## 1. Introduction



First let us delimit our field. The examples with which we began are
traditionally called "indicative conditionals". There are
also "subjunctive" or "counterfactual"
conditionals like "Tom would have cooked the dinner if Mary had
not done so", "We would have been home by ten if the train
had been on time". Counterfactuals will be the subject of a
separate entry, and theories addressing them will not be discussed
here. That there is some difference between indicatives and
counterfactuals is shown by pairs of examples like "If Oswald
didn't kill Kennedy, someone else did" and "If
Oswald hadn't killed Kennedy, someone else would have":
you can accept the first yet reject the second (Adams (1970)). That
there is not a huge difference between them is shown by examples like
the following: "Don't go in there", I say, "If
you go in you will get hurt". You look sceptical but stay
outside, when there is large crash as the roof collapses. "You
see", I say, "if you had gone in you would have got hurt.
I told you so." 



It is controversial how best to classify conditionals. According to
some theorists, the forward-looking "indicatives" (those
with a "will" in the main clause) belong with the
"subjunctives" (those with a "would" in the
main clause), and not with the other "indicatives". (See
Gibbard (1981, pp. 222-6), Dudman (1984, 1988), Bennett (1988).
Bennett (1995) changed his mind. Jackson (1990) defends the
traditional view.) The easy transition from typical
"wills" to "woulds" is indeed a datum to be
explained. Still, straightforward statements about the past, present
or future, to which a conditional clause is attached -- the
traditional class of indicative conditionals -- do (in my view)
constitute a single semantic kind. The theories to be discussed do not
fare 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 be
on conditional statements and what they express -- conditional
beliefs; but we will consider which of the theories we have examined
extends most naturally to these other kinds of conditional.



Three kinds of theory will be discussed. In SS2 we compare
truth-functional and non-truth-functional accounts of the truth
conditions of conditionals. In SS3 we examine what is called the
suppositional theory: that conditional judgements essentially involve
suppositions. On development, it appears to be incompatible with
construing conditionals as statements with truth conditions. SS4
looks at some responses from advocates of truth conditions. In SS5
we consider the problem for the suppositional theory of complex
sentences with conditional parts. In SS6 we consider a wider
variety of conditional speech acts and propositional attitudes.



Where we need to distinguish between different interpretations, we
write "\(A \supset B\)" for the truth-functional
conditional, "\(A \rightarrow B\)" for a
non-truth-functional conditional and "\(A \Rightarrow B\)"
for the conditional as interpreted by the suppositional theory; and
for brevity we call protagonists of the three theories Hook, Arrow and
Supp, 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 specifying
the meaning of a complex sentence in terms of the meanings of its
parts, is to specify the truth conditions of the complex sentence, in
terms of the truth conditions of its parts. A semantics of this kind
yields an account of the validity of arguments involving the complex
sentence, given the conception of validity as necessary preservation
of truth. Throughout this section we assume that this approach to
conditionals is correct. Let \(A\) and \(B\) be two sentences such as
"Ann is in Paris" and "Bob is in Paris". Our
question will be: are the truth conditions of "If \(A,
B\)" of the simple, extensional, truth-functional kind, like
those 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, like
those of "\(A\) because \(B\)", "\(A\) before
\(B\)", "It is possible that \(A\)"? That is, are
they 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 to
Frege's new logic (1879). It was taken up enthusiastically by
Russell (who called it "material implication"),
Wittgenstein in the *Tractatus*, and the logical positivists,
and it is now found in every logic text. It is the first theory of
conditionals which students encounter. Typically, it does not strike
students as *obviously* correct. It is logic's first
surprise. Yet, as the textbooks testify, it does a creditable job in
many circumstances. And it has many defenders. It is a strikingly
simple theory: "If \(A, B\)" is false when \(A\) is true
and \(B\) is false. In all other cases, "If \(A, B\)" is
true. It is thus equivalent to "\({\sim}(A \amp{\sim}B)\)"
and to "\({\sim}A\) or \(B\)". "\(A \supset
B\)" has, by stipulation, these truth conditions.



*If* "if" is truth-functional, this is the right
truth function to assign to it: of the sixteen possible
truth-functions of \(A\) and \(B\), it is the only serious candidate.
First, it is uncontroversial that when \(A\) is true and \(B\) is
false, "If \(A, B\)" is false. A basic rule of inference
is modus ponens: from "If \(A, B\)" and \(A\), we can
infer \(B\). 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 \(A, B\)" is
*sometimes* true when \(A\) and \(B\) are respectively (true,
true), or (false, true), or (false, false). "If it's a
square, 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 the
conditional is *determined* by the truth values of its parts
-- it follows that a conditional is *always* true when its
components have these combinations of truth values.



Non-truth-functional accounts agree that "If \(A, B\)" is
false when \(A\) is true and \(B\) is false; and they agree that the
conditional is sometimes true for the other three combinations of
truth-values for the components; but they deny that the conditional is
always true in each of these three cases. Some agree with the
truth-functionalist that when \(A\) and \(B\) are both true, "If
\(A, B\)" must be true. Some do not, demanding a further
relation between the facts that \(A\) and that \(B\) (see Read
(1995)). This dispute need not concern us, as the arguments which
follow depend only on the feature on which non-truth-functionalists
agree: that when \(A\) is false, "If \(A, B\)" may be
either true or false. For instance, I say (\*) "If you touch that
wire, 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 are
insulated, and so forth. Robert Stalnaker's (1968) account is of
this 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 that
possible situation.



Let \(A\) and \(B\) be two logically independent propositions. The
four lines below represent the four incompatible logical possibilities
for 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), and
non-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 truth
values are possible for the corresponding assignment of truth values
to \(A\) and \(B\). For instance, line 4, column (iv), represents two
possibilities for \(A, B\), If \(A, B\), (F, F, T) and (F, F, F).




Truth-Functional Interpretation|  |  |  | (i) | (ii) | (iii) |
|  | \(A\) | \(B\) | \(A \supset B\) | \({\sim}A \supset B\) | \(A \supset{\sim}B\) |
| 1. | T | T | T | T | F |
| 2. | T | F | F | T | T |
| 3. | F | T | T | T | T |
| 4. | F | F | T | F | T |


  



Non-Truth-Functional Interpretation|  |  |  | (iv) | (v) | (vi) |
|  | \(A\) | \(B\) | \(A \rightarrow B\) | \({\sim}A \rightarrow B\) | \(A \rightarrow{\sim}B\) |
| 1. | T | T | T | T/F | F |
| 2. | T | F | F | T/F | T |
| 3. | F | T | T/F | T | T/F |
| 4. | F | F | T/F | F | T/F |


### 2.2 Arguments for Truth-Functionality



The main argument points to the fact that minimal knowledge that the
truth-functional truth condition is satisfied is enough for knowledge
that 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 of
them is red. That's enough to know that if \(x\) isn't
red, \(y\) 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 not red.




Suppose you start off with no information about which of the four
possible combinations of truth values for \(A\) and \(B\) obtains. You
then acquire compelling reason to think that either \(A\) or \(B\) is
true. You don't have any stronger belief about the matter. In
particular, you have no firm belief as to whether \(A\) is true or
not. 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. You
have 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 at
column (ii). Eliminate line 4 and line 4 only, and you have eliminated
the only possibility in which "\({\sim}A \supset B\)" is
false. You know enough to conclude that "\({\sim}A \supset
B\)" is true.



The non-truth-functionalist (call her Arrow) gets this wrong. Look at
column (v). Eliminate line 4 and line 4 only, and some possibility of
falsity remains in other cases which have not been ruled out. By
eliminating just line 4, you do not thereby eliminate these further
possibilities, incompatible with line 4, in which "\({\sim}A
\rightarrow B\)" is false.



The same point can be made with negated conjunctions. You discover for
sure that \({\sim}(A \amp B)\), but nothing stronger than that. In
particular, you don't know whether \(A\). You rule out line 1,
nothing more. You may justifiably infer that if \(A, {\sim}B\). Hook
gets this right. In column (iii), if we eliminate line 1, we are left
only with cases in which "\(A \supset{\sim}B\)" is true.
Arrow gets this wrong. In column (vi), eliminating line 1 leaves open
the possibility that "\(A \rightarrow{\sim}B\)" is
false.



The same argument renders compelling the thought that if we eliminate
*just* \(A \amp{\sim}B\), nothing stronger, i.e., we
don't eliminate \(A\), then we have sufficient reason to
conclude that if \(A, B\).



Here is a second argument in favour of Hook, in the style of Natural
Deduction. 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 \({\sim}B\). So
by 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 Negation
Elimination, we can derive "If \(A\), then \(C\)" from
"\({\sim}(A \amp{\sim}C)\)".



Conditional Proof seems sound: "From \(X\) and \(Y\), it follows
that \(Z\). So from \(X\) it follows that if \(Y, Z\)". Yet
*for no reading of "if" which is stronger than the
truth-functional reading is CP valid* -- at least this is so
if we treat "&" and "\({\sim}\)" in the
classical way and accept the validity of the inference: (I) \({\sim}(A
\amp{\sim}B)\); \(A\); therefore \(B\). Suppose CP is valid for some
interpretation of "If \(A, B\)". Apply CP to (I), and we
get \({\sim}(A \amp{\sim}B)\); 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 to
Hook, the falsity of \(A\) is sufficient for the truth of "If
\(A, B\)". Look at the last two lines of column (i). In every
possible situation in which \(A\) is false, "\(A \supset
B\)" is true. Can it be right that the falsity of "She
touched the wire" entails the truth of "If she touched the
wire she got a shock"? 



Hook might respond as follows. How do we test our intuitions about the
validity of an inference? The direct way is to imagine that we know
for sure that the premise is true, and to consider what we would then
think 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 to
play, and we have no practice in assessing them. The direct intuitive
test is, therefore, silent on whether "If \(A, B\)"
follows from \({\sim}A\). If our smoothest, simplest, generally
satisfactory theory has the consequence that it does follow, perhaps
we should learn to live with that consequence.



There may, of course, be further consequences of this feature of
Hook's theory which jar with intuition. That needs
investigating. But, Hook may add, even if we come to the conclusion
that "\(\supset\)" does not match perfectly our
natural-language "if", it comes close, and it has the
virtues of simplicity and clarity. We have seen that rival theories
also have counterintuitive consequences. Natural language is a fluid
affair, and we cannot expect our theories to achieve better than
approximate 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 primary
concern was to construct a system of logic, formulated in an idealized
language, which was adequate for mathematical reasoning. If "\(A
\supset B\)" doesn't translate perfectly our
natural-language "If \(A, B\)", but plays its intended
role, so much the worse for natural language.



