SORITES, ISSN 1135-1349
Issue #14 -- October 2002. Pp. 7-15
Copyright © by SORITES and Simon Salzedo
On the Analysis of Conditionals
There is some analogy between the conditional and the material implication (MI) which is defined by a two-valued, binary truth table to be false in the case of a true antecedent and a false consequent and to be true otherwise. Yet the assertibility of a conditional is not always coextensive with the truth of the equivalent MI. In particular, neither the falsity of an antecedent nor the truth of a consequent are sufficient to warrant the assertion of a conditional in which they feature, yet either guarantees the truth of such an MI. Such worries have lead many to despair of saying just what the link is between conditionals and material implications.
It has also been claimed that there are two types of conditionals each of which requires a different analysis. For example, it is said that
are fundamentally different which can be judged from the fact that someone who asserted 1) could easily deny 2). Therefore, it is argued, we need analyses which will allow us to call 1) true while calling 2) false. Thus many have despaired of understanding conditionals as one type of assertion.
My aim in this essay is to overcome the dual despairs just described by defending the following claims:
Following Grice, I hold that there is a `Cooperative Principle' in conversation, that each contribution should be appropriately informative, true, backed by evidence, relevant, clear and perhaps other things. Where the application of this principle to what is said in some utterance entails some unsaid proposition, that unsaid proposition is a conversational implicature of the utterance. An important feature of conversational implicatures is that they may be cancelled by the speaker or by their context. Cancellable implicatures can also be attached to particular forms of utterance, which may be termed generalised conversational implicatures.
An utterance which breaks the Cooperative Principle, or which has false implicatures, is not assertible without some indication of its peculiarity. Most true sentences are not assertible most of the time because they are not relevant or not the most informative available. In any given situation, though, there will be many possible utterances which would not break any part of the Cooperative Principle and which are assertible.
Where all that the speaker wishes to convey is some assertion, a conditional with that assertion as the consequent or its negation as the antecedent will not normally be assertible. This is for the simple reason that neither the truth of the consequent nor the falsity of the antecedent follow from the conditional by normal truth-functional rules. Even though such a conditional is true, it is generally not assertible in this context because it is not appropriately informative. An exception to this generalisation demonstrates how it works: «If this is justice then I am a banana» succeeds in conveying the unassertibility of its antecedent because its consequent is known by all parties to the conversation to be not merely false, but chosen for the obviousness of its falsity.
Conditionals suggest some connection between antecedent and consequent. More specifically, I want to propose that «If A then C» carries a generalised conversational implicature of «If not A then not C». As an example, consider how «the cradle will rock anyway» can act as a qualification (which cancels the implicature) when conjoined with «if the wind blows the cradle will rock» or as an objection to the same sentence when not qualified.
Conditionals like 1) are asserted when it is appropriate within the conversation to discuss the consequences of the antecedent. Either the antecedent is believed to be true or supposed to be so for the purposes of a reductio. Such conditionals accordingly implicate that their antecedents are assertible. Consider the contrast between «If Booth killed Lincoln, and I think he did,...» and «If Booth killed Lincoln, though I am not saying that he did,...». The conjunctions in these and many similar sentences are not comfortably interchanged because «though» or «but» are only appropriate for indicating the cancellation of an implicature or another change of course. This implicature is easily cancelled. For example, a heavy stress on the «if» in the above sentences may be enough to remove it, allowing «If Booth killed Lincoln, and I am not saying that he did,...»
Conditionals like 2), by contrast, implicate that their antecedents are not assertible. 2) is assertible in situations where there is an assumption in the conversation that Booth did kill Lincoln. This has been disputed by, for example, Appiah (see Appiah p164), who suggests that «Had the butler done it, there would have been blood in the pantry» can be part of an argument that the butler did it. But even here there is a presumption of the butler's innocence, the implicature of which is cancelled by a later stage of the argument. In isolation, the sentence suggests that the butler did not do it and that there was no blood in the pantry. What is missing from Appiah's argument is the conversational context for his example.
In deciding whether a conditional like 1) or one like 2) is more appropriate, what is important is not the speaker's belief regarding the antecedent's truth, but his belief about its assertibility in the given context. Often, the two will coincide; but where the speaker is seeking to overthrow a presumption of his listeners, he may adopt that presumption to present an argument by reductio. This is the most likely context for Appiah's example.
More generally, when a conditional «If A then C» is asserted, some presumption about A's assertibility has been adopted. The tense or mood words in the conditional indicate which presumption is in force, that is, whether A is or is not assertible in that conversational context. Thus, «The butler did not do it. If the butler had done it, there would have been blood in the pantry.» is a more natural utterance than «The butler did it. If the butler had done it, there would have been blood in the pantry.» The latter utterance demands the appendage «and there was blood in the pantry» to cancel the implicature of its second sentence and thus resolve that sentence's conflict with its first. The former utterance stands alone without confusion.
