The Logic of Indicative Conditionals
Abstract and Keywords
Expounds, mainly on the basis of Adams's work, the logic of indicative conditionals—with some help from his adaptation of Venn diagrams. Failure for indicatives of or‐to‐if, contraposition, transitivity, and antecedent strengthening. Controversy over whether modus ponens holds for indicatives. Independent indicatives identified and snubbed.
53. Probabilistic Validity9. The Logic of Indicative Conditionals53. Probabilistic Validity
Here is a valid argument form:
As long as there is no malpractice with tenses or with ‘he’, the premisses of any instance of this cannot be true while the conclusion is false. What if F = funny? When you say things of the form ‘x is funny’, you do not attribute a property, funniness, to the subject; there is no such property. What you do rather is to express your own amusedness, thereby saying something that lacks a truth value (§ 42). Yet the validity of the above argument form guarantees the virtue of this argument:
The first thing he did was F.
The first thing he did was to introduce himself.
∴ His introduction of himself was F.
Its virtue is not classical or truth‐conditional validity; it does not consist in the fact that the premisses cannot be true when the conclusion is false. Rather, it consists in the fact that someone who believes the second premiss and has a frame of mind apt for asserting the first also has a frame of mind apt for asserting the conclusion.
The first thing he did was funny.
The first thing he did was to introduce himself.
∴ His introduction of himself was funny.
Thus, some sentences that lack truth values can enter into logical arguments just as though they had them. These are sentences that are deprived of truth values by their use of a predicate such as ‘is funny’, ‘is wrong’, ‘is disgusting’, or the like. Some of their logic is just that of any predicating sentences, and the lack of truth values does not affect it.
(p. 128 ) Where indicative conditionals are concerned, however, the case changes. They are deprived of truth values by their form, not by their mere inclusion of some special word. The only wider class of sentences on whose logical back they can ride are ones occurring in the propositional calculus. This, for example, is universally valid: and instances of it where Q or R is an indicative conditional are all right. Much of the propositional calculus cannot be applied to indicative conditionals because there are limits to how far they can be embedded in larger propositional structures (§ 39); but when the embedding is all right, so are the classically valid inferences. The validity of the above argument form, for instance, ensures that a frame of mind apt for asserting ‘The economy is slumping and if X is as stupid as he seems then we are in trouble’ is apt for asserting ‘If X is as stupid as he seems then we are in trouble’.
That is plain sailing. But a problem arises when we try to evaluate an argument using an indicative conditional. I mean by this that the conditional's form contributes to the workings of the argument, rather than the conditional's occurring as a mere substitution instance of a propositional variable. Consider:
There is nothing on whose back this can ride. Nor can we ask whether it is classically (truth‐conditionally) valid, for that requires being such that when its premisses are true so is its conclusion, whereas the first premiss of this cannot be true (or false). Well, according to Adams** (§ 47), the first premiss can be true if the second is, and in just that case the conclusion is true; so that Adams** endorses Modus Ponens, and thus the above argument, as classically valid! But that is an isolated, opportunistic application of classical notions to inferences using indicative conditionals, and it holds only on the basis of something that is not universally accepted. The general question about how arguments using indicative conditionals can be valid still stands.
If X is as stupid as he seems, then we are in trouble.
X is as stupid as he seems.
∴ We are in trouble.
A variant of this question is: Given that indicative conditionals lack truth values, how can there be truths about what they logically entail and are entailed by? Consider this threesome:
(1) A logically entails C
(2) A→ C
(3) A⊃ C
The present chapter will outline Adams's solution to this problem, relying mainly on his 1975 book. He has extended and refined this work in his 1996, and further still in his impressive Primer of Probability Logic (1998). Aiming to give readers the kind of help I needed when I first encountered Adams's work, I shall—greatly aided by Dorothy Edgington—spell things out quite slowly.
The main insight that Adams brought to this matter is best stated with help from ‘uncertainty’—a technical term used in probability studies. A proposition's uncertainty is its improbability, which equals 1 minus its probability. In symbols: U(Q) = P(¬ Q) = 1−P(Q). Although uncertainty can be simply reduced to probability through this equation, we sometimes find it helpful to invoke it separately—as in the present topic.
Now make the acquaintance of the notion of probabilistic validity. To get a feel for it, reflect that no instance of a classically valid argument form allows falsity to enter along the way from premisses to conclusion. Then think of a probabilistically valid form as one no instance of which allows improbability to enter along the way. In other words, it is true by definition that in a probabilistically valid argument the uncertainty of the conclusion cannot exceed the sum of the uncertainties of the premisses. (Adams 1975: 1–2 is misleading about this; treat those pages with caution.)
It is worthwhile to note that an argument can be probabilistically valid even though each premiss is highly probable and its conclusion utterly improbable. The so‐called lottery paradox shows this. A million people have entered an unrigged lottery whose rules guarantee that someone will win. Now we conduct an inference with a million premisses of the form
for every value of i from 1 through 1,000,000, together with one further premiss:
Entrant ei will not win the lottery,
(p. 130 ) From this it follows by classically valid logic that
The lottery will be won ⊃ It will be won by one of e1, . . . , e1,000,000.
This conclusion is false; but so is one premiss, so classical validity is not infringed. What about probabilistic validity? Well, each premiss is enormously probable, while the conclusion has a probability = 0; but the argument is probabilistically valid, because the uncertainty of the conclusion is not greater than—and happens to be equal to—the sum of the uncertainties of the premisses. One million one‐millionths equals 1, which is the uncertainty of the conclusion. No extra improbability has filtered into the argument along the way. What enabled us to get from premisses that are severally probable to an improbable conclusion was the sheer number of the premisses.
The lottery will not be won.