For the purpose of doing mathematics, Frege's judgement was
probably correct. The main defects of \(\supset\) don't show up
in mathematics. There are some peculiarities, but as long as we are
aware of them, they can be lived with. And arguably, the gain in
simplicity and clarity more than offsets the oddities.



The oddities are harder to tolerate when we consider conditional
judgements about empirical matters. The difference is this: in
thinking about the empirical world, we often accept and reject
propositions with degrees of confidence less than certainty. "I
think, but am not sure, that \(A\)" plays no central role in
mathematical thinking. We can, perhaps, ignore as unimportant the use
of indicative conditionals in circumstances in which we are
*certain* that the antecedent is false. But we cannot ignore
our use of conditionals whose antecedent we 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 your partner is about to go away; not
"If I do I'll manage by telepathy". "I think
John spoke to Mary; if he didn't he wrote to her"; not
"If he didn't he shot her". Hook's theory has
the unhappy consequence that *all* conditionals with unlikely
antecedents are likely to be true. To think it likely that \({\sim}A\)
is to think it likely that a sufficient condition for the truth of
"\(A \supset B\)" obtains. Take someone who thinks that
the Republicans won't win the election \(({\sim}R)\), and who
rejects the thought that if they do win, they will double income tax
\((D)\). According to Hook, this person has grossly inconsistent
opinions. For if she thinks it's likely that \({\sim}R\), she
must think it likely that at least one of the propositions,
\(\{{\sim}R, D\}\) is true. But that is just to think it likely that
\(R \supset D\). (Put the other way round, to reject \(R \supset D\)
is to accept \(R \amp{\sim}D\); for this is the only case in which \(R
\supset D\) is false. How can someone accept \(R \amp{\sim}D\) yet
reject \(R\)?) Not only does Hook's theory fit badly the
patterns of thought of competent, intelligent people. It cannot be
claimed that we would be better off with \(\supset\). On the contrary,
we would be intellectually disabled: we would not have the power to
discriminate between believable and unbelievable conditionals whose
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 Hook
all conditionals with true consequents are true: from \(B\) it follows
that \(A \supset B\). This is perhaps less obviously unacceptable: if
I'm sure that \(B\), and treat \(A\) as an epistemic
possibility, I must be sure that if \(A, B\). Again the problem
becomes vivid when we consider the case when I'm only nearly
sure, but not quite sure, that \(B\). I think \(B\) *may* be
false, and will be false if certain, in my view unlikely,
circumstances obtain. For example, I think Sue is giving a lecture
right now. I don't think that if she was seriously injured on
her way to work, she is giving a lecture right now. I reject that
conditional. But on Hook's account, the conditional is false
only if the consequent is false. I think the consequent is true: I
think a sufficient condition for the truth of the conditional
obtains.


### 2.4 Grice's Pragmatic Defence of Truth-Functionality



H. P. Grice famously defended the truth-functional account, in his
William James lectures, "Logic and Conversation",
delivered in 1967 (see Grice (1989); see also Thomson (1990)). There
are many ways of speaking the truth yet misleading your audience,
given the standard to which you are expected to conform in
conversational exchange. One way is to say something weaker than some
other relevant thing you are in a position to say. Consider
disjunctions. I am asked where John is. I am sure that he is in the
pub, and know that he never goes near libraries. Inclined to be
unhelpful but not wishing to lie, I say "He is either in the pub
or in the library". My hearer naturally assumes that this is the
most precise information I am in a position to give, and also
concludes from the truth (let us assume) that I told him "If
he's not in the pub he's in the library". The
conditional, like the disjunction, according to Grice, is true if
he's in the pub, but misleadingly asserted on that ground. 



Another example, from David Lewis (1976, p. 143): "You
won't eat those and live", I say of some wholesome and
delicious mushrooms--knowing that you will now leave them alone,
deferring to my expertise. I told no lie--for indeed you
don't eat them--but of course I misled you.



Grice drew attention, then, to situations in which a person is
*justified in believing* a proposition, which would
nevertheless be an unreasonable thing for the person to *say*,
in normal circumstances. His lesson was salutary and important. He is
right, I think, about disjunctions and negated conjunctions. Believing
that John is in the pub, I can't consistently
*disbelieve* "He's either in the pub or the
library"; 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 I
know you won't eat them. But it is implausible that the
difficulties with the truth-functional conditional can be explained
away in terms of what is an inappropriate conversational remark. They
arise at the level of belief. Thinking that John is in the pub, I may
without irrationality disbelieve "If he's not in the pub
he's in the library". Thinking you won't eat the
mushrooms, I may without irrationality reject "If you eat them
you will die". As facts about the norms to which people defer,
these claims can be tested. A good enough test is to take a
co-operative person, who understands that you are merely interested in
her opinions about the propositions you put to her, as opposed to what
would be a reasonable remark to make, and note which conditionals she
assents to. Are we really to brand as illogical someone who dissents
from both "The Republicans will win" and "If the
Republicans win, income tax will double"?



The Gricean phenomenon is a real one. On anyone's account of
conditionals, there will be circumstances when a conditional is
justifiably believed, but is liable to mislead if stated. For
instance, I believe that the match will be cancelled, because all the
players have 'flu. I believe that whether or not it rains, the
match will be cancelled: if it rains, the match will be cancelled, and
if it doesn't rain, the match will be cancelled. Someone asks me
whether the match will go ahead. I say, "If it rains, the match
will be cancelled". I say something I believe, but I mislead my
audience -- why should I say that, when I think it will be
cancelled whether or not it rains? This does not demonstrate that Hook
is correct. Although I believe that the match will be cancelled, I
don't believe that if all the players make a very speedy
recovery, 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 has
six 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 if
it's a pentagon, it has six sides. 



Another example, due to Gibbard (1981, pp. 235-6): of a glass
that had been held a foot above the floor, you say (having left the
scene) "If it broke if it was dropped, it was fragile".
Intuitively this seems reasonable. But by Hook's lights, if the
glass 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 assert
certain conditionals which (by Hook's lights) we have reason to
believe true. In the above two cases, the problem is reversed: there
are compounds of conditionals which we confidently assert and accept
which, by Hook's lights, we do not have reason to believe
true.



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



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's
equilateral; therefore, either, if it's a triangle it's
equilateral, or, if it's equiangular it's
equilateral". (I owe this example to Alberto Mura.)



The above examples are not a problem for Arrow. But other cases of
embedded conditionals count in the opposite direction. Here are two
sentence 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: "If
Mary comes then if John doesn't have to leave early we will play
Bridge"; "If Mary comes and John doesn't have to
leave early we will play Bridge". "If they were outside
and it rained, they got wet"; "If they were outside, then
if 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 conditions
stronger than \(\supset\) does Import-Export hold. Assume
Import-Export holds for some reading of "if". The key to
the 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 logical
truth. 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 in
which 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. We
still have to consider Jackson's defence of Hook, and
Stalnaker's response to the problem about non-truth-functional
truth conditions raised in SS2.2. These are deferred to SS4,
because they depend on the considerations developed in SS3.


## 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 to
believe, or to be more or less certain, that \(B\) if \(A\) --
that John cooked the dinner if Mary didn't, that you will
recover if you have the operation, and so forth. How do you make such
a judgement? You suppose (assume, hypothesise) that \(A\), and make a
hypothetical 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 are
> both in doubt as to \(p\), they are adding \(p\) hypothetically to
> their stock of knowledge, and arguing on that basis about \(q\);
> ... they are fixing their degrees of belief in \(q\) given \(p\)
> (1929, p. 247).
> 



A suppositional theory was advanced by J. L. Mackie (1973, chapter 4).
See also David Barnett (2006). Peter Gardenfors's work
(1986, 1988) could also come under this heading. But the most fruitful
development of the idea (in my view) takes seriously the last part of
the above quote from Ramsey, and emphasises the fact that conditionals
can 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 \(B\): we
may 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 that
degrees of closeness to certainty can be quantified: 100% certain, 90%
certain, etc.; and we can turn to probability theory for what Ramsey
called the "logic of partial belief". There we find a
well-established, indispensable concept, "the conditional
probability of \(B\) given \(A\)". It is to this notion that
Ramsey refers by the phrase "degrees of belief in \(q\) given
\(p\)".



It is, at first sight, rather curious that the best-developed and most
illuminating suppositional theory should place emphasis on uncertain
conditional judgements. If we knew the truth conditions of
conditionals, we would handle uncertainty about conditionals in terms
of a general theory of what it is to be uncertain of the truth of a
proposition. But there is no consensus about the truth conditions of
conditionals. It happens that when we turn to the theory of uncertain
judgements, we find a concept of conditionality in use. It is worth
seeing what we can learn from it.



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



> 
> The probability that two ... events will both happen is ...
> the probability of the first [multiplied by] the probability of the
> second *on the supposition that* the first happens [my
> emphasis].
> 



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 red
balls have a black spot (so the probability that a ball with a black
spot 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% of
70%, i.e. 42%. 



Ramsey, arguing that "degrees of belief" should conform to
probability theory, stated the same "fundamental law of partial
belief":



> 
> 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 on
conditionals, and about 80% certain that you will pass, on the
supposition that it is on conditionals. So you are about 40% certain
that the test will be on conditionals and you will pass. 



