Take a sentence in the indicative mood, suitable for making astatement: “We’ll be home by ten”, “Tom cookedthe dinner”. Attach a conditional clause to it, and you have asentence which makes a conditional statement: “We’ll behome by ten if the train is on time”, “If Marydidn’t cook the dinner, Tom cooked it”. A conditionalsentence “If \(A, C\)” or “\(C\) if \(A\)”thus has two contained sentences or sentence-like clauses. \(A\) iscalled the antecedent, \(C\) the consequent. If you understand \(A\)and \(C\), and you have mastered the conditional construction (as weall do at an early age), you understand “If \(A, C\)”.What does “if” mean? Consulting the dictionary yields“on condition that; provided that; supposing that”. Theseare adequate synonyms. But we want more than synonyms. A theory ofconditionals aims to give an account of the conditional constructionwhich explains when conditional judgements are acceptable, whichinferences involving conditionals are good inferences, and why thislinguistic construction is so important. Despite intensive work ofgreat ingenuity, this remains a highly controversial subject.

• 1. Introduction
• 2. Truth Conditions for Indicative Conditionals
  • 2.1 Two Kinds of Truth Condition
  • 2.2 Arguments for Truth-Functionality
  • 2.3 Arguments Against Truth-Functionality
  • 2.4 Grice’s Pragmatic Defence of Truth-Functionality
  • 2.5 Compounds of Conditionals: Problems for Hook and Arrow
• 3. The Suppositional Theory
  • 3.1 Conditional Belief and Conditional Probability
  • 3.2 Validity
• 4. Truth Conditions Revisited
  • 4.1 Nearest Possible Worlds
  • 4.2 A Special Assertability Condition
  • 4.3 Restrictors and the Strict Conditional
  • 4.4 Heuristics and Semantics
• 5. Compounds of Conditionals: Problems for the Suppositional Theory, and Some Solutions
• 6. Other Conditional Speech Acts and Propositional Attitudes
• Bibliography
  • General Overviews
  • Other Cited Works
• Academic Tools
• Other Internet Resources
• Related Entries

1. Introduction

First let us delimit our field. The examples with which we began aretraditionally called “indicative conditionals”. There arealso “subjunctive” or “counterfactual”conditionals like “Tom would have cooked the dinner if Mary hadnot done so”, “We would have been home by ten if the trainhad been on time”. Counterfactuals are the subject of a separateentry, and theories addressing them will not be discussed here (seethe entry oncounterfactuals). That there is some difference between indicatives and counterfactualsis shown by pairs of examples like “If Oswald didn’t killKennedy, someone else did” and “If Oswald hadn’tkilled Kennedy, someone else would have”: you can accept thefirst yet reject the second (Adams (1970)). That there is not a hugedifference between them is shown by examples like the following:“Don’t go in there”, I say, “If you go in youwill get hurt”. You look sceptical but stay outside, when thereis large crash as the roof collapses. “You see”, I say,“if you had gone in you would have got hurt. I told youso.”

It is controversial how best to classify conditionals. According tosome theorists, the forward-looking “indicatives” (thosewith a “will” in the main clause) belong with the“subjunctives” (those with a “would” in themain clause), and not with the other “indicatives”. (SeeGibbard (1981, pp. 222–6), Dudman (1984, 1988), Bennett (1988).Bennett (1995) changed his mind. Jackson (1990) defends thetraditional view.) The easy transition from typical“wills” to “woulds” is indeed a datum to beexplained. Still, straightforward statements about the past, presentor future, to which a conditional clause is attached — thetraditional class of indicative conditionals — do (in my view)constitute a single semantic kind. The theories to be discussed do notfare better or worse when restricted to a particular subspecies.

As well as conditional statements, there are conditional commands,promises, offers, questions, etc.. As well as conditional beliefs,there are conditional desires, hopes, fears, etc.. Our focus will beon conditional statements and what they express — conditionalbeliefs; but we will consider which of the theories we have examinedextends most naturally to these other kinds of conditional.

Three kinds of theory will be discussed. In §2 we comparetruth-functional and non-truth-functional accounts of the truthconditions of conditionals. In §3 we examine what is called thesuppositional theory: that conditional judgements essentially involvesuppositions. On development, it appears to be incompatible withconstruing conditionals as propositions with truth conditions. §4looks at some responses from advocates of truth conditions. In §5we consider the problem for the suppositional theory of complexsentences with conditional parts, and some proposed solutions to theproblem. In §6 we consider a wider variety of conditional speechacts and propositional attitudes.

Where we need to distinguish between different interpretations, wewrite “\(A \supset B\)” for the truth-functionalconditional, “\(A \rightarrow B\)” for anon-truth-functional conditional and “\(A \Rightarrow B\)”for the conditional as interpreted by the suppositional theory; andfor brevity we call protagonists of the three theories Hook, Arrow andSupp, respectively. We use “\({\sim}\)” for negation.

2. Truth Conditions for Indicative Conditionals

2.1 Two Kinds of Truth Condition

The generally most fruitful, and time-honoured, approach to specifyingthe meaning of a complex sentence in terms of the meanings of itsparts, is to specify the truth conditions of the complex sentence, interms of the truth conditions of its parts. A semantics of this kindyields an account of the validity of arguments involving the complexsentence, given the conception of validity as necessary preservationof truth. Throughout this section we assume that this approach toconditionals is correct. Let \(A\) and \(B\) be two sentences such as“Ann is in Paris” and “Bob is in Paris”. Ourquestion will be: are the truth conditions of “If \(A,B\)” of the simple, extensional, truth-functional kind, likethose of “\(A\) and \(B\)”, “\(A \text{ or }B\)” and “It is not the case that \(A\)”? That is,do the truth values of \(A\) and of \(B\) determine the truth value of“If \(A, B\)”? Or are they non-truth-functional, likethose of “\(A\) because \(B\)”, “\(A\) before\(B\)”, “It is possible that \(A\)”? That is, arethey such that the truth values of \(A\) and \(B\) may, in some cases,leave open the truth value of “If \(A, B\)”?

The truth-functional theory of the conditional was integral toFrege’s new logic (1879). It was taken up enthusiastically byRussell (who called it “material implication”),Wittgenstein in theTractatus, and the logical positivists,and it is now found in every logic text. It is the first theory ofconditionals which students encounter. Typically, it does not strikestudents asobviously correct. It is logic’s firstsurprise. Yet, as the textbooks testify, it does a creditable job inmany circumstances. And it has many defenders. It is a strikinglysimple theory: “If \(A, B\)” is false when \(A\) is trueand \(B\) is false. In all other cases, “If \(A, B\)” istrue. It is thus equivalent to “\({\sim}(A \amp{\sim}B)\)”and to “\({\sim}A\) or \(B\)”. “\(A \supsetB\)” has, by stipulation, these truth conditions.

If “if” is truth-functional, this is the righttruth function to assign to it: of the sixteen possibletruth-functions of \(A\) and \(B\), it is the only serious candidate.First, it is uncontroversial that when \(A\) is true and \(B\) isfalse, “If \(A, B\)” is false. A basic rule of inferenceis modus ponens: from “If \(A, B\)” and \(A\), we caninfer \(B\). If it were possible to have \(A\) true, \(B\) false and“If \(A, B\)” true, this inference would be invalid.Second, it is uncontroversial that “If \(A, B\)” issometimes true when \(A\) and \(B\) are respectively (true,true), or (false, true), or (false, false). “If it’s asquare, it has four sides”, said of an unseen geometric figure,is true, whether the figure is a square, a rectangle or a triangle.Assuming truth-functionality — that the truth value of theconditional isdetermined by the truth values of its parts— it follows that a conditional isalways true when itscomponents have these combinations of truth values.

Non-truth-functional accounts agree that “If \(A, B\)” isfalse when \(A\) is true and \(B\) is false; and they agree that theconditional is sometimes true for the other three combinations oftruth-values for the components; but they deny that the conditional isalways true in each of these three cases. Some agree with thetruth-functionalist that when \(A\) and \(B\) are both true, “If\(A, B\)” must be true. Some do not, demanding a furtherrelation between the facts that \(A\) and that \(B\) (see Read(1995)). This dispute need not concern us, as the arguments whichfollow depend only on the feature on which non-truth-functionalistsagree: that when \(A\) is false, “If \(A, B\)” may beeither true or false. For instance, I say (*) “If you touch thatwire, you will get an electric shock”. You don’t touch it.Was my remark true or false? According to the non-truth-functionalist,it depends on whether the wire is live or dead, on whether you areinsulated, and so forth. Robert Stalnaker’s (1968) account is ofthis type: consider a possible situation in which you touch the wire,and which otherwise differs minimally from the actual situation. (*)is true (false) according to whether or not you get a shock in thatpossible situation.

Let \(A\) and \(B\) be two logically independent propositions. Thefour lines below represent the four incompatible logical possibilitiesfor the truth values of \(A\) and \(B\). “If \(A, B\)”,“If \({\sim}A, B\)” and “If \(A, {\sim}B\)”are interpreted truth-functionally in columns (i)–(iii), andnon-truth-functionally (when their antecedents are false) in columns(iv)–(vi). The non-truth-functional interpretation we write“\(A \rightarrow B\)”. “T/F” means both truthvalues are possible for the corresponding assignment of truth valuesto \(A\) and \(B\). For instance, line 4, column (iv), represents twopossibilities for \(A, B\), If \(A, B\), (F, F, T) and (F, F, F).

2.2 Arguments for Truth-Functionality

The main argument points to the fact that minimal knowledge that thetruth-functional truth condition is satisfied is enough for knowledgethat if \(A, B\). Suppose there are two balls in a bag, labelled \(x\)and \(y\). All you know about their colour is that at least one ofthem is red. That’s enough to know that if \(x\) isn’tred, \(y\) is red. Or: all you know is that they are not both red.That’s enough to know that if \(x\) is red, \(y\) is notred.

Suppose you start off with no information about which of the fourpossible combinations of truth values for \(A\) and \(B\) obtains. Youthen acquire compelling reason to think that either \(A\) or \(B\) istrue. You don’t have any stronger belief about the matter. Inparticular, you have no firm belief as to whether \(A\) is true ornot. You have ruled out line 4. The other possibilities remain open.Then, intuitively, you are justified in inferring that if \({\sim}A,B\). Look at the possibilities for \(A\) and \(B\) on the left. Youhave eliminated the possibility that both \(A\) and \(B\) are false.So if \(A\) is false, only one possibility remains: \(B\) is true.

The truth-functionalist (call him Hook) gets this right. Look atcolumn (ii). Eliminate line 4 and line 4 only, and you have eliminatedthe only possibility in which “\({\sim}A \supset B\)” isfalse. You know enough to conclude that “\({\sim}A \supsetB\)” is true.

The non-truth-functionalist (call her Arrow) gets this wrong. Look atcolumn (v). Eliminate line 4 and line 4 only, and some possibility offalsity remains in other cases which have not been ruled out. Byeliminating just line 4, you do not thereby eliminate these furtherpossibilities, incompatible with line 4, in which “\({\sim}A\rightarrow B\)” is false.

The same point can be made with negated conjunctions. You discover forsure that \({\sim}(A \amp B)\), but nothing stronger than that. Inparticular, you don’t know whether \(A\). You rule out line 1,nothing more. You may justifiably infer that if \(A, {\sim}B\). Hookgets this right. In column (iii), if we eliminate line 1, we are leftonly with cases in which “\(A \supset{\sim}B\)” is true.Arrow gets this wrong. In column (vi), eliminating line 1 leaves openthe possibility that “\(A \rightarrow{\sim}B\)” isfalse.

The same argument renders compelling the thought that if we eliminatejust \(A \amp{\sim}B\), nothing stronger, i.e., wedon’t eliminate \(A\), then we have sufficient reason toconclude that if \(A, B\).

Here is a second argument in favour of Hook, in the style of NaturalDeduction. The rule of Conditional Proof (CP) says that if \(Z\)follows from premises \(X\) and \(Y\), then “If \(Y, Z\)”follows from premise \(X\). Now the three premises \({\sim}(A \amp B),A\) and \(B\) entail a contradiction. So, by Reductio Ad Absurdum,from \({\sim}(A \amp B)\) and \(A\), we can conclude \({\sim}B\). Soby CP, \({\sim}(A \amp B)\) entails “If \(A, {\sim}B\)”.Substitute “\({\sim}C\)” for \(B\), and we have a proof of“If \(A\), then \({\sim}{\sim}C\)” from “\({\sim}(A\amp{\sim}C)\)”. And provided we also accept Double NegationElimination, we can derive “If \(A\), then \(C\)” from“\({\sim}(A \amp{\sim}C)\)”.

Conditional Proof seems sound: “From \(X\) and \(Y\), it followsthat \(Z\). So from \(X\) it follows that if \(Y, Z\)”. Yetfor no reading of “if” which is stronger than thetruth-functional reading is CP valid — at least this is soif we treat “&” and “\({\sim}\)” in theclassical way and accept the validity of the inference: (I) \({\sim}(A\amp{\sim}B)\); \(A\); therefore \(B\). Suppose CP is valid for someinterpretation of “If \(A, B\)”. Apply CP to (I), and weget \({\sim}(A \amp{\sim}B)\); therefore if \(A, B\), i.e., \(A\supset B\) entails if \(A, B\).

