Conditionals and Counterfactuals

The word "if" is apparently one of the least significant in the English language, but it proves very difficult to define. This essay considers the definition posed by the ancient Greek philosophers, and by two philosophers of the twentieth century, and shows the inadequacy of each. I suggest that misuse of the word has led to some major misunderstandings, and I attempt to show how we can avoid such misuse.

INTRODUCTION

"If the switch is down the lamp will be alight." "If John lives in London he must live in England." "If the train is late I shall complain." "If Peter had come he would have won a prize." "If you had never been born I should now be the boss."

Statements of this form are called "Conditionals". Each consists of two propositions, the first, the "if" proposition, being known as the "antecedent", and the second as the "consequent". We frequently make assertions of this type, but we seldom ask ourselves exactly what they mean. Several subtle problems are associated with them, and philosophical confusion can arise from their misuse.

The last two of the examples quoted in the above paragraph are of a fundamentally different type from the others, in that the antecedent is known to be false; we know that Peter did not come, and that the person we are talking to has been born. Such conditionals are called "Counterfactuals", or "Subjunctive Conditionals", and I shall attempt to show that in some circumstances they are indeed meaningless.

Any conditional can be expressed in the form "If p then q", where p and q are assumed to be propositions which are bivalent, i.e. to be either true or false. We are not concerned with propositions which can show degrees of truth. For brevity and clarity I shall write "p -> q" to stand for "if p then q"; we say that p implies q, and that the relationship is an example of "implication". Thus p may stand for "John lives in London", while q may stand for "John lives in England". Clearly p -> q is true in this case, while q -> p is not. Notice that each of these relationships is itself a proposition, whose truth or falsehood depends upon the relationship that exists between p and q.

CONDITIONALS IN ANCIENT GREECE

The basic principles of deductive logic were laid down by the Greek philosophers nearly 2400 years ago. Like many aspects of their teaching, this represented a remarkable achievement in its day, but the reverence which it inspired outlived its usefulness, and hindered the development of logic for many centuries. The treatment given to this concept of "implication" illustrates well the deleterious effect of their teaching. To the Greeks, "implication" was a relationship wholly analogous to "conjunction" and "disjunction". The conjunction of two propositions is the new proposition arising by interposing AND between them, their disjunction is obtained by interposing OR, while implication arises when IMPLIES is interposed. Thus we may say "Athens is in Greece AND Rome is in Crete" (which is false), or "Athens is in Greece OR Rome is in Crete" (which is true), or "Athens is in Greece IMPLIES Rome is in Crete" (which is false).

The meaning of each of these three relationships is most clearly shown by a "truth table". For brevity we write "1" for TRUE and "0" for FALSE. The first table below shows that "p AND q" is true only when p and q are each true, and the second that "p OR q" is true unless p and q are both false.

At first sight the third table also seems plausible. It purports to illustrate (in modern notation) the ancient Greek interpretation of the proposition "If p then q", and which has come to be known in modern times as "material implication". When p is false this relationship tells us nothing about q, which may be either true or false, as the first two rows of the table indicate. When p is true, however, then q must be true, as the fourth row shows. With p true and q not true, then the relationship p -> q must be in error, as is shown in the third row.

But to clarify the position, let us consider a simple arithmetical example. Suppose x represents any whole number, and we define p and q as follows:

The observant reader will notice here that, strictly, p and q are not propositions, but "propositional functions" (as defined by Russell, see [1]), for they incorporate the variable x rather than a specific number such as 9, or 10 (say). But each can be converted into a proposition by assigning a particular value for x, and this enables us to obtain different combinations of truth and falsehood without having to redefine p and q. To get the four rows of the above truth tables we shall take the following values: (i) x = 11, (ii) x = 15, (iii) x = 16, and (iv) x = 18. Substituting these values for x in the above definitions of p and q will be seen to give, respectively, propositions whose truth or falsehood form the four rows of the truth tables quoted earlier. Let us see what the first table now tells us:

(11 is divisible by 2) AND (11 is divisible by 3) is FALSE
(15 is divisible by 2) AND (15 is divisible by 3) is FALSE
(16 is divisible by 2) AND (16 is divisible by 3) is FALSE
(18 is divisible by 2) AND (18 is divisible by 3) is TRUE

The reader will be easily satisfied that this is all correct. It is also easily seen that these x values serve to verify the second table, and illustrate the meaning of "p OR q".

Now see what the third table gives:

(11 is divisible by 2) IMPLIES (11 is divisible by 3) is TRUE
(15 is divisible by 2) IMPLIES (15 is divisible by 3) is TRUE
(16 is divisible be 2) IMPLIES (16 is divisible by 3) is FALSE
(18 is divisible by 2) IMPLIES (18 is divisible by 3) is TRUE

With the possible exception of the third row, this is all nonsense. Even the last row, the one that should really contain the essence of the p -> q relationship, does not say what it should. The two propositions in brackets are both true, but the implication itself is not. It is not because 18 is divisible be 2 that it also divides by 3.