Accepting Ramsey's suggestion that "if",
"given that", "on the supposition that" come
to the same thing, writing "\(\mathbf{p}(B)\)" for
"degree of belief in \(B\)", and "\(\mathbf{p}\_A
(B)\)" for "degree of belief in \(B\) given \(A\)",
and rearranging the basic law, we have:

 

\[ 
\mathbf{p}(B \text{ if } A) = \mathbf{p}\_A (B) = 
 \frac{\mathbf{p}(A \amp B)}{\mathbf{p}(A)}, \text{ provided } \mathbf{p}(A) \text{ is not } 0.
\]


Call a set of mutually exclusive and jointly exhaustive propositions a
partition. The lines of a truth table constitute a partition.
One's degrees of belief in the members of a partition, idealized
as precise, should sum to 100%. That is all there is to the claim that
degrees of belief should have the structure of probabilities. Consider
a 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 displayed
geometrically, 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 another
vertical 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.




|  |  |  |
| --- | --- | --- |
| \(A \amp B\) | \(A \amp{\sim}B\) | \({\sim}A\) |



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



How does X evaluate "If \(A, B\)"? She assumes that \(A\),
that is, hypothetically eliminates \({\sim}A\). In the part of the
partition that remains, in which \(A\) is true, \(B\) is four times as
likely as \({\sim}B\); that is, on the assumption that \(A\), it is
four to one that \(B: \mathbf{p}(B\) if \(A)\) is 80%,
\(\mathbf{p}({\sim}B\) if \(A)\) is 20%. Equivalently, as \(A \amp B\)
is four times as likely as \(A \amp{\sim}B, \mathbf{p}(B\) if \(A)\)
is 4/5, or 80%. Equivalently, \(\mathbf{p}(A \amp B)\) is 4/5 of
\(\mathbf{p}(A)\). In non-numerical terms: you believe that if \(A,
B\) to the extent that you think that \(A \amp B\) 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 \(A, B\). In this case, your
\(\mathbf{p}(A \amp{\sim}B) = 0\).



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
\(\mathbf{p}(A)\) is not 0. On this approach, indicative conditionals
only have a role when the thinker takes \(A\) to be an epistemic
possibility. If you take yourself to know for sure that Ann is in
Paris, you don't go 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 something as an epistemic possibility, temporarily, to
comply with the epistemic state of the hearer. When playing the
sceptic, there are not many limits on what you *can*, at a
pinch, take as an epistemic possibility - as not already ruled
out. But there are some limits, 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 to
give a high value to \(\mathbf{p}(A \supset B)\). How does
\(\mathbf{p}(A \supset B)\) compare with \(\mathbf{p}\_A (B)\)? In two
special cases, they are equal: first, if \(\mathbf{p}(A \amp{\sim}B) =
0\) (and \(\mathbf{p}(A)\) is not 0), \(\mathbf{p}(A \supset B) =
\mathbf{p}\_A (B) = 1\) (i.e. 100%). Second, if \(\mathbf{p}(A) =
100\)%, \(\mathbf{p}(A \supset B) = \mathbf{p}\_A (B) =
\mathbf{p}(B)\). In all other cases, \(\mathbf{p}(A \supset B)\) is
greater than \(\mathbf{p}\_A (B)\). To see this we need to compare
\(\mathbf{p}(A \amp{\sim}B)\) and \(\mathbf{p}(A
\amp{\sim}B)/\mathbf{p}(A)\). Consider again the partition \(\{A \amp
B, A \amp{\sim}B, {\sim}A\}. \mathbf{p}(A \amp{\sim}B)\) is a smaller
proportion of the whole space than it is of the \(A\)-part -- the
part of the space in which \(A\) is true -- except in the special
cases in which \(\mathbf{p}(A \amp{\sim}B) = 0\), or
\(\mathbf{p}({\sim}A) = 0\). So, except in these special cases,
\(\mathbf{p}\_A ({\sim}B)\) is greater than \(\mathbf{p}(A
\amp{\sim}B)\). Now \(\mathbf{p}(A \supset B) = \mathbf{p}({\sim}(A
\amp{\sim}B))\); and \(\mathbf{p}(A \amp{\sim}B) + \mathbf{p}({\sim}(A
\amp{\sim}B)) = 1\). Also \(\mathbf{p}\_A (B) + \mathbf{p}\_A ({\sim}B)
= 1\). So from \(\mathbf{p}\_A ({\sim}B) \gt \mathbf{p}(A
\amp{\sim}B)\) it follows that \(\mathbf{p}(A \supset B) \gt
\mathbf{p}\_A (B)\).



Hook and the suppositional theorist (call her Supp) come spectacularly
apart when \(\mathbf{p}({\sim}A)\) is high and \(\mathbf{p}(A \amp
B)\) is much smaller than \(\mathbf{p}(A \amp{\sim}B)\). Let
\(\mathbf{p}({\sim}A) = 90\)%, \(\mathbf{p}(A \amp B) = 1\)%,
\(\mathbf{p}(A \amp{\sim}B) = 9\)%. \(\mathbf{p}\_A (B) = 10\)%.
\(\mathbf{p}(A \supset B) = 91\)%. For instance, I am 90% certain that
Sue won't be offered the job \(({\sim}O)\), and think it only
10% likely that she will decline the offer \((D)\) if it is made, that
is \(\mathbf{p}\_O (D) = 10\)%. \(\mathbf{p}(O \supset D) =
\mathbf{p}({\sim}O \text{ or } (O \amp D)) = 91\)%.



Now let us compare Hook, Arrow, and Supp with respect to two questions
raised in SS2.


* Question 1. You are certain that \({\sim}(A \amp{\sim}B)\), but
not 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 \(A. \mathbf{p}\_A (B)
= 1\).



Arrow: no, not necessarily. For "\(A \rightarrow B\)" may
be false when \(A \amp{\sim}B\) is false. With just the information
that \(A \amp{\sim}B\) is false, I should not be certain that if \(A,
B\).
* Question 2. If you think it likely that \({\sim}A\), might you
still think it unlikely that if \(A, B\)?


Hook: no. "\(A \supset B\)" is true in all the possible
situations in which \({\sim}A\) is true. If I think it likely that
\({\sim}A\), I think it likely that a sufficient condition for the
truth 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 goes
to \({\sim}A\) leaves open the question whether or not \(A \amp B\) is
more probable than \(A \amp{\sim}B\). If \(\mathbf{p}(A \amp{\sim}B)\)
is greater than \(\mathbf{p}(A \amp B)\), 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 \(A\) is
false. 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 to
both questions. In this she differs from both Hook and Arrow.
Supp's way of assessing conditionals is incompatible with the
truth-functional way (they answer Question 2 differently); and
incompatible with stronger-than-truth-functional truth conditions
(they answer Question 1 differently). It follows that Supp's way
of assessing conditionals is incompatible with the claim that
conditionals have truth conditions of any kind. \(\mathbf{p}\_A (B)\)
does not measure the probability of the truth of any proposition.
Suppose it did measure the probability of the truth of some
proposition \(A\*B\). Either \(A\*B\) is entailed by "\(A \supset
B\)", or it is not. If it is, it is true whenever \({\sim}A\) is
true, and hence cannot be improbable when \({\sim}A\) is probable.
That is, it cannot agree with Supp in its answer to Question 2. If
\(A\*B\) is not entailed by "\(A \supset B\)", it may be
false when \({\sim}(A \amp{\sim}B)\) is true, and hence certainty that
\({\sim}(A \amp{\sim}B)\) (in the absence of certainty 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 the
following as an expository, heuristic device, a harmless fiction.
Imagine a partition as carved into a large finite number of
equally-probable chunks, such that the propositions with which we are
concerned are true in an exact number of them. The probability of any
proposition is the proportion of chunks in which it is true. The
probability of \(B\) on the supposition that \(A\) is the proportion
*of the* \(A\)*-chunks* (the chunks in which \(A\) is
true) which are \(B\)-chunks. With some misgivings, I succumb to the
temptation to call these chunks "worlds": they are equally
probable, mutually incompatible and jointly exhaustive epistemic
possibilities, enough of them for the propositions with which we are
concerned to be true, or false, at each world. The heuristic value is
that judgements of probability and conditional probability then
translate into statements about proportions.



Although Supp and Hook give the same answer to Question 1, their
reasons are different. Supp answers "yes" *not*
because a proposition, \(A\*B\), is true whenever \(A \amp{\sim}B\) is
false; but because \(B\) is true in all the "worlds" which
matter for the assessment of "If \(A, B\)": the
\(A\)-worlds. Although Supp and Arrow give the same answer to Question
2, their reasons are different. Supp answers "yes", not
because a proposition \(A\*B\) may be false when \(A\) is false; but
because the fact that most worlds are \({\sim}A\)-worlds is irrelevant
to whether most *of the* \(A\)*-worlds* are
\(B\)-worlds. To judge that \(B\) is true *on the supposition
that* \(A\) is true, it turns out, is not to judge that
something-or-other, \(A\*B\), is true.



By a different argument, David Lewis (1976) was the first to prove
this remarkable result: there is no proposition \(A\*B\) such that, in
all probability distributions, \(\mathbf{p}(A\*B) = \mathbf{p}\_A (B)\).
Conditional probability does not measure the probability of the truth
of any proposition. If a conditional has truth conditions, one should
believe it to the extent that one thinks it is probably true. If Supp
is correct, that one believes "If \(A, B\)" to the extent
that one thinks it probable that \(B\) on the supposition that \(A\),
then this is not equivalent to believing some proposition to be
probably true. Hence, it appears, if Supp is right, conditionals
shouldn't be construed as having truth conditions at all. A
conditional judgement involves two propositions, which play different
roles. One is the content of a supposition. The other is the content
of a judgement made under that supposition. They do not combine to
yield a single proposition which is judged to be likely to be true
just when the second is judged likely to be true on the supposition of
the first.



(Lewis called his proofs "triviality results", because the
conclusions are avoided only in a trivial probability space which is
incapable of giving positive probability to more than two incompatible
propositions--for instance, is incapable of giving positive
probability to \(A \amp B, A \amp{\sim}B\), and \({\sim}A\). The name
is widely used in the literature. For recent examples see Khoo and
Mandelkern (2019) and Charlow (2019).)



Note: ways of restoring truth conditions, compatible with Supp's
thesis, are considered in SS5.


### 3.2 Validity



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



First consider classically valid (that is, necessarily
truth-preserving) arguments which don't involve conditionals. We
use them in arguing from contingent premises about which we are often
less than completely certain. The question arises: how certain can we
be of the conclusion of the argument, given that we think, but are not
sure, that the premises are true? Call the improbability of a
statement one minus its probability. Adams showed this: if (and only
if) an argument is valid, then in no probability distribution does the
improbability of its conclusion exceed the sum of the improbabilities
of its premises. Call this the Probability Preservation Principle
(PPP).