2.3 Arguments Against Truth-Functionality

The best-known objection to the truth-functional account, one of the“paradoxes of material implication”, is that according toHook, the falsity of \(A\) is sufficient for the truth of “If\(A, B\)”. Look at the last two lines of column (i). In everypossible situation in which \(A\) is false, “\(A \supsetB\)” is true. Can it be right that the falsity of “Shetouched the wire” entails the truth of “If she touched thewire she got a shock”?

Hook might respond as follows. How do we test our intuitions about thevalidity of an inference? The direct way is to imagine that we knowfor sure that the premise is true, and to consider what we would thenthink about the conclusion. Now when we know for sure that\({\sim}A\), we have no use for thoughts beginning “If \(A\),…”. When you know for sure that Harry didn’t do it,you don’t go in for “If Harry did it …”thoughts or remarks. In this circumstance conditionals have no role toplay, and we have no practice in assessing them. The direct intuitivetest is, therefore, silent on whether “If \(A, B\)”follows from \({\sim}A\). If our smoothest, simplest, generallysatisfactory theory has the consequence that it does follow, perhapswe should learn to live with that consequence.

There may, of course, be further consequences of this feature ofHook’s theory which jar with intuition. That needsinvestigating. But, Hook may add, even if we come to the conclusionthat “\(\supset\)” does not match perfectly ournatural-language “if”, it comes close, and it has thevirtues of simplicity and clarity. We have seen that rival theoriesalso have counterintuitive consequences. Natural language is a fluidaffair, and we cannot expect our theories to achieve better thanapproximate fit. Perhaps, in the interests of precision and clarity,in serious reasoning we should replace the elusive “if”with its neat, close relative, \(\supset\) .

This was no doubt Frege’s attitude. Frege’s primaryconcern was to construct a system of logic, formulated in an idealizedlanguage, which was adequate for mathematical reasoning. If “\(A\supset B\)” doesn’t translate perfectly ournatural-language “If \(A, B\)”, but plays its intendedrole, so much the worse for natural language.

For the purpose of doing mathematics, Frege’s judgement wasprobably correct. The main defects of \(\supset\) don’t show upin mathematics. There are some peculiarities, but as long as we areaware of them, they can be lived with. And arguably, the gain insimplicity and clarity more than offsets the oddities.

The oddities are harder to tolerate when we consider conditionaljudgements about empirical matters. The difference is this: inthinking about the empirical world, we often accept and rejectpropositions with degrees of confidence less than certainty. We can,perhaps, ignore as unimportant the use of indicative conditionals incircumstances in which we arecertain that the antecedent isfalse. But we cannot ignore our use of conditionals whose antecedentwe think is likely to be false. We use them often, accepting some,rejecting others. “I think I won’t need to get in touch,but if I do, I shall need a phone number”, you say as yourpartner is about to go away; not “If I do I’ll manage bytelepathy”. “I think John spoke to Mary; if hedidn’t he wrote to her”; not “If he didn’t heshot her”. Hook’s theory has the unhappy consequence thatall conditionals with unlikely antecedents are likely to betrue. To think it likely that \({\sim}A\) is to think it likely that asufficient condition for the truth of “\(A \supset B\)”obtains. Take someone who thinks that the Republicans won’t winthe election \(({\sim}R)\), and who rejects the thought that if theydo win, they will double income tax \((D)\). According to Hook, thisperson has grossly inconsistent opinions. For if she thinks it’slikely that \({\sim}R\), she must think it likely that at least one ofthe propositions, \(\{{\sim}R, D\}\) is true. But that is just tothink it likely that \(R \supset D\). (Put the other way round, toreject \(R \supset D\) is to accept \(R \amp{\sim}D\); for this is theonly case in which \(R \supset D\) is false. How can someone accept\(R \amp{\sim}D\) yet reject \(R\)?) Not only does Hook’s theoryfit badly the patterns of thought of competent, intelligent people. Itcannot be claimed that we would be better off with \(\supset\). On thecontrary, we would be intellectually disabled: we would not have thepower to discriminate between believable and unbelievable conditionalswhose antecedent we think is likely to be false.

Arrow does not have this problem. Her theory is designed to avoid it,by allowing that “\(A \rightarrow B\)” may be false when\(A\) is false.

The other paradox of material implication is that according to Hookall conditionals with true consequents are true: from \(B\) it followsthat \(A \supset B\). This is perhaps less obviously unacceptable: ifI’m sure that \(B\), and treat \(A\) as an epistemicpossibility, I must be sure that if \(A, B\). Again the problembecomes vivid when we consider the case when I’m only nearlysure, but not quite sure, that \(B\). I think \(B\)may befalse, and will be false if certain, in my view unlikely,circumstances obtain. For example, I think Sue is giving a lectureright now. I don’t think that if she was seriously injured onher way to work, she is giving a lecture right now. I reject thatconditional. But on Hook’s account, the conditional is falseonly if the consequent is false. I think the consequent is true: Ithink a sufficient condition for the truth of the conditionalobtains.

2.4 Grice’s Pragmatic Defence of Truth-Functionality

H. P. Grice famously defended the truth-functional account, in hisWilliam James lectures “Logic and Conversation”, deliveredin 1967 (see Grice (1989); see also Thomson (1990)). There are manyways of speaking the truth yet misleading your audience, given thestandards to which you are expected to conform in conversationalexchange. One way is to say something weaker than some other relevantthing you are in a position to say. Consider disjunctions. I am askedwhere John is. I am sure that he is in the pub, and know that he nevergoes near libraries. Inclined to be unhelpful but not wishing to lie,I say “He is either in the pub or in the library”. Myhearer naturally assumes that this is the most precise information Iam in a position to give, and also concludes from the truth (let usassume) that I told him “If he’s not in the pub he’sin the library”. The conditional, like the disjunction,according to Grice, is true if he’s in the pub, but misleadinglyasserted on that ground.

Another example, from David Lewis (1976, p. 143): “Youwon’t eat those and live”, I say of some wholesome anddelicious mushrooms—knowing that you will now leave them alone,deferring to my expertise. I told no lie—for indeed youdon’t eat them—but of course I misled you.

Grice drew attention, then, to situations in which a person isjustified in believing a proposition, which wouldnevertheless be an unreasonable thing for the person tosay,in normal circumstances. His lesson was salutary and important. He isright, I think, about disjunctions and negated conjunctions. Believingthat John is in the pub, I can’t consistentlydisbelieve “He’s either in the pub or thelibrary”; if I have any epistemic attitude to this proposition,it should be one of belief, however inappropriate for me to assert it.Similarly for “You won’t eat those and live” when Iknow you won’t eat them. But it is implausible that thedifficulties with the truth-functional conditional can be explainedaway in terms of what is an inappropriate conversational remark. Theyarise at the level of belief. Thinking that John is in the pub, I maywithout irrationality disbelieve “If he’s not in the pubhe’s in the library”. Thinking you won’t eat themushrooms, I may without irrationality reject “If you eat themyou will die”. As facts about the norms to which people defer,these claims can be tested. A good enough test is to take aco-operative person, who understands that you are merely interested inher opinions about the propositions you put to her, as opposed to whatwould be a reasonable remark to make, and note which conditionals sheassents to. Are we really to brand as illogical someone who dissentsfrom both “The Republicans will win” and “If theRepublicans win, income tax will double”?

The Gricean phenomenon is a real one. On anyone’s account ofconditionals, there will be circumstances when a conditional isjustifiably believed, but is liable to mislead if stated. Forinstance, I believe that the match will be cancelled, because all theplayers have flu. I believe that whether or not it rains, the matchwill be cancelled: if it rains, the match will be cancelled, and if itdoesn’t rain, the match will be cancelled. Someone asks mewhether the match will go ahead. I say, “If it rains, the matchwill be cancelled”. I say something I believe, but I mislead myaudience — why should I say that, when I think it will becancelled whether or not it rains? This does not demonstrate that Hookis correct. Although I believe that the match will be cancelled, Idon’t believe that if all the players make a very speedyrecovery, the match will be cancelled.

2.5 Compounds of Conditionals: Problems for Hook and Arrow

\({\sim}(A \supset B)\) is equivalent to \(A \amp{\sim}B\).Intuitively, you may safely say, of an unseen geometric figure,“It’s not the case that if it’s a pentagon, it hassix sides”. But by Hook’s lights, you may well be wrong;for it may not be a pentagon, and in that case it is true that ifit’s a pentagon, it has six sides.

Another example, due to Gibbard (1981, pp. 235–6): of a glassthat had been held a foot above the floor, you say (having left thescene) “If it broke if it was dropped, it was fragile”.Intuitively this seems reasonable. But by Hook’s lights, if theglass was not dropped, and was not fragile, the conditional has a true(conditional) antecedent and false consequent, and is hence false.

Grice’s strategy was to explain why we don’t assertcertain conditionals which (by Hook’s lights) we have reason tobelieve true. In the above two cases, the problem is reversed: thereare compounds of conditionals which we confidently assert and acceptwhich, by Hook’s lights, we do not have reason to believetrue.

Another bad result is that according to Hook, the following is a validargument:

If \(A \amp B, C\); therefore, either, if \(A, C\), or, if \(B,C\).

Even in mathematics, this looks wrong. Said of an unseen plane figure:“If it’s a triangle and it’s equiangular, it’sequilateral; therefore, either, if it’s a triangle it’sequilateral, or, if it’s equiangular it’sequilateral”. (I owe this example to Alberto Mura.)

The above examples are not a problem for Arrow. But other cases ofembedded conditionals count in the opposite direction. Here are twosentence forms which are, intuitively, equivalent:

1. If \((A \amp B), C\).
2. If \(A\), then if \(B, C\).

(Following Vann McGee (1989) I’ll call the principle that (i)and (ii) are equivalent the Import-Export Principle, or“Import-Export” for short.) Try any example: “IfMary comes then if John doesn’t have to leave early we will playBridge”; “If Mary comes and John doesn’t have toleave early we will play Bridge”. “If they were outsideand it rained, they got wet”; “If they were outside, thenif it rained, they got wet”. For Hook, Import-Export holds.(Exercise: do a truth table, or construct a proof.) Gibbard (1981, pp.234–5) has proved that for no conditional with truth conditionsstronger than \(\supset\) does Import-Export hold. AssumeImport-Export holds for some reading of “if”. The key tothe proof is to consider the formula

1. If \((A \supset B)\) then (if \(A,B)\).

By Import-Export, (1) is equivalent to

2. If \(((A \supset B) \amp A))\) then \(B\).

The antecedent of (2) entails its consequent. So (2) is a logicaltruth. So by Import-Export, (1) is a logical truth. On any reading of“if”, “if \(A, B\)” entails \((A \supset B)\).So (1) entails

3. \((A \supset B) \supset\) (if \(A, B)\).

So (3) is a logical truth. That is, there is no possible situation inwhich its antecedent \((A \supset B)\) is true and its consequent (if\(A, B)\) is false. That is, \((A \supset B)\) entails “If \(A,B\)”.

Neither kind of truth condition has proved entirely satisfactory. Westill have to consider Jackson’s defence of Hook, andStalnaker’s response to the problem about non-truth-functionaltruth conditions raised in §2.2. These are deferred to §4,because they depend on the considerations developed in §3.

3. The Suppositional Theory

3.1 Conditional Belief and Conditional Probability

Let us put truth conditions aside for a while, and ask what it is tobelieve, or to be more or less certain, that \(B\) if \(A\) —that John cooked the dinner if Mary didn’t, that you willrecover if you have the operation, and so forth. How do you make sucha judgement? You suppose (assume, hypothesise) that \(A\), and make ahypothetical judgement about \(B\), under the supposition that \(A\),in the light of your other beliefs. Frank Ramsey put it like this:

If two people are arguing “If \(p\), will \(q\)?” and areboth in doubt as to \(p\), they are adding \(p\) hypothetically totheir stock of knowledge, and arguing on that basis about \(q\);… they are fixing their degrees of belief in \(q\) given \(p\)(1929, p. 247).

This claim of Ramsey’s has come to be known as “the RamseyTest”.

A suppositional theory was advanced by J. L. Mackie (1973, chapter 4).See also David Barnett (2006). Peter Gärdenfors’s work(1986, 1988) could also come under this heading. But the most fruitfuldevelopment of the idea (in my view) takes seriously the last part ofthe above quote from Ramsey, and emphasises the fact that conditionalscan be accepted with different degrees of closeness to certainty.Ernest Adams (1965, 1966, 1975) has developed such a theory.

When we are neither certain that \(B\) nor certain that \({\sim}B\),there remains a range of epistemic attitudes we may have to \(B\): wemay be nearly certain that \(B\), think \(B\) more likely than not,etc.. Similarly, we may be certain, nearly certain, etc. that \(B\)given the supposition that \(A\). Make the idealizing assumption thatdegrees of closeness to certainty can be quantified: 100% certain, 90%certain, etc.; and we can turn to probability theory for what Ramseycalled the “logic of partial belief”. There we find awell-established, indispensable concept, “the conditionalprobability of \(B\) given \(A\)”. It is to this notion thatRamsey refers by the phrase “degrees of belief in \(q\) given\(p\)”.

