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Chapter 2

propositions ♦ Deductive soundness

56 Logic: so deductive validity

Our attempts to engage in critical thinking are sometimes frustrating. This is often because, even when we feel certain that there is something wrong with an argument, we find it hard to explain exactly what is wrong with it. Often this is frustration with ourselves; but it can easily look like frustration with the person giving the argument (it can certainly be interpreted as such by that person!) One of the primary aims of training in critical thinking is to learn concepts and techniques which will help us to express clearly what is wrong with an argument, thereby dispelling that frustration. By helping us to assess arguments more efficiently, this helps us in the pursuit of truth. But also, by becoming more articulate in our criticisms, we become less frustrated, and thereby less bad-tempered. This can help to smooth out our relationships with other people (whereas you might have thought that improving your skill at critical thinking would make you into a disagreeable, captious quibbler).

This frustration derives from two sources: First, confronted with an argument, we find it hard to hold the whole thing clearly before our mind's eye; we find it hard to say exactly what reasoning it is that we think must be mistaken. Second, even when we do succeed in laying the argument out before us clearly, we find it hard to describe or explain what is wrong with it.

• The first issue is addressed by techniques and strategies for argument reconstruction: the representation of arguments in standard form, so as to give us a clear and comprehensive view of them.

• The second issue is addressed by techniques and concepts of argument assessment: the determination of whether or not arguments provide good reasons for accepting their conclusions.

We discuss practical details of argument reconstruction in Chapter 5. This chapter and the next are mostly concerned with argument assessment. Ordinarily, we speak of arguments as being good or bad, strong or weak, valid or invalid, sound or unsound, persuasive or unpersuasive, intelligent or stupid, without having a clear idea of what we mean by these terms, and without clearly distinguishing their meanings. So not only are we vague when we use one of these terms to criticise the argument; our attempts to explain ourselves by means of the others are still vague. Thus our primary task in this chapter and the next is to explain the basic logical concepts in terms of which assessment is carried out -validity, soundness and inductive force.

You may be surprised that detailed discussion of argument assessment precedes the detailed discussion of argument reconstruction in Chapter 5; surely you have to reconstruct an argument before you can assess it? In fact, it is slightly less straightforward than that: although the final assessment of an argument must await its reconstruction, good reconstruction practice must be informed by a good grasp of the concepts used in assessment. The purpose of the next section is to explain this important point.

The Principle of Charity

An argument is a system of propositions: a set of premises advanced in support of a conclusion. People succeed in expressing the propositions they have in mind with varying degrees of clarity. In addition, an argument may depend upon premises that the arguer does not state at all, but which he or she is implicitly assuming.

For example, if someone argues: 'Sally is taking drugs; therefore she is breaking the law', the arguer is probably using the rather vague term 'drugs' in the narrow sense of 'recreational drugs', or perhaps in the sense of 'narcotics'. In the wider sense of 'drugs' that includes medicinal drugs, this would obviously be a bad argument. Furthermore, the arguer is assuming, without explicitly stating, that it is illegal to take such drugs. So two sorts of thing are left implicit in this argument: first, the arguer assumes a more precise meaning than is explicitly expressed by the word 'drugs'; second, the arguer fails to make explicit all the facts from which he or she infers the conclusion. A premise is left implicit.

Since the purpose of argument reconstruction is to determine exactly what argument has been given, it follows that part of the task of argument reconstruction is to clarify what the arguer actually said, and to supplement what the arguer actually said (to make explicit what was merely implicit in the arguer's statements). That is, we try to represent the argument in such a way as to create a perfect match between the propositions upon which the argument actually depends and the sentences which represent the argument in standard form. Two important consequences follow from this:

♦ The sentences we use in a reconstruction of the argument need not be the very same sentences used by the arguer in giving their argument. We may employ sentences which more clearly or precisely express the propositions that constitute the argument.

♦ Our reconstructed version of the argument may contain premises which are not expressed by any of the sentences actually used by the arguer.

Argument reconstruction is essentially a task of interpretation. What we are trying to reconstruct, to represent as clearly as we can, is a certain train of thought, of reasoning - however well or badly the arguer may have succeeded in expressing it. This cannot be an exact science. It cannot be mechanical or foolproof. It calls for judgement, a critical but sympathetic eye or ear, and even a certain degree of intuition, of understanding of people - of the ways people tend to think in given sets of circumstances, and of some typical ways in which people fail to express themselves clearly.

Nevertheless, the process can be undertaken in a systematic way, and there are general guidelines to follow. One of the most general of these is what we call the Principle of Charity, which we now explain.