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


1. if \(A\) entails \(B, \mathbf{p}(A) \le \mathbf{p}(B)\)
2. \(\mathbf{p}(A \text{ or } B) = \mathbf{p}(A) + \mathbf{p}(B) -
\mathbf{p}(A \amp B) \le \mathbf{p}(A) + \mathbf{p}(B)\)
3. For all \(n, \mathbf{p}(A\_1\) or ... or \(A\_n) \le
\mathbf{p}(A\_1) + \cdots + \mathbf{p}(A\_n)\)



Suppose \(A\_1, \ldots, A\_n\) entail \(B\). Then \({\sim}B\) entails
\({\sim}A\_1\) or ... or \({\sim}A\_n\). Therefore
\(\mathbf{p}({\sim}B) \le \mathbf{p}({\sim}A\_1) + \cdots +
\mathbf{p}({\sim}A\_n)\): the improbability of the conclusion of a
valid argument cannot exceed the sum of the improbabilities of the
premises. 



The result is useful to know: if you have two premises of which you
are at least 99% certain, they entitle you to be at least 98% certain
of a conclusion validly drawn from them. Of course, if you have 100
premises each at least 99% certain, your conclusion may have zero
probability. That is the lesson of the "Lottery Paradox".
Still, Adams's result vindicates deductive reasoning from
uncertain premises, provided that they are not too uncertain, and
there are not too many of them.



So far, we have a very useful consequence of the classical notion of
validity. Now Adams extends this consequence to arguments involving
conditionals. Take a language with "and",
"or", "not" and "if" -- but
with "if" occurring only as the main connective in a
sentence. (We put aside compounds of conditionals.) Take any argument
formulated in this language. Consider any probability function over
the sentences of this argument which assigns non-zero probability to
the antecedents of all conditionals -- that is, any assignment of
numbers to the non-conditional sentences which conforms to the
Partition Principle, and to the conditional sentences which conforms
to Supp's thesis: \(\mathbf{p}(B\) if \(A) = \mathbf{p}\_A (B) =
\mathbf{p}(A \amp B)/\mathbf{p}(A)\). Let the improbability of the
conditional "If \(A, B\)" be \(1 - \mathbf{p}\_A (B)\).
*Define* a valid argument as one such that there is no
probability function in which the improbability of the conclusion
exceeds the sum of the improbabilities of the premises. And a nice
logic emerges, which is now well known. It is the same as
Stalnaker's logic over this domain (see SS4.1). There are
rules of proof, a decision procedure, consistency and completeness can
be proved. See Adams (1998 and 1975).



I shall write the conditional which satisfies Adams's criterion
of validity "\(A \Rightarrow B\)".
We have already seen that in all distributions, \(\mathbf{p}\_A (B) \le
\mathbf{p}(A \supset B)\). Therefore, \(A \Rightarrow B\) entails \(A
\supset B\): it cannot be the case that the former is more probable
than the latter. Call a non-conditional sentence a factual sentence.
If an argument has a factual conclusion, and is classically valid with
the conditional interpreted as \(\supset\), it is valid with the
conditional interpreted as the stronger \(\Rightarrow\). The following
patterns of inference are therefore valid:

 

\[\begin{align}
A; A \Rightarrow B; &\text{ so } B \text{ (modus ponens)} \\
A \Rightarrow B; {\sim}B; &\text{ so } {\sim}A \text{ (modus tollens)} \\
A \text{ or } B; A \Rightarrow C; B \Rightarrow C; &\text{ so } C.
\end{align}\]


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



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

 

\[
B; \text{ so } A \Rightarrow B.
\]


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

 

\[
{\sim}A; \text{ so } A \Rightarrow B.
\]


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

 

\[
{\sim}(A \amp B); \text{ so } A \Rightarrow{\sim}B
\]


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

 

\[
A \text{ or } B; \text{ so } {\sim}A \Rightarrow B.
\]


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

 

\[
A \Rightarrow B; \text{ so } (C \amp A) \Rightarrow B
\text{ (strengthening of the antecedent).}
\]


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



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

 

\[
A \Rightarrow B; B \Rightarrow C; \text{ so } A \Rightarrow C.
\]


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



We saw in SS2.2 that Conditional Proof (CP) is invalid for any
conditional stronger than \(\supset\). It is invalid in Adams's
logic. For instance, "\({\sim}(A \amp B)\); \(A\); so
\({\sim}B\)" is valid. It contains no conditionals. Any
necessarily truth-preserving argument satisfies PPP. If I'm
close 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 I
must be close to certain that I won't be injured. But, as we
saw, "\({\sim}(A \amp B)\); 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 an
*assumption*. What is the difference between the roles of a
premise, 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 not
immediately clear what it would be to treat a conditional, construed
according to Supp, as an assumption: to assume something, as
ordinarily understood, is to assume that it is true; and conditionals
are not being construed as ordinary statement of fact. But we could
approximate the idea of taking the premises as assumptions, by
treating them, hypothetically, as certainties. Treating the premises
thus would be to require of a valid argument that it preserve
certainty: that there must be no probability distributions in which
all the premises (conditional or otherwise) are assigned probability 1
and the conclusion is assigned probability less than 1. Call this the
certainty-preservation principle (CPP).



The conception of validity we have been using (PPP) takes as central
the fact that premises may be accepted with degrees of confidence less
than certainty. Now, anything which satisfies PPP satisfies CPP. And
for argument involving only factual propositions, the converse is also
true: the same class of arguments necessarily preserves truth,
necessarily preserves certainty and necessarily preserves probability
in the sense of PPP. But arguments involving conditionals can satisfy
CPP without satisfying PPP. The invalid argument forms above do
preserve certainty: if you assign probability 1 to the premises, then
you are constrained to assign probability 1 to the conclusion (in all
probability distributions in which the antecedent of any conditional
gets non-zero probability). But they do not preserve high probability.
They do not satisfy PPP. If at least one premise falls short of
certainty by however small an amount, the conclusion can plummet to
zero.



The logico-mathematical fact behind this is the difference in logical
powers 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: almost
all \(A\)-worlds are \(B\)-worlds but no \(C \amp A\)-world is a
\(B\)-world. If all \(A\)-worlds are \(B\)-worlds and all \(B\)-worlds
are \(C\)-worlds, then all \(A\)-worlds are \(C\)-worlds. But we can
have: all \(A\)-worlds are \(B\)-worlds, almost all \(B\)-worlds are
\(C\)-worlds, yet no \(A\)-world is a \(C\)-world; just as we can
have, all kiwis are birds, almost all birds fly, but no kiwi
flies.



Someone might react as follows: "All I want of a valid argument
is that it preserve certainty. I'm not bothered if an argument
can have premises close to certain and a conclusion far from certain,
as long as the conclusion is certain when the premises are
certain".



We *could* use the word "valid" in such a way that
an argument is valid provided it preserves certainty. If our interest
in logic is confined to its application to mathematics or other a
priori matters, that is fine. Further, when our arguments do not
contain conditionals, if we have certainty-preservation,
probability-preservation comes free. But if we use conditionals when
arguing about contingent matters, then great caution will be required.
Unless we are 100% certain of the premises, the arguments above which
are invalid on Adams's criterion guarantee nothing about what
you are entitled to think about the conclusion. The line between 100%
certainty and something very close is hard to make out: it's not
clear how you tell which side of it you are on. The epistemically
cautious might admit that they are never, or only very rarely, 100%
certain of contingent conditionals. So it would be useful to have
another category of argument, the "super-valid", which
preserves high probability as well as certainty. Adams has shown us
which arguments (on Supp's reading of "if") are
super-valid.



Continuing to restrict our attention to the case where the antecedent
has non-zero probability, this argument-form preserves certainty:
A\(\supset\)B; so A\(\Rightarrow\)B. The
converse inference is uncontroversial. So if we were just concerned
with certainty preservation, Hook and Supp would be equivalent. But
they are far from equivalent for uncertain beliefs: the former can be
arbitrarily 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 in
evidence. Nor was Lewis's result that conditional probabilities
are not probabilities of the truth of a proposition. (Adams expressed
scepticism about truth conditions for conditionals, but the question
was still open.) Stalnaker's (1968) semantics for conditionals
was an attempt to provide truth conditions which were compatible with
Ramsey's and Adams's thesis about conditional belief. (See
also Stalnaker (1970), where the probabilistic aspects of his proposal
are developed.) That is, he sought truth conditions for a proposition
\(A\gt B\) (his notation) such that \(\mathbf{p}(A\gt B)\) must equal
\(\mathbf{p}\_A (B)\): 



> 
> Now that we have found an answer to the question, "How do we
> decide whether or not we believe a conditional statement?"
> [Ramsey's and Adams's answer] the problem is to make the
> transition from belief conditions to truth conditions; ... . The
> concept of a *possible world* is just what we need to make the
> transition, since a possible world is the ontological analogue of a
> stock of hypothetical beliefs. The following ... is a first
> approximation to the account I shall propose: Consider a possible
> world in which \(A\) is true and otherwise differs minimally from the
> actual 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 of
its conclusion cannot exceed the sum of the improbabilities of the
premises. The latter was the criterion Adams used in constructing his
logic. So Stalnaker's logic for conditionals must agree with
Adams's over their common domain. And it does. The argument
forms we showed to be invalid in Adams's logic (SS3.2) are
invalid on Stalnaker's semantics. For instance, the following is
possible: in the nearest possible world in which you strike the match,
it lights; in the nearest world in which you dip the match in water
and strike it, it doesn't light. So Strengthening fails. (By
"nearest world in which ..." I mean the possible
world 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: it
can 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 Ann
did not cook it, Mary cooked it.



Stalnaker (1975) argued that although (\*) is invalid, it is
nevertheless 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 \(w\), a
world, \(w'\), the nearest (most similar) world to \(w\) at which
\(A\) is true. "If \(A, B\)" is true at \(w\) iff \(B\) is
true at F\((A, w)\), i.e. at \(w'\). "If \(A, B\)" is true
simpliciter iff \(B\) is true at the nearest \(A\)-world to the actual
world. (However, we do not know which world is the actual
world--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 to
the actual world is the actual world itself, so in this case "If
\(A, B\)" is true iff \(B\) is also true. The selection function
does substantive work only when \(A\) is false.



Stalnaker's theory is intended to apply to counterfactuals and
indicative conditionals alike, but in the case of indicative
conditionals, he claims, the selection function is subject to a
pragmatic constraint, set in the framework of the dynamics of
conversation. At any stage in a conversation, many things are taken
for granted by speaker and hearer, i.e. many possibilities are taken
as already ruled out. The remaining possibilities are live. He calls
the set of worlds which are not ruled out -- the live
possibilities -- the context set. For indicative conditionals,
antecedents are typically live possibilities, and we focus on that
case. The pragmatic constraint for indicative conditionals says that
if the antecedent \(A\) is compatible with the context set (i.e. true
at some worlds in the context set) then for any world \(w\) in the
context set, the nearest \(A\)-world to \(w\) -- i.e. the world
picked out by the selection function -- is also a member of the
context set. Roughly, if \(A\) is a live possibility (i.e. not already
ruled out), then for any world \(w\) which is a live possibility, the
nearest \(A\)-world to \(w\) is also a live possibility. Or: things
which are taken to be epistemically possible count as closer to
actuality than things which are not.