It is, at first sight, rather curious that the best-developed and mostilluminating suppositional theory should place emphasis on uncertainconditional judgements. If we knew the truth conditions ofconditionals, we would handle uncertainty about conditionals in termsof a general theory of what it is to be uncertain of the truth of aproposition. But there is no consensus about the truth conditions ofconditionals. It happens that when we turn to the theory of uncertainjudgements, we find a concept of conditionality in use. It is worthseeing what we can learn from it.

The notion of conditional probability entered probability theory at anearly stage because it was needed to compute the probability of aconjunction. Thomas Bayes (1763) wrote:

The probability that two … events will both happen is …the probability of the first [multiplied by] the probability of thesecondon the supposition that the first happens [myemphasis].

A simple example: a ball is picked at random. 70% of the balls are red(so the probability that a red ball is picked is 70%). 60% of the redballs have a black spot (so the probability that a ball with a blackspot is picked, on the supposition that a red ball is picked, is 60%).The probability that a red ball with a black spot is picked is 60% of70%, i.e. 42%.

Ramsey, arguing that “degrees of belief” should conform toprobability theory, stated the same “fundamental law of partialbelief”:

Degree of belief in \((p\) and \(q) =\) degree of belief in \(p\times\) degree of belief in \(q\) given \(p\). (1926, p. 77)

For example, you are about 50% certain that the test will be onconditionals, and about 80% certain that you will pass, on thesupposition that it is on conditionals. So you are about 40% certainthat the test will be on conditionals and you will pass.

Accepting Ramsey’s suggestion that “if”,“given that”, “on the supposition that” cometo the same thing, writing “\(\bp(B)\)” for “degreeof belief in \(B\)”, and “\(\bp_A (B)\)” for“degree of belief in \(B\) given \(A\)”, and rearrangingthe basic law, we have:

(Note: another common notation for “degree of belief in \(B\)given \(A\)” is “\(\bp(B\mid A\)”.)

Call a set of mutually exclusive and jointly exhaustive propositions apartition. The lines of a truth table constitute a partition.One’s degrees of belief in the members of a partition, idealizedas precise, should sum to 100%. That is all there is to the claim thatdegrees of belief should have the structure of probabilities. Considera partition of the form \(\{A \amp B, A \amp{\sim}B, {\sim}A\}\).Suppose someone X thinks it 50% likely that \({\sim}A\) (hence 50%likely that \(A), 40\)% likely that \(A \amp B\), and 10% likely that\(A \amp{\sim}B\). Think of this distribution as displayedgeometrically, as follows. Draw a long narrow horizontal rectangle.Divide it in half by a vertical line. Write “\({\sim}A\)”in the right-hand half. Divide the left-hand half with anothervertical line, in the ratio 4:1, with the larger part on the left.Write “\(A \amp B\)” and “\(A \amp{\sim}B\)”in the larger and smaller cells respectively.

(Note that as \(\{A \amp B, A \amp{\sim}B, {\sim}A\}\) and \(\{A,{\sim}A\}\) are both partitions, it follows that \(\bp(A) = \bp(A \ampB) + \bp(A \amp{\sim}B)\).)

How does X evaluate “If \(A, B\)”? She assumes that \(A\),that is, hypothetically eliminates \({\sim}A\). In the part of thepartition that remains, in which \(A\) is true, \(B\) is four times aslikely as \({\sim}B\); that is, on the assumption that \(A\), it isfour to one that \(B: \bp(B\) if \(A)\) is 80%, \(\bp({\sim}B\) if\(A)\) is 20%. Equivalently, as \(A \amp B\) is four times as likelyas \(A \amp{\sim}B, \bp(B\) if \(A)\) is 4/5, or 80%. Equivalently,\(\bp(A \amp B)\) is 4/5 of \(\bp(A)\). In non-numerical terms: youbelieve that if \(A, B\) to the extent that you think that \(A \ampB\) is nearly as likely as \(A\); or, to the extent that you think \(A\amp B\) is much more likely than \(A \amp{\sim}B\). If you think \(A\amp B\) is as likely as \(A\), you are certain that if \(A, B\). Inthis case, your \(\bp(A \amp{\sim}B) = 0\).

Go back to the truth table. You are wondering whether if \(A, B\).Assume \(A\). That is, ignore lines 3 and 4 in which \(A\) is false.Ask yourself about the relative probabilities of lines 1 and 2.Suppose you think line 1 is about 100 times more likely than line 2.Then you think it is about 100 to 1 that \(B\) if \(A\).

Note: these thought-experiments can only be performed when \(\bp(A)\)is not 0. On this approach, indicative conditionals only have a rolewhen the thinker takes \(A\) to be an epistemic possibility. If youtake yourself to know for sure that Ann is in Paris, you don’tgo in for “If Ann is not in Paris …” thoughts(though of course you can think “If Ann had not been in Paris…”). In conversation, you can pretend to take somethingas an epistemic possibility, temporarily, to comply with the epistemicstate of the hearer. When playing the sceptic, there are not manylimits on what youcan, at a pinch, take as an epistemicpossibility – as not already ruled out. But there are somelimits, as Descartes found. Is there a conditional thought that begins“If I don’t exist now …”?

On Hook’s account, to be close to certain that if \(A, B\) is togive a high value to \(\bp(A \supset B)\). How does \(\bp(A \supsetB)\) compare with \(\bp_A (B)\)? In two special cases, they are equal:first, if \(\bp(A \amp{\sim}B) = 0\) (and \(\bp(A)\) is not 0),\(\bp(A \supset B) = \bp_A(B) = 1.\) Second, if \(\bp(A)=1\), \(\bp(A\supset B) = \bp_A (B) = \bp(B)\)). In all other cases, \(\bp(A\supset B)\) is greater than \(\bp_A (B)\). To see this we need tocompare \(\bp(A \amp{\sim}B)\) and \(\bp(A \amp{\sim}B)/\bp(A)\).Except in the special cases mentioned above, when they are equal, thelatter is greater than the former. \((A\amp{\sim}B)\) is true in alarger proportion of the part of the space in which \(A\) is true,than it is of the whole space.) \(\bp(A \supset B) = 1 - \bp(A\amp{\sim}B)\). \(\bp_A (B) = 1 - \bp(A \amp{\sim}B)/\bp(A)\). So\(\bp(A \supset B) \gt \bp_A (B)\).

Hook and the suppositional theorist (call her Supp) come spectacularlyapart when \(\bp({\sim}A)\) is high and \(\bp(A \amp B)\) is muchsmaller than \(\bp(A \amp{\sim}B)\). Let \(\bp({\sim}A) = 90\)%,\(\bp(A \amp B) = 1\)%, \(\bp(A \amp{\sim}B) = 9\)%. \(\bp_A (B) =10\)%. \(\bp(A \supset B) = 91\)%. For instance, I am 90% certain thatSue won’t be offered the job \(({\sim}O)\), and think it only10% likely that she will decline the offer \((D)\) if it is made, thatis \(\bp_O (D) = 10\)%. \(\bp(O \supset D) = \bp({\sim}O \text{ or }(O \amp D)) = 91\)%.

Now let us compare Hook, Arrow, and Supp with respect to two questionsraised in §2.

• Question 1. You are certain that \({\sim}(A \amp{\sim}B)\), butnot certain that \({\sim}A\). Should you be certain that if \(A, B\)?

  Hook: yes. Because “\(A \supset B\)” is true whenever \(A\amp{\sim}B\) is false.

  Supp: yes. Because \(A \amp B\) is as likely as \(A. \bp_A (B) =1\).

  Arrow: no, not necessarily. For “\(A \rightarrow B\)” maybe false when \(A \amp{\sim}B\) is false. With just the informationthat \(A \amp{\sim}B\) is false, I should not be certain that if \(A,B\).

• Question 2. If you think it likely that \({\sim}A\), might youstill think it unlikely that if \(A, B\)?

  Hook: no. “\(A \supset B\)” is true in all the possiblesituations in which \({\sim}A\) is true. If I think it likely that\({\sim}A\), I think it likely that a sufficient condition for thetruth of “\(A \supset B\)” obtains. I must, therefore,think it likely that if \(A, B\).

  Supp: yes. We had an example above. That most of my probability goesto \({\sim}A\) leaves open the question whether or not \(A \amp B\) ismore probable than \(A \amp{\sim}B\). If \(\bp(A \amp{\sim}B)\) isgreater than \(\bp(A \amp B)\), I think it’s unlikely that if\(A, B\). That’s compatible with thinking it likely that\({\sim}A\).

  Arrow: yes. “If \(A, B\)” may be false when \(A\) isfalse. And I might well think it likely that that possibility obtains,i.e. unlikely that “If \(A, B\)” is true.

Supp has squared the circle: she gets the intuitively right answer toboth questions. In this she differs from both Hook and Arrow.Supp’s way of assessing conditionals is incompatible with thetruth-functional way (they answer Question 2 differently); andincompatible with stronger-than-truth-functional truth conditions(they answer Question 1 differently). It follows that Supp’s wayof assessing conditionals is incompatible with either kind of truthcondition. \(\bp_A (B)\) does not measure the probability of the truthof any proposition. Suppose it did measure the probability of thetruth of some proposition \(A*B\). Either \(A*B\) is entailed by“\(A \supset B\)”, or it is not. If it is, it is truewhenever \({\sim}A\) is true, and hence cannot be improbable when\({\sim}A\) is probable. That is, it cannot agree with Supp in itsanswer to Question 2. If \(A*B\) is not entailed by “\(A \supsetB\)”, it may be false when \({\sim}(A \amp{\sim}B)\) is true,and hence certainty that \({\sim}(A \amp{\sim}B)\) (in the absence ofcertainty that \({\sim}A)\) is insufficient for certainty that\(A*B\); it cannot agree with Supp in its answer to Question 1.

To make the point in a slightly different way, let me adopt thefollowing as an expository, heuristic device, a harmless fiction.Imagine a partition as carved into a large finite number ofequally-probable chunks, such that the propositions with which we areconcerned are true in an exact number of them. The probability of anyproposition is the proportion of chunks in which it is true. Theprobability of \(B\) on the supposition that \(A\) is the proportionof the \(A\)-chunks (the chunks in which \(A\) istrue) which are \(B\)-chunks. With some misgivings, I succumb to thetemptation to call these chunks “worlds”: they are equallyprobable, mutually incompatible and jointly exhaustive epistemicpossibilities, enough of them for the propositions with which we areconcerned to be true, or false, at each world. The heuristic value isthat judgements of probability and conditional probability thentranslate into statements about proportions.

Although Supp and Hook give the same answer to Question 1, theirreasons are different. Supp answers “yes”notbecause a proposition, \(A*B\), is true whenever \(A \amp{\sim}B\) isfalse; but because \(B\) is true in all the “worlds” whichmatter for the assessment of “If \(A, B\)”: the\(A\)-worlds. Although Supp and Arrow give the same answer to Question2, their reasons are different. Supp answers “yes”, notbecause a proposition \(A*B\) may be false when \(A\) is false; butbecause the fact that most worlds are \({\sim}A\)-worlds is irrelevantto whether mostof the \(A\)-worlds are\(B\)-worlds. To judge that \(B\) is trueon the suppositionthat \(A\) is true, it turns out, is not to judge that someproposition, \(A*B\), is true.

By a different argument, David Lewis (1976) was the first to provethis remarkable result: there is no proposition \(A*B\) such that, inall probability distributions, \(\bp(A*B) = \bp_A (B)\). Conditionalprobability does not measure the probability of the truth of anyproposition. If a conditional has truth conditions, one should believeit to the extent that one thinks it is probably true. If Supp iscorrect, that one believes “If \(A, B\)” to the extentthat one thinks it probable that \(B\) on the supposition that \(A\),then this is not equivalent to believing some proposition to beprobably true. Hence, it appears, if Supp is right, conditionalsshouldn’t be construed as propositions with truth conditions. Aconditional judgement involves two propositions, which play differentroles. One is the content of a supposition. The other is the contentof a judgement made under that supposition. They do not combine toyield a single proposition which is judged to be likely to be truejust when the second is judged likely to be true on the supposition ofthe first.

Lewis called his proofs “triviality results”, because theconclusions are avoided only in a trivial probability space which isincapable of giving positive probability to more than two incompatiblepropositions—for instance, is incapable of giving positiveprobability to \(A \amp B, A \amp{\sim}B\), and \({\sim}A\). The nameis widely used in the literature. For recent examples see Khoo andMandelkern (2019) and Charlow (2019). Another recent discussion of thesuppositional theory, and some difficulties that have been raised forit, is Ciardelli and Ommundsen (2022).

Note: a way of restoring truth conditions (but not classicalpropositions), compatible with Supp’s thesis, is considered in§5.

3.2 Validity

Ernest Adams, in two articles (1965, 1966) and a subsequent book(1975), gave a theory of the validity of arguments involvingconditionals as construed by Supp. He taught us something importantabout classically valid arguments as well: that they are, in a specialsense to be made precise, probability-preserving. This property can begeneralized to apply to arguments with conditionals. The valid onesare those which, in the special sense, preserve probability orconditional probability.

First consider classically valid (that is, necessarilytruth-preserving) arguments which don’t involve conditionals. Weuse them in arguing from contingent premises about which we are oftenless than completely certain. The question arises: how certain can webe of the conclusion of the argument, given that we think, but are notsure, that the premises are true? Call the improbability of aproposition one minus its probability. Adams showed this: if (and onlyif) an argument is valid, then in no probability distribution does theimprobability of its conclusion exceed the sum of the improbabilitiesof its premises. Call this the Probability Preservation Principle(PPP).