Now, probabilistic validity is something an argument can have even if it uses indicative conditionals in premisses or conclusion. Although these cannot be relied upon to have truth values, they can always have probabilities; and we can ask of a given argument form whether in any instance of it the conclusion could have a level of uncertainty (for a given person at a given time) that exceeded the sum of the uncertainties (for that person at that time) of the several premisses. With this concept at our disposal, we can evaluate logical arguments using indicative conditionals, despite their lack of truth values and despite there being no more general logic upon whose back these arguments can ride.
Before going along that path, let us see how probabilistic validity behaves in arguments that are candidates also for classical validity because all the sentences in them have truth values. We start with the theorem labelled 10 in § 21 above:
We proved this from the axioms of probability logic, and from it we can derive the important theorem that among arguments from a single premiss, whatever is classically is also probabilistically valid:
If Q entails R, then P(Q) ≤ P(R).
In one‐premiss arguments, therefore, probabilistic validity will take care of itself, tagging along safely on the heels of classical validity. What about reasoning from more than one premiss? An argument of the form Q, S ∴ R can be expressed as one of the form Q&S ∴ R, and then our previous result holds: if the argument is classically valid then it is also probabilistically so—U(R) ≤ U(Q&S). In real life, however, we often move to conclusions from premisses that are accorded subjective probabilities separately rather than in a single conjunctive lump. In studies of classical validity, ‘Each premiss is true’ stands or falls with (p. 131 ) ‘The conjunction of all the premisses is true’. Replace ‘true’ by ‘probable’ and the equivalence no longer holds. So we need to be able to relate the uncertainty of the conclusion of a classically valid argument to the uncertainties (plural) of its premisses taken separately.
First result: If Q entails R, U(R) ≤ U(Q).
We do this with the aid of:
that is, the uncertainty of a conjunction never exceeds the sum of the uncertainties of its conjuncts. To see why, bear in mind that just now we are attending only to propositions that can be true; and the probability of such a proposition is its probability of being true, its uncertainty the probability of its being false. Now, the probability that Q&S is false obviously comes from the probability that Q is false and the probability that S is false. It cannot exceed the sum of those two probabilities, because if it did there would be a chance for Q&S to be false although Q and S were each true. This leads to the ‘second result’, given above. We cannot strengthen it to the equality U(Q&S) = U(Q) + U(S), because there can be overlap between the chances of Q's being false and the chances of S's being false, and a mere summing of the separate uncertainties would double count the ones in the overlap. It can easily happen, indeed, that U(Q) + U(S) > 1, but it cannot be that U(R) > 1. For me, U(I shall live to be 90) ≈ 0.95 and U(I shall write a book on Hegel) ≈ 0.99; but it cannot be that for me U(I shall live to be 90 and write a book on Hegel) ≈ 1.94. Thus, the sum of the uncertainties of the conjuncts need not equal the uncertainty of the conjunction, but sets an upper bound to it.
Second result: U(Q & S) ≤ U(Q) + U(S),
Put together the first and second results displayed above, and we get:
We can easily generalize this to something holding for any number of premisses:
Third result: If Q, S entail R, then U(R) ≤ U(Q) + U(S).
So we have the important general result that in a classically valid argument the uncertainty of the conclusion cannot exceed the sum of the uncertainties of the premisses. This means that any classically valid argument is also probabilistically valid.
If Q, S, . . . , X entail R, then U(R) ≤ U(Q) + U(S) + . . . U(X).
What Adams found, then, is that any argument that is classically or truth‐conditionally valid also has another virtue, probabilistic validity. An argument using indicative conditionals cannot have the former of these virtues, but it can have the latter, and that is a basis upon which we can evaluate it, as I shall often do in the present chapter.
(p. 132 ) As a preliminary exercise, let us consider an argument which I have held over from § 40, concerning the thesis If‐and:
Adams (1975: 33) can be read as objecting to this on the following grounds. If If‐and is universally correct, then it holds when A = C. That equates C → (B→ C) with (C&B) → C. But the latter of these is necessary; so according to If‐and the former is necessary also, in which case the ‘argument from the consequent’— ‐is valid, and Adams is rightly sure that it is not. For many values of B and C, a rational person can be much more uncertain of B→ C than of C.
A → (B→ C) is logically equivalent to (A&B) → C.
Does that imply that C → (B→ C) is not necessary? If it does, then we must deny that C&B→ C is necessary or else deny the equivalence of the two—that is, deny If‐and. The threat of losing If‐and is fairly dire, given how well it stands up to philosophical probing, and given Stalnaker's and Hájek's formal arguments for it.
Well, as a start on clearing up this mess, I argue that C → (B→ C) is necessary, which nips in the bud the threat to If‐and. Because this is an indicative conditional, its necessity cannot consist in its being true at all worlds; rather, it must consist in a rational requirement that every instance of it be accorded probability = 1. The way to test this is through the Ramsey procedure: to a rational belief system, add P(C) = 1 for some C, adjust minimally to make room for that, and then see what the value is of P(B→ C) in the resultant system of beliefs. Well, any system in which P(C) = 1 had better be one in which P(B→ C) = 1 also. If not, then someone could rationally give C some chance of being false if B is true, while giving it no chance at all of being false. This is incoherent. Another way to see this is through the ratio formula which equates P(B→ C) with P(B&C) ÷ P(B). Because P(C) = 1, P(B&C) = P(B), and so our value for P(B→ C) simplifies to P(B) ÷ P(B), which = 1.
So we have excellent reason to regard C → (B→ C) as probabilistically necessary, to coin a phrase; as of course is B&C → C also. Where (what seems to be) Adams's argument goes astray is in its final move from the necessity of C → (B→ C) to the validity of C ∴ B→ C. It is valid to go from the (truth‐conditional) necessity of a material conditional to the (truth‐conditional) validity of the corresponding argument form; but there is a definite mistake involved in going from the (probabilistic) necessity of an indicative conditional to the probabilistic validity of the corresponding argument form. Probabilistic necessity has to be understood through the Ramsey procedure, which always starts by supposing (p. 133 ) the antecedent to have probability 1. Probabilistic validity, on the other hand, is defined in terms of a relation of probabilities across the whole range; it does not just ask how B→ C fares when P(C) = 1. It also asks whether P(B→ C) could be lower than P(C) when the latter has values other than 1. Clearly it could; but when P(C) = 1 the picture changes, and P(B→ C) is locked at 1 also.
