by Corry Shores
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[Bracketed commentary and boldface are my own. Please forgive my typos, as proofreading is incomplete. I am not trained in logic, so please consult and trust the original text, which is wonderful.]
Logic: A Very Short Introduction
Conditionals: What’s in an If?
Conditionals are of the form, “if a then c,” or a→c. The first term is the antecedent, and the second, the consequent. Conditionals are false only if the antecedent is true and the consequent false, and they are true for all other value assignments. But there are many difficulties regarding conditionals, and some of which call into question the universal applicability of these value-assignments. For example, according to the truth table for conditionals, when the antecedent is false, then the whole conditional is true, regardless of whether or not the consequent is true. This means that the following two conditionals should both be true: “If Italy is part of France, Rome is in France” and “If Italy is part of France, Beijing is in France”. But intuitively, the second one seems false. So conditionals are not truth-functional, since a lot depends on the meanings of the terms. In order to evaluate them, we can use possible worlds, like with modal operators: “the conditional a→c is true in some situation, s, just if c is true in every one of the possible situations associated with s in which a is true; and it is false in s if c is false in some possible situation associated with s in which a is true.” Since Rome is by definition in Italy, that means in no possible world would it not be in France, were Italy to be in France. So that is why the first sentence is true. However, since Beijing is by definition a city in China and not a city of Italy, then in some possible worlds Beijing will not be in France, were Italy to be in France. And that is why the second sentence is false. Another problem with conditionals has to do with ¬(a&¬c), which has the same truth table as a→c, and in fact is called the material conditional and is symbolized as a⊃c. But although we might think that we can infer a→c from ¬(a&¬c), this is not in fact a valid inference, and we can show this using the possible worlds analysis. The important difference between a→c and ¬(a&¬c) is that a→c involves the relevance of a to c, where there is no such relevance implied in ¬(a&¬c). For this reason we can think of situations where a→c will be false but ¬(a&¬c) will technically true, thereby invalidating the inference. There are other cases too of inferences using conditionals that seem valid, and yet there are troubling counter-examples that call their validity into question.
[In the previous chapter, we discussed modal logic, and we used formulations like, “if it is true to say that I will be involved in an accident then it cannot fail to be the case that I will be involved”. Also recall the inference called “modus ponens”, where you say, “if a then b,” then you assert a (is true), which allows you to infer b (is true).]
Recall that a conditional is a sentence of the form ‘if a then c’, which we are writing as a→c. Logicians call a the antecedent of the conditional, and c the consequent. We also noted that one of the most fundamental inferences concerning the conditional is modus ponens: a, a→c/c.
Priest explains that although conditionals are “fundamental to much of our reasoning” and “[t]hey have been studied in logic ever since the earliest times,” they nonetheless are also “are deeply puzzling” (47).
Priest will now show one reason why conditionals can be so puzzling. We will take an example of a conditional: “if you miss the bus, you will be late.” [Now, imagine that you miss the bus. You will be late. Think of this now as a conjunction of facts. Imagine two things: you miss the bus and you are late. It would then seem to be false to make the following conjunction: you miss the bus and you are still on time (that is, not late). We can formulate this as, “it is not the case that both you miss the bus and you are not late”. And we can symbolize this as:
What is interesting is that although this is a conjunction and in English uses ‘and’, we can also consider it as a sort of conditional with another symbol.]
Suppose, for example, that someone informs you that if you miss the bus, you will be late. You can infer from this that it is false that you will miss the bus and not be late. Conversely, if you know that a→c, it would seem that you can infer ¬(a&¬c) (it is not the case that a and not c). Suppose, for example, that someone informs you that if you miss the bus, you will be late. You can infer from this that it is false that you will miss the bus and not be late. Conversely, if you know that ¬(a&¬c), it would seem that you can infer a→c from this. Suppose, for example, that someone tells you that you won’t go to the movies without spending money (it’s not the case that you go to the movies and do not spend money). You can infer that if you go to the movies, you will spend money. |
¬(a&¬c) is often written as a⊃c, and called the material conditional. Thus, it would appear that a→c and a⊃c mean much the same thing.
[For reference, this again is the truth table for conditionals that we saw already.
Priest then shows the truth table for a⊃c. and says, “it is a simple exercise, which I leave to you, to show that this is as follows:”
(based on Priest 48)
Perhaps we might work out the steps for ¬(a&¬c) in the following way.
As we can see, the material conditional ⊃ and the conditional → have the same truth tables.
] [The next idea is a bit complicated, and I may misconstrue it. Priest will note some oddities in the truth table. We first consider the first and third rows. They say that if c is T, then the whole implication is true. But that would suggest it does not matter what you say for the a, since it can be false and yet the implication will be true. We begin by giving a strange example for the third row. Our c value will be T, which means it does not matter what we put for the a value. Regardless, the whole implication will be true. But Priest gives an example where the a sentence is directly contradictory to the c sentence. So how can such a contradiction still be true, even if all we need is for c to be true? The third and fourth rows suggest that if the a value is false, then it does not matter what we say for the c value, since in all cases the whole implication will be true. He then gives an example for row four where both the a and the c are false, which makes the implication seem false rather than true.]