The proof of PPP rests on the Partition Principle — that theprobabilities of the members of a partition sum to 100% —nothing else, beyond the fact that if \(A\) entails \(B, \bp(A\amp{\sim}B) = 0\). Here are three consequences:

1. if \(A\) entails \(B, \bp(A) \le \bp(B)\)
2. \(\bp(A \text{ or } B) = \bp(A) + \bp(B) - \bp(A \amp B) \le\bp(A) + \bp(B)\)
3. For all \(n, \bp(A_1\) or … or \(A_n) \le \bp(A_1) + \cdots+ \bp(A_n)\)

Suppose \(A_1, \ldots, A_n\) entail \(B\). Then \({\sim}B\) entails\({\sim}A_1\) or … or \({\sim}A_n\). Therefore \(\bp({\sim}B)\le \bp({\sim}A_1) + \cdots + \bp({\sim}A_n)\): the improbability ofthe conclusion of a valid argument cannot exceed the sum of theimprobabilities of the premises.

The result is useful to know: if you have two premises of which youare at least 99% certain, they entitle you to be at least 98% certainof a conclusion validly drawn from them. Of course, if you have 100premises each at least 99% certain, your conclusion may have zeroprobability. That is the lesson of the “Lottery Paradox”.Still, Adams’s result vindicates deductive reasoning fromuncertain premises, provided that they are not too uncertain, andthere are not too many of them.

So far, we have a very useful consequence of the classical notion ofvalidity. Now Adams extends this consequence to arguments involvingconditionals. Take a language with “and”,“or”, “not” and “if” — butwith “if” occurring only as the main connective in asentence. (We put aside compounds of conditionals.) Take any argumentformulated in this language. Consider any probability function overthe sentences of this argument which assigns non-zero probability tothe antecedents of all conditionals — that is, any assignment ofnumbers to the non-conditional sentences which conforms to thePartition Principle, and to the conditional sentences which conformsto Supp’s thesis: \(\bp(B\) if \(A) = \bp_A (B) = \bp(A \ampB)/\bp(A)\). Let the improbability of the conditional “If \(A,B\)” be \(1 - \bp_A (B)\).Define a valid argument asone such that there is no probability function in which theimprobability of the conclusion exceeds the sum of the improbabilitiesof the premises. And a nice logic emerges, which is now well known. Itis the same as Stalnaker’s logic over this domain (see§4.1). There are rules of proof, a decision procedure,consistency and completeness can be proved. See Adams (1998 and1975).

I shall write the conditional which satisfies Adams’s criterionof validity“\(A \Rightarrow B\)”.We have already seen that in all distributions, \(\bp_A (B) \le \bp(A\supset B)\). Therefore, \(A \Rightarrow B\) entails \(A \supset B\):it cannot be the case that the former is more probable than thelatter. Call a non-conditional sentence a factual sentence. If anargument has a factual conclusion, and is classically valid with theconditional interpreted as \(\supset\), it is valid with theconditional interpreted as the stronger \(\Rightarrow\). The followingpatterns of inference are therefore valid:

We cannot consistently have their premises highly probable and theirconclusion highly improbable.

Arguments with conditional conclusions, however, may be valid when theconditional is interpreted as the weaker \(A \supset B\), but invalidwhen it is interpreted as the stronger \(A \Rightarrow B\). Here aresome examples.

I can consistently be close to certain that Sue is lecturing rightnow, while thinking it highly unlikely that if she had a heart attackon her way to work, she is lecturing just now.

You can consistently be close to certain that the Republicanswon’t win, while thinking it highly unlikely that if they winthey will double income tax.

I can consistently be close to certain that it’s not the casethat I will be hit by a bomb and injured today, while thinking ithighly unlikely that if I am hit by a bomb, I won’t beinjured.

As I think it is very likely to rain tomorrow, I think it’s verylikely to be true that it will rain or snow tomorrow. But I thinkit’s very unlikely that if it doesn’t rain, it willsnow.

I can think it’s highly likely that if you strike the match, itwill light; but highly unlikely that if you dip it in water and strikeit, it will light.

Strengthening is a special case of transitivity, in which the missingpremise is a tautology: if \(C \amp A\) then \(A\); if \(A, B\); so if\(C \amp A, B\). So transitivity also fails:

Adams gave this example (1966): I can think it highly likely that ifJones is elected, Brown will resign immediately afterwards; I can alsothink it highly likely that if Brown dies before the election, Joneswill be elected; but I do not think it at all likely that if Browndies before the election, Brown will resign immediately after theelection!

We saw in §2.2 that Conditional Proof (CP) is invalid for anyconditional stronger than \(\supset\). It is invalid in Adams’slogic. For instance, “\({\sim}(A \amp B)\); \(A\); so\({\sim}B\)” is valid. It contains no conditionals. Anynecessarily truth-preserving argument satisfies PPP. If I’mclose to certain that I won’t be hit by a bomb and injured,and close to certain that I will be hit by a bomb, then Imust be close to certain that I won’t be injured. But, as wesaw, “\({\sim}(A \amp B)\); so \(A \Rightarrow{\sim}B\)”is invalid. Yet we can get the latter from the former by CP.

Why does CP fail on this conception of conditionals? After all,Supp’s idea is to treat the antecedent of a conditional as anassumption. What is the difference between the roles of apremise, and of the antecedent of a conditional in the conclusion?

The antecedent of the conditional is indeed treated as an assumption.On this conception of validity, the premises are not, primarily,treated as assumptions. We also make inferences from beliefs,including beliefs which are less than certain. Indeed, it is notimmediately clear what it would be to treat a conditional, construedaccording to Supp, as an assumption: to assume something, asordinarily understood, is to assume that it is true; and conditionalsare not being construed as ordinary statement of fact. But we couldapproximate the idea of taking the premises as assumptions, bytreating them, hypothetically, as certainties. Treating the premisesthus would be to require of a valid argument that it preservecertainty: that there must be no probability distributions in whichall the premises (conditional or otherwise) are assigned probability 1and the conclusion is assigned probability less than 1. Call this thecertainty-preservation principle (CPP).

The conception of validity we have been using (PPP) takes as centralthe fact that premises may be accepted with degrees of confidence lessthan certainty. Now, anything which satisfies PPP satisfies CPP. Andfor argument involving only factual propositions, the converse is alsotrue: the same class of arguments necessarily preserves truth,necessarily preserves certainty and necessarily preserves probabilityin the sense of PPP. But arguments involving conditionals can satisfyCPP without satisfying PPP. The invalid argument forms above dopreserve certainty: if you assign probability 1 to the premises, thenyou are constrained to assign probability 1 to the conclusion (in allprobability distributions in which the antecedent of any conditionalgets non-zero probability). But they do not preserve high probability.They do not satisfy PPP. If at least one premise falls short ofcertainty by however small an amount, the conclusion can plummet tozero.

The logico-mathematical fact behind this is the difference in logicalpowers between “All” and “Almost all”. If all\(A\)-worlds are \(B\)-worlds (and there are some \(C \amp A\)-worlds)then all \(C \amp A\)-worlds are \(B\)-worlds. But we can have: almostall \(A\)-worlds are \(B\)-worlds but no \(C \amp A\)-world is a\(B\)-world. If all \(A\)-worlds are \(B\)-worlds and all \(B\)-worldsare \(C\)-worlds, then all \(A\)-worlds are \(C\)-worlds. But we canhave: all \(A\)-worlds are \(B\)-worlds, almost all \(B\)-worlds are\(C\)-worlds, yet no \(A\)-world is a \(C\)-world; just as we canhave, all kiwis are birds, almost all birds fly, but no kiwiflies.

Someone might react as follows: “All I want of a valid argumentis that it preserve certainty. I’m not bothered if an argumentcan have premises close to certain and a conclusion far from certain,as long as the conclusion is certain when the premises arecertain”.

Wecould use the word “valid” in such a way thatan argument is valid provided it preserves certainty. If our interestin logic is confined to its application to mathematics or other apriori matters, that is fine. Further, when our arguments do notcontain conditionals, if we have certainty-preservation,probability-preservation comes free. But if we use conditionals whenarguing about contingent matters, then great caution will be required.Unless we are 100% certain of the premises, the arguments above whichare invalid on Adams’s criterion guarantee nothing about whatyou are entitled to think about the conclusion. The line between 100%certainty and something very close is hard to make out: it’s notclear how you tell which side of it you are on. The epistemicallycautious might admit that they are never, or only very rarely, 100%certain of contingent conditionals. So it would be useful to haveanother category of argument, the “super-valid”, whichpreserves high probability as well as certainty. Adams has shown uswhich arguments (on Supp’s reading of “if”) aresuper-valid.

Continuing to restrict our attention to the case where the antecedenthas non-zero probability, this argument-form preserves certainty:A\(\supset\)B; soA\(\Rightarrow\)B. Theconverse inference is uncontroversial. So if we were just concernedwith certainty preservation, Hook and Supp would be equivalent. Butthey are far from equivalent for uncertain beliefs: the former can bearbitrarily close to 1 while the latter is 0.

4. Truth Conditions Revisited

4.1 Nearest Possible Worlds

Adams’s theory of validity emerged in the mid-1960s.“Nearest possible worlds” theories were not yet inevidence. Nor was Lewis’s result that conditional probabilitiesare not probabilities of the truth of a proposition. (Adams expressedscepticism about truth conditions for conditionals, but the questionwas still open.) Stalnaker’s (1968) semantics for conditionalswas an attempt to provide truth conditions which were compatible withRamsey’s and Adams’s thesis about conditional belief. (Seealso Stalnaker (1970), where the probabilistic aspects of his proposalare developed.) That is, he sought truth conditions for a proposition\(A\gt B\) (his notation) such that \(\bp(A\gt B)\) must equal \(\bp_A(B)\):

Now that we have found an answer to the question, “How do wedecide whether or not we believe a conditional statement?”[Ramsey’s and Adams’s answer] the problem is to make thetransition from belief conditions to truth conditions; … . Theconcept of apossible world is just what we need to make thetransition, since a possible world is the ontological analogue of astock of hypothetical beliefs. The following … is a firstapproximation to the account I shall propose: Consider a possibleworld in which \(A\) is true and otherwise differs minimally from theactual world.“If \(A\), then \(B\)” is true (false)just in case \(B\) is true (false) in that possible world. (1968,pp. 33–4)

If an argument is necessarily truth-preserving, the improbability ofits conclusion cannot exceed the sum of the improbabilities of thepremises. The latter was the criterion Adams used in constructing hislogic. So Stalnaker’s logic for conditionals must agree withAdams’s over their common domain. And it does. The argumentforms we showed to be invalid in Adams’s logic (§3.2) areinvalid on Stalnaker’s semantics. For instance, the following ispossible: in the nearest possible world in which you strike the match,it lights; in the nearest world in which you dip the match in waterand strike it, it doesn’t light. So Strengthening fails. (By“nearest world in which …” I mean the possibleworld which is minimally different from the actual world in which… .)

Conditional Proof fails for Stalnaker’s semantics. “\(A\)or \(B\); \({\sim}A\); so \(B\)” is of course valid. But (*)“\(A\) or \(B\), therefore \({\sim}A\gt B\)” is not: itcan be true that Ann or Mary cooked the dinner (for Ann cooked it);yet false that in the nearest world to the actual world in which Anndid not cook it, Mary cooked it.

Stalnaker (1975) argued that although (*) is invalid, it isnevertheless a “reasonable inference” when “\(A\) or\(B\)” is assertable, that is, in a context in which \({\sim}A\amp{\sim}B\) has been ruled out but \({\sim}A \amp B\) and \(A\amp{\sim}B\) remain open possibilities.

Stalnaker’s semantics uses a “selection function”,F, which selects, for any proposition \(A\) and any world \(w\), aworld, \(w'\), the nearest (most similar) world to \(w\) at which\(A\) is true. “If \(A, B\)” is true at \(w\) iff \(B\) istrue at F\((A, w)\), i.e. at \(w'\). “If \(A, B\)” is truesimpliciter iff \(B\) is true at the nearest \(A\)-world to the actualworld. (However, we do not know which world is the actualworld—there are many candidates compatible with our knowledge.To be sure that if \(A, B\), we need to be sure that whichever world\(w\) is a candidate for actuality, \(B\) is true at the nearest\(A\)-world to \(w\).) If \(A\) is true, the nearest \(A\)-world tothe actual world is the actual world itself, so in this case “If\(A, B\)” is true iff \(B\) is also true. The selection functiondoes substantive work only when \(A\) is false.

Stalnaker’s theory is intended to apply to counterfactuals andindicative conditionals alike, but in the case of indicativeconditionals, he claims, the selection function is subject to apragmatic constraint, set in the framework of the dynamics ofconversation. At any stage in a conversation, many things are takenfor granted by speaker and hearer, i.e. many possibilities are takenas already ruled out. The remaining possibilities are live. He callsthe set of worlds which are not ruled out — the livepossibilities — the context set. For indicative conditionals,antecedents are typically live possibilities, and we focus on thatcase. The pragmatic constraint for indicative conditionals says thatif the antecedent \(A\) is compatible with the context set (i.e. trueat some worlds in the context set) then for any world \(w\) in thecontext set, the nearest \(A\)-world to \(w\) — i.e. the worldpicked out by the selection function — is also a member of thecontext set. Roughly, if \(A\) is a live possibility (i.e. not alreadyruled out), then for any world \(w\) which is a live possibility, thenearest \(A\)-world to \(w\) is also a live possibility. Or: thingswhich are taken to be epistemically possible count as closer toactuality than things which are not.