54. A→ C and Modus Ponens9. The Logic of Indicative Conditionals54. A→ C and Modus Ponens
Probabilistic validity is not everything. The form of argument A, A⊃ C ∴ C is probabilistically valid; it has to be, because it is classically valid. Still, A⊃ C is unsatisfactory for use in Modus Ponens, because one may accept it for a reason that will vanish if one comes to accept A. In so far as the point of indicative conditionals is their use in Modus Ponens, they must have a built‐in guarantee that they can safely be thus used. We know what provides it. In Jackson's terminology, it is the fact that when you accept A→ C the consequent C is, for you, robust with respect to A. If that is the whole essence of A→ C—something that cannot be dispensed with but need not be added to—we get the result that P(A→ C) = π (C/A), which then leads by a winding road to Adams's view that indicative conditionals are inference tickets rather than propositions. But I repeat myself.
What spoils A⊃ C for use in Modus Ponens is that it can be accepted because A is rejected. Its big defect will therefore be visible only when P(A) is low—and not always then. This is an informal way of putting something that Adams presents more technically in his thesis that the uncertainty of an indicative conditional equals the uncertainty of the corresponding material conditional divided by the probability of its antecedent (1975: 3–4). Let us trace out the proof of this important theorem. We start with a certain quantity: By the interrelations of U and P this is equivalent to which by the Ratio Formula is equivalent to which, by the interrelation of U and P, is equivalent to which, by the definition of ⊃, is equivalent to (p. 134 ) Go in a jump from top to bottom and you get:
QED. Thus, the higher the value of P(A) the nearer the two probabilities—of the indicative and the material conditional—are to being equal. That is why it is only when P(A) is low that A⊃ C comes to grief when used in Modus Ponens. We shall soon see Adams putting this powerfully to use.
Fourth result: U(A→ C) = U(A⊃ C) ÷ P(A),
The move from the first to the second line in the above argument assumes that A→ ¬ C is the contradictory of A→ C. So indeed it is, because, for given values of A and C and a given rational person at a time, the probabilities of the two add up to 1. Using the Equation and the Ratio Formula, it is easy to prove arithmetically that P(A→ C) + P(A→ ¬ C) = 1. It is also intuitively evident that it is right.
55. Adams's Use of Venn Diagrams9. The Logic of Indicative Conditionals55. Adams's Use of Venn Diagrams
Adams ingeniously adapts Venn diagrams to say things about the probabilistic validity of inference forms. Venn diagrams are standardly used to handle truth‐conditional validity only, and their basic features are topological. We represent propositions by ovals within a rectangle (the latter representing the limits on what is possible, or representing the necessary proposition). Then we can represent ‘P entails Q’ by putting P's oval entirely within Q's; ‘P rules out Q’ by putting their ovals outside one another, not touching; ‘P neither entails nor rules out Q’ by giving their ovals a common portion that does not exhaust either oval; and so on.
Think of each point on the rectangle as representing one possible world. Then each perimeter encloses a class of possible worlds and thus represents the proposition that is true at exactly those worlds. Thus, if ovals P and Q have some overlap though neither contains the other, this means that there are worlds at which P&Q is true, worlds at which P& ¬ Q is true, and worlds at which ¬ P& Q is true. If some of the containing region (‘the rectangle’) lies outside both ovals, then at some worlds ¬ P& ¬ Q is true.
Although this is all purely topological, having only to do with what is inside or outside what, it has some metrical side‐effects. The main one is the fact that if Q entails R and not vice versa, Q's oval is smaller than R's, as in Fig. 2. Venn diagrams give no significance to how much bigger one oval is than another. Adams adapts them by adding further significance to metrical features of the drawings: he lets the size of a proposition's oval represent how probable it is in the belief scheme of the person in question (1975: 9–10).
(p. 135 ) For example,‘Q rules out R and is more probable than it (but is not certain)’ will be represented by Adams through something like Fig. 3. He attaches the same significance to the size of the region outside an oval: an oval and the remainder of the rectangle add up to the rectangle, just as P(Q) + P(¬ Q) = 1.
In a diagram of the sort Adams highlights, P(A→ C) is represented by what proportion of the A oval overlaps the C oval. This can be derived from our two equations—P(A→ C) = π (C/A) = P(A&C) ÷ P(A)—but I shall come at it informally, without relying on the Ratio Formula. The probability of A→ C is high for someone if he thinks that most chances of A's being true are also chances of C's being true. Thus, the diagram of his thought puts only a little of the A oval outside the C oval, putting most of it inside. So the value of his probability for A→ C equals the proportion of the A oval lying within the C oval.
(p. 136 ) This fits what we independently know about probabilities of conditionals. For example, that P(A→ C) + P(A→ ¬ C) = 1; the part of the A region lying inside C together with the part lying inside ¬ C add up to the whole of the A region. This holds, no matter what the size is of the A and C ovals.
Another result illustrated by Adams's diagrams is that if P(C) = 1 then P(A→ C) = 1. This is certainly correct so far as the diagrams go: if P(C) = 1 then C is represented by the entire rectangle, and thus no part of A's region can lie outside it. As I showed in § 40, the Ramsey procedure endorses this. I am perfectly certain of C, and want to consider whether A→ C; I pretend to be sure of A, and then adjust the rest of my beliefs so as to restore harmony within the whole. If my most conservative way of doing this lowers P(C), then I was not perfectly certain of C's truth after all, because it turns out that a discovery that I might make would make me less than perfectly certain of C. Accepting A→ C on the ground that P(C) = 1 is accepting it as a special sort of non‐interference conditional (§ 50).
It is also satisfactory that Adams's diagrams cannot represent P(A→ C) when P(A) = 0. If P(A) = 0, then A has no area, so it cannot be diagrammed. This is just what we want.