The implicature regarding the assertibility of «Booth killed Lincoln» is, I have argued, the most important difference between conditionals like 1) and those like 2). The question of taxonomy should therefore be approached by considering that implicature. Such an approach, restricted to the generalised implicatures of the different forms of conditional, reveals a continuum of tense and mood combinations. Examples of the most discussed such combinations might be:
6) is more or less neutral, as future events can be discussed without any presumption about whether they will actually occur. Since the first part of 7) is the same as that of 6), the test we are using reveals the same neutrality. This is reasonable because conversations about general rules normally presume that the antecedent is true in some but not all instances which are being considered in the conversation.
If 6) lies between 3) and 8) in the sense that I have suggested then the debate about with which of the two groups it should be placed can be seen to be misguided. The debatability of the point is, however, not surprising.
In this taxonomy, the implicature is decisive, and the generalised implicature of an isolated sentence may be overridden by elements of the context. And the neutral 6) may be used when an assumption about the antecedent's assertibility is determined by its context. The tense and mood words are not themselves definitive of the type of conditional; they merely indicate it in the absence of other signals.
For formal purposes, the conditional is translated into an MI with a defined truth table; the truth of the antecedent and consequent are assessed and the truth value of the MI falls out from the truth table.
In non-formal settings a hearer who questions an assertion questions its assertibility. This may be on any of several grounds two of which are that the speaker does not or should not believe it to be true. That is, the speaker is lying or is mistaken. But while falsity leads to non-assertibility, truth does not lead to assertibility; put another way, truth is a necessary but not sufficient condition of assertibility.
A different ground for questioning an assertion is that it is misleading. An assertion may be misleading if it makes conversational implicatures which are false. In this way, 2) is often misleading when asserted by a speaker who believes that Booth did not kill Lincoln because it implicates, in the absence of other information, that Booth did kill Lincoln.
When we do wish to assess the truth of a conditional in an informal context, something very similar to the formal case takes place so long as the antecedent is presumed to be true. Thus, if Booth did not in fact kill Lincoln, then both 1) and 2) share the truth value of «somebody else killed Lincoln».
If, however, Booth did kill Lincoln, then it is less obvious what would determine the truth of 1) and 2). We can imagine two people who disagree about the truth of 1) or of 2). Then suppose that the falsity of «Booth did not kill Lincoln» was established to the satisfaction of both of them. Now their debate over 1) or 2) would take on a different character. It is about opinions and no longer about facts. They agree that Booth killed Lincoln and so they agree that somebody else did not. Any remaining argument is not so much over the truth of 1) or 2), but over the evidence for those sentences. The fact that such a debate is possible is, therefore, not an objection to the claim that all conditionals with false antecedents are true. In the same way, we can stipulate that all conditionals with true consequents are true without thereby closing off the chance of understanding why only some of them are assertible. The line drawn here between opinion and fact may seem arbitrary. But there are distinctions to be made between misleading, misjudging and lying which alternative schemes would struggle to accommodate.
As noted above (Section I), it is often claimed that conditionals like 1) and 2) are different in some fundamental other than an implicature about their antecedent.Foot note 1_4 But the differences between 1) and 2) are due to the different attitude towards «Booth did not kill Lincoln» of which the implicature is an expression.
With 1), the conversation is on the basis that in the actual world Booth did not kill Lincoln. We can then consider the fact as to whether or not someone else killed Lincoln. This fact about the past determines the truth of the conditional. In 2), there is a presumption that Booth did kill Lincoln. We must then consider the question of how things would be different for other people killing Lincoln if they had been different for Booth doing so. There is no fact about the past or present (or even future) available to help us here and we do not look for one. Where the antecedent is presumed false, we almost invariably lose interest in the truth of the conditional. We swap truth for other standards of assessment and the reason for the change is our assumption of the unassertibility of the antecedent.
There is a further question which may be raised here about whether conditionals like 2) have truth conditions at all. An approach which allows some other elements than truth in assertibility is essential both to explain the non-assertibility of statements whose truth no one would deny (many theorems of mathematics or formal logic, for example, are assertible in only a tiny number of situations) and to explain the wrongness of misleading someone without telling a direct lie. Yet we cannot ignore the fact that truth-conditional falsity always bars assertibility. «Booth did not kill Lincoln, and neither did anyone else» is a clear denial of 2) just because it entails the falsity of 2). This suggests ascribing to all conditionals the truth conditions of MI which can be seen as necessary but not sufficient for their assertibility. That the implicature of a false antecedent carried by some conditionals is enough to direct the hearer to considerations other than truth does not mean that there is no use for truth conditions. First, if the antecedent of a conditional like 2) turns out to be true, then the truth of the conditional is decided in the usual, truth functional way. Secondly, in formalising we are always interested in truth, and it would be churlish to refuse to allow this notion of truth to decide the matter when there is no conflicting interest.