The proposition expressed by “If \(A, B\)” is the set ofworlds \(w\) such that the nearest \(A\)-world to \(w\) is a\(B\)-world. The ordering of worlds, by the pragmatic constraint,depends on the conversational setting. As different possibilities arelive in different conversational settings, a different proposition maybe expressed by “If \(A, B\)” in different conversationalsettings. Thus, the truth-conditions of conditionals arecontext-dependent, depending on which possibilities the speaker andhearer have ruled out.

Let us transpose this to the one-person case: I am talking to myself,i.e. thinking — considering whether if \(A, B\). The context setis the set of worlds compatible with what I take for granted, i.e. theset of worlds not ruled out, i.e. the set of worlds which areepistemically possible for me. Let \(A\) be epistemically possible forme. Then the pragmatic constraint requires that for any world in thecontext set, the nearest \(A\)-world to it is also in the context set.Provided you and I have different bodies of information, theproposition I am considering when I consider whether if \(A, B\) maywell differ from the proposition you would express in the same words:the constraints on nearness differ; worlds which are near for me maynot be near for you.

This enables Stalnaker to avoid the argument againstnon-truth-functional truth conditions given in §2.2. The argumentwas as follows. There are six incompatible logically possiblecombinations of truth values for \(A, B\) and \({\sim}A\gt B\). Westart off with no firm beliefs about which obtains. Now we eliminatejust \({\sim}A \amp{\sim}B\), i.e. establish \(A\) or \(B\). Thatleaves five remaining possibilities, including two in which“\({\sim}A\gt B\)” is false. So we can’t be certainthat \({\sim}A\gt B\) (whereas, intuitively, one can be certain of theconditional in these circumstances). Stalnaker replies: wecan’t, indeed, be certain that the proposition we were wonderingabout earlier is true. But we are now in a new context: \({\sim}A\amp{\sim}B\)-worlds have been ruled out (but \({\sim}A \ampB\)-worlds remain). We now express a different proposition by“\({\sim}A\gt B\)”, with different truth conditions,governed by a new nearness relation. As all our live\({\sim}A\)-worlds are \(B\)-worlds (none are \({\sim}B\)-worlds), weknow that the new proposition is true.

This sensitivity of the proposition expressed by “If \(A,B\)” to what is taken for granted by speaker and hearer, or tothe epistemic state of the thinker, is somewhat unnatural. One usuallydistinguishes between the content of what is said and the differentepistemic attitudes one may take to that same content. Someoneconjectures that if Ann isn’t home, Bob is. We are entirelyagnostic about this. Then we discover that at least one of them is athome (nothing stronger). We now accept the conditional. It seems morenatural to say that we now have a different attitude to the sameconditional thought, that \(B\) on the supposition that \({\sim}A\).It does not seem that the content of our conditional thought haschanged. And if there are conditional propositions, it seems morenatural to say that we now take to be true what we were previouslywondering about. There does not seem to be independent motivation forthinking the content of the proposition has changed.

Also, Stalnaker’s argument is restricted to the special casewhere we take the \({\sim}A \amp{\sim}B\)-possibilities to be ruledout. Consider a case when, starting out agnostic, we become close tocertain, but not quite certain, that \(A\) or \(B\) — say webecome about 95% certain that \(A\) or \(B\), and are about 50%certain that \(A\). According to Supp, we are entitled to be quiteclose to certain that if \({\sim}A, B\) — 90% certain in fact.(If \(\bp(A \text{ or } B) = 95\)% and \(\bp(A) = 50\)%, then\(\bp({\sim}A \amp B) = 45\)%. Now \(\bp({\sim}A \amp{\sim}B) = 5\)%.So, on the assumption that \({\sim}A\), it’s 45:5, or 9:1, that\(B\).) In this case, no additional possibilities have been ruled out.There are \({\sim}A \amp{\sim}B\)-worlds as well as \({\sim}A \ampB\)-worlds which are permissible candidates for being nearest.Stalnaker has not told us why we should think it likely, in this case,that the nearest \({\sim}A\)-world is a \(B\)-world.

Uncertain conditional judgements create difficulties for allpropositional theories. As we have seen, it is easy to constructprobabilistic counterexamples to Hook’s theory; and it is easyto do so for the variant of Stalnaker’s theory according towhich “If \(A, B\)” is true iff \(B\) is true atall nearest \(A\)-worlds (as Lewis (1973) holds forcounterfactuals). (It is very close to certain that if you toss thecoin ten times, you will get at least one head; but it is certainlyfalse that the consequent is true atall nearestantecedent-worlds.) It is rather harder for Stalnaker’s theory,because nearness is so volatile, and also because it is not fullyspecified. But here is a putative counterexample: the short straws.(An example of this type I learned from James Studd.)

You are to pick a straw from a collection of 100 straws. From theangle you see them—end on—they all look the same; and theyare the same, except for length. 90 are of length 10 cm, 1 is 11 cm,and 9 are 20 cm. Consider this conditional, about the straw that willbe picked:

(*) If it’s over 10 cm, it’s less than 15 cm.

Intuitively, (*) is 10% likely: of those over 10 cm, one is under 15cm and nine are not. But on Stalnaker’s theory, (*) appears tobe 91% likely: it’s 90% likely that it is not over 10 cm, inwhich case, in the world most similar to the actual world in which itis over 10 cm, it is 11 cm, i.e. less than 15 cm. And we add another1% for the case in which it is 11 cm, hence under 15 cm.

(The point is simpler for the counterfactual: “If it had beenover 10 cm, it would have been less than 15 cm”: judging bysimilarity to the actual world, this seems true; but intuitively it isonly 10% likely.)

The example casts doubt on whether any notion of similarity, orminimal difference from the actual world, is the right notion forunderstanding conditionals, as opposed to taking a probabilitydistribution over the various possible antecedent worlds.

There is also the question, for Stalnaker, of the uniquenessassumption—that there is a unique closest antecedent-world.Stalnaker (1981, pp. 87–91) discusses this, and proposes to usethe machinery of supervaluations when there is no unique nearestworld: the conditional is true iff true whichever of the candidatesfor nearest the selection function selects, false iff false for allsuch selections, otherwise it is indeterminate—neither true norfalse. As the uniqueness assumption often fails, a great manyconditionals will just get the verdict indeterminate. For instance, Iam considering whether, (*) if I pick a red ball, it will have a blackspot. 90% of the red balls have black spots. Merely to be told that(*) is indeterminate, is less helpful than being told it is 90%likely.

Does making the proposition expressed by the conditionalcontext-dependent escape Lewis’s result that a conditionalprobability is not the probability of the truth of any proposition?Lewis showed that there is no proposition \(A*B\) such that in everybelief state \(\bp(A*B) = \bp_A (B)\). He did not rule out that inevery belief state there is some proposition or other, \(A*B\), suchthat \(\bp(A*B) = \bp_A (B)\). However, in the wake of Lewis,Stalnaker himself proved a stronger result, for his conditionalconnective: the equation \(\bp(A\gt B) = \bp_A (B)\) cannot hold forall propositions \(A, B\) in a single belief state. If it holds for\(A\) and \(B\), we can find two other propositions, \(C\) and D(truth-functional compounds of \(A, B\) and \(A\gt B)\) for which,demonstrably, it does not hold. (See Stalnaker’s letter to vanFraassen published in van Fraassen (1976, pp. 303–4), Gibbard(1981, pp. 219–20), and Edgington (1995, pp. 276–8).

It was Gibbard (1981, pp. 231–4) who showed just how sensitiveto epistemic situations Stalnaker’s truth conditions would be.Later (1984, ch. 6), reacting to Gibbard, Stalnaker seemed moreambivalent about whether conditional judgements express propositions.But he still takes his original theory to be a serious candidate(Stalnaker 2005, 2019), and this remains an influential theory. Hiswork has inspired others to develop related theories:context-dependent theories are currently popular; the fact thatStalnaker kept the probabilistic considerations aside in his 1968paper led others to develop the Ramsey Test for all-or-nothingbeliefs—see e.g. Gärdenfors (1986); another close relativeof Stalnaker’s semantics, due to Richard Bradley, is discussedbelow in §5, and some other similar theories are brieflymentioned there.

4.2 A Special Assertability Condition

Frank Jackson holds that “If \(A, B\)” has the truthconditions of “\(A \supset B\)”, i.e. “\({\sim}A\)or \(B\)”; but it is part of its meaning that it is governed bya special rule of assertability. “If” is assimilated towords like “but”, “nevertheless” and“even”. “\(A\) but \(B\)” has the same truthconditions as “\(A\) and \(B\)”, yet they differ inmeaning: “but” is used to signal a contrast between \(A\)and \(B\). When \(A\) and \(B\) are true and the contrast is lacking,“\(A\) but \(B\)” is true but inappropriate. Likewise,“Even John can understand this proof” is true when Johncan understand this proof, but inappropriate when John is aworld-class logician.

According to Jackson, in asserting “If \(A, B\)” thespeaker expresses his belief that \(A \supset B\), and also indicatesthat this belief is “robust” with respect to theantecedent \(A\). In Jackson’s early work (1979, 1980)“robustness” was explained thus: the speaker would notabandon his belief that \(A \supset B\) if he were to learn that\(A\). This, it was claimed, amounted to the speaker’s having ahigh probability for \(A \supset B\) given \(A\), i.e. for\(({\sim}A\) or \(B)\) given \(A\), which is just to have a highprobability for \(B\) given \(A\). Thus, assertability goes byconditional probability. Robustness was meant to ensure that anassertable conditional is fit for modus ponens. Robustness is notsatisfied if you believe \(A \supset B\) solely on the grounds that\({\sim}A\). Then, if you discover that \(A\), you will abandon yourbelief in \(A \supset B\) rather than conclude that \(B\).

Jackson came to realise, however, that there are assertableconditionals which one would not continue to believe if one learnedthe antecedent. I say “If Reagan worked for the KGB, I’llnever find out” (Lewis’s example (1986, p. 155)). Myconditional probability for consequent given antecedent is high. Butif I were to discover that the antecedent is true, I would abandon theconditional belief, rather than conclude that I will never find outthat the antecedent is true. So, in Jackson’s later work (1987),robustness with respect to \(A\) is simply defined as \(\bp_A (A\supset B)\) being high, which is trivially equivalent to \(\bp_A(B)\) being high. In most cases, though, the earlier explanation willhold good.

What do we need the truth-functional truth conditions for? Do theyexplain the meaning of compounds of conditionals? According toJackson, they do not (1987, p. 129). We know what “\(A \supsetB\)” means, as a constituent in complex sentences. But“\(A \supset B\)” does not mean the same as “If \(A,B\)”. The latter has a special assertability condition. And histheory has no implications about what, if anything, “if \(A,B\)” means when it occurs, unasserted, as a constituent in alonger sentence.

(Here his analogy with “but” etc. fails. “But”can occur in unasserted clauses: “Either he arrived on time butdidn’t wait for us, or he never arrived at all” (see Woods(1997, p. 61)). It also occurs in questions and commands: “Shutthe door but leave the window open”. “Does anyone wanteggs but no ham?”. “But” means “and incontrast”. Its meaning is not given by an “assertabilitycondition”.)

Do the truth-functional truth conditions explain the validity ofarguments involving conditionals? Not in a way that accords well withintuition, we have seen. Jackson claims that our intuitions are atfault here: we confuse preservation of truth and preservation ofassertability (1987, pp. 50–1).

Nor is there any direct evidence for Jackson’s theory. Nobodywho thinks the Republicans won’t win treats “If theRepublicans win, they will double income tax” as inappropriatebut probably true, in the same category as “Even Gödelunderstood truth-functional logic”. Jackson is aware of this. Heseems to advocate an error theory of conditionals: ordinary linguisticbehaviour fits the false theory that there is a proposition \(A*B\)such that \(\bp(A*B) = \bp_A (B)\) (1987, pp. 39–40). If this ishis view, he cannot hold that his own theory is a psychologicallyaccurate account of what people do when they use conditionals. Perhapsit is an account of how weshould use conditionals, and wouldif we were free from error: weshould accept that “Ifthe Republicans win they will double income tax” is probablytrue when it is probable that the Republicans won’t win. Wouldwe gain anything from following this prescription? It is hard to seethat we would: we would deprive ourselves of the ability todiscriminate between believable and unbelievable conditionals whoseantecedents we think false.

For Jackson’s more recent thoughts on conditionals see hispostscript (1998, pp. 51–54). See also Edgington (2009) andJackson’s reply (2009, pp. 463–6).