We have just said that the reconstruction of arguments is often a task of surmising what the arguer had in mind, and was trying to express. Our main evidence for this, naturally, is the specific words actually used by the arguer. Beyond this, we look to various sorts of facts about the context or circumstances in which the person employed the words that he or she did. For example, consider this argument:

But he is still in Paris! Therefore, he cannot possibly be in St Petersburg by tomorrow.

Of course nowadays St Petersburg is only a few hours from Paris by aeroplane. In the context of today, someone's being in Paris today would not prevent their being in St Petersburg tomorrow. If someone were to give a similar argument regarding some presently living person - the Russian president Vladimir Putin for example - then we would be puzzled. Since everyone is aware of jet travel, nobody thinks that it is impossible to get from Paris to St Petersburg in one day. So in the context of today, we would have to inquire further to discover what the arguer thinks is preventing Mr Putin's journey. But suppose these words were given by someone in 1807, referring to Napoleon. Then surely the arguer would be assuming that it is not possible to get from Paris to St Petersburg at such a speed. That certainly would have been an appropriate assumption in Napoleon's day, so much so that it would have gone without saying. The fastest way to travel then was by horse.

Such facts pertaining to the context in which the argument is given, together with the specific words used by the person, will constitute the total evidence you have for reconstructing the argument. In some cases, the context is known, and makes it obvious what the arguer was implicitly assuming. In other cases, we may have to learn more about the context; this happens especially when interpreting historical documents.

In other cases, however, we may learn all the relevant contextual factors, yet it remains possible to represent the person's argument in more than one way. And it may happen that one reconstruction represents the argument as a good one, another as a bad one. In such a case, which reconstruction should you prefer? Which should you advance as the reconstruction of the argument?

It depends upon your purpose. If you are hoping to convince others that the person is wrong, you are most likely to succeed if you represent it as a bad one. Indeed, this is a very common ploy. If your aim is to defeat your opponent - or to make it seem as if you have defeated them - then success is more likely if you attack a weakened form of your opponent's argument. In a context like that of a public debate, this is often a good strategy to some extent. For what you are trying to do is to appear, to the audience, to get the upper hand. By representing your opponent's position as weaker than it really is, you are more likely to appear to be the victor. You also put your opponent on the defensive, forcing him to scramble, saying things like 'that's not what I meant'. If your aim is to persuade, or to appear to be the victor, then you may be well-advised to choose the weaker version, especially if your audience is not aware that a stronger version is available.

However, if what matters to you is whether or not the conclusion of the person's argument is true, then you should choose the best representation of the argument.

Why? Suppose you are wondering whether some particular proposition is true. You are wondering, for example, whether increasing taxes for the wealthy would lead to a rise in unemployment. Suppose further that you honestly have no idea whether or not this is true; you cannot see any reasons at all either in favour or against this proposition. Now suppose that someone attempts to persuade you that this proposition is true by giving you an argument for it. But you find that this argument admits of being reconstructed in either of two ways. On one reconstruction, the argument is good, that is, it provides a good reason for accepting the proposition as true. On the other reconstruction, however, it is no good at all; you find that the reasons you have represented the person as giving in favour of this proposition do not support it at all. Suppose you decide on this latter representation of the argument, the one which represents it as bad. Can you now conclude that the proposition is true, or that it is false? Since, reconstructed that way, the argument was no good, you certainly cannot conclude, on the basis of it, that the proposition is true. But nor can you conclude that the proposition is false. The fact that someone has given a bad argument for some proposition is not, in itself, a reason to reject the proposition as false. For example, someone might argue that since three is a lucky number, there will not be a third world war. That is a bad argument; it gives you no reason to believe that there will not be a third world war. But (fortunately!) its being a bad argument provides no reason to believe that there will be a third world war. In short, the fact that someone has given a bad argument for the proposition in question leaves you in precisely the same position as you were when you started. If you began with no evidence either for or against the proposition, then your position is unchanged - you've no reason to accept the proposition as true, and none to reject it as false.

Suppose, then, that you accept the first reconstruction of the argument. Since this constitutes a good argument, you are now in a different position; now you do have some indication as to whether or not the proposition is true. In particular, you have a reason for its being true. On this first reconstruction, then, you represent the person as having made a useful contribution to the debate. You now have reasons that you lacked before. Thus, insofar as we engage in critical thinking - insofar as our interest is in discovering the truth of things, and not just in persuading or refuting people - we are most interested to discover good arguments, not bad ones. So we should always choose the best reconstruction of a given argument. That way, we discover reasons for accepting or rejecting particular propositions, advancing the cause of knowledge. This is an application of the Principle of Charity.