As well as fitting the facts about how the probabilities of conditionals relate to other probabilities, Adams's diagrams also highlight certain structural features of his theory about indicative conditionals. I shall present two.
(1) The diagrams dramatize the idea that indicative conditionals, because they do not always have truth values, are not propositions and do not correspond to sets of possible worlds. Like Venn's original diagrams, Adams's adaptation of them represents a proposition by a region within the rectangle; but it does not represent an indicative conditional in this way or in any other. If we demand ‘Show us which parts or features of a diagram of Adams's represent A→ C’, there is no answer; all that is represented is something of the form P(A→ C) = n. This reflects the fact that Adams's theory tells us what indicative conditionals are by telling us what it is to accord probabilities to them. The fundamental reality is not A→ C but rather P(A→ C), and Adams represents this by a feature of the diagram, not by a part of it.
Two people can differ in how likely they take a single indicative conditional to be, which suggests that there is a detachable item—the conditional—for them to disagree about. But the suggestion is false. If you attach a higher probability to A→ C than I do, this means that you and I differ in the values we assign to π (C/A), which is a ratio between two probabilities. The reality underlying our talk about our different attitudes to ‘the same conditional’ is a difference in our levels of belief in one or both of two genuine propositions, A&C and A. The ratio (p. 137 ) between your probabilities for those differs from the ratio between mine; and that is the whole story. It concerns those two propositions, not any item that can be called the conditional A→ C. (If the Ratio Formula is wrong, so is what I have just said; but I could replace ‘ratio between two probabilities’ by ‘relation between two probabilities', the relation being defined by the Ramsey test; and the basic story would be the same.)
(2) Adams's diagrams cannot represent compounds in which conditionals are embedded. They can represent A and C and B, and they can bring A→ C into the picture indirectly, by representing its probability; but they cannot represent B & (A→ C), because they cannot represent A→ C.
It might be thought unsatisfactory that Adams's theory and his diagrams forbid even so simple a construction as ¬ (A→ C), but really it is not a loss. The theory does let us say what a person accepts in rejecting A→ C, namely A→ ¬ C.
56. Adams and Venn: Comparisons and Contrasts9. The Logic of Indicative Conditionals56. Adams and Venn
The representation of probability by region size in Adams's diagrams accords with the slight metrical significance of Venn diagrams. If Q entails R but not vice versa, then in a Venn diagram Q's oval coincides with or lies inside R's, and is therefore not larger than the latter. In such a case, a logically idealized thinker will accord to Q a probability no larger than P(R)—this is because of § 21’s theorem 10:
with which we are now familiar. So we have P(Q) ≤ P(R); and a diagram of Adams's sort represents this by making Q's region no larger than R's.
If Q entails R, then P(Q) ≤ P(R),
Again, if Q entails R then all of Q's region overlaps R's in both diagrams. According to Adams's reading of his diagram, this says that P(Q→ R) = 1; which is right if that entailment holds. In this case, Q→ R is an independent conditional—that is, one in which the consequent logically or causally follows from the antecedent alone (§ 7)—and for that reason I keep it at arm's length. Still, this point of Adams's is sound: it does no harm to apply probability logic to conditionals that are secured by logic alone, saying for example that when A entails C the probability of A→ C is 1. You can easily derive that from the Ratio Formula.
Adams's diagrams do not merely add to Venn ones—they alter them in one respect. In Venn diagrams the containing rectangle represents the necessary proposition, or the set of all possible worlds, and regions within it represent contingent propositions, ones that conflict with some possibilities. In Adams's (p. 138 ) diagrams, on the other hand, the rectangle represents any proposition to which the person accords probability 1, and regions within it represent propositions of whose truth the person is less than certain—propositions that conflict with something that is, for him, epistemically possible. According to Regularity (§ 26), an ideally rational believer assigns probability = 1 only to absolutely necessary propositions; in which case the two sorts of diagram do after all attach the same significance to the rectangle. If Regularity is wrong, however, we have here a difference.
It is an intelligible and inoffensive one, however: what the whole rectangle means in the Venn diagram is at least isomorphic with what it means in the Adams adaptation. Here are two examples of this. (1) Venn: If Q1 and Q2 are both necessary, they can be substituted for one another salva veritate in any statement about entailment, necessity or possibility. Adams: If P(Q1)= P(Q2)= 1, those two propositions can be substituted for one another in any statement about that person's absolute or conditional probabilities; for example, it follows that π (R/Q1) = π (R/Q2), that P(R&Q1)= P(R&Q2), and so on. (2) Venn: In many contexts a necessary proposition can be dropped—because if N is necessary then N&Q is equivalent to Q, so is N⊃ Q, and so on. Adams: If P(N) = 1, then N can be dropped from various compounds without affecting the person's credence levels: if for him P(N) = 1, then for him P(N&Q) = P(Q), and so does P(N⊃ Q), and so on.
Venn diagrams can naturally be interpreted in terms of worlds, as I did recently: if Q entails R and not vice versa, then every Q‐world is an R‐world and not vice versa, and this is represented by a diagram in which Q's oval lies inside R's. Let us see whether we can extend this idea to Adams's diagrams and to the associated notion of probability. To do so, it seems, we must associate the size of a region in one of Adams's diagrams with how many worlds it represents; which raises a difficulty if there are infinitely many possible worlds. We might deal with this by finding some way of dividing an infinity of worlds into a finite set of clumps, and then counting those (§ 101). Alternatively, we can contend that in any realistic treatment of subjective probability we have only finitely many worlds to deal with. We have seen Hájek take this line in his 1989 argument against the Equation (§ 31).