4.3 Restrictors and the Strict Conditional

Angelika Kratzer’s work on conditionals has been veryinfluential in linguistics, and also in philosophy. Her articles haveappeared, reworked, as a book,Modals and Conditionals(2012). Kratzer’s inspiration comes from a paper by David Lewis,“Adverbs of Quantification” (1975). Lewis’s paper isabout the analysis of sentences containing adverbs such asalways,never,usually,often,seldom …, sentences such as “The fog usuallylifts before noon here” and “Caesar seldom awoke beforedawn”. After considering and rejecting some alternatives, Lewisintroduces “restriction by if-clauses”: he proposes thatthere is a use of if-clauses whose function is to restrict the rangeof cases to which the operator or quantifier applies. First paraphrasethe sentences: “Usually if there is fog here, it lifts beforenoon.” “Seldom if Caesar awoke, it was before dawn.”(Lewis’s target sentences do not have “if” in theirsurface structure, but they could have had: the theory also applies tosentences like “Usually, if Mary visits, she brings herdog”.) The “if” restricts the “usually”to the occurrences of fog here, or of Mary’s visits, and the“seldom” to Caesar’s awakenings. These sentences arenot to be construed as applying an adverb to a conditionalproposition. The adverb applies to the main clause, its scoperestricted by the if-clause. Thus Lewis:

[T]heif of our restrictive if-clauses should not be regardedas a sentence connective. It has no meaning apart from the adverb itrestricts. Theif inalways if …,sometimes if …, and the rest is on a par with thenon-connectiveand inbetween … and …,with the non-connectiveor inwhether … or…, or with the non-connectiveif intheprobability that … if…. It serves merely to mark anargument-place in a polyadic construction. (Lewis 1975 reprinted inLewis 1998 pp. 14–15)

Lewis’s final example is particularly interesting, especiallybecause this paper was written at much the same time as his proof thatconditional probabilities are not to be construed as probabilities ofconditional propositions.

Lewis has three different accounts of “if”: he followsJackson in claiming that the “if” of indicativeconditionals is the truth-functional “if”, with a specialrule of assertability (see Lewis 1986 pp. 152–6); there is hisfamous account of the “if” of counterfactual conditionals(Lewis 1973); and there is this use of “if” as arestrictor.

Kratzer’s idea is that this last account of “if” asa restrictor should be applied to all conditionals. Consider firstconditionals which contain a modal term: “If it’s not inthe kitchen it must be in the bathroom/might be in the bathroom/isprobably in the bathroom”. By analogy with Lewis, she arguesthat these are not to be construed as attaching a modal term to aconditional proposition; rather, they are to be construed as attachinga modal term to the main clause, the scope of the modal term beingrestricted by the conditional clause.

But what of a simple conditional which does not contain a modaloperator, such as “If it’s not in the kitchen it is in thebathroom” — what Kratzer calls the “bareconditional”? Here is her famous remark:

The history of the conditional is the history of a syntactic mistake.There is no two-placeif … then connective in thelogical forms of natural languages.If-clauses are devicesfor restricting the domains of operators.Bare conditionals haveunpronounced modal operators [my emphasis]. EpistemicMUST is one option. (Kratzer (1986), quoted from Kratzer(2012) p. 106)

There is much in common between the restrictor-view of conditionalsand the suppositional view. A supposition also restricts one’sclaim to the case in which the antecedent is true. The strength ofyour conditional belief is measured by how probable you judge theconsequent, on the assumption that the antecedent is satisfied; andthis is not the same as thinking a conditional proposition is probablytrue. Recall Lewis’s remark about “the probability that… if …”. Kratzer’s treatment of modalconditionals may be seen as a generalization to other modalities ofthis treatment of “Probably, if \(A\), \(B\)”.

However, Kratzer’s treatment of the “bareconditional” is controversial: at the level of semanticstructure, there really are no such things — apparent bareconditionals contain an “unpronounced modal operator”. Ifthe modal operator is an epistemic “must”, as shesuggests, bare conditionals are a species of strict conditional— something like “all live \(A\)-possibilities are\(B\)-possibilities”. Here are some problems.

First, the suppositional theory, like propositional theories, holdsthat different people, or the same person at different times, can holddifferent epistemic attitudes to the same conditional thought –that Jane will accept if she is offered the job, etc., i.e., can becloser or further from certain about it. It is hard to see how thatcan be the case if there really are no bare conditionals at the levelof semantic structure.

Second, it is hard to see that there are any alternatives to hersuggestion that the unpronouced modal operator in a bare conditionalis the epistemic “must”. It wouldn’t do to interpreta bare “if \(A\), \(B\)” as “Probably, if \(A\),\(B\)”. For the former is incompatible with \(A \amp {\sim}B\),and the latter is not.

Third, sticking with the epistemic “must” as theunpronounced modal operator leads to difficulties. Consider thefollowing exchange:

A: Will Jane accept if she is offered the job?
B: No, it is certainly not the case that she will accept if offeredthe job [for not all offer-possibilities are accept-possibilities].But she might well accept if she is offered the job.

B’s remark sounds contradictory. (Stalnaker (1981 p. 100) madeessentially the same point, about counterfactuals, comparing his viewwith Lewis’s: Lewis’s counterfactuals go plain false whennotall closest \(A\)-worlds are \(B\)-worlds, although someare.) Thus, while the restrictor view has some plausibility, itstreatment of the “bare conditional” as a modalisedproposition is problematic.

Perhaps there is room for a compromise. The suppositional theory is anaccount of conditional thinking: you suppose \(A\), and consider \(B\)under that supposition. Kratzer (being a linguist) is more interestedin conditional speech. It is compatible with the suppositional theorythat you should only assert or assent to the bare “if \(A\),\(B\)” when all salient \(A\)-possibilities are\(B\)-possibilities. (The standards for salience may depend on thecontext.) Otherwise, you should qualify your remark, perhaps with“probably” or “I think”. On this view ofassertion, it is fine to take an assertion of the bare conditional asgoverned by epistemic necessity. What is not fine is to take anassertion which breaks this rule to be plain false--for it still mightbe highly probable. I say “If you buy the lottery ticket, youwill lose”. I should have qualified my assertion, butnevertheless it has a high probability. Similarly, the unconditionalassertion (after the ticket has been bought), “You willlose”, should have been qualified, but it is very probably true.

Other philosophers have also defended the view that indicativeconditionals are context-dependent strict conditionals, withoutadopting Kratzer’s restrictor view. According to Anthony Gillies(2009), a context determines a set of possibilities compatible withthe relevant information in the context. “If \(A, C\)” istrue at a context iff all relevant \(A\)-possibilities are\(C\)-possibilities, false otherwise. William Lycan (2001), similarly,claims that “If \(A, C\)” is true iff all real andrelevant \(A\)-events are \(C\)-events. Context-dependent strictconditionals are also defended by Daniel Rothschild (2013, 2015). Thedifficulty mentioned above remains: these theories often make theconditional certainly false, when intuitively it is may be highlyprobable.

4.4 Heuristics and Semantics

Timothy Williamson (2020) accepts that the suppositionalprocedure—suppose the antecedent, and on that basis come to ajudgement about the consequent—is our fundamental, primarymethod of conditional judgement, an essential part of our cognitiveequipment. As uncertainty is often involved, he accepts that it isappropriate to theorize about this process in terms of conditionalprobabilities. This is what he calls theheuristics ofconditionals (not to be confused with semantics). Heuristics are partof our cognitive and psychological apparatus, “fast andfrugal”, immensely useful and valuable, but typically imperfect.In this case, they lead to logical problems, indeed inconsistencies,he argues (ch. 3), broadly of the same kind as Lewis’s (1976)and other arguments that a conditional probability does not measurethe probability of truth of a proposition. (See section 3.1 above; seealso Edgington (2021) for a discussion of Williamson).

Also, he argues, this primary heuristic is sometimes at odds with asecondary heuristic—acquiring conditional beliefs by testimony.Often this is unproblematic, but, as Gibbard (1981) showed, two peoplewith different background knowledge can flawlessly come to oppositejudgements about whether if \(A, B\). They then convey theirjudgements to a third person, who trusts them, but who cannot acceptboth judgements, using the suppositional procedure.

Here is Williamson’s example: there has been an accident at adodgy nuclear power plant. Several warning lights are connected to adetector behind the nuclear core. When the detector is working anddetects overheating, each light is red. When the detector is workingand does not detect overheating, each light is green. When thedetector is not working, each light is red or green at random,independently of the others. A competent engineer, East, sees only theeast light, which is red, and says

If the detector is working, the core is overheating.

Another competent engineer, West, not in contact with East, sees onlythe west light, which is green, and says

If the detector is working, the core is not overheating.

Each sends a report of their judgement to the controller, who truststhem both, But the controller cannot take over both conditionaljudgements, using the Suppositional Procedure: supposing that thedetector is working, she cannot give a high value to both “thecore is overheating” and “the core is notoverheating”.

This is a glitch in the suppositional procedure, as far as testimonyis concerned, but it is perfectly understandable why it arises, andwhat one should do about it. When a reliable and trustworthy persontells you a straight factual proposition, and you believe them, thereis no relevant ignorance involved. If someone tells you a conditional,there is virtually always relevant ignorance – why else wouldthey have said “if”? And it can happen, as here, that twopeople with different combinations of knowledge and ignorance cancome, flawlessly, to opposite conclusions. The saving grace is thatthe suppositional conditional entails the material conditional: Eastis committed to “The detector is working \(\supset\) the core isoverheating”, i.e. either the detector is not working or thecore is overheating. West is committed to “The detector isworking \(\supset\) the core is not overheating,” i.e., eitherthe detector is not working or the core is not overheating. From bothjudgement it follows that the detector is not working.

It is worth stressing that the above is an unusual case. Itdoesn’t follow that in general, it is only the materialconditional that is passed on by testimony. Suppose a trusted medicalexpert tells you that it is unlikely that if you get the virus, youwill be seriously ill. You do not infer from this that you are likelyto get the virus, as you should if you read the conditional as amaterial conditional (for if the antecedent were unlikely, thematerial conditional would be probably true). If the expert went on tosay “And anyway, it is unlikely that you will get thevirus”, on the material-conditional view, she would havecontradicted herself.

The case of East and West is one of the reasons Williamson gives forhis claim that thesemantics of the conditional is besttreated as the truth function. The semantics is not something whichevery speaker knows, or is readily available to them. We do not learnto use “if” via the truth table. And we know that nosemantic theory of the conditional isobviously correct! Theprobabilities generated by the semantics are typically higher, andnever lower than the probabilities generated by our basicsuppositional procedure for assessing conditionals. Nevertheless, heargues, the truth-functional semantics does the best job ofrationalizing our overall practice.

Williamson gives other examples of useful but imperfect heuristics.One concerns vagueness, and the so-called “toleranceprinciples”, such as “If \(n\) seconds after noon isnoonish, \(n{+}1\) seconds after noon is noonish”. That is auseful rule of thumb, though not all of its instances can be true. Forthe epistemicist, one instance is false, but we don’t knowwhich. On other views, all are at least very close to clearly true,but not all are clearly true. Another example is the problem the LiarParadox presents for the principle that it is true that \(P\) if andonly if \(P\). In both cases, the exceptions are rare and can normallybe overlooked. He also mentions the heuristics involved in perceptualjudgements, normally very reliable, which sometimes lead us astray.“Humans predictably resort to fast and frugal heuristics,reliable enough under normal conditions, but not perfectlyreliable” he says, (p. 265).

However, in the case of uncertain conditionals, it is not easy toaccept that the heuristics are “reliable enough under normalconditions”, when combined with the truth-functional semantics.The following equation shows how the two ways of assessingconditionals are related.

When the conditional is certain, the suppositional procedure and thetruth function agree. They also agree in the relatively uninterestingcase in which the antecedent is certain. In all other cases, thetruth-functional conditional gets a higher value than thesuppositional conditional, and the difference between them can bearbitrarily large. Many examples have already been given (see§§2.3, 2.5, 3.1, 3.2). Another simple example: how likely isit that if the die lands an even number, it lands 6? Most people,rightly in my view, will answer 1/3. If the conditional is a materialimplication, the answer is 2/3: if it lands 1, 3, 5 or 6, theconditional is true. As already mentioned, all conditionals whoseantecedents are improbable, are probable as judged by the truthfunction.

Another example: we are planning a trip in a couple of days, andwondering whether

(*) if it snows the night before, the road will be impassable.

The probability of snow is around 0.5; and we reckon (on thesuppositional procedure), it’s around 0.2 that the road will beimpassable if it snows. According to the truth function, theconditional gets 0.6. Then, as forecasts are updated, the probabilityof snow decreases. Nothing else changes. On the truth-functionalsemantics, the probability of our conditional goes up as theprobability of snow goes down: when the probability of snow goes downto 0.25, the probability of (*) is 0.8 on the truth-functionalreading, although it remains at 0.2 on the suppositional approach.Williamson says that our suppositional procedure errs “on theside of caution” (p. 104) by generating lower probability valuesthan the truth-functional conditional. It is hard to see why, as theprobability of snow decreases, we become more risk-averse, and makelarger errors on the side of caution.

This example, and the above equation, bring out another big differencebetween \(\bp(A \supset B)\) and \(\bp_A(B)\). The former isultra-sensitive to the probability of \(A\). Keeping \(\bp_A(B)\)constant, any change to \(\bp(A)\) changes \(\bp(A \supset B)\).Conditional probabilities, on the other hand, typically do not changewhen the only relevant change is a change to \(\bp(A)\). (Rareapparent counterexamples usually involve some subtle unnoticedadditional change.) That is why they are so useful in accounts of howbeliefs should change when new evidence is required –conditionalization and Jeffrey conditionalization (Jeffrey 1985). Andour conditional judgements also seem to have this property of beingunchanged when all that changes is the probability of the antecedent.In contrast, \(\bp(A \supset B)\), while well defined, appears to be aform of judgement of little use.