The proposition expressed by "If \(A, B\)" is the set of
worlds \(w\) 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 are
live in different conversational settings, a different proposition may
be expressed by "If \(A, B\)" in different conversational
settings. Thus, the truth-conditions of conditionals are
context-dependent, depending on which possibilities the speaker and
hearer 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 set
is the set of worlds compatible with what I take for granted, i.e. the
set of worlds not ruled out, i.e. the set of worlds which are
epistemically possible for me. Let \(A\) be epistemically possible for
me. Then the pragmatic constraint requires that for any world in the
context set, the nearest \(A\)-world to it is also in the context set.
Provided you and I have different bodies of information, the
proposition I am considering when I consider whether if \(A, B\) may
well differ from the proposition you would express in the same words:
the constraints on nearness differ; worlds which are near for me may
not be near for you.



This enables Stalnaker to avoid the argument against
non-truth-functional truth conditions given in SS2.2. The argument
was as follows. There are six incompatible logically possible
combinations of truth values for \(A, B\) and \({\sim}A\gt B\). We
start off with no firm beliefs about which obtains. Now we eliminate
just \({\sim}A \amp{\sim}B\), i.e. establish \(A\) or \(B\). That
leaves five remaining possibilities, including two in which
"\({\sim}A\gt B\)" is false. So we can't be certain
that \({\sim}A\gt B\) (whereas, intuitively, one can be certain of the
conditional in these circumstances). Stalnaker replies: we
can't, indeed, be certain that the proposition we were wondering
about earlier is true. But we are now in a new context: \({\sim}A
\amp{\sim}B\)-worlds have been ruled out (but \({\sim}A \amp
B\)-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), we
know 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 to
the epistemic state of the thinker, is somewhat unnatural. One usually
distinguishes between the content of what is said and the different
epistemic attitudes one may take to that same content. Someone
conjectures that if Ann isn't home, Bob is. We are entirely
agnostic about this. Then we discover that at least one of them is at
home (nothing stronger). We now accept the conditional. It seems more
natural to say that we now have a different attitude to the same
conditional thought, that \(B\) on the supposition that \({\sim}A\).
It does not seem that the content of our conditional thought has
changed. And if there are conditional propositions, it seems more
natural to say that we now take to be true what we were previously
wondering about. There does not seem to be independent motivation for
thinking the content of the proposition has changed.



Also, Stalnaker's argument is restricted to the special case
where we take the \({\sim}A \amp{\sim}B\)-possibilities to be ruled
out. Consider a case when, starting out agnostic, we become close to
certain, but not quite certain, that \(A\) or \(B\) -- say we
become about 95% certain that \(A\) or \(B\), and are about 50%
certain that \(A\). According to Supp, we are entitled to be quite
close to certain that if \({\sim}A, B\) -- 90% certain in fact.
(If \(\mathbf{p}(A \text{ or } B) = 95\)% and \(\mathbf{p}(A) = 50\)%,
then \(\mathbf{p}({\sim}A \amp B) = 45\)%. Now \(\mathbf{p}({\sim}A
\amp{\sim}B) = 5\)%. 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 \({\sim}A \amp B\)-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 all
propositional theories. As we have seen, it is easy to construct
probabilistic counterexamples to Hook's theory; and it is easy
to do so for the variant of Stalnaker's theory according to
which "If \(A, B\)" is true iff \(B\) is true at
*all* nearest \(A\)-worlds (as Lewis (1973) holds for
counterfactuals). (It is very close to certain that if you toss the
coin ten times, you will get at least one head; but it is certainly
false that the consequent is true at *all* nearest
antecedent-worlds.) It is rather harder for Stalnaker's theory,
because nearness is so volatile, and also because it is not fully
specified. 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 the
angle you see them--end on--they all look the same; and they
are 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 will be 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
15 cm and nine are not. But on Stalnaker's theory, (\*)
appears to be 91% likely: it's 90% likely that it is not over
10 cm, in which case, in the world most similar to the actual
world in which it is over 10 cm, it is 11 cm, i.e. less than
15 cm. And we add another 1% for the case in which it is
11 cm, hence under 15 cm.



(The point is simpler for the counterfactual: "If it had been
over 10 cm, it would have been less than 15 cm":
judging by similarity to the actual world, this seems true; but
intuitively it is only 10% likely.)



The example casts doubt on whether any notion of similarity, or
minimal difference from the actual world, is the right notion for
understanding conditionals, as opposed to taking a probability
distribution over the various possible antecedent worlds.



There is also the question, for Stalnaker, of the uniqueness
assumption--that there is a unique closest antecedent-world.
Stalnaker (1981, pp. 87-91) discusses this, and proposes to use
the machinery of supervaluations when there is no unique nearest
world: the conditional is true iff true whichever of the candidates
for nearest the selection function selects, false iff false for all
such selections, otherwise it is indeterminate--neither true nor
false. As the uniqueness assumption often fails, a great many
conditionals will just get the verdict indeterminate. For instance, I
am considering whether, (\*) if I pick a red ball, it will have a black
spot. 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 conditional
context-dependent escape Lewis's result that a conditional
probability is not the probability of the truth of any proposition?
Lewis showed that there is no proposition \(A\*B\) such that in every
belief state \(\mathbf{p}(A\*B) = \mathbf{p}\_A (B)\). He did not rule
out that in every belief state there is some proposition or other,
\(A\*B\), such that \(\mathbf{p}(A\*B) = \mathbf{p}\_A (B)\). However, in
the wake of Lewis, Stalnaker himself proved a stronger result, for his
conditional connective: the equation \(\mathbf{p}(A\gt B) =
\mathbf{p}\_A (B)\) cannot hold for all 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 van Fraassen 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 sensitive
to epistemic situations Stalnaker's truth conditions would be.
Later (1984, ch. 6), reacting to Gibbard, Stalnaker seemed more
ambivalent 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. His
work has inspired others to develop related theories:
context-dependent theories are currently popular (see below,
SS4.3); the fact that Stalnaker kept the probabilistic
considerations aside in his 1968 paper led others to develop the
Ramsey test for all-or-nothing beliefs--see e.g. Gardenfors
(1986); and another close relative of Stalnaker's semantics, due
to Richard Bradley (2012), is discussed below in SS5.


### 4.2 A Special Assertability Condition



Frank Jackson holds that "If \(A, B\)" has the truth
conditions of "\(A \supset B\)", i.e. "\({\sim}A\)
or \(B\)"; but it is part of its meaning that it is governed by
a special rule of assertability. "If" is assimilated to
words like "but", "nevertheless" and
"even". "\(A\) but \(B\)" has the same truth
conditions as "\(A\) and \(B\)", yet they differ in
meaning: "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 John
can understand this proof, but inappropriate when John is a
world-class logician. 



According to Jackson, in asserting "If \(A, B\)" the
speaker expresses his belief that \(A \supset B\), and also indicates
that this belief is "robust" with respect to the
antecedent \(A\). In Jackson's early work (1979, 1980)
"robustness" was explained thus: the speaker would not
abandon his belief that \(A \supset B\) if he were to learn that
\(A\). This, it was claimed, amounted to the speaker's having a
high probability for \(A \supset B\) given \(A\), i.e. for
\(({\sim}A\) or \(B)\) given \(A\), which is just to have a high
probability for \(B\) given \(A\). Thus, assertability goes by
conditional probability. Robustness was meant to ensure that an
assertable conditional is fit for modus ponens. Robustness is not
satisfied if you believe \(A \supset B\) solely on the grounds that
\({\sim}A\). Then, if you discover that \(A\), you will abandon your
belief in \(A \supset B\) rather than conclude that \(B\).



Jackson came to realise, however, that there are assertable
conditionals which one would not continue to believe if one learned
the antecedent. I say "If Reagan worked for the KGB, I'll
never find out" (Lewis's example (1986, p. 155)). My
conditional probability for consequent given antecedent is high. But
if I were to discover that the antecedent is true, I would abandon the
conditional belief, rather than conclude that I will never find out
that the antecedent is true. So, in Jackson's later work (1987),
robustness with respect to \(A\) is simply defined as \(\mathbf{p}\_A
(A \supset B)\) being high, which is trivially equivalent to
\(\mathbf{p}\_A (B)\) being high. In most cases, though, the earlier
explanation will hold good.



What do we need the truth-functional truth conditions for? Do they
explain the meaning of compounds of conditionals? According to
Jackson, they do not (1987, p. 129). We know what "\(A \supset
B\)" 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 his
theory has no implications about what, if anything, "if \(A,
B\)" means when it occurs, unasserted, as a constituent in a
longer sentence.



(Here his analogy with "but" etc. fails. "But"
can occur in unasserted clauses: "Either he arrived on time but
didn't wait for us, or he never arrived at all" (see Woods
(1997, p. 61)). It also occurs in questions and commands: "Shut
the door but leave the window open". "Does anyone want
eggs but no ham?". "But" means "and in
contrast". Its meaning is not given by an "assertability
condition".)



Do the truth-functional truth conditions explain the validity of
arguments involving conditionals? Not in a way that accords well with
intuition, we have seen. Jackson claims that our intuitions are at
fault here: we confuse preservation of truth and preservation of
assertability (1987, pp. 50-1).



Nor is there any direct evidence for Jackson's theory. Nobody
who thinks the Republicans won't win treats "If the
Republicans win, they will double income tax" as inappropriate
but probably true, in the same category as "Even Godel
understood truth-functional logic". Jackson is aware of this. He
seems to advocate an error theory of conditionals: ordinary linguistic
behaviour fits the false theory that there is a proposition \(A\*B\)
such that \(\mathbf{p}(A\*B) = \mathbf{p}\_A (B)\) (1987, pp.
39-40). If this is his view, he cannot hold that his own theory
is a psychologically accurate account of what people do when they use
conditionals. Perhaps it is an account of how we *should* use
conditionals, and would if we were free from error: we *should*
accept that "If the Republicans win they will double income
tax" is probably true when it is probable that the Republicans
won't win. Would we gain anything from following this
prescription? It is hard to see that we would: we would deprive
ourselves of the ability to discriminate between believable and
unbelievable conditionals whose antecedents we think false.