57. Four Probabilistically Invalid Argument Forms9. The Logic of Indicative Conditionals57. Four Probabilistically Invalid Forms
With illustrative help from his diagrams, Adams defends various theses about the logic of indicative conditionals. So far, I have mentioned only a feature (p. 139 ) shared by A→ C and A⊃ C, namely supporting Modus Ponens (though see § 61); but Adams's theory commits him to several logical differences between those two. Above all, these two forms of argument: which would both be classically valid if → were ⊃, are not probabilistically valid and so are not endorsed in Adams's theory. The latter does imply that when my value for P(A) = 0, I should have no value for P(A→ C); and that when my value for P(C) = 1, my value for P(A→ C) should be 1. But nothing follows about the value for P(A→ C) when P(A) is low but > 0, or when P(C) is high but < 1. Edgington (1995a: 71) has a helpful discussion of the probabilistic failure of those two forms of argument, which she acknowledges having learned about from Ellis 1973. See also Adams 1975: 11–12.
Those two ‘paradoxes of material implication’ are unpopular anyway, frequently being adduced as refuting the horseshoe analysis. Adams's theory rejects some other proposed argument forms as well, however, and since these are not as obviously absurd as are the notorious two, his denial of them is not so obviously a merit in his theory. Here are the four main ones:
1. A∨ C ∴ ¬ A→ C (Or‐to‐if)
2. A→ C ∴ ¬ C→ ¬ A (Contraposition)
3. (A→ B), (B→ C) ∴ A→ C (Transitivity)
4. A→ C ∴ (A&B) → C (Antecedent Strengthening)
(i) A∨ C (given)
(ii) T→ (A∨ C) (follows from i)
(iii) (T& ¬ A) → (A∨ C) (from ii by Antecedent Strengthening)
(iv) ((T & ¬ A) & (A∨ C)) → C (logical truth, because the antecedent entails C)
(v) (T& ¬ A) → C (from iii and iv by Limited Transitivity)
(vi) ¬ A→ C (equivalent to v).
The four argument forms have struck many people as intuitively acceptable, Adams says, going so far as to call them ‘intuitively rational’. He accepts an onus to explain this. Presumably, nobody will get the impression that a given argument form is valid just because it has many valid instances, for that is true of almost every argument form—for example, the form P, Q ∴ R. Adams, however, marks off a determinate, probabilistically valid subset of instances of the four forms, and suggests reasons why people using the forms tend to produce instances belonging to this subset (1975: 18–19). One detail in his reasoning is left obscure, but it comes clear—though still not simple—in Edgington's version (1995b: 285), which I shall follow.
Consider any classically valid argument whose conclusion is a material conditional—express it as ‘A⊃ C’ for short. Being classically valid, the argument must also be probabilistically valid, which means that:
(1) U(A⊃ C) ≤ the sum of the uncertainties of the premisses.
(2) The sum of the uncertainties of the ⊃‐premisses ≤ the sum of the uncertainties of the →‐premisses.
(3) U(A⊃ C) ≤ the sum of the uncertainties of the →‐premisses.
(p. 141 ) Now, suppose we replace ⊃ by → in the conclusion. This takes us from a weak conditional to one that may be stronger; so it may be a move from a lesser to a greater uncertainty; so the probabilistic validity of this new argument is no longer guaranteed. Thus, we cannot say:
But now let us recall our ‘fourth result’ in § 54 above, namely that U(A→ C) = U(A⊃ C) ÷ P(A), which means that the greater P(A) is, the nearer P(A→ C) is to P(A⊃ C). From this something important follows, namely: if X is a probabilistically valid argument for A⊃ C, then to the extent that P(A) is high X is also a probabilistically secure agument for A→ C. So Adams can infer this remarkable result (the name and this wording are mine):
U(A→ C) ≤ the sum of the uncertainties of the →‐premisses.
Probabilistic security is a matter of degree; how probabilistically secure an argument is depends, by definition, on how little the uncertainty of its conclusion can exceed the sum of the uncertainties of the premisses. (There is no concept of ‘classical security’ related in the analogous way to classical validity. That is because, whereas probabilistic validity depends on a quantitative notion, classical validity does not.)
Security Thesis: If X is an argument whose conclusion is an indicative conditional A→ C, and if what results from replacing → by ⊃ throughout X is a classically valid inference, then X is probabilistically secure to the extent that P(A) is high.
Adams applies this to the four challenged argument forms, each of which is probabilistically secure to the extent that the antecedent of the conclusion is probable. The counterexamples to them—applications in which the premisses are probable for someone for whom the conclusion is not—all involve conclusions with improbable antecedents. This leads him to the following:
This would help to explain why the four challenged argument forms are probabilistically secure in ‘much real life reasoning’. In attending to them individually, we shall find that the Security Thesis fits every case, so that the ‘plausible hypothesis’ could explain why each form of argument feels good to most people most of the time. We shall find that it can usually be explained in other ways as well.
Plausible hypothesis: Recognition that a conclusion's antecedent is not too improbable is a ‘tacit premise’ in much real life reasoning which appears to be of a pattern which is not universally probabilistically [valid]. (1975: 19)
(p. 142 ) 58. Or‐To‐If9. The Logic of Indicative Conditionals58. Or‐To‐If
The Or‐to‐if pattern of argument is not generally valid. I might assign a high probability to A∨ C because I am fairly sure of A, in which case the disjunction gives me no reason to assign a high probability to C given ¬ A. I think it fairly likely that (The North Koreans started the Korean war ∨ Some Martians started the American civil war); but this is only because I am pretty sure the North Koreans started their war. This does not incline me to think that if they didn't then some Martians fired on Fort Sumter. Fig. 4 shows a diagram of Adams's sort illustrating a case where P(A∨ C) is high only because P(A) is high, and where P(¬ A→ C) is low. It is easy to show through the Ratio Formula that A∨ C can be more probable than ¬ A→ C. You will find, as the diagram suggests, that counterexamples will be ones where ¬ A—the antecedent of the argument's conclusion—is not highly probable.
This fits Adams's Security Thesis, but in explaining the apparent acceptability of Or‐to‐if (1975: 19–21) he does not directly rely on the ‘plausible hypothesis’, appealing instead to Gricean conversational implicature, in the manner of § 18, to explain the illusion that A∨ C ∴ ¬ A→ C is probabilistically valid.
