraint amounts to the assumption that the quantifier appropriate to the conditional is monotonic increasing. The powerset conception builds such an assumption into the truth condition, for it in effect assumes that the quantifier relevant to the conditional is all. It holds iff the conditional obeys weakening of the consequent. Weakening of the Consequent (W): A B & B = C A C

57 Neighborhood Additional Constraints The quantification constraint and the associated conservativity constraint, C, and monotonicity, M, are arguably the only constraints essential to the test and inferential role conceptions of the conditional. Those and only those properties are essential to conditionality.

58 Neighborhood Additional Constraints Most conditionals in the literature, however, are much stronger, validating many inference patterns inessential to conditionals as such. There may moreover be good arguments for imposing additional constraints, especially in certain contexts or as analyses of certain constructions. It is therefore important to explore additional algebraic constraints.

59 Neighborhood Downward Closure A constraint assumed in the powerset conception is downward closure. It corresponds to the assumption that the quantifier appropriate to the conditional is, in the language of Barwise and Cooper 1981, antipersistent (as all is). Downward Closure (D): X Y & Y g(z) X g(z)

60 Neighborhood Downward Closure The pointwise conception makes this attractive, for if something is true of a set of test conditions, it is tempting to think that it is true of each subset of that set. On the neighborhood conception, however, it is much less plausible. That a property holds of a neighborhood does not entail that it holds of each subset of the neighborhood.

61 Neighborhood Strengthening of the Antecedent Given certain assumptions about the selection function f, D holds iff the conditional validates the inference pattern: Strengthening of the Antecedent (SA): A = B & B C A C

62 Neighborhood Strengthening of the Antecedent That too would constitute an argument against D in many contexts, since SA is one of the inference patterns well known to be invalid for counterfactual conditionals. The Lewis and Asher-Morreau conditionals obey D, but do not validate SA, however, so the requisite assumptions about f are clearly quite strong. Assume that f([b], w) g[c] and [A] [B]. Given D, we can obtain f([a], w) g[c] if we have f([a], w) f([b], w). The constraint on f required to validate SA is thus the monotonicity of f: [A] [B] f w [A] f w [B]

63 Neighborhood Boundary Conditions on g Another set of constraints in effect impose boundary conditions on the conditional, telling us how g(x) and X relate to each other. Again, they are assumed in the powerset conception. Together with downward closure, they moreover entail that conception. Anchoring (ANCH): X g(x)

64 Neighborhood Boundary Conditions on g This seems plausible on the powerset conception; the things in X have the property associated with X. It seems somewhat less attractive in a more general setting; the neighborhood X does not necessarily itself have the property associated with it.

65 Neighborhood Seriality It is useful to assume a principle of seriality: Seriality (S): X Y X g(y)

66 Neighborhood Seriality This says that What if? questions always have answers. S, C, and M entail ANCH. Someone who maintains C can resist ANCH, therefore, but must give up M or S to do so.

67 Neighborhood Idempotence ANCH has some connection with idempotence, the validity of A A. But it suffices for it only in the company of two other assumptions: downward closure, which gives us Y X Y g(x), and boundedness, which yields f w [A] [A].

68 Neighborhood Boundary Conditions The following boundary condition limits g(x) to subsets of X. It says, in other words, that the property associated with a set holds only of subsets of that set. In a quantificational setting, this forces the relevant quantifier to be all. It is thus optional on a pointwise conception.

69 Neighborhood Boundary Conditions On a neighborhood conception, it says that the property associated with a neighborhood applies only to parts of that neighborhood. It thus specifies the property associated with a neighborhood as being a part of that neighborhood. That is one interpretation of the associated property admittedly, a fairly salient one but not by any means the only possible one. Limitation (L): g(x) X

70 Neighborhood Boundary Conditions These boundary conditions are very powerful. L entails g(x) (X) while ANCH and D together entail (X) g(x). ANCH, L, and D thus yield the powerset conception, which entails every constraint we have so far considered. Since ANCH follows from C, M, and S, {C, M, S, L, D} suffice for the powerset conception.

71 Neighborhood Theorem: C, M, S, L, D all Theorem: If g satisfies C, M, S, L, and D, then X g(y) iff X Y. Proof. Assume g satisfies C, M, S, L, and D. The right-to-left direction: Say X Y. By S, there is some Z such that Y g(z). By C, Y g(y Z). By M, Y g(y). But then, by D, X g(y). To proceed left-to-right: Assume that X g(y). By L, g(y) Y; so X Y.

72 Neighborhood C, L, D Since M and S seem plausible for any reasonable account of the conditional at least, any account based on inferential role and the test conception this suggests that one can move away from the assimilation of if to all only by denying C, L, or D.

73 Neighborhood Denying C The first is the most radical, amounting to the denial that a conditional is any kind of quantifier over cases, and rejecting the inference from A B to A (A B).

74 Neighborhood Denying L The second amounts to holding that If A then B can hold even if the A-worlds, or the selected A-worlds, are not all B-worlds. It thus amounts to the idea that a conditional can be true even though the antecedent could be true while the consequent was false. Lewis and Stalnaker conditionals already allow that possibility for worlds that are not among the closest A-worlds; Asher and Morreau already allow it for worlds that are not A-normal. Probabilistic and threshold conditionals adopt this strategy.

75 Neighborhood Denying D The third is to reject downward closure. Lewis, Stalnaker, Asher, and Morreau get the effect of rejecting downward closure by employing the selection function. Once the inference form in general is rejected, however, what s the argument for keeping it in the underlying logic?

76 Neighborhood Weaker Quantifiers There are candidates for quantifiers that would have these properties and might serve as ways of interpreting conditionals: most, the vast majority of, almost all, and, perhaps most plausibly, the null quantifier of generics such as Birds fly.