For Jackson's more recent thoughts on conditionals see his
postscript (1998, pp. 51-54). See also Edgington (2009) and
Jackson's reply (2009, pp. 463-6). 


### 4.3 Restrictors and the Strict Conditional



Angelika Kratzer's work on conditionals has been very
influential in linguistics, and also in philosophy. Her articles have
appeared, 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 is
about the analysis of sentences containing adverbs such as
*always*, *never*, *usually*, *often*,
*seldom* ..., sentences such as "The fog usually
lifts before noon here" and "Caesar seldom awoke before
dawn". After considering and rejecting some alternatives, Lewis
introduces "restriction by if-clauses": he proposes that
there is a use of if-clauses whose function is to restrict the range
of cases to which the operator or quantifier applies. First paraphrase
the sentences: "Usually if there is fog here, it lifts before
noon." "Seldom if Caesar awoke, it was before dawn."
(Lewis's target sentences do not have "if" in their
surface structure, but they could have had: the theory also applies to
sentences like "Usually, if Mary visits, she brings her
dog".) 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 are
not to be construed as applying an adverb to a conditional
proposition. The adverb applies to the main clause, its scope
restricted by the if-clause. Thus Lewis: 



> 
> [T]he *if* of our restrictive if-clauses should not be regarded
> as a sentence connective. It has no meaning apart from the adverb it
> restricts. The *if* in *always if* ...,
> *sometimes if* ..., and the rest is on a par with the
> non-connective *and* in *between ... and ...*,
> with the non-connective *or* in *whether ... or
> ...*, or with the non-connective *if* in *the
> probability that ... if*. It serves merely to mark an
> argument-place in a polyadic construction. (Lewis 1975 reprinted in
> Lewis 1998 pp. 14-15)
> 



Lewis's final example is particularly interesting, especially
because this paper was written at much the same time as his proof that
conditional probabilities are not to be construed as probabilities of
conditional propositions. 



Lewis has three different accounts of "if": he follows
Jackson in claiming that the "if" of indicative
conditionals is the truth-functional "if", with a special
rule of assertability (see Lewis 1986 pp. 152-6); there is his
famous account of the "if" of counterfactual conditionals
(Lewis 1973); and there is this use of "if" as a
restrictor. 



Kratzer's idea is that this last account of "if" as
a restrictor should be applied to all conditionals. Consider first
conditionals which contain a modal term: "If it's not in
the kitchen it must be in the bathroom/might be in the bathroom/is
probably in the bathroom". By analogy with Lewis, she argues
that these are not to be construed as attaching a modal term to a
conditional proposition; rather, they are to be construed as attaching
a modal term to the main clause, the scope of the modal term being
restricted by the conditional clause. 



But what of a simple conditional which does not contain a modal
operator, such as "If it's not in the kitchen it is in the
bathroom" -- what Kratzer calls the "bare
conditional"? Here is her famous remark: 



> 
> The history of the conditional is the history of a syntactic mistake.
> There is no two-place *if ... then* connective in the
> logical forms of natural languages. *If*-clauses are devices
> for restricting the domains of operators. *Bare conditionals have
> unpronounced modal operators* [my emphasis]. Epistemic
> *MUST* is one option. (Kratzer (1986), quoted from Kratzer
> (2012) p. 106)
> 



There is much in common between the restrictor-view of conditionals
and the suppositional view. A supposition also restricts one's
claim to the case in which the antecedent is true. The strength of
your conditional belief is measured by how probable you judge the
consequent, on the assumption that the antecedent is satisfied; and
this is not the same as thinking a conditional proposition is probably
true. Recall Lewis's remark about "the probability that
... if". Kratzer's treatment of modal conditionals
may be seen as a generalization to other modalities of this treatment
of "Probably, if \(A, C\)". 



However, Kratzer's treatment of the "bare
conditional" is controversial: at the level of semantic
structure, there really are no such things -- apparent bare
conditionals contain an "unpronounced modal operator". If
the modal operator is an epistemic "must", as she
suggests, bare conditionals are a species of strict conditional
-- something like "all live \(A\)-possibilities are
\(C\)-possibilities". 



This proposal has difficulty handling the fact that one may adopt
epistemic attitudes to a conditional of varying degrees of closeness
to certainty. I may be close to certain, but not completely certain,
that Jane will accept if she is offered the job, that if I have the
operation I will be cured, etc. Not all the relevant
\(A\)-possibilities are \(C\)-possibilities. On this proposal, in
these circumstances the conditionals are clearly, definitely false,
and should be completely rejected, and hence not something one should
be close to certain of. This point holds for any kind of strict
conditional -- any kind of "must". Stalnaker (1981 p.
100) made essentially the same point, about counterfactuals, comparing
his view with Lewis's. On a strict-conditional account, the
following exchange should be in order: 



> 
> A: Will Jane accept if she is offered the job?
>   
> 
> B: No, it is certainly not the case that she will accept if offered
> the 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 (ibid.) attributes to Thomasson the closely related point:




> 
> A: Will Jane accept if she is offered the job?
>   
> 
> B: I believe so, but she might not.
> 



Although B's reply seems sensible, it is defective on the
strict-conditional proposal: not all offer-possibilities are
accept-possibilities. The conditional is clearly false. One should not
believe something which one judges to be clearly false. 



Kratzer will reply that "I believe Jane will accept if she
offers the job" introduces a different modality: restricting
attention to the offer-possibilities, I believe (but am not certain)
that she accepts. Nevertheless, she cannot allow that one can take
different epistemic attitudes to the same conditional thought. In this
respect her view differs from the Ramsey-Adams approach, as well
as the propositional view.



Nor would it do to make the unpronounced modal operator in bare
conditionals "probably"; for one can be certain that it is
probable that if \(A, C\), without being certain that if \(A, C\).
This point is made in more detail by Edgington (1995, pp.
292-3). Thus, while the restrictor view has some plausibility,
its treatment of the "bare conditional" as a modalised
proposition is problematic. 



Other philosophers have also defended the view that indicative
conditionals are context-dependent strict conditionals, without
adopting Kratzer's restrictor view. According to Anthony Gillies
(2009), a context determines a set of possibilities compatible with
the relevant information in the context. "If \(A, C\)" is
true 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 and
relevant \(A\)-events are \(C\)-events. Context-dependent strict
conditionals are also defended by Daniel Rothschild (2013, 2015). The
difficulty mentioned above remains: on these accounts, a conditional
may be certainly false, yet probable. 


### 4.4 Heuristics and Semantics



I shall discuss only briefly a recently published book by Timothy
Williamson, *Suppose and Tell* (2020). Williamson accepts that
the suppositional procedure--suppose the antecedent, and on that
basis come to a judgement about the consequent--is our
fundamental, primary method of conditional judgement, an essential
part of our cognitive equipment. As uncertainty is often involved, he
accepts that it is appropriate to theorize about this process in terms
of conditional probabilities. This is what he calls the
*heuristics* of conditionals (not to be confused with
semantics). Heuristics are part of our cognitive and psychological
apparatus, "fast and frugal", immensely useful and
valuable, but typically imperfect. In this case, they lead to subtle
logical problems, he argues (ch. 3), broadly of the same kind as
Lewis's triviality results--indeed, to inconsistencies. An
assessment of these arguments will have to wait for another occasion.
Nevertheless, he claims, they are useful and valuable for our everyday
conditional thinking. 



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



Here is Gibbard's example: two henchmen, Zack and Jack, observe
a poker game, between Sly Pete and Mr Stone. Zack sees Stone's
hand, and signals its contents to Pete. He knows that Pete won't
call unless he has the winning hand. Jack sees both hands, and sees
that Pete does not have the winning hand. The room is cleared.
Outside, Zack hands a note to the boss which says "If Pete
called, he won". Jack hands a note to the boss which says
"If Pete called, he lost". Boss trusts them both, and
concludes that Pete did not call. But Boss cannot simply take over
both conditionals as suppositional judgements on the supposition that
Pete called: if one gets a high value, the other gets a low value.
(Nor can Boss think that in the nearest world in which Pete called, he
lost and he won.) Williamson has several variants of examples of this
structure. (As some have pointed out, Gibbard's example is
perhaps not perfectly symmetric, as Jack's judgement might seem
the most securely based; but plenty of perfectly symmetric examples
have since been given.) 



The truth-functional, material conditional turns out to be useful
here. The suppositional conditional entails the truth-functional
conditional. Zack is committed to "Pete called \(\supset\) Pete
won", i.e., either Pete didn't call, or he won. Jack is
committed to "Pete called \(\supset\) Pete lost", i.e.,
either Pete didn't call, or he lost. From both, it
straightforwardly follows that Pete didn't call. The
truth-functional conditional is the strongest *proposition*
which gets transmitted by conditional testimony. Testimony, at its
best, concerns the transmission of facts; and we can always resort to
the truth-functional conditional as a fact that gets transmitted by a
reliable conditional statement, when problems arise from the differing
background knowledge of our informants (as suppositionalists can
agree). 



This is one of the reasons why Williamson argues that the
*semantics* of the conditional is best treated as the truth
function. The semantics is not something which every speaker knows, or
is readily available to them. We do not learn to use "if"
via the truth table. And we know that no semantic theory of the
conditional is *obviously* correct! The probabilities generated
by the semantics are typically higher, and never lower than the
probabilities generated by our basic suppositional procedure for
assessing conditionals. Nevertheless, he argues, the truth-functional
semantics does the best job of rationalizing our overall practice.




Williamson gives other examples of useful but imperfect heuristics.
One concerns vagueness, and the so-called "tolerance
principles", such as "If \(n\) seconds after noon is
noonish, \(n+1\) seconds after noon is noonish". That is a
useful rule of thumb, though not all of its instances can be true. For
the epistemicist, one instance is false, but we don't know
which. On other views, all are at least very close to clearly true,
but not all are clearly true. Another example is the problem the Liar
Paradox presents for the principle that it is true that \(P\) if and
only if \(P\). In both cases, the exceptions are rare and can normally
be overlooked. He also mentions the heuristics involved in perceptual
judgements, normally very reliable, which sometimes lead us astray.
"Humans predictably resort to fast and frugal heuristics,
reliable enough under normal conditions, but not perfectly
reliable" he says, (p. 265). 