Adams offers a counterexample to Or‐to‐if (1975: 11–12). Here is one that I like better. Let A = ‘There will be snow in Buffalo in 2009’ and let C = ‘A woman will be elected President of the USA in 2008’. I give a high probability to A∨ C, and a low one to ¬ A→ C. Notice that again the Security Thesis is confirmed.
(p. 143 ) 59. Contraposition9. The Logic of Indicative Conditionals59. Contraposition
Contraposition—the argument form —is not virtuous in any theory giving primacy to the Ramsey test, as Adams's does. The fact that for me C is highly probable given A does not guarantee that ¬ A is highly probable for me given ¬ C.
Appiah presents a nice clean example of the failure of Contraposition, but it too easily admits of being read as involving subjunctive conditionals, which makes it un‐ideal for its announced purpose (1985: 171–2). Adams (1988) offers counterexamples like the following, which Jackson has also used. It may be reasonable for me to say (1) ‘If he does not live in Paris, he lives somewhere in France’, whereas it would be absurd to say (2) ‘If he does not live anywhere in France, he lives in Paris', which is 1’s contrapositive. One may object to such examples:
This is not conclusive. It offers a reasonable expansion of 1, but does not show what is wrong with accepting 1 just as it stands.
1 really means ‘If he does not live in Paris, he lives elsewhere in France’, meaning ‘ . . . in France outside Paris’; and the contrapositive of that is all right. It says: ‘If he does not live in France outside Paris, he lives in Paris.’
Still, the proposed counterexample is suspect, because in it P(A→ C) is high and P(¬ C→ ¬ A) = 0. If Contraposition as such is not probabilistically valid, there ought to be cases where one probability is high and the other is low but not zero. If the only apparent counterexamples we could find assigned 0 to the contrapositive, this would look like a discovery not about Contraposition but rather about this class of examples. I say the same about Adams's diagram 3 and verbal example at 1975: 13–14.
Better counterexamples to Contraposition are easy to devise. To get a sense of how to construct them, consider Fig. 4 illustrating the failure of Or‐to‐if. It also diagrams the case where P(A→ ¬ C) is high and P(C→ ¬ A) low.
Suppose a textual scholar's researches into the newly discovered play Hate's Labour's Won leads her to the following position:
(p. 144 ) This scholar, then, has a high probability for Beaumont → Not‐Fletcher and a low one (but not zero) for Fletcher → Not‐Beaumont. In applying Fig. 4 to this, Abe ‘Beaumont had a hand in the work’ and C be ‘Fletcher had a hand in it’.
Probably Beaumont wrote it alone; some slight but not quite negligible evidence points to Fletcher's having a hand in it also; there is almost no chance that Fletcher contributed while Beaumont did not. The slight Fletcherian indications in the work nearly all suggest the kind of work Fletcher did with Beaumont at his elbow; it is hardly thinkable that Fletcher should have written this on his own.
You may notice that in the invalid inference from Beaumont → Not‐Fletcher to Fletcher → Not‐Beaumont the antecedent of the conclusion—namely ‘Fletcher had a hand in the work’—has a low probability. This is illustrated by the diagram, and predicted by Adams's Security Thesis.
In this case the protagonist gives high credence to ‘If Beaumont was involved in this work, then Fletcher wasn't’ and to ‘If Beaumont was not involved in the work, Fletcher wasn't’. This lets us refute Contraposition in a dramatic way. The person rationally accepts A→ ¬ C and ¬ A→ ¬ C, giving a high credence to each; if Contraposition were valid, the person should accept both C→ ¬ A and C→ A—which is contradictory.
If our literary scholar is pretty confident that Fletcher was not involved in the writing of Hate's Labour's Won, why should she bother with conditionals on this topic? Here is one of several possible answers. The scholar announces that she doubts that Fletcher was involved, a colleague says: ‘Well, if Beaumont had no hand in it, I can see the case for thinking that Fletcher hadn't either; but if Beaumont was involved . . . ’ and she breaks in: ‘Even if Beaumont was involved in the writing of Hate's Labour's Won, Fletcher wasn't.’
We have seen Contraposition fail through counterexamples and through Adams's diagrams. It can also be shown to fail using the Ratio Formula and the Equation; you may enjoy devising these proofs for yourself.
Why do people tend to think that Contraposition is all right? Well, the Security Thesis tells us that A→ ¬ C ∴ C→ ¬ A is secure to the extent that P(C) is not low; so what we need to explain is why people are apt to think of A→ ¬ C as asserted only when P(C) is not low, that is, when P(¬ C) is low. The best explanation is Gricean: there will usually be no point in asserting A→ ¬ C where P(¬ C) is high, because the stronger ¬ C can be asserted without much loss of confidence. That is why in my Beaumont example I supposed a conversation in which a second person invited the protagonist to assert A→ C, thereby making it pointful for her to counter‐assert A→ ¬ C.
Adams explains the phenomenon differently. When someone asserts ‘If A, ¬ C’, it is natural for hearers to think that P(¬ C) is not high for her, he argues, because if it were high she would more naturally say ‘Even if A, ¬ C’. That would indeed be more natural, as witness my own ‘Even if Beaumont was involved. . . ’. This does not explain much, however; and Adams's handling of it seems to assume that ‘If A, C’ and ‘Even if A, C’ differ from one another more than they (p. 145 ) really do. I shall defend a view about how they differ in Chapter 17. Gricean pragmatics play a large role in Adams's discussion of these matters in his 1983: 293–4.