But this is odd. It means that if c is true in a situation (first and third rows), so is a→c. This hardly seems right. It is true, for example, that Canberra is the federal capital of Australia, but the conditional ‘If Canberra is not the federal capital of Australia, Canberra is the federal capital of Australia’ seems plainly false. Similarly, the truth table shows us that if a is false (third and fourth rows), a→c is true. But this hardly seems right either. The conditional ‘If Sydney is the federal capital of Australia, then Brisbane is the federal capital’ also appears patently false. What has gone wrong?
[Recall what we said about modal operators. We noted that when we add them to sentences, they may or may not alter their truth value. It depends on the situation, and so it is not a change that happens with mechanical regularity. Thus modal operators are not truth functions. Priest will then give two examples of conditionals. In the first, the a will be false and the c will be false, but because of the meanings of the terms, the conditional will still be true. In the second example, likewise the a will be false and the c will be false, but this time the conditional will be false. This demonstrates that conditionals also are not truth functional, since the same input values do not always give the same output values.]
What these examples seem to show is that → is not a truth function: the truth value of a→c is not determined by the truth values of a and c. Both of ‘Rome is in France’ and ‘Beijing is in France’ are false; but it’s true that:
If Italy is part of France, Rome is in France.
While it’s false that:
If Italy is part of France, Beijing is in France.
Priest then wonders, how do conditionals work [since we do not know in a mechanically consistent way what determines the output values]?
He says that one way we can explain the workings of conditionals is by using the “machinery of possible worlds” that we saw in the prior chapter. [It seems the reasoning is as follows. Consider the first sentence: “If Italy is part of France, Rome is in France.” We consider all possible worlds where Italy becomes a part of France. In all of them, Rome would have to also become a part of France. For, it is a part of Italy and thereby becomes a part of France when Italy becomes a part of France. (So perhaps we might say, the conditional is true if in no possible world can it be untrue). What about “If Italy is part of France, Beijing is in France”? We can imagine some possible worlds where Beijing does in fact become a part of France in conjunction with Italy’s becoming a part of France. But surely we can think of possible worlds where China is entirely unaffected by Italy’s incorporation into France. Thus, it is not necessarily true. (And so we might say, the conditional is false if in some possible world it is false).]
Consider the last two conditionals. In any possible situation in which Italy had become incorporated into France, Rome would indeed have been in France. But there are possible situations in which Italy was incorporated in France, but this had no effect on China at all. So Beijing was still not in France. This suggests that the conditional a→c is true in some situation, s, just if c is true in every one of the possible situations associated with s in which a is true; and it is false in s if c is false in some possible situation associated with s in which a is true.
[The next idea I do not follow adequately. Let us first work on the point just made: “the conditional a→c is true in some situation, s, just if c is true in every one of the possible situations associated with s in which a is true; and it is false in s if c is false in some possible situation associated with s in which a is true.” We first note that we are not determining that a→c is necessarily true, but just that it is true. And we are saying that we can determine that a→c is true in a situation if c is true in all associated worlds where a is true. On this basis, we will say that the inference modus ponens is valid. Priest will also say, “suppose that a and a→c are true in some situation, s. Then c is true in all situations associated with s in which a is true.” This seems to follow from the stipulation. We are saying that a→c is true in a situation, and the stipulation says that c will be true in all associated situations where a is true. But we still have not yet gotten to modus ponens. For, we know that in this situation a is true and a→c. But we do not yet know that c is true. The next step in the reasoning is tricky. We do not yet know that c is true in our ‘actual’ situation. But we do know it is true in the other possible ones. The next step in the reasoning is to say that an actual situation should be counted among the possible ones. And this “possible” rendition of an actual situation is identical to the actual one in all respects regarding its facts and truth value assignments. Whatever we say of the possible version we can say of the actual, since they are identical. Now, if c is true in all possible situations related to the actual one, including the possible situation that is identical to the actual one, then it must also be true for the actual version as well.]
This gives a plausible account of →. For example, it shows why modus ponens is valid – at least on one assumption. The assumption is that we count s itself as one of the possible situations associated with s. This seems reasonable: anything that is actually the case in s is surely possible. Now, suppose that a and a→c are true in some situation, s. Then c is true in all situations associated with s in which a is true. But s is one of those situations, and a is true in it. Hence, so is c, as required.
[We already established that ¬(a&¬c) and a→c have the same truth tables. But the question we had was, can we make the following inference:
? We then test for validity it seems by seeing if in our situation it can be that the premise is true and the conclusion false. But we will make those determinations by way of possible worlds which can determine the values in our world. Now, one way that the premise can be true is if a is false. That makes the conjunction false, regardless of c’s value. And the negation of that false conjunction then becomes true. Moreover, if we stick just to the truth tables, the conclusion would be true, since, if you recall from the table, any time the a value is false, the whole implication will be true. But just on the basis of a true conjunction we do not know what the a value is in the other worlds. (Things were different for the conditional, where by knowing that a→c is true in a situation means that we know c is true in associated situations where a is true. For conjunctions we do not seem to be able to draw any further conclusions about the terms’ values in the other words, and thus they can be assigned either way.) If in another world a is true but c is false, then ¬(a&¬c) will be true, but a→c will be false.