However, in the case of uncertain conditionals, it is hard to accept
that the heuristics are "reliable enough under normal
conditions", when combined with the truth-functional semantics.
When the conditional is certain, the suppositional procedure and the
truth function agree. They also agree in the relatively uninteresting
case in which the antecedent is certain. In all other cases, the
truth-functional conditional gets a higher value than the
suppositional conditional, and the difference between them can be
arbitrarily large. Many examples have already been given (see
SSSS2.3, 2.5, 3.1, 3.2). Another simple example: how likely is
it 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 material
implication, the answer is 2/3: if it lands 1, 3, 5 or 6, the
conditional is true. As already mentioned, all conditionals whose
antecedents are improbable, are probable as judged by the truth
function. 



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



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



The probability of snow is around 0.5; and we reckon (on the
suppositional procedure), it's around 0.2 that the road will be
impassable if it snows. According to the truth function, the
conditional gets 0.6. Then, as forecasts are updated, the probability
of snow decreases. Nothing else changes. On the truth-functional
semantics, the probability of our conditional goes up as the
probability of snow goes down: when the probability of snow goes down
to 0.25, the probability of (\*) is 0.8 on the truth-functional
reading, although it remains at 0.2 on the suppositional approach.
Williamson says that our suppositional procedure errs "on the
side of caution" (p. 104) by generating lower probability values
than the truth-functional conditional. It is hard to see why, as the
probability of snow decreases, we become more risk-averse, and make
larger errors on the side of caution. 


## 5. Compounds of Conditionals: Problems for the Suppositional Theory



A common complaint against Supp's theory is that if conditionals
do not express propositions with truth conditions, we have no account
of the behaviour of compound sentences with conditionals as parts (see
e.g. Lewis (1976, p. 142)). Probability theory is no help: conditional
probabilities never occur inside wider constructions. However, no
theory has an intuitively adequate account of compounds of
conditionals: we saw in SS2.4 that there are compounds which Hook
gets wrong; and compounds which Arrow gets wrong. Grice's and
Jackson's defences of Hook focus on what more is needed to
justify the *assertion* of a conditional, beyond belief that it
is true. This is no help when it occurs, unasserted, as a constituent
of a longer sentence, as Jackson accepts. And with negations of
conditionals and conditionals in antecedents, we saw, the problem is
reversed: we assert conditionals which we would not believe if we
construed them truth-functionally.



Some followers of Adams have tried to show that when a sentence with a
conditional subsentence is intelligible, it can be paraphrased, at
least in context, by a sentence without a conditional subsentence. For
instance, 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 some
constructions are rarer, and harder to understand, and more peculiar,
than would be expected if conditionals had truth conditions and
embedded 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 (SS4.3)
also involves paraphrase, so that conditional propositions are not
embedded in adverbs and operators.) But it would be better to have a
systematic solution to this problem, and there have several
attempts.



The first attempt to construct a theory of compounds of conditionals,
compatible with Supp's thesis, 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 good measure of one's degree of belief in a
conditional. To deal with compounds of conditionals, he proposed a
three-valued semantics for the conditional--true, false,
undefined. "If \(A, B\)" is true if \(A \amp B\), false if
\(A \amp{\sim}B\), and lacks a truth value--is undefined--if
\({\sim}A\). He called these semantic entities "conditional
events" or "tri-events". The probability of a
conditional is not the probability of its truth (which is just the
probability of \(A \amp B)\), but the probability of its truth given
that it is either true or false, which is just the conditional
probability of \(B\) given \(A\). Then de Finetti gave truth tables,
which accommodate compounds of conditionals. A conjunction is true iff
both conjuncts are true, false iff at least one conjunct is false,
otherwise undefined. A disjunction is true iff at least one disjunct
is true, false if both disjuncts are false, otherwise undefined.
Negation takes true to false, false to true, undefined to undefined. A
conditional with a false or undefined antecedent is undefined. And the
probability of a compound is also the probability 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 what happens if
\(A\) is false. But there are costs. On this account, to
believe/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 no
fault in a conditional that it has a false antecedent. I say "If
you press that button there will be an explosion". A disaster is
avoided, because, fortunately, my remark was not true. One might say
the normative dimension of truth has been lost. We have to give up the
equivalence between "If \(A, B\)" and "It is true
that 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 embedded
conditionals are not plausible. Mother says "If it doesn't
rain tomorrow we'll go to the beach, and if it rains we'll
go to the cinema". This rightly inspires confident expectations.
But on the present theory, as one or the other conditional has a false
antecedent, the conjunction cannot be true. Yet one or the other
conditional might be false, due to some unlikely contretemps, such as
illness, in which case the conjunction is false. So the probability
that it is true, given that it is either true or false, is 0. The
children are 99% confident that if \({\sim}R, B\), and 99% confident
that if \(R, C\), but, on this account, they should be 0% confident
that (if \({\sim}R, B)\) & (if \(R, C)\). (McGee (1989) raises
this objection to the account.)



A second approach gives "semantic values" to conditionals
as follows: \(1 (=\) true) if \(A \amp B\); \(0 (=\) false) if \(A
\amp{\sim}B\); \(\mathbf{p}\_A (B)\) if \({\sim}A\). See van Fraassen
(1976), McGee (1989), Jeffrey (1991), Stalnaker and Jeffrey (1994),
Sanfilippo et al. (2020). Thus we have a belief-relative three-valued
entity. Its probability is its "expected value". For
instance, I'm to pick a ball from a bag. 50% of the balls are
red. 80% of the red balls have black spots. Consider "If I pick
a red ball \((R)\) it will have a black spot \((B)\)".
\(\mathbf{p}\_R (B) = 80\)%. If \(R \amp B\), the conditional gets
semantic 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 a refinement of Stalnaker's truth conditions. Is the
nearest \(R\)-world a \(B\)-world or not? Well, if I actually
don't pick a red ball, there isn't any difference, in
nearness to the actual 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 into those in which "If \(R, B\)" is
true and those in which it is false. Instead the conditional gets
value 80% in all of them. The expected value of "If \(R,
B\)" is

 

\[\begin{align}
(\mathbf{p}(R \amp B) &\times 1) + (\mathbf{p}(R \amp{\sim}B) \times 0) +
(\mathbf{p}({\sim}R) \times 0.8)) \\
&= (0.4 \times 1) + (0.1 \times 0) + (0.5 \times 0.8) \\
&= 0.8 \\
&= \mathbf{p}\_R (B).
\end{align}\]


Ways of handling compounds of conditionals have been proposed on the
basis of these semantic values.



Some of the difficulties of the former account are avoided. Necessary
conditionals like "If \(A \amp B\), then \(A\)" are
true--get value 1. Conjunctions of the form (if \(A, B)\) &
(if \({\sim}A, C)\) do not all get 0. Indeed, conditionals this form
always get the value \(\mathbf{p}\_A (B).\mathbf{p}\_{{\sim}A}(C)\).
(The expected value of the conjunction if \(\mathbf{p}(A \amp
B).\mathbf{p}\_{{\sim}A}(C) + \mathbf{p}({\sim}A \amp C).\mathbf{p}\_A
(B)\), which simplifies to \(\mathbf{p}\_A
(B).\mathbf{p}\_{{\sim}A}(C)\).) Indeed, the stronger result holds,
that if the two antecedents are incompatible, the value of the
conjunction is the product of the two conditional probabilities. So
our above example about rain, beach and cinema indeed gets a high
value.



However, this result for conjunctions of conditionals gives some
implausible results. An example due to Mark Lance (1991) concerns a
werewolf, such that it's 50% likely that it is our area tonight.
If it is, it will kill everyone outside. "If John went out, he
was killed" gets 0.5. But "If John went out the back door,
he was killed, and if John went out the front door, he was
killed" gets 0.25 on this proposal, whereas it should still get
0.5. Another example, due to Richard Bradley: I must pick one of two
urns, only one of which contains the prize. "If I pick the left
urn, I'll win" gets 0.5. "If I pick the right urn,
I'll win" gets 0.5. On this proposal, "If I 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 get 0.



The flaw in this proposal is that the conditional gets the same value
in every possible situation in which the antecedent is false. In our
second example, in the "pick left and win" situation,
"If pick right, win" deserves 0, not 0.5, despite its
false antecedent, and in the "pick left and lose"
situation, "If pick right, win" deserves 1, not 0.5. In
the werewolf example, in the "goes out front door and is
killed" situation, "If back door, killed" deserves
1, not 0.5, despite its false antecedent. This point is made by
Bradley (2012), who has a different approach.



Bradley's account can be seen as a modification of
Stalnaker's (1968) theory discussed in SS4.1: 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 true in that possible world. First modification:
Bradley abandons the notion of similarity, or of minimal difference,
in favour of a probability distribution over the candidate A-worlds.
The example of the short straws in SS4.1 shows that this is a good
idea. Second modification: conditionals are not propositions. This is
a suppositional theory. Conditionals involve two propositions which
play different roles, one a supposition, one a judgement within its
scope. They cannot be represented by the set of worlds in which they
are true. Indeed, conditionals are not ingredients of
worlds--they are cross-world entities. Bradley proposes that the
conditional "if \(A, B\)" can be represented by the set of
*pairs* of worlds, \(\langle w\_i, w\_j\rangle\) such that, if
\(w\_i\) is actual, and \(w\_j\) is the "nearest"
\(A\)-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: "nearest" does not mean "most similar".
There is no ordering of worlds for similarity. It means the world that
would be actual if \(A\) were true. Often, we do not know which world
that is--hence the probability distribution over candidates.
Sometimes it might even be indeterminate which world that is, but
reflection again on the "short straws" example in
SS4.1 shows that the probabilities are still in order.



Two types of uncertainty, Bradley notes, are involved in assessing a
conditional--uncertainty about the facts--about which world
is actual; and uncertainty about what would be the case if some
supposition, which may be false, were true. These combine in a joint
probability distribution over the set of ordered pairs.