Indeed with this interpretation, whenever p is false, p -> q is true, and whenever q is true, p -> q is again true. So if the moon is made of green cheese then it follows both that it is made of chocolate (which is false), and that it is made of volcanic rock (which is true). And if Shakespeare wrote Hamlet (or even if he didn't), it follows that two and two make four. Modern logicians would never have subscribed to such confusion were they not still in awe of the old Greeks who devised this interpretation in the fourth century BC.

We see that p -> q is not the sort of proposition whose truth value depends upon the truth values of p and q, as do those of p AND q and of p OR q. A truth table is not an appropriate way to illustrate this relationship, for it suggests incorrectly that the truth value of an implication changes according to the truth values of its two component propositions. But if we know that p implies q, then we know it because of some other argument, and it remains true irrespective of whether p and q are true. If John lives in London then this implies that he lives in England, and this implication remains true even if John does not live in London.

AN ANALYSIS

How then should we define the concept of implication, "if p then q", and yet avoid a meaningless or anomalous formula? We note firstly that the word "if" is merely a part of language, a tool used by humans for communicating and storing information. Any anomalies that we detect tell us nothing about the outside world; they certainly do not mean that Nature herself shows inconsistency. Like any tool, language must be used correctly or it may cause damage. So what rules should we obey when using this form of language if the resulting expressions are to be meaningful? The essential requirement is that p must be a proposition which can conceivably be true and conceivably false. In fact, p will often be not a proposition but a propositional function which can take many values, some of which are true and some false. So we can say, "If 2x = 6 then ... ", which is meaningful whatever numerical value we give to x, although it is true for only one value of x. Or we can say, "If a man lives in London ...", which is meaningful if we substitute any man's name in place of "man", but true for only some of these. The "if" clause divides the whole class of men into two groups, those who live in London and those who do not. Alternatively, p may be a particular case of such a propositional function, like "John lives in London". For John himself, the proposition is either true or false, but we think of John as a particular member of the class of men, for some of whom it is true and for some false. The meaning of the implication is now clear. The group of men for whom the antecedent is true (those living in London), lies wholly within the group for which the consequent is true (those living in England). For a conditional to be meaningful, the "if" clause must divide some class of individuals into two groups, for one of which it is true and for the other it is false. And for all individuals in the first group, the "implied" clause must also be true.

When p is not a proposition of this type, but is necessarily always true or necessarily always false, we ought not to use p as the antecedent of a conditional. Thus we should not say, "If two and two make four...", or, "If two and two make five...". To do so is to abuse our language and to risk talking nonsense.

There are, however, two exceptions to this rule. If a proposition must necessarily be true, or necessarily false, but we do not know which, then we may legitimately use it as an antecedent. Thus we may say, "If 97486 divides exactly by 1234 ...". In fact this proposition is true, but people who do not know this may reasonably utter the phrase as part of a conditional. Secondly, even when we know whether a proposition is true or false we can temporarily feign ignorance in order to pursue an argument by reductio ad absurdum. Thus we can begin a proof that there is no highest prime number by asserting, "If there is a highest prime number, then ...". (Notice that we do not say, "If there were a highest prime number", for this would reveal our knowledge that there is not.) We discuss this topic more fully below.

One consequence of the uniformity of Nature is that frequently one particular state of affairs is always associated with some other particular state of affairs. And one aspect of human intelligence is our ability to notice such associations, and to understand and explain them. So we often know that the truth of a proposition p necessitates the truth of some other proposition q. There are many different ways in which we can become aware of such associations. If 2x = 6, then our understanding of mathematics allows to assert that x = 3. If a switch is "on", our understanding of electrical circuitry decrees that the lamp will be alight. Knowledge of our own intentions enables us to predict that the late arrival of the train will be followed by our own complaint; and our knowledge of geography convinces us that the fact of John's living in London entails his living in England. In cases where we do not understand the reason, we can still often be sure one fact entails another because similar situations in the past have always shown such an association. So we can say, "If you drink that substance you will be poisoned", even though we do not know the chemical and biological reactions which make it true.

To summarise the conclusions of the above paragraphs, we have the following rules which must be satisfied if we are to assert legitimately, "If p then q".

(i) It must be conceivable for the antecedent p to be true, and conceivable that it be false (with a few exceptions in special cases).
(ii) Some elements of our knowledge or intelligence convince us that the truth of p necessarily entails the truth of q.
(iii) Our assertion carries no information concerning the implication of the falsehood of p.
(iv) The truth of the assertion depends upon the relationship of p to q, and not upon their truth or falsehood in particular cases.