60. Transitivity and Antecedent Strengthening9. The Logic of Indicative Conditionals60. Transitivity
The literature is full of examples purporting to show that Transitivity fails for indicative conditionals, but most come from the borderline territory where indicative conditionals are apt to be thought of as though they were subjunctives. That Transitivity fails for the latter is, as will appear in § 65, easy to see and to understand. It is trickier to devise good examples showing Transitivity to fail for indicative conditionals that do not risk being thought of as subjunctives; but it can be done. A farmer, thinking about the state of things on his farm, believes—though not with complete certainty—that the gate into the turnip field is closed and that his cows have not entered that field. He accepts the conditional:
because the least adjustment to his belief system that will admit ‘The cows are among the turnips’ into it is the dropping of his belief that the gate was closed. Other adjustments would also do the job—lowering his trust in the gate not to fall down, in the cows not to push it down, and so on—but they are more radical than this one. This farmer also accepts:
If (A) the cows are in the turnip field, (B) the gate has been left open,
He accepts that consequent given that antecedent, because he thinks that the cows are not in the turnip field. Apply Transitivity to those two indicative conditionals—each accepted by one person at one time—and you get:
If (B) the gate to the turnip field has been left open, (C) the cows have not noticed the gate's condition.
which it would be stupid for the farmer to accept.
If (A) the cows are in the turnip field, (C) they have not noticed the gate's condition,
It is a fairly straightforward task to establish mathematically—using the Equation and the Ratio Formula—that Transitivity is not probabilistically valid. It is also an interesting exercise to devise a diagram of Adams's sort illustrating the failure of Transitivity. Fig 5 does this, showing a case where P(A→ B) and P(B→ C) are high while P(A→ C) is low. I prefer it to Adams's diagram for this purpose (1975: 16), because in his P(A→ B) = 1, and P(A→ C) = 0. Mine, allowing that the cows might have inattentively drifted through the open gate, gives a probability < 1 and > 0 to each of the three conditionals. (p. 146 )
A To explain why Transitivity seems valid, Adams suggests that in inferences that seem to have the form this is really an ellipsis, and what the speaker or writer means is This is probabilistically valid, Adams argues, and so inferences that sound like transitivity of → really are all right but have the form not of Transitivity but of something weaker (1975: 22). Why should speakers tend to say A→ C when they mean (B&A) → C? Adams could answer this in Gricean terms: someone who asserts a conditional, thereby inviting us to think about B given A, is behaving badly if he goes straight on to say other things about B while silently cancelling the supposition of A.
Adams has shown me that Antecedent Strengthening entails Transitivity, and vice versa; so it is not surprising that the failures of these two principles are similar. The parallel can be seen by adapting the counterexample I brought against one so that it counts also against the other. The farmer accepts ‘If the gate to the turnip field has been left open, the cows have been inattentive to the gate's condition’, but he does not accept ‘If the gate to the turnip field has been left open and the cows have got in, they have been inattentive’ etc.
Here, as with Transitivity, it is easy to work out arithmetically why Antecedent Strengthening fails. The diagram for such failures is also straightforward. Try it for yourself: all you need is for a large portion of A's oval to overlap C's and a small proportion of (A&B)'s region to overlap C's.
(p. 147 ) Of the four principles selected for discussion, Antecedent Strengthening is the most richly endowed with obvious counterexamples. Think of any indicative conditional that you accept and regard as contingent, and you will quickly see how to strengthen its antecedent so that the result is no longer acceptable. Take our old friend ‘If Booth didn't kill Lincoln, someone else did [kill Lincoln]’ (§ 1), and strengthen its antecedent to yield this: ‘If Booth didn't kill Lincoln, the person he killed being not Lincoln but a double, then someone else did kill Lincoln.’ There are so many clear counterexamples to Antecedent Strengthening that we do not have to explain why it usually seems valid—it doesn't! Still, things can be said about circumstances under which the acceptability of A→ C is evidence for the acceptability of (A&B) → C, and at his 1975: 25–8 Adams says them.
Stalnaker (1984: 123–4) has offered a direct argument against Antecedent Strengthening and thus against Transitivity, which entails it. Counterexamples have value, but ultimately this direct argument should suffice. If Antecedent Strengthening is valid, Stalnaker points out, then we get this special case: as long as A does not entail C. Stalnaker argues that this cannot be right because (A& ¬ C) → C is intolerable: any conditional whose antecedent is inconsistent with its consequent must be false, except perhaps in the special case where the antecedent is itself impossible. So Antecedent Strengthening implies that every contingent indicative conditional—every one in which A does not entail C—is false because it entails something false. This seems to be decisive.
The failure of Antecedent Strengthening for indicative conditionals is one symptom of the non‐monotonic nature of any reasoning where probability is involved (§ 32).
61. Modus Ponens9. The Logic of Indicative Conditionals61. Modus Ponens
Although I have written throughout as though Modus Ponens were plainly valid for indicative conditionals, this has been challenged by McGee and Lycan, and perhaps by others. I shall discuss three of their reasons.
One of Lycan's objections to Modus Ponens rests on the undisputed fact that Antecedent Strengthening fails for indicative conditionals, so that P(A→ C) may be very high for a person for whom, at the same time, P((A&B) → C) is low. Or, in Lycan's terms, A→ C can be true for someone for whom (A&B) → C is false. Because of this, unrestricted Modus Ponens does fail for →; that is, the form of Modus Ponens that says that from A→ C and anything that entails A it is legitimate to infer C.
(p. 148 ) Putting flesh on those bones: while I have a high probability for A→ C, I am in a condition where it is rational for me, if I come to accept P(A) = 1 without any other change in my belief system except ones that follow from that, to attach a high probability to C. (Understand this to be silently qualified to allow for the Thomason examples (§ 12).) But it may not be rational for me to make P(C) high conditional on adding to my belief system not only A but also some further item B that I have not inferred from the addition of A.
It is good to be reminded that when → is in question, the only legitimate form of Modus Ponens is the restricted one in which the second premiss is just exactly the antecedent of the conditional, and not any old thing that entails it. Objection: ‘This is not a form of Modus Ponens, but something else. In so describing it you are just refusing to face a refutation.’ I shall not argue about the label. What matters is that indicative conditionals are of service largely as inference tickets, and that this involves their being legitimately usable in the manner I have described. The thesis that Modus Ponens fails for them does not interfere with that. It merely reminds us that the ticket is not valid if you bring carry‐on luggage. I shall continue to use the terminology of ‘(un)restricted Modus Ponens’ for (in)valid forms of this inference pattern.