There is a further reason for observing the Principle of Charity, which has more to do with ethics than with logic. When you give an argument, you may or may not succeed in expressing yourself clearly, but you do want your listener to try to understand you. If your listener impatiently seizes upon your words in order to refute your argument as swiftly as possible without taking the trouble to understand you, naturally you feel ill-used, that the person is not being fair to you. You think it wrong, unjust to be treated that way. If so then we ought to try to be equally receptive to others - to try to understand them, rather than be too eager to refute them or discredit them. When people give arguments, they almost always have some reason or other for what they are saying (although, of course, sometimes people do try to persuade us of things - especially to do things like buy Coke - without actually trying to give us good reasons). People are very seldom completely illogical. But they are seldom very well-practised at expressing their reasons clearly either, and often they are not so interested in clarity as in persuasion or eloquence. Still, beneath it all, they will usually have genuine reasons of some sort in mind, so it seems only right and proper that we should try to bring them to light, to understand what the person is really trying to say. If we do not attempt this, then we are not really doing the person justice; we are not being as receptive to his or her attempts at communication, as we would surely wish others to be to ours.

The Principle of Charity, however, has a certain limit, beyond which the nature of what we are doing changes somewhat: If our task is to reconstruct the argument actually intended by the person, then we must not go beyond what, based upon the evidence available to us, we may reasonably expect the arguer to have had in mind. Once we go beyond this point then we are no longer in the business of interpreting their argument. Instead, we have become the arguer.

If our concern is with how well a particular person has argued, then we should not overstep this boundary. However, if our concern is simply with the truth of the matter in question, then to overstep this boundary is perfectly all right. It often happens that, in reconstructing an argument, we hit upon another, similar or related argument for the same conclusion which is better than the one we are reconstructing. If what concerns us is simply to find the best arguments on either side of an issue, then we will want to give a representation of this better argument.

Truth

If your aim is to give the best possible reconstruction of an argument, then you have to know something of what makes an argument good or bad. Fortunately, logic gives us some very clear answers as to what does make arguments good and bad.

The fundamental concept of logic is the concept of truth.1 For one thing, the overarching concern of the critical thinker is typically with the truth, or lack of it, of the conclusions of arguments. Further, truth is the

1 The great German logician Gottlob Frege - who is generally agreed to be the inventor of the modern science of logic - said that the Laws of Logic are really the 'Laws of Truth', in something like the way that the Laws of Physics are the laws of the physical world.

concept in terms of which the logician attempts to explain everything else. We thus begin our discussion of the concepts of logic by saying a little bit more about this uniquely important concept.

Many people are put off by the word 'truth'. This is usually the symptom of a philosophical worry that one cannot speak simply of 'truth'. One might worry that perhaps there is no one truth: that what is true for one person or group need not be true for another person or group. Or one may worry that truth is in some way beyond us, unapproachable by mere fallible human beings. But for our purposes, we can leave aside those sorts of abstruse philosophical worries as irrelevant. As noted in Chapter 1, the way in which the logician uses the word 'truth' is really very simple and down-to-earth. Properly understood, the word should not invite those sorts of controversies.

Consider the following proposition:

This proposition is true. What does it mean to say that this proposition is true? It means, simply, that that is the way things are. To say that that proposition is true is to say nothing more than that, yes, fish do live in water. Thus consider the proposition that says that (A) is true:

(B) It is true that fish live in water.

(A) and (B) are equivalent in the sense that, necessarily, if (A) is true then so is (B), and if (B) is true then so is (A). In other words, to say that it is true that fish live in water comes to the same thing as saying that fish live in water. Used this way - which is all that is needed for logic or critical thinking - the word 'true' is no more mysterious than the words occurring in the sentence 'Fish live in water'. In this sense, you cannot doubt that there is 'really' such a thing as truth, or that truth is knowable, any more than you can doubt that fish live in water, or that the sky is blue, or that the Earth is bigger than a grapefruit. For these are all known truths.

Discomfort with the word 'true' is sometimes due to a failure to distinguish truth from belief. If John says 'Fish live in water', then he does, of course, show that he believes that fish live in water (presumably he knows that fish live in water). Likewise, if Mary now refers to what John said, and says 'That's true', then she also shows that she believes that fish live in water. Despite their having done so by different means, both John and Mary have asserted the proposition that fish live in water. Mary, unlike John, has used the word 'true'. But they have asserted the same proposition; they have expressed the same belief. Yet clearly the truth of this proposition has nothing to do with what Mary or anyone else believes. That depends only on how things stand as regards fish, and what fish do does not depend upon what people think. So despite the fact that Mary has used the word 'true' to assert something, the truth of what she asserts does not depend on her beliefs in any way.

Of course, what Mary believes depends on her, and it is possible that people could have different beliefs as regards fish. But that has no effect on fish (we will, however, return to this issue in the final chapter).

The reverse side of this is that to say that a proposition is false is just to deny it - in this case, for example, some misinformed person who thought that snakes are fish might say, 'That's false; not all fish live in water'. 'It is false that fish live in water' is equivalent to 'Fish do not live in water'.