ERNEST ADAMS

A very different interpretation is presented by E. W. Adams in his book "The Logic of Conditionals" (see [2]). He regards a proposition as possessing not only a truth value (true or false), but also a probability. He gives no definition of probability, but presumably he assumes the usual philosopher's definition of "degree of rational belief". He also overlooks the dependence of probability on a person's state of knowledge, or on what is "given". He supposes that a proposition can have an intrinsic probability, independently of anyone's knowledge or ignorance.

Adams writes (p.2), "The probability of a proposition is the same as the probability that it is true". Now as it stands this is a valueless truism; presumably what he means is, "The probability of a propositional function is the same as the probability that its particular instances are true propositions". But I reject this approach, for it involves a distortion of the true meaning of "probability". Consider the statement "All men die before reaching the age of 100". If, in fact, 99.9% of all men die before they are 100, then the probability that the statement is true, taking "all men" as our sample space, is 0.999. So Adams claims that the probability of the statement itself is 0.999, and this I reject. The statement makes an assertion which actually is false; it is untrue that all men die before age 100. If one must assign to it a probability, the value is 0.

In describing the purpose of his book, Adams states on p.3 that his fundamental assumption is: "The probability of an indicative conditional of the form 'if A is the case then B is' is a conditional probability". Adams writes "A => B" to represent this particular interpretation, and reiterates his assertion by quoting the well-known formula for conditional probability:

I reject this as explained, because a conditional assertion is either true of false. If it is true only in some cases, then it is false, and describing this degree of falsity as a probability is an inappropriate use of the term. He calls his principle an "assumption", but it is really a "definition" of a probability in a context where probability cannot be applied. If the "probability of a conditional" has any meaning, then it must, like the "truth value of a conditional", be an intrinsic characteristic of that conditional, and must not depend upon the probability or the truth of the individual propositions, p and q.

Consider the statement, "If a man was born on a Friday, then he will die on a Friday". Adams would claim that the probability of this assertion is 1/7. But the antecedent does not imply the consequent, and so the assertion is false. The essence of a conditional is implication; often we can find reasons for the antecedent necessarily implying the consequent, and in other cases we rely on induction to convince ourselves of the implication. In the present case there is no implication, and it is an error to quote a probability value.

Adams does show (by obscure and doubtful reasoning) that assigning probabilities to conditionals is inconsistent with assigning truth values to them. He accepts the former and rejects the latter, but I maintain it is the latter we should accept.

He considers also (p.4) the assignment of probabilities to conditionals on the basis of the material implication which we defined earlier. He finds that the probability defined in this manner is never less than that defined in his previously quoted manner, and dismisses it because of the "notorious fallacies of the material implication". I dismiss it for the same reasons that I quoted above for the dismissal of p(A=>B).

Adams' book considers many other aspects of conditionals and counterfactuals, but I feel it makes little contribution to the solution of outstanding problems.

COUNTERFACTUALS

It is not surprising that many people hold strong views on the question of abortion, and that these views differ so widely. On the one hand we have those who maintain that a human foetus from the moment of conception is a human being, and that it is always wrong to take a human life. On the other hand there are those who believe that life begins at birth; while still unborn the embryo is unconscious, unfeeling and unknowing, and deserves no more respect than we show to a fish or a fly, whose life we do not hesitate to terminate when we can thereby contribute to human welfare and comfort. Between these two positions are many shades of opinion; some would allow abortion if the life of the mother were in danger, or her well-being, or her happiness, or even just her convenience. Others would allow it up to four weeks after conception, or eight, or eighteen. This position is clearly a compromise made by people who find the question difficult. While such middle views are perhaps the most commonly held, they are less easy to justify than either of the extreme positions.

It may be thought that those who would forbid abortion under any circumstances are motivated solely by their concern for human life in general. But it seems likely that, consciously or unconsciously, another thought lies behind their belief; it must have occurred to them that, if their mother had had an abortion, then they themselves would not exist. In effect, for them the whole universe would not have existed. Each of us seems to view the world through a small window in space and time; its duration is the length of our life from birth to death, and its spatial extent is limited by our travels during life. If our life is terminated before birth that window never exists. It is fatuous for others to tell us how small is one person's contribution to the world, and that if we had not been born, billions of others still would have been, and the world would have been much the same. From our viewpoint, if we had not been born the world would not have existed.

There is clearly something wrong with this argument. Your mother undergoing an abortion is only one of a million other causes that would have resulted in your not being born. The process of conception is itself very unreliable, and may well not have resulted in the union of that particular sperm and egg which led to you being born. Or your father or your mother might have died in infancy, or a grandparent or great grandparent, or any one of your ancestors right back to antiquity. Or perhaps one of the apes in whose line of descent you lie may have died before procreating, or any one of countless other possible accidents had befallen your genealogy. If any one of these links in the chain had been broken, then however many other people might exist, you would not. You are indeed lucky! The probability of your escaping all these hazards must have been infinitesimal; it was millions of times more likely that you would not have been born.