Lycan rejects even restricted Modus Ponens. He holds that indicative conditionals have truth values, and that someone who accepts A→ C accepts something true if C obtains in every relevant A‐situation that she can envisage. So it could happen that in accepting A→ C she accepts something true although A is true and C false, the latter in an A‐situation she could not have envisaged; in which case Modus Ponens would lead her from truth to falsity (Lycan 1993: 417–18).
Lycan's exposition of his view is not ideally clear, and some of its features suggest that he is backing off from the bold thesis that T→ F can be true. But I think it is there, as it seems to be also—though again in a somewhat muffled form—when he revisits this territory in his 2001: 48–72.
To be swayed by this, I would have to retract large portions of my Chapters 3–8, and I am not about to do that.
A third objection to Modus Ponens—again to the restricted form of it—is an argument that was mentioned in Adams 1975: 33, taken up and endorsed by McGee 1985, and heralded by Lycan as furnishing a ‘triumph for my semantics . . . the greater for being entirely unanticipated’ (1993: 425). During the 1980 Presidential campaign, in which Reagan was leading Carter, with Republican Anderson a distant third, it was highly reasonable, McGee says, to accept
(1) If a Republican wins the election, then if it's not Reagan who wins it will be Anderson,
(2) A Republican will win.
(3) If it's not Reagan who wins, it will be Anderson,
I deny that it was reasonable to accept 1. Any informed person back then, faced with 1, would have replied: ‘Look, if a Republican wins, it will be Reagan.’ Every such person had a belief system such that, if ‘A Republican will win’ was fed into it and minimal adjustments made, the resulting belief state accorded no probability at all to any conditional starting ‘If Reagan doesn't win . . . ’ except for independently true bores such as ‘If Reagan doesn't win, then somebody won't win’.
Certain known facts about the election process, conjoined with ‘A Republican will win’ and ‘Reagan will not win’, entail ‘Anderson will win’. But this merely illustrates how far logical entailment is from ordinary indicative conditionals.
There is also a shorter route to the same conclusion about McGee's argument. Relying on If‐and (§ 40), we can maintain that 1 is equivalent to ‘If a Republican wins the election and it's not Reagan who wins, it will be Anderson’, which should be rejected because of zero‐intolerance.
In rejecting this argument of McGee's, I do not want to demean his substantial project (1989) of extending Adams's work to cover a range of embeddings of conditionals. I do not report on this, because I have not mastered it; but it does not look negligible.
62. Independent Indicatives9. The Logic of Indicative Conditionals62. Independent Indicatives
Having completed my main account of indicative conditionals, I can discuss how dependent conditionals relate to independent ones within the indicative group (§ 7). I shall confine myself to logically true independent conditionals. For causal and moral ones, the main outlines of the story are the same, but with tedious complications of detail.
I have been building on Ramsey's bedrock idea about what goes into someone's finding A→ C to be more or less probable, namely, letting P(A) = 1 interact with the rest of his belief system, and seeing what the effect this has on the latter's probability for C. Where he accepts A→ C as a logical truth, identity does not interact with the rest of his belief system: he gets C out of A purely through logical principles. In this case there is no source for subjectivity (except in so far as people may have different opinions about what is logically true), nor is there (p. 150 ) anything to block Antecedent Strengthening, Transitivity, and the rest from being valid. Furthermore, with logically true conditionals—as handled by a logically idealized person, for whom P(A) = 1 if A is necessary—the concept of probability is flattened. The person's probability for the conditional itself will always be 1, and the equation P(A) ≤ P(C) will always hold. There will be no use for probabilities other than 0 and 1.
These facts are plain enough, and beyond dispute. Still, a non‐trivial question arises about how to handle them theoretically. In the following, I shall assume that when we are dealing with a logically true conditional, we are considering its behaviour in the hands of someone—call him ‘Bright’—who knows for sure that it is logically true. Now, compare two things said or thought by Bright:
(1) If she reports that payment to the income tax authorities, I shall be in trouble.
(2) If that man is a cannibal, he is not a vegetarian.
We shall do better to suppose that each person means 2 to express his high value for P(C) given A, this being the outcome of the relevant Ramsey test. It happens that in Bright's mind the Ramsey test for 2 is a peculiarly simple affair, and Bright knows this; whereas for Dim the test has the usual complexities. Thus we can capture the real difference between the two without descending to the idea that 2 is ambiguous.
This explains why independent indicatives can be zero‐tolerant (§ 23). The zero‐intolerance of dependent indicative conditionals comes from the fact that if I am utterly sure that not‐A, I have no disciplined way of adjusting my other particular beliefs to make room for P(A) = 1. In performing the Ramsey test, it is understood that one's most general principles are held steady without adjustment, (p. 151 ) and one's beliefs about particular matters of fact (and perhaps about less general matters of fact) are adjusted; the pressure for adjustment comes from the high value for P(A), and the most general principles are the conduits along which the pressure is transmitted. With an independent conditional, the added P(A) = 1 sends pressure straight through the logical conduits to generate P(C) = 1; no adjustment of other beliefs comes into it; so there is nothing to make zero‐intolerance kick in.
‘In a context where it is understood to be logic all the way, isn't it irresistibly tempting to understand indicative conditionals as entailment statements?’ If you find it so, give in; but not by postulating sentence ambiguity as something we must allow, and—connected with that—not presenting this diagnosis as a primary bearer of theoretical load. The Ramsey‐based account that Adams gives carries all the weight; but in a special kind of case where the story flattens itself into something simpler, and this fact is clear to all concerned, then it does no harm to suppose that this shared understanding enters into what the person's conditional sentences mean in this context.