But, recall our stipulation, “the conditional a→c is true in some situation, s, just if c is true in every one of the possible situations associated with s in which a is true; and it is false in s if c is false in some possible situation associated with s in which a is true.” So even in our own world, a→c is false in accordance with this criteria.
Thus since in our own world the premise is true but the conclusion false, it is not a valid inference.]
Going back to the argument with which we started, we can now see where it fails. The inference on which the argument depends is:
And this is not valid. For example, if a is F in some situation, s, this suffices to make the premiss true in s. But this tells us nothing about how a and c behave in the possible situations associated with s. It could well be that in one of these, say s′, a is true and c is not, like this: |
So a→c is not true at s.
[But recall our prior example where this inference seemed to hold. “Suppose, for example, that someone tells you that you won’t go to the movies without spending money (it’s not the case that you go to the movies and do not spend money). You can infer that if you go to the movies, you will spend money.” Priest then gives a counter example, but the reasoning is tricky for me. It seems the idea is the following. We begin by assuming: it is not the case that you will go to the movies, and also, you will not spend money: ¬(g&¬m). We next make two other assumptions, namely, that tonight the movies are free and also that we will not go to them anyway. We might now say: ‘but then the original statement, “you cannot go to the movies without spending money” no longer seems to apply. For, now the movies are free’. But it can still apply. If we do not go to the movies, then the g is false in ¬(g&¬m). That means the whole negated conjunction is true. So even if the movies are free, if we do not go, it is still true that: “you cannot go to the movies without spending money” (“it’s not the case that you go to the movies and do not spend money”). Although this negated conjunction is true, the conditional is not, namely, that “if you go to the movies you will spend money”. For, as we said, it is a free movie night.]
What about the example we had earlier, where you are informed that you won’t go to the movies without spending money. Didn’t the inference seem valid in this case? Suppose you know that you won’t go to the movies without spending money: ¬(g&¬m). Are you really entitled to conclude that if you go to the movies you will spend money: g→m? Not necessarily. Suppose you are not going to go to the movies, come what may, even if admission is free that night. (There is a programme on the television that is much more interesting.) Then you know that it is not true that you will go (¬g), and so that it is not true that you will go and not spend money: ¬(g&¬m). Are you then entitled to infer that if you go you will spend money? Certainly not: it may be a free night.
The next point seems to be about relevance. Recall the above situation where we said that “it’s not the case that you go to the movies and do not spend money” was true because actually we are not going to the movies anyway. A person would not normally make such a statement if they knew you were not going, because then who cares whether or not you spend money? Of course you will not anyway. Instead, if someone tells you this, it matters that there be an important connection between g being true and m being true. So therefore, if someone tells us this, even though we cannot logically infer g→m, we can still conclude that this is what is meant by the statement (50-51).
Priest then notes how we often make correct inferences based on context and relevance, even though the inferences are not made deductively. Unlike implication in the sense of conditionals, this inductive sort of inferring is called “conversational implicature.”
Suppose, for example that I ask someone how to get my computer to do something or other, and they reply ‘There is a manual on the shelf’. I infer that it is a computer manual. This does not follow from what was actually said, but the remark would not have been relevant unless the | manual was a computer manual, and people are normally relevant in what they say. Hence, I can conclude that it is a computer manual from the fact that they said what they did. The inference is not a deductive one. After all, the person could have said this, and it not be a computer manual. But the inference is still an excellent inductive inference. It is of a kind usually called conversational implicature.
Priest now addresses another problem with how we have so far characterized conditionals. He will first give two arguments that are variations on a certain form. The examples will seem valid intuitively, and thus the forms will seem valid. Then afterward he gives other examples which fit the form, but they seem intuitively invalid. He leaves it to the reader to think about the matter, and it is left unsettled.
The first form is:
And its example is:
If you go to Rome you will be in Italy.
If you go to Italy, you are in Europe.
Hence, if you go to Rome, you will be in Europe.
But then he gives this counter-example:
If Smith dies before the election, Jones will win.
If Jones wins the election, Smith will retire and take her pension.
Hence, if Smith dies before the election, she will retire and take her pension.
[Perhaps the problem here has to do with the relevance between the two premises, but I am not sure.]
The second form is:
And its example is:
If x is greater than 10 then x is greater than 5.
Hence, if x is greater than 10 and less than 100, then x is greater than 5.
But its counter-example is:
If Smith jumps from the top of a tall precipice, she will die from the fall. Hence, if Smith jumps from the top of a tall precipice and wears a parachute, she will die from the fall.
These tricky examples demonstrate just how contentious the topic of conditionals is in logic (54).
[The following is quotation.]
Main Idea of the Chapter
● a→b is true in a situation, s, just if b is true in every situation associated with s where a is true.
(quoted from Priest, 54, boldface his)
Priest, Graham. Logic: A Very Short Introduction. Oxford: Oxford University, 2000.