What has gone wrong with our reasoning? Its essence is that if your mother had had an abortion then you would not exist; or if your grandmother, or ... Each of these is an example of a counterfactual, a conditional statement with a false antecedent, an antecedent which you know must be false, for were it not so you would not exist and could have no thoughts. What can such a counterfactual possibly mean? We saw above that any conditional statement has an antecedent which conceivably can be either true or false. It separates the true cases from the false, and tells us that when the antecedent is true then so is the consequent. But it tells us nothing about the cases in which it is false; here we know that the antecedent is false, and so the statement has nothing at all to say. It is meaningless.

DAVID LEWIS

The generally accepted meaning of counterfactual statements is in terms of "possible worlds", as laid out carefully by David Lewis in his book "Counterfactuals" (see [3]). Lewis begins the book with the assertion that "If kangaroos had no tails, they would topple over", and interprets this as referring to a possible world where kangaroos indeed had no tails, but which is otherwise as similar as it can be to our own world. He accepts that this imagined world must differ from our own in many other respects; for example, kangaroos' genetic coding must be different, as must the trails they leave in the sand. But I would like to suggest that these worlds would be so different from ours that they are not possible at all. Why would the kangaroos have developed differently? Because of different environmental conditions. Why were these conditions different? Because the earth had formed differently. Why ... ? Pursuing the argument backwards we must eventually reach a point beyond which we cannot go. There is only one world, and only one history of the world. If it requires the idea of a different world in order to explain the meaning of a counterfactual, then that world is an impossible one. Counterfactuals, like other conditionals, must spring from an implication whose reason we can understand, and not from imagining another world. No doubt a student of anatomy and one of mechanics could together account for the fact that a kangaroo without a tail would topple over; they need not concern themselves with the question of whether such a creature can exist.

In the light of this, my criterion for conditional statements to be meaningful is actually less stringent than Lewis'. I maintain that a world in which kangaroos have no tails is impossible, but it is nevertheless conceivable. We can hold in the mind a picture of such an animal without this picture being destroyed by its inconsistencies. I reject a conditional only when its antecedent is logically impossible, as in the abortion case, or physically impossible, as when it directly infringes the laws of nature.

We saw above that even such impossible antecedents as these must be accepted when we wish to promote an argument by reductio ad absurdum, but this does not really contradict the present thesis. In presenting such an argument, we may indeed know our initial premise to be false, but we must not allow this to influence the chain of reasoning, or the argument will be circular. It is therefore allowable to pretend that the premise is true; in effect we pretend to be unaware of the conclusion to which it leads; we make it appear that the argument is leading us to it for the first time. In this case we would not be aware of the falsehood of the premise until we reach the contradiction. In general we must not say, "If I had not been born ...". However, it is legitimate to use the phrase in a reductio argument: "If I had not been born I would not be writing this paragraph. But I am writing this paragraph, so I have been born".

TIME TRAVEL

Our final example is more frivolous. Many writers of science fiction, and not a few scientists, have contemplated the possibility of travelling backwards in time. But what would happen, they ask, if you travelled back, and killed your own grandmother before your mother had been born? This paradox has often been advanced as a proof that time travel is impossible.

Now I do believe that backwards time travel is indeed impossible, but my reasoning has nothing to do with grandmothers or murderers; it is decreed by the Second Law of Thermodynamics. If someone from the twentieth century could go back to the nineteenth, what would we mean in asserting that this nineteenth century incarnation did indeed represent the same person as his twentieth century one? We would describe the two as the same person only if he took with him some memories or records from the 1900's and displayed them in the 1800's, and this would involve a violation of the Second Law. So time travel into the past is ruled out as physically impossible, and for this reason our counterfactual, "If someone from the twentieth century could go back ...", is meaninglass. A full discussion of the reasons memories and records can be only of past events, and never of the future, will be found in the author's book:

But ignoring this argument, and imagining time travel to be possible, where is the problem? If a woman were murdered in the 1800's before having offspring, she could not possibly become a grandmother. If her murderer claimed to be her grandchild, he must be lying. So here is a further reason for declaring the quoted counterfactual to be meaningless. Its antecedent is logically impossible in the same way that the antecedent in the abortion argument is impossible.

The story is no more mystifying than the person who says, "This statement is untrue". We have only ourselves to blame for the contradictions that arise when we misuse language, and conditionals with impossible antecedents are an abuse of language.

1. B. Russell, Introduction to Mathematical Philosophy, (Allen & Unwin, 1919)

2. E. Adams, The Logic of Conditionals (1975)

3. D. Lewis, Counterfactuals (1973)