AdamsFoot note 1_5 points out that there are truth-functionally valid inferences of which the premises may be assertible while the conclusion, a conditional, is not. According to Adams, his examples are not convincingly explained by conversational implicature. Based on the account given above, Adams' examples admit of the following explanations.
(i) Adams takes two sentences, A and B:
A. It will not rain in Berkeley next year.
B. It will rain in Berkeley next year.
He says that we would not infer «If A then B» from not A or from B. In this case, A and B entail each other's negation. There is on my version of the conversational theory a particular objection to the assertion of «If A then not A» which is that it has an implicature of «If not A then not not A» which entails A which is false. This is on top of the question of why we should want to make such an inference. Outside a book of logic puzzles it is hard to imagine being interested in the question of whether to accept that conditional or its negation beyond deciding between A and not A.
(ii) Adams says that for the following A and B we might assert «If B then not A» but would not infer the contraposition, «If A then not B»:
A. There will be a terrific cloudburst tomorrow.
B. It will rain tomorrow.
There is here an additional premise, «If A then B,» which is implicit in any discussion of cloudbursts. If we do assert «If B then not A» we mean to convey «not A» and this is done by the normal truth-functional rules according to which «not A» follows from the two premises. Given «not A», we would not assert «If A then not B» despite its literal truth because the presumption of not A leads us to consider other things than truth. We have no interest in making purely logical deductions about the consequences of false antecedents. In these cases we move on to consider other evidence than facts and other standards of assessment than truth.
(iii) Adams says that we do not reason from «A or B» to «If not A then B» with the following A,B:
A. It will rain in Berkeley next year.
B. It will snow in Berkeley next year.
The implicature of connection which comes with the conditional is one reason why not. «If not A then B» implicates «If A then not B» and so we have «A or B but not both» which is most likely to be unassertible in this case. A related reason is the implicit premise «A» which turns «If not A then B» into a conditional the truth of which derives from the falsity of its antecedent and so is not a good standard of assessment.
(iv) Adams has two examples of hypothetical syllogism. In the first these sentences are used:
A. Smith will die before the election.
B. Jones will win the election.
C. Smith will retire after the election.
Adams notes that we do not move from the premises «If A then B» and «If B then C» to «If A then C». The reason is simple: the two premises are not assertible together in any one context. Once the question of Smith dying before the election is raised, we would not assert «If B then C» until we had decided that «not A» was assertible. The high probability of each of the premises is not an issue in deciding whether to assert them. In conversation, the important considerations for judging conditionals for which the facts are not (yet) available are the ones which are raised in the conversation or are taken as read by the parties.
In his other example, Adams has
A. Jones will study.
B. Jones will pass.
C. Jones will graduate.
Here we can move from «If A then B» and «If B then C» to «If A then C». The reason is not Adams', which is that the first premise is implicit in the second. It is that there is no conflict here. The truth functional rules do apply when there are no inconvenient implicatures or antecedents presumed to be false.
(v) Finally, with A,B,C as for the Smith and Jones election example, Adams says that we cannot reason from «If (A or B) then C» to «If A then C», noting that the premise is, although highly probable, intuitively absurd. This absurdity is what Adams claims cannot be explained by implicature.
But on the conversational theory, high probability is not even a prima facie reason for assertion. «If (A or B) then C», when «If A then not C» is assertible, is odd because it is uninformative with no commensurate gain in clarity or brevity. What is meant is «If B then C» and if we do not introduce probability, then no urge to assert «If (A or B) then C» emerges.
An objection to such explanations is that we do not always assert the stronger, or more informative, of particular options. Jackson, for example, says that «If the sun goes out of existence in ten minutes time, then the earth will be plunged into darkness in about eighteen minutes time» is assertible despite being only marginally more probable than the negation of its antecedent which would be strongerFoot note 1_6. But we can consider the consequences of antecedents which we take to be false regardless of how probable their falsity is. And in any case, with the implicature of connection between consequent and antecedent, a conditional is not always less informative, even though it may be logically weaker, than the negation of its antecedent. The cases where a conditional is unassertible due to being uninformative are those where the implicatures which go with the conditional are not appropriate, perhaps because the negation of the antecedent is all that is meant. For Jackson's example, there will be some occasions on which the conditional will be assertible, and others on which the negation of the antecedent will be. What decides between these types of occasion is the content of what is being discussed. Assertions should be relevant as well as informative.