He accepts centering: if \(A\) is true, the "nearest"
\(A\)-world is the 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 \(B\) is
false; at \(w\_3, A\) is false. These generate the following
possibilities for the conditional if \(A, B\).




|  |  |  |  |
| --- | --- | --- | --- |
|  |  |  | If \(A, B\) |
| \(w\_1\) | \(A,B\) | \(\langle w\_1,w\_1\rangle\) | T |
| \(w\_2\) | \(A,{\sim}B\) | \(\langle w\_2,w\_2\rangle\) | F |
| \(w\_3\) | \({\sim}A\) | \(\langle w\_3,w\_1\rangle\) | T |
| \(w\_3\) | \({\sim}A\) | \(\langle w\_3,w\_2\rangle\) | F |



The probabilities of these four lines sum to 1.



The first two lines are the cases in which the antecedent is true, so
in those the "nearest" world is the actual world. If on
the other hand \(w\_3\) is actual, that does not tell us whether the
nearest \(A\)-world is \(w\_1\), in which case the conditional is true,
or \(w\_2\), in which case the conditional is false.



The crucial rule governing this non-propositional entity is this: the
probability of "if \(A, B\)" given \(A\), is the same as
the probability of "if \(A, B\)" given \({\sim}A\); the
probability of the conditional is independent of its antecedent. This
guarantees that \(\mathbf{p}\)(if \(A, B) = \mathbf{p}\_A (B)\).



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



With this machinery, the contents of conjunctions, disjunctions and
negations of conditionals are given in the usual way by intersection,
union and complements of the contents of the component sentences. When
a sentence has two conditionals, with two antecedents, such as those
of the form \((A\Rightarrow B) \amp({\sim}A\Rightarrow C)\), their
semantics requires not ordered pairs but ordered triples, \(\langle
w\_i, w\_j, w\_k\rangle\), such that, if \(w\_i\) is actual and \(w\_j\) is
the nearest \(A\)-world, and wk is the nearest \({\sim}A\)-world, the
conditional is true. As it is perfectly proper to give probabiity 0 to
the possibility: "I pick right and win, and in the nearest world
in which I pick left, I win", the problem 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's
probabilistic criterion of validity is demonstrable. If B is
true/false at all \(A\)-worlds, \(A\Rightarrow B\) is
straightforwardly true/false. Plenty of others may be
straightforwardly true/false, whether or not we know this. The many
uncertain conditionals come out with the right probability--the
conditional probability of \(B\) given \(A\). The construction is not
easy to work with, but it is a sort of possibility proof--that
the embeddings objection can be met. (Bradley also has a suggested
approach to conditionals within conditionals, but this has not been
worked out in detail.)



It is not ad hoc or unheard-of to claim that some kinds of content
cannot be represented by a set of worlds--the set of worlds in
which they are true. Some examples: to capture the content of
indexical thoughts using "I" and "now", we
need the richer notion of a "centred world"--an
ordered triple of a world, and individual and a time (see Lewis 1979).
Gibbard (1990) proposes that the content of a normative judgement can
be represented by a set of ordered pairs \(\langle w, n\rangle\) where
w is a world and n is a set of norms. Moss (2018) argues that the
contents of probability judgements are not propositions but sets of
probability spaces. And Bacon (2018) argues that the contents of vague
thoughts cannot be represented by a set of worlds. (He calls them
propositions nevertheless--that is a verbal issue--but they
are not propositions in the sense that is relevant here.)



I shall end this section by discussing a notorious example of Vann
McGee's (1985)--a counterexample to modus ponens. Before
Reagan's first election, Reagan was hot favourite, a second
Republican, Anderson, was a complete outsider, and Carter was lagging
well behind Reagan. Consider first


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



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


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



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



Suppose I'm close to certain (say, 90% certain) that Reagan will
win. 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 believe
that if Reagan doesn't win, Carter will win. As these opinions
seem sensible, we have a prima facie counterexample to modus ponens: I
accept (2) and (3), but reject (4). Truth conditions or not, valid
arguments 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 will
win". As I'm 90% certain that Reagan will win, I must
accept 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. For
Stalnaker,


5. If a Republican wins, then if Reagan doesn't win, Carter
will win



is true. To assess (5), we need to consider the nearest world in which
a Republican wins (call it \(w)\), and ask whether the conditional
consequent is true at \(w\). At \(w\), almost certainly, it is Reagan
who wins. We need now to consider the nearest world to \(w\) in which
Reagan does not win. Call it \(w'\). In \(w'\), almost certainly,
Carter wins.



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



Supp can save modus ponens by denying that the argument is really of
that form. "\(A\Rightarrow B\); \(A\); so \(B\)" is
demonstrably valid when \(A\) and \(B\) are propositions. For
instance, if \(\mathbf{p}(A) = 90\)% and \(\mathbf{p}\_A (B) = 90\)%
the lowest possible value for \(\mathbf{p}(B)\) is 81%. The
"consequent" of (2), "If Reagan doesn't win,
Anderson will win", is not a proposition. The argument is really
of the form "If \(A \amp B\), then \(C\); \(A\); so if \(B\)
then \(C\)". This argument form is invalid (Supp and Stalnaker
agree). It is one of the many argument forms which do preserve
certainty, but do not preserve high probability. Take the case where
\(C = A\), and we have "If \(A \amp B\) then \(A\); \(A\); so if
\(B\) then \(A\)". The first premise is a tautology and falls
out as redundant; and we are left with "\(A\); so if \(B\) then
\(A\)". We have already seen that this is invalid: I can think
it very likely that Sue is lecturing right now, without thinking that
if she was seriously injured on her way to work, she is lecturing
right now.



I have put this in the terms of the kind of suppositionalist who
eliminates the embedded conditional by paraphrase. If we do develop a
systematic account of the embedded conditional, it is surely a
desideratum that (1) and (2) are equivalent. In that case, modus
ponens is fine for propositions, but not when applied to a consequent
which is not a proposition.



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


## 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 conditional
commands, questions, offers, promises, bets, etc.. "If he
calls" plays the same role in "If he calls, what shall I
say?", "If he calls, tell him I'm out" and
"If he calls, Mary will be pleased". Which of our theories
extends to these other kinds of conditional? 



One believes that \(B\) to the extent that one thinks \(B\) more
likely than not \(B\); according to Supp, one believes that \(B\) if
\(A\) to the extent that one believes that \(B\) under the supposition
that \(A\), i.e. to the extent that one thinks \(A \amp B\) more
likely than \(A \amp{\sim}B\); and there is no proposition \(X\) such
that one must believe \(X\) more likely than \({\sim}X\), just to the
extent that one believes \(A \amp B\) more likely than \(A
\amp{\sim}B\). Conditional desires appear to be like conditional
beliefs: to desire that \(B\) is to prefer \(B\) to \({\sim}B\); to
desire 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 very
small 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 most
likely way for \(({\sim}W\) or \(W \amp T)\) to be true is that I
don't win the prize. Nor is my conditional desire satisfied if I
don't win but in the nearest possible world in which I win, you
tell 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 a
position to make a conditional commitment to \(B\): to assert that
\(B\), conditionally upon \(A\). If \(A\) is found to be true, my
conditional 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 said
nothing. Suppose I say "If you press that switch, there will be
an explosion", and my hearer takes me to have made a conditional
assertion of the consequent, one which will have the force of an
assertion of the consequent if she presses the button. Provided she
takes me to be trustworthy and reliable, she thinks that if she
presses the switch, the consequent is likely to be true. That is, she
acquires a reason to think that if she presses it, there will be an
explosion; and hence a reason not to press it.



Conditional commands can, likewise, be construed as having the force
of a command of the consequent, conditional upon the
antecedent's being true. The doctor says to the nurse in the
emergency ward, "If the patient is still alive in the morning,
change the dressing". Considered as a command to make
Hook's conditional true, this is equivalent to "Make it
the case that either the patient is not alive in the morning, or you
change the dressing". The nurse puts a pillow over the
patient's face and kills her. On the truth-functional
interpretation, the nurse can claim that he was carrying out the
doctor's order. Extending Jackson's account to conditional
commands, the doctor said "Make it the case that either the
patient is not alive in the morning, or you change the
dressing", and indicated that she would still command this if
she knew that the patient would be alive. This doesn't help. The
nurse who kills the patient still carried out an order. Why should the
nurse be concerned with what the doctor would command in a
counterfactual situation?



Hook will reply to the above argument about conditional commands that
we need to appeal to pragmatics. Typically, for any command,
conditional or not, there are tacitly understood reasonable and
unreasonable ways of obeying it; and killing the patient is to be
tacitly understood as a totally unreasonable way of making the
truth-functional conditional true -- as, indeed, would be
changing the dressing in such an incompetent way that you almost
strangle the patient in the process. The latter clearly is obeying the
command, but not in the intended manner. But it is stretching
pragmatics rather far to say the same of the former. To take a less
dramatic example, at Fred's request, the Head of Department
agrees to bring it about that he gives the Kant lectures if his
appointment is extended. She then puts every effort into making sure
that his appointment is not extended. Is it plausible to say that this
is doing what she was asked to do, albeit not in the intended way?



Extending Stalnaker's account to conditional commands, "If
it rains, take your umbrella" becomes "In the nearest
possible world in which it rains, take your umbrella". Suppose I
have forgotten your command or alternatively am inclined to disregard
it. However, it doesn't rain. In the nearest world in which it
rains, I don't take my umbrella. On Stalnaker's account, I
disobeyed you. Similarly for conditional promises: on this analysis I
could break my promise to go to the doctor if the pain gets worse,
even if the pain gets better. This is wrong: conditional commands and
promises are not requirements on my behaviour in other possible
worlds.



Among conditional questions we can distinguish those in which the
addressee is presumed to know whether the antecedent is true, and
those in which he is not. In the latter case, the addressee is being
asked to suppose that the antecedent is true, and give his opinion
about the consequent: "If it rains, will the match be
cancelled?". In the former case -- "If you have been
to London, did you like it?" -- he is expected to answer
the consequent-question if the antecedent is true. If the antecedent
is false, the question lapses: there is no conditional belief for him
to express. "Not applicable" as the childless might write
on a form which asks "If you have children, how many children do
you have?". You are not being asked how many children you have
in the nearest possible world in which you have children. Nor is it
permissible to answer "17" on the grounds that "I
have children \(\supset\) I have 17 children" is true. Nor are
you being asked what you would believe about the consequent if you
came to believe that you did have children.



Widening our perspective to include these other conditionals tends to
confirm Supp's view. Any propositional attitude can be held
categorically, or under a supposition. Any speech act can be performed
unconditionally, or conditionally upon something else. Our uses of
"if", on the whole, seem to be better and more uniformly
explained without invoking conditional propositions.