5. Compounds of Conditionals: Problems for the Suppositional Theory, and Some Solutions

A common complaint against Supp’s theory is that if conditionalsdo not express propositions with truth conditions, we have no accountof the behaviour of compound sentences with conditionals as parts (seee.g. Lewis (1976, p. 142)). Probability theory is no help: conditionalprobabilities never occur inside wider constructions. For instance,probability theory does not have formulas which represent theprobability of (\(B\) on the supposition that \(A\), and \(D\) on thesupposition that \(C\)), or \(\bp({}_A B \amp {}_C D)\). However, notheory has an intuitively adequate account of compounds ofconditionals: we saw in §2.4 that there are compounds which Hookgets wrong; and compounds which Arrow gets wrong. Grice’s andJackson’s defences of Hook focus on what more is needed tojustify theassertion of a conditional, beyond belief that itis true. This is no help when it occurs, unasserted, as a constituentof a longer sentence, as Jackson accepts. And with negations ofconditionals and conditionals in antecedents, we saw, the problem isreversed: we assert conditionals which we would not believe if weconstrued them truth-functionally.

Some followers of Adams have tried to show that when a sentence with aconditional subsentence is intelligible, it can be paraphrased, atleast in context, by a sentence without a conditional subsentence. Forinstance, they read “It’s not the case that if \(A\),\(B\)” as “If \(A\), it’s not the case that\(B\)”, and “If \(A\), then if \(B\), \(C\)” as“If \(A \amp B\), \(C\)”. They also point out that someconstructions are rarer, and harder to understand, and more peculiar,than would be expected if conditionals had truth conditions andembedded in a standard way. See Appiah (1985, pp. 205–10),Gibbard (1981, pp. 234–8), Edgington (1995, pp. 280–4),Woods (1997, pp. 58–68 and 120–4); see also Jackson (1987,pp. 127–37). (Note that the Lewis-Kratzer strategy (§4.3)also involves paraphrase, so that conditional propositions are notembedded in adverbs and operators.) But it would be better to have asystematic solution to this problem, and there have severalattempts.

Before turning to systematic solutions, let us look at a notoriousexample of Vann McGee’s (1985)—a counterexample to modusponens. Before Reagan’s first election, Reagan was hotfavourite, a second Republican, Anderson, was a complete outsider, andCarter was lagging well behind Reagan. Consider first

1. If a Republican wins and Reagan does not win, then Anderson willwin.

As these are the only two Republicans in the race, (1) isunassailable. Now consider

2. If a Republican wins, then if Reagan does not win, Anderson willwin.

We read (2) as equivalent to (1), hence also unassailable.

Suppose I’m close to certain (say, 90% certain) that Reagan willwin. Hence I am close to certain that

3. A Republican will win.

But I don’t believe

4. If Reagan does not win, Anderson will win.

I’m less than 1% certain that (4). On the contrary, I believethat if Reagan doesn’t win, Carter will win. As these opinionsseem sensible, we have a prima facie counterexample to modus ponens: Iaccept (2) and (3), but reject (4). Truth conditions or not, validarguments obey the probability-preservation principle. I’m 100%certain that (2), 90% certain that (3), but less than 1% certain that(4).

Hook saves modus ponens by claiming that I must accept (4). For Hook,(4) is equivalent to “Either Reagan will win or Anderson willwin”. As I’m 90% certain that Reagan will win, I mustaccept this disjunction, and hence accept (4). Hook’s reading of(4) is, of course, implausible.

Arrow saves modus ponens by claiming that, although (1) is certain,(2) is not equivalent to (1), and (2) is almost certainly false. ForStalnaker,

5. If a Republican wins, then if Reagan doesn’t win, Carterwill win

is true. To assess (5), we need to consider the nearest world in whicha Republican wins (call it \(w)\), and ask whether the conditionalconsequent is true at \(w\). At \(w\), almost certainly, it is Reaganwho wins. We need now to consider the nearest world to \(w\) in whichReagan does not win. Call it \(w'\). In \(w'\), almost certainly,Carter wins.

Stalnaker’s reading of (2) is implausible; intuitively, weaccept (2) as equivalent to (1), and do not accept (5).

Supp can save modus ponens by denying that the argument is really ofthat form. “\(A\Rightarrow B\); \(A\); so \(B\)” isdemonstrably valid when \(A\) and \(B\) are propositions. Forinstance, if \(\bp(A) = 90\)% and \(\bp_A (B) = 90\)% the lowestpossible value for \(\bp(B)\) is 81%. The “consequent” of(2), “If Reagan doesn’t win, Anderson will win”, isnot a proposition. The argument is really of the form “If \(A\amp B\), then \(C\); \(A\); so if \(B\) then \(C\)”. Thisargument form is invalid (Supp and Stalnaker agree). It is one of themany argument forms which do preserve certainty, but do not preservehigh probability. Take the case where \(C = A\), and we have “If\(A \amp B\) then \(A\); \(A\); so if \(B\) then \(A\)”. Thefirst premise is a tautology and falls out as redundant; and we areleft with “\(A\); so if \(B\) then \(A\)”. We have alreadyseen that this is invalid: I can think it very likely that Sue islecturing right now, without thinking that if she was seriouslyinjured on her way to work, she is lecturing right now.

Compounds of conditionals are a hard problem for everyone. It isdifficult to see why this should be so if conditionals are ordinarypropositions with truth conditions.

Let us turn to systematic attempts to construct a theory of compoundsof conditionals, compatible with the suppositional view. The firstattempt is due to Bruno de Finetti (1936), who, shortly after Ramsey,independently developed a theory of probability as degree of belief,and like Ramsey, saw that conditional probability seemed a goodmeasure of one’s degree of belief in a conditional. To deal withcompounds of conditionals, he proposed a three-valued semantics forthe conditional—true, false, undefined. “If \(A\),\(B\)” is true if \(A \amp B\), false if \(A \amp {\sim}B\), andlacks a truth value—is undefined—if \({\sim}A\). He calledthese semantic entities “conditional events” or“tri-events”. The probability of a conditional is not theprobability of its truth (which is just the probability of \(A \ampB\)), but the probability of its truth given that it is either true orfalse, which is just the conditional probability of \(B\) given \(A\).Then de Finetti gave truth tables, which accommodate compounds ofconditionals. A conjunction is true iff both conjuncts are true, falseiff at least one conjunct is false, otherwise undefined. A disjunctionis true iff at least one disjunct is true, false if both disjuncts arefalse, otherwise undefined. Negation takes true to false, false totrue, undefined to undefined. A conditional with a false or undefinedantecedent is undefined. And the probability of a compound is also theprobability that it is true, given that it is either true or false.For work in this tradition see Milne (1997). See also Belnap (1970)and McDermott (1996).

The idea has a certain appeal. A conditional, if \(A\), \(B\),involves the supposition that \(A\). It tells us nothing about whathappens if \(A\) is false. But there are costs. On this account, tobelieve/assert a conditional is not to believe/assert that it is true.It is no fault in a conditional that it is not true, for it is nofault in a conditional that it has a false antecedent. I say “Ifyou press that button there will be an explosion”. A disaster isavoided, because, fortunately, my remark was not true. One might saythe normative dimension of truth has been lost. We have to give up theequivalence between “If \(A, B\)” and “It is truethat if \(A, B\)”. Even a necessary conditional such as“If \(A \amp B\), then \(A\)” can fail to be true.Validity cannot be preservation of truth, for if it were, “If\(A, B\); so \(A \amp B\)” would be a valid argument.

Further, some of the results this theory delivers for embeddedconditionals are not plausible. Mother says “If it doesn’train tomorrow we’ll go to the beach, and if it rains we’llgo to the cinema”. This rightly inspires confident expectations.But on the present theory, as one or the other conditional has a falseantecedent, the conjunction cannot be true. Yet one or the otherconditional might be false, due to some unlikely contretemps, such asillness, in which case the conjunction is false. So the probabilitythat it is true, given that it is either true or false, is 0. Thechildren are, say, 99% confident that if it doesn’t rain, theygo to the beach, and 99% confident that if it rains, they will go tothe cinema, but, on this account, they should be 0% confident of theconjunction of these two conditionals. (McGee (1989) raises thisobjection to the account.)

A second approach (apparently also due to De Finetti but unpublished)gives values to conditionals as follows: \(1 (=\) true) if \(A \ampB\); \(0 (=\) false) if \(A \amp{\sim}B\); \(\bp_A (B)\) if\({\sim}A\). See van Fraassen (1976), McGee (1989), Jeffrey (1991),Stalnaker and Jeffrey (1994), Sanfilippo et al. (2020). Thus we have abelief-relative three-valued entity. Its probability is its“expected value”. It is suggested that this can be thoughtof as the fair price for a conditional bet which pays 1 if \(A \ampB\), 0 if \(A \amp {\sim}B\), and give your money back if ~A. Forinstance, I’m to pick a ball from a bag. 50% of the balls arered. 80% of the red balls have black spots. Consider “If I picka red ball \((R)\) it will have a black spot \((B)\)”. \(\bp_R(B) = 80\)%. If \(R \amp B\), the conditional gets value 1, if \(R\amp{\sim}B\), it gets semantic value 0. What does it get if\({\sim}R\)? One way of motivating this approach is to treat it as arefinement of Stalnaker’s truth conditions. Is the nearest\(R\)-world a \(B\)-world or not? Well, if I actually don’t picka red ball, there isn’t any difference, in nearness to theactual world, between the worlds in which I do; but 80% of them are\(B\)-worlds. Select an \(R\)-world at random; then it’s 80%likely that it is a \(B\)-world. So “If \(R, B\)” gets 80%if \({\sim}R\). You don’t divide the \({\sim}R\)-worlds intothose in which “If \(R, B\)” is true and those in which itis false. Instead the conditional gets value 80% in all of them. Theexpected value of “If \(R, B\)” is

Ways of handling compounds of conditionals have been proposed on thebasis of these values.

Some of the difficulties of the former account are avoided. Necessaryconditionals like “If \(A \amp B\), then \(A\)” aretrue—get value 1. Conjunctions of the form (if \(A, B)\) &(if \({\sim}A, C)\) do not all get 0. Indeed, conditionals this formalways get the value \(\bp_A (B).\bp_{{\sim}A}(C)\). (The expectedvalue of the conjunction is \(\bp(A \amp B).\bp_{{\sim}A}(C) +\bp({\sim}A \amp C).\bp_A (B)\), which simplifies to \(\bp_A(B).\bp_{{\sim}A}(C)\).) Indeed, the stronger result holds, that ifthe two antecedents are incompatible, the value of the conjunction isthe product of the two conditional probabilities. So our above exampleabout rain, beach and cinema indeed gets a high value.

However, this result for conjunctions of conditionals gives someimplausible results. An example due to Mark Lance (1991) concerns awerewolf, such that it’s 50% likely that it is our area tonight.If it is, it will kill everyone outside. “If John went out, hewas killed” gets 0.5. But “If John went out the back door,he was killed, and if John went out the front door, he waskilled” gets 0.25 on this proposal, whereas it should still get0.5. Another example, due to Richard Bradley (2012): I must pick oneof two urns, only one of which contains the prize. “If I pickthe left urn, I’ll win” gets 0.5. “If I pick theright urn, I’ll win” gets 0.5. On this proposal, “IfI pick the left urn I’ll win, and if I pick the right urn,I’ll win” gets 0.25, whereas it surely deserves to get0.

The flaw in this proposal is that the conditional gets the same valuein every possible situation in which the antecedent is false. In oursecond example, in the “pick left and win” situation,“If pick right, win” deserves 0, not 0.5, despite itsfalse antecedent, and in the “pick left and lose”situation, “If pick right, win” deserves 1, not 0.5. Inthe werewolf example, in the “goes out front door and iskilled” situation, “If back door, killed” deserves1, not 0.5, despite its false antecedent.

This proposal still has defenders, who are attracted to the idea ofthe expected value of a conditional bet, and are prepared tocountenance the above results. See Sanfilippo et al (2020), Cantwell(2022), Gilio et al. (2024).

A third approach is due to Richard Bradley (2012). Bradley’saccount can be seen as a modification of Stalnaker’s (1968)theory discussed in §4.1: consider a possible world in which\(A\) is true, and otherwise differs minimally from the actual world;“If \(A\), then \(B\)” is true just in case \(B\) is truein that possible world. First modification: Bradley abandons thenotion of similarity, or of minimal difference, in favour of aprobability distribution over the candidate \(A\)-worlds. The exampleof the short straws in §4.1 shows that this is a good idea.Second modification: conditionals are not propositions. This is asuppositional theory. Conditionals involve two propositions which playdifferent roles, one a supposition, one a judgement within its scope.They cannot be represented by the set of worlds in which they aretrue. Indeed, conditionals are not ingredients of worlds—theyare cross-world entities. Bradley proposes that the conditional“if \(A\), \(B\)” can be represented by the set ofpairs of worlds, \(\langle w_i, w_j\rangle\) such that, if\(w_i\) is actual, and \(w_j\) is the “potential”\(A\)-world, i.e. the world that would be actual if \(A\) were true,the conditional would be true (because \(B\) is true at \(w_j)\).(Note: Bradley still uses the term “nearest” for the worldthat would be actual if \(A\) were true; but, as there is no orderingof worlds for closeness or similarity, I prefer the term“potential”.) Often, we do not know which world would beactual if \(A\) were true—hence the probability distributionover candidates. Sometimes it might even be indeterminate which worldthat is, but reflection again on the “short straws”example in §4.1 shows that the probabilities are still in order.(Note, the case of indeterminacy is not mentioned by Bradley, but isdiscussed in Edgington (2023).)