Appiah ( Ch8) also adduces an example against the principle of asserting what is more informative rather than what is less. He suggests that I can say «John or Mary will arrive soon» while believing that it is far more likely to be John. The circumstances of this utterance are that either one of the couple is required for a meeting. While the assertion is logically weaker than «John will arrive soon», it is more relevant because the interest is in the arrival of «John or Mary» as an entity. In another context in which John and Mary have different roles in the meeting, Appiah's sentence would be misleading due to being uninformative, because it would then be relevant to consider whether or not John will arrive soon.
Other conditionals which have been thought problematic for a unified conversational account are those like «Your essay is a bit short this week, if you don't mind me saying so» or «If you are hungry, there are biscuits on the table.» These illustrate well that English conditionals deal in assertibility and not just truth. If you do mind or are not hungry, then I would not assert the respective consequents of those conditionals. I do not withdraw my belief in or assent to the consequents, only my assertion of them.
The account I have given of conditionals may be criticised for vagueness. Other accounts like those of Adams or Lewis give more specific criteria for the assertibility of conditionals in terms of probability and spheres of possible worlds respectively.
Adams suggests that inferences should be assessed by the principle that it should be impossible for the premises to be probable and the conclusion to be improbable. The probability of a conditional, «if A then C», is held to be the probability of C given A, p(C/A), which is p(A & C)/p(A). Where this probability is high, broadly speaking, the conditional is assertible. This account is only meant to apply to conditionals like 1).
The attraction of this account is due to its attempt to make clearer the nature of the connection between antecedent and consequent which is indicated by the use of a conditional. It makes the connection into something within the grasp of logicians. However, it takes it out of the grasp of the users of English conditionals. Competent speakers do assess assertions for relevance, informativeness and support by appropriate evidence (even though they may be unable to describe these practices of assessment in much detail), but hardly ever for high probability. The reason why they do not poses another problem for a probabilistic scheme. In so far as the account is restricted to conditionals about past events, all probabilities involved are 1 or 0, and the criteria do not diverge from truth-conditional criteria. In so far as we are considering future events or general laws or counterfactual events, it is hard to see how the relevant probabilities could be calculated short of providing a complete description of the universe at the relevant time along with a probabilistic theory of changes to it through time. It might be neater if our practice of assessing evidence was based on measuring or intuiting probability, but for most purposes it is not plausible to say that it is.
Lewis suggests that conditionals like 2) are true just when the relevant MI holds at all of the nearest possible worlds to the actual world. This is a plausible working out of the circumstances in which these conditionals are assertible because it retains the vagueness of the conversational view. It seems attractive because it provides a formal framework for their manipulation which avoids the perception of awkwardness involved in counting all conditionals like 2) as true so long as their antecedents are false.
The central difficulty with possible worlds is that it is no easy matter to specify which worlds are closer. The nearest world in which Booth did not kill Lincoln is that in which what is changed from the actual world is just that which the parties to the conversation recognise as needing to be changed. We do not have to build in our minds a fully working, consistent model of a possible world. We just notice those aspects which interest us for the moment. Attributes of the possible worlds are therefore inescapably context relative. Closeness of worlds can be felt within a conversation, but cannot be objectively measured by logicians outside it. For this reason, possible worlds provide only a framework for analysis, and not much content.
Our ability to use conditionals with impossible antecedents is an additional problem for Lewis's approach. We can use sentences like «If Godel's Incompleteness Theorems had been false, then formalism would have been more attractive». This use is not explicable within Lewis's framework, which must be limited to conditionals with antecedents presumed to be false, but which are possible.
There is also a danger of a regress in analysing conditionals by possible worlds. Full descriptions of worlds could not be given without using conditionals like 2) which bring in a new infinity of possible worlds and so more conditionals and so on.
Assertibility is best viewed as being a property of an utterance by a particular person in the context of a particular conversation. It should be ascribed to conditionals which do not breach the Cooperative Principle and which do not carry uncancelled implicatures which are unassertible. Truth is one of the requirements of the cooperative principle and is ascribed to each conditional as it would be for its equivalent MI.
The conditionals which are relevantly like 2) are those the antecedents of which are assumed in the conversation to be false. Because of this assumption, there is a special problem in saying what kind of evidence is appropriate for supporting such conditionals as there is no fact which could decide them. The solution to this problem would be a description of our practice of assessing such statements for support, coherence or plausibility. This practice does not reduce to a formal analysis and it is therefore not possible to use a description of it to replace the MI notion of truth. The attempt is also misguided because assertibility depends on other factors as well as truth.
I have not given the full content of an analysis of conditionals here. What I have tried to do is to suggest that the Gricean conversational framework is adequate to contain such an analysis and that the temptation towards more precise, but restricted, analyses should be resisted. All conditionals, like other assertions, must adhere to the Cooperative Principle and this is the basis for their assertibility. The unusual implicature of the conditional form is of a connection between the antecedent and consequent. I have suggested that this connection should be specified as being that «If A then C» implicates «If not A then not C».