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Another example is PI of the last example of the preceding section of this chapter. Conditionals can also be expressed in other ways, however. For example, the following statements express the very same proposition as the example just given; they represent exactly the same connection between rain and barometric pressure:

► It is raining only if the barometric pressure is low.

► Either the barometric pressure is low, or it is not raining.

► It is not raining unless the barometric pressure is low.

► If the barometric pressure is not low, then it is not raining.

► It is not raining if the barometric pressure is not low.

► The barometric pressure is low if it is raining.

► There cannot be rain without low barometric pressure.

You may find the first one puzzling, because in it the word 'if' precedes the bit about the barometric pressure rather than the bit about the rain. But with a little reflection it is not difficult to see that it means the same as 'If it is raining then the barometric pressure is low'. In general, a conditional is a compound proposition consisting of two parts, each of which is itself a proposition, where these two parts are joined by some connecting words (they are called 'logical connectives') such as 'if-then', 'either-or', 'unless', or 'only-if', or something similar. Sometimes the presence of two whole propositions is somewhat concealed, as in the last example.

Especially notable is the second example, using the words 'either-or'. Wherever two statements are joined by 'either-or', or just by 'or', to form a compound statement, the compound is equivalent to a statement using 'if-then', and vice versa. But to pass from the version using 'or' to the version using 'if-then', or vice versa, we have to insert the word 'not'. For example, the following pairs of statements are equivalent:3

3 A complication is that the word 'or1 is used in either of two ways, known as the 'inclusive1 and 'exclusive1 senses. In the inclusive sense, 'P or Q1 means that either P is true, or Q is true, or they are both true; so the compound sentence is false only if both P and Q are false. This is the sense intended if one says something like 'United will lose if either Beckham receives a red card or Keane receives a red card1. In the exclusive sense, 'P or Q1 means that either P is true or Q is true, but not both. This is the sense normally intended if one says something like 'Either you go to bed now or I will not read you a story1 (the child would rightly feel cheated if he or she goes to bed immediately but doesn't get a story). A conditional statement 'If P then Q1 is equivalent only to 'Either not-P or Q1 only in the inclusive sense of 'or1. Where it is clear that the exclusive sense is intended, the reconstructed argument should contain conditionals running in both directions. A sentence such as 'Either the murder took place here or there was an accident here1, intended in the exclusive sense, would be represented as 'If no accident took place here then the murder took place here1 and 'If an accident took place here then the murder did not take place

Rangers will win or Celtic will win.

If Rangers do not win then Celtic will win.

Either Jane will practise diligently or she will fail her exam.

If Jane does not practise diligently then she will fail her exam.

However they are joined, what a conditional says is that if one proposition is true then another one is true also. Usually in logic this relation is represented by a single device, an arrow:

It is raining —> The barometric pressure is low.

The one from which the arrow points is called the antecedent; one to which the arrow points is called the consequent, for obvious reasons. We will use this terminology a lot, so you should memorise it.

Now, here is tricky and important point. In one way or another, the examples above all express the very same conditional proposition. They all express the same relation between rain and barometric pressure. Thus the antecedent and consequent in all the above examples are the same -in all of them, the logical antecedent is the proposition that it is raining, and the logical consequent is the proposition that the barometric pressure is low. In this sense, the fact that one proposition the antecedent and another the consequent of a conditional statement is a matter of the logic of the statements. It is not a matter of the grammar of the sentences. It does not matter in what order, in the whole sentence, the two smaller sentences occur; what matters is the logical relationship asserted by the sentence. You can see this from the fact that in the sentences, the bit about rain sometimes occurs before, sometimes after the bit about barometric pressure.

Our example of a conditional is a true statement. Some conditionals are false, such as 'If the barometric pressure is low, then it is raining' (since, sometimes, the barometric pressure is not low but it is not raining). A conditional is said to be true or false, rather than valid or invalid. For a conditional is not itself an argument. A conditional is one proposition that comprises two propositions as parts, joined by 'if-then' or a similar device. An argument cannot be just one proposition. It needs at least two. The following, however, would be an argument:

It is raining. Therefore, the barometric pressure is low.

This is not a conditional, but an argument composed of two propositions. Moreover, this argument actually asserts that it is raining, and that the barometric pressure is low. A person giving it would actually be asserting those things. Not so for the corresponding conditional: To say 'If it is raining then the barometric pressure is low' is not itself to assert that it is raining, or that the barometric pressure is low. People sometimes make a mistake on this point; we sometimes witness conversations like this:

Mary: If Edna gets drunk, then her graduation party will be a mess.

Jane: Why do you say Edna's going to get drunk? You're always so unfair to her.

Mary: I didn't say Edna is going to get drunk, I said if Edna gets drunk ...

Jane: Well, you implied it.

Jane: Don't try to get out of it! You always think the worst of Edna, and you're always trying to take back what you say!