Two types of uncertainty, Bradley notes, are involved in assessing aconditional—uncertainty about the facts—about which worldis actual; and uncertainty about what would be the case if somesupposition, which may be false, were true. These combine in a jointprobability distribution over the set of ordered pairs.

He accepts centering: if \(A\) is true, the potential \(A\)-world isthe actual world, and the conditional is true iff \(B\) is true.

Here is a simple model. There are just three possible worlds: at\(w_1\), \(A\) and \(B\) are true; at \(w_2, A\) is true and \(B\) isfalse; at \(w_3, A\) is false. These generate the followingpossibilities for the conditional if \(A\), \(B\).

The probabilities of these four lines sum to 1. [Note: world are beingconstrued as coarse-grained. We divide logical space into a partition,fine enough for the present purpose, and call the members of thepartition “worlds”, as probability theorists oftendo.]

The first two lines are the cases in which the antecedent is true, soin those the potential \(A\)-world is the actual world. If on theother hand \(w_3\) is actual, that does not tell us whether thepotential \(A\)-world is \(w_1\), in which case the conditional istrue, or \(w_2\), in which case the conditional is false. Thus thereare four possible pairs of worlds.

The crucial rule governing this non-propositional entity is this: theprobability of “if \(A\), \(B\)” given \(A\), is the sameas the probability of “if \(A\), \(B\)” given \({\sim}A\);the probability of the conditional is independent of its antecedent.This guarantees that \(\bp(\text{if } A, B)\) is \(\bp_A(B)\).

The non-propositional nature of the conditional is essential here.Suppose we just redescribe the four lines above as four possibleworlds, four ways the world might be, in two of which the conditionalis true—as Stalnaker did. And suppose we start off thinking eachof the four is equally likely. Then we learn \(({\sim}A \amp B)\): thefirst line goes out. We learn nothing other than that. \(\bp({}_A B)\)is now 0. But \(\bp(\text{if } A, B)\) is not 0: the third lineremains a possibility, and we haven’t eliminated that. (Indeed,if probabilities change by conditionalization, the third line now hasprobability 1/3.) In short, no two contingent propositions areprobabilistically independent in all probability distributions. Butthe conditional—not a proposition—is stipulated to beindependent of \(A\). On Bradley’s theory, having learned that\(({\sim}A \amp {\sim}B)\), \(\bp(\text{if } A, B) = 0\); the thirdline gets 0; should \(A\) turn out to be false, it’s false thatif \(A\), \(B\).

(To avoid confusion, let me say something about conditionalization.This notion does not concern the analysis of conditional judgements.It is the thesis that if you learn for certain a proposition \(A\),and nothing else of relevance, your new probability for anyproposition \(B\) should be your old probability for \(B\) given\(A\), \(\bp({}_AB)\). Equivalently, on learning \(A\), You assign 0to the worlds in which \(A\) is false, and “renormalise”:keep the relative probabilities of the worlds in which \(A\) is truethe same as they were before. Bradley’s thesis is that whilethis is fine when \(A\) and \(B\) are propositions, it is not the wayconditional judgements change given new information. Rather, we mustkeep the probability of the conditional given \(A\), equal to itsprobability given \({\sim}A\), and this is incompatible withconditionalization.)

With this machinery, the contents of conjunctions, disjunctions andnegations of conditionals are given in the usual way by intersection,union and complements of the contents of the component sentences. Whena sentence has two conditionals, with two antecedents, such as thoseof the form \((A\Rightarrow B) \amp({\sim}A\Rightarrow C)\), theirsemantics requires not ordered pairs but ordered triples, \(\langlew_i, w_j, w_k\rangle\), such that, if \(w_i\) is actual and \(w_j\) isthe potential \(A\)-world, and \(w_k\) is the potential\({\sim}A\)-world, the conditional is true. As it is perfectly properto give probability 0 to the possibility: “I pick right and win,and in the potential world in which I pick left, I win”, theproblem for the previous proposal is avoided.

This is a remarkable achievement. Probability is probability of truth.Validity is necessary preservation of truth, and so Adams’sprobabilistic criterion of validity is demonstrable. If \(B\) istrue/false at all \(A\)-worlds, “If \(A\), \(B\)” isstraightforwardly true/false. Plenty of others may bestraightforwardly true/false, whether or not we know this. Forinstance, returning to the case of the two urns one and only one ofwhich contains the prize, “If I pick left I’ll win”is true iff the prize is in the left urn, whichever urn I pick). Themany uncertain conditionals come out with the rightprobability—the conditional probability of \(B\) given \(A\).The construction is not easy to work with, but it shows that asystematic account of compounds of conditionals can give plausibleanswers. Returning to Williamson’s distinction between semanticsand heuristics, there will be many heuristics we can use withoutresort to the semantics. Suppose it’s 90% likely that if itdoesn’t rain we will go to the beach, and 90% likely that if itrains we will go to the cinema. Then the conjunction of these twoconditionals must be at least 80% likely (and not more than 90%likely).

It is not ad hoc or unheard-of to claim that some kinds of contentcannot be represented by a set of worlds—the set of worlds inwhich they are true. Some examples: to capture the content ofindexical thoughts using “I” and “now”, weneed the richer notion of a “centred world”—anordered triple of a world, an individual and a time (see Lewis 1979).Gibbard (1990) proposes that the content of a normative judgement canbe represented by a set of ordered pairs \(\langle w, n\rangle\) where\(w\) is a world and \(n\) is a set of norms. Sarah Moss (2018) arguesthat the contents of probability judgements are not propositions butsets of probability spaces. Andrew Bacon (2018) argues that thecontents of vague thoughts cannot be represented by a set of worlds.(He calls them propositions nevertheless—claiming that is averbal issue—but they are not propositions in the sense that isrelevant here.)

There are other recent theories which share many of the features asBradley’s. Bacon (2015) adapts Stalnaker, using probabilities inplace of similarity relations, and makes the conditionalcontext-sensitive to the evidence on which it is based. Paolo Santorio(2022) developed “path semantics” for indicativeconditionals – also a modification of Stalnaker semantics. Hisobject in that paper was to show that this is compatible with theRamsey Test for outright beliefs. In another paper Goldstein andSantorio (2021) construct a probabilistic version of the theory, andargue that it satisfies Ramsey’s thesis: \(\bp(\text{if } A, B)= \bp_A(B)\). Again, conditionals are more complex than classicalpropositions, and conditional degrees of belief do not change byconditionalization. Justin Khoo’s book (2022) is based on theidea that conditionals encode inferential dispositions, but he alsoclaims that they can be represented in a Stalnaker-like semantics,with the same consequences: they are not classical propositions,conditional beliefs do not change by conditionalization, the“Triviality Results” are avoided and Ramsey’s thesisis again maintained. I am unable to discuss these theories in detailhere, but the convergence of these approaches is worthy of note.

6. Other Conditional Speech Acts and Propositional Attitudes

As well as conditional beliefs, there are conditional desires, hopes,fears, etc.. As well as conditional statements, there are conditionalcommands, questions, offers, promises, bets, etc.. “If hecalls” plays the same role in “If he calls, what shall Isay?”, “If he calls, tell him I’m out” and“If he calls, Mary will be pleased”. Which of our theoriesextends to these other kinds of conditional?

One believes that \(B\) to the extent that one thinks \(B\) morelikely than not \(B\); according to Supp, one believes that \(B\) if\(A\) to the extent that one believes that \(B\) under the suppositionthat \(A\), i.e. to the extent that one thinks \(A \amp B\) morelikely than \(A \amp{\sim}B\); and there is no proposition \(X\) suchthat one must believe \(X\) more likely than \({\sim}X\), just to theextent that one believes \(A \amp B\) more likely than \(A\amp{\sim}B\). Conditional desires appear to be like conditionalbeliefs: to desire that \(B\) is to prefer \(B\) to \({\sim}B\); todesire that \(B\) if \(A\) is to prefer \(A \amp B\) to \(A\amp{\sim}B\); there is no proposition \(X\) such that one prefers\(X\) to \({\sim}X\) just to the extent that one prefers \(A \amp B\)to \(A \amp{\sim}B\). I have entered a competition and have a verysmall chance of winning. I express the desire that if I win the prize\((W)\), you tell Fred straight away \((T)\). I prefer \(W \amp T\) to\(W \amp{\sim}T\). I do not necessarily prefer \((W \supset T)\) to\({\sim}(W \supset T)\), i.e. (\({\sim}W\) or \((W \amp T)\)) to \(W\amp{\sim}T\). For I also want to win the prize, and much the mostlikely way for (\({\sim}W\) or \((W \amp T)\)) to be true is that Idon’t win the prize. Nor is my conditional desire satisfied if Idon’t win but in the nearest possible world in which I win, youtell Fred straight away.

If I believe that \(B\) if \(A\), i.e. (according to Supp) think \(A\amp B\) much more likely than \(A \amp{\sim}B\), this puts me in aposition to make a conditional commitment to \(B\): to assert that\(B\), conditionally upon \(A\). If \(A\) is found to be true, myconditional assertion has the force of an assertion of \(B\). If \(A\)is false, there is no proposition that I asserted. I did, however,express my conditional belief — it is not as though I saidnothing. Suppose I say “If you press that switch, there will bean explosion”, and my hearer takes me to have made a conditionalassertion of the consequent, one which will have the force of anassertion of the consequent if she presses the button. Provided shetakes me to be trustworthy and reliable, she thinks that if shepresses the switch, the consequent is likely to be true. That is, sheacquires a reason to think that if she presses it, there will be anexplosion; and hence a reason not to press it.

Conditional commands can, likewise, be construed as having the forceof a command of the consequent, conditional upon theantecedent’s being true. The doctor says to the nurse in theemergency ward, “If the patient is still alive in the morning,change the dressing”. Considered as a command to makeHook’s conditional true, this is equivalent to “Make itthe case that either the patient is not alive in the morning, or youchange the dressing”. The nurse puts a pillow over thepatient’s face and kills her. On the truth-functionalinterpretation, the nurse can claim that he was carrying out thedoctor’s order. Extending Jackson’s account to conditionalcommands, the doctor said “Make it the case that either thepatient is not alive in the morning, or you change thedressing”, and indicated that she would still command this ifshe knew that the patient would be alive. This doesn’t help. Thenurse who kills the patient still carried out an order. Why should thenurse be concerned with what the doctor would command in acounterfactual situation?

Hook will reply to the above argument about conditional commands thatwe need to appeal to pragmatics. Typically, for any command,conditional or not, there are tacitly understood reasonable andunreasonable ways of obeying it; and killing the patient is to betacitly understood as a totally unreasonable way of making thetruth-functional conditional true — as, indeed, would bechanging the dressing in such an incompetent way that you almoststrangle the patient in the process. The latter clearly is obeying thecommand, but not in the intended manner. But it is stretchingpragmatics rather far to say the same of the former. To take a lessdramatic example, at Fred’s request, the Head of Departmentagrees to bring it about that he gives the Kant lectures if hisappointment is extended. She then puts every effort into making surethat his appointment is not extended. Is it plausible to say that thisis doing what she was asked to do, albeit not in the intended way?

Extending Stalnaker’s account to conditional commands, “Ifit rains, take your umbrella” becomes “In the nearestpossible world in which it rains, take your umbrella”. Suppose Ihave forgotten your command or alternatively am inclined to disregardit. However, it doesn’t rain. In the nearest world in which itrains, I don’t take my umbrella. On Stalnaker’s account, Idisobeyed you. Similarly for conditional promises: on this analysis Icould break my promise to go to the doctor if the pain gets worse,even if the pain gets better. This is wrong: conditional commands andpromises are not requirements on my behaviour in other possibleworlds.

Among conditional questions we can distinguish those in which theaddressee is presumed to know whether the antecedent is true, andthose in which he is not. In the latter case, the addressee is beingasked to suppose that the antecedent is true, and give his opinionabout the consequent: “If it rains, will the match becancelled?”. In the former case — “If you have beento London, did you like it?” — he is expected to answerthe consequent-question if the antecedent is true. If the antecedentis false, the question lapses: there is no conditional belief for himto express. “Not applicable” as the childless might writeon a form which asks “If you have children, how many children doyou have?”. You are not being asked how many children you havein the nearest possible world in which you have children. Nor is itpermissible to answer “17” on the grounds that “Ihave children \(\supset\) I have 17 children” is true. Nor areyou being asked what you would believe about the consequent if youcame to believe that you did have children.

Widening our perspective to include these other conditionals tends toconfirm Supp’s view. Any propositional attitude can be heldcategorically, or under a supposition. Any speech act can be performedunconditionally, or conditionally upon something else. Our uses of“if”, on the whole, seem to be better and more uniformlyexplained without invoking conditional propositions.

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