What Jane seems not to understand is that a conditional does not assert either its antecedent or its consequent. An argument asserts its premises and its conclusion, but Mary is not arguing that the party is going to be a mess; she is only saying that it will be if Edna gets drunk (perhaps because she knows that Edna doesn't handle alcohol very well). Strictly speaking, she is not even suggesting that Edna is likely to get drunk (of course, she would not have brought the whole thing up if she did not think there was some danger of Edna's getting drunk).

Finally, it is important to recognise that many arguments have conditional conclusions - that is, conclusions which are themselves conditionals. For example:

P1) If Labour does not change its platform, it will not attract new supporters.

P2) If Labour does not attract new supporters, it will lose the next election.

C) If Labour does not change its platform, then it will lose the next election.

Here both premises as well as the conclusion are conditionals. This particular pattern is very common, and is called a 'chain' argument. A chain of conditionals is set up, like a row of dominoes. What the argument is saying is that if the antecedent of PI comes true, then the consequent of P2 will come true. Chain arguments can have any number of links.

Deductive soundness

Normally, if you wish to assess an argument, you wish to do so because you wonder whether or not the conclusion is true. You want to know whether the arguer has given you a reason for thinking that the conclusion is true. Now if you find that the argument is invalid, then you know that the conclusion could be false, even if the premises are true. Therefore the reasons given by the arguer - the premises - would not suffice to establish the conclusion, even if they are true. But suppose you find that the argument is valid. Then there are two possibilities:

A One or more of the premises are (actually) false.

B All of the premises are (actually) true.

Now, as illustrated by examples 2 and 5 on p. 53 knowing that the argument is valid is not enough to show you that the conclusion is true. In order to determine that, you need a further step: you must determine the truth-values of the premises. You might already know them. But if you don't, then of course, logic is no help. If one of the premises is that the Octopus is a fish, then unless you know already, you have to consult a book or ask an ichthyologist. Suppose now that you have done this, and what you have is a case of (A), i.e. one (or more) of the premises is false. In that case, you can draw no conclusion as to the truth-value of the conclusion (as illustrated by arguments 4 and 5 on p. 53, a valid argument with one or more false premises may have either a true or a false conclusion). But now suppose you find the argument to be a case of 1 -that is, you have found it to be a valid argument with true premises. Eureka! For now, according to the definition of validity, you may infer reliably that the conclusion of the argument is true. The argument has accomplished its purpose; it has demonstrated its conclusion to be true. We call this a deductively sound argument. Argument 1 above about Janet Baker, for example, is a deductively sound argument.

To say that an argument is deductively sound is to say:

The argument is valid, and all its premises are (actually)

true.

This reveals the importance of the concept of validity. Given the definition of validity, it follows from the definition that the conclusion of a deductively sound argument must be true. There cannot be a deductively sound argument with a false conclusion.

An argument which is not deductively sound - which has one or more false premises, or is invalid, or both - is said to be deductively unsound. Deductive soundness, like validity, pertains to whole arguments, and not to single propositions.

It is also important to recognise what follows if you happen to know that the conclusion of an argument is not true. Suppose someone gives you an argument, the conclusion of which is that there are platypuses at the local zoo. And suppose you know that this is not true. You know that the local zoo has no platypuses. Therefore you know that this argument is not deductively sound. You should make it clear to yourself why this is so; check the definitions again if this is not clear.

If you really do know - and do not merely have the opinion - that there are no platypuses at the zoo, then you know it is possible to give a deductively sound argument for that conclusion. But there cannot be deductively sound arguments on both sides of the issue. For deductively sound arguments have true conclusions. If there were deductively sound arguments on both sides of this issue, it would follow that there are platypuses at the zoo, and also that there are not, which is impossible. This is important to recognise, because frequently we do say that there can be 'good' arguments on both sides of a given issue (especially a controversial one); we say this, perhaps, out of a wish to show respect for different opinions, or simply to express our own indecision over the issue. But in saying this, we cannot mean that there are deductively sound arguments on both sides of an issue. Later, we will explain in exactly what sense there can be 'good' arguments on both sides of an issue (for, to be sure, there can be).

If we know that the conclusion of an argument is false, then we know that the argument is deductively unsound. What follows from that? Look at the definition of deductive soundness. If the argument is deductively unsound, it follows that either the argument has (at least) one false premise, or the argument is invalid (or perhaps both - perhaps it is invalid and it has one or more false premises). Suppose then that you determine the argument to be valid. Then you know that at least one premise must be false. On the other hand, suppose that you find that the argument is invalid. What can you conclude about the truth-values of the premises? Nothing! For you know that an invalid argument may have either true premises or false premises.

Similarly, if you perhaps do not know (for you are not certain), but you do believe, or hope, that the conclusion of a given argument is false, then you must run through the same procedure. Suppose you merely hope that there are no platypuses at the zoo, because you fear they would not be happy there (this might be reasonable; platypuses are notoriously shy, so they might not like to be put on display). Then you must hope that the argument is deductively unsound, in which case you must hope to find either that the argument is invalid, or that it has a false premise. This is the sort of thing you might do if you were a courtroom barrister, hoping to refute the opposing side's arguments. You might want to show that the prosecution's argument for your client's guilt is unsound.

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