Sometimes we will speak of the truth-value of a proposition. This just means the truth of the proposition, if it is true, or its falsity, if it is false. There are two truth-values, true and false. For example we can say that the truth-value of 'Fish live in water' is truth, that that of 'Fish live in the sky' is falsity, and that the truth-value of 'It is now Tuesday' must always differ from that of 'It is now Friday'. To ask 'what is the truth-value of that proposition?' is the same as asking whether or not that proposition is true.

A question might have occurred to you: If to say that a proposition is true amounts to the same as asserting it, then why do we have the terms 'is true' and 'is false'? What is their purpose? Why are they not just redundant, superfluous appendages? One reason is convenience; saying 'that's true' is quick and easy, like saying 'yes', or nodding one's head. But a more important reason is that we sometimes want to generalise about propositions in terms of truth and falsity. That is, we sometimes wish to speak about true or false propositions in general, without specifying any propositions in particular. This is crucial in the formal study of logic, but less technical examples are no less important. For example, we have characterised critical thinking as aiming at truth. This means that we undertake it because we want to know whether capitalism is the fairest economic system, whether so-and-so committed the crime, whether the danger of war is increasing or decreasing . . . and so on, for everything we might want to know. That critical thinking aims at truth is a generalisation that sums this up.

Deductive validity

We also need this sort of generalisation in order to define the important concept of deductive validity, to which we now turn. For brevity, we will sometimes call it simply 'validity'. In studying it, you should forget whatever the word might mean to you ordinarily. We mean logical validity, the concept of validity that concerns logic, the study of reasoning. Consider the following arguments:

A P1) The Prime Minister's dog is infested with fleas. P2) Fleas are bacteria.

C) The Prime Minister's dog is infested with bacteria.

B P1) Colette owned a dog.

P2) All French Bulldogs are dogs.

C) Colette owned a French Bulldog.

Argument A speaks of 'The Prime Minister's dog', but it is not made clear who that is, for we are given no indication of when, or even in what country, this argument was given. So we have no idea what dog, if any, has been referred to. Furthermore, P2 of A would be false under any circumstances in which the argument might have been given - fleas are insects, not bacteria. But scrutinise these arguments carefully. You can easily recognise that there is something right about A, and something wrong with B. The conclusion of A does follow from its premises, and the conclusion of B does not follow from its premises. What you are recognising is that A is valid, and that B is invalid.

Now what does this mean? What, exactly, are you seeing when you see that A is valid and B is invalid? Consider A. When you recognise its validity, you do not need to know whether or not PI or C of A is true or even precisely which dog of which Prime Minister of which country is intended; nor do you care about the fact that P2 of A is positively false. For what you are seeing is that if the premises of A were true, then the conclusion would be have to be true as well. In short, it would be impossible for the premises to be true but the conclusion false. The truth of the premises, in any possible or imaginable situation, would necessitate the truth of the conclusion. If fleas were bacteria, and the dog being referred to were infested with fleas, then it would, in that case, be infested with bacteria.

On the other hand, consider B. When you recognise that it is not a valid argument, what you are recognising is that even if the premises were true, it would still be possible for the conclusion to be false. The conclusion does not follow. Whether or not the premises are in fact true, it would be possible or conceivable for the premises to be true and the conclusion false.

Now as it happens, the premises of B are true. The French author Colette did have a dog, and of course French Bulldogs are dogs. Indeed, the conclusion of B is also true; Colette's dog was a French Bulldog. But that is beside the point, so far as validity is concerned. It may be true that Colette had a French Bulldog, but this does not follow merely from the fact that she had a dog (along with the fact that French Bulldogs are dogs). A person given only the premises of the argument, and lacking any further information about Colette, would be in no position to infer that Colette had a French Bulldog. If this point seems strange, remember that you saw that the argument is invalid before you knew the truth-values of its premises or of its conclusion. And that is as it should be. You can tell that an argument is valid or not without knowing the truth-values of the propositions it comprises, because the validity of an argument (or lack thereof), does not depend upon the actual truth-values of those propositions.

This shows you that the concept of validity pertains to the connection between the premises and conclusion of an argument, not their truth-values considered individually. This is indeed the crucial lesson about the concept of validity: it pertains to whole arguments (more exactly: it pertains to inferences; extended arguments may contain more than one inference, and each one is subject to being valid or invalid).

Thus, it should be clear that it would be nonsense, to say of a single proposition, that it is valid. That would be like saying, of a single word, that it rhymes (a rhyme requires a relation between words). By the same token, it would be nonsense to say, of an argument, that it is true. That would be like saying, of an entire jigsaw-puzzle, that it doesn't fit (this could be said of some or even of every piece, but not of the puzzle itself). A single proposition can be true or false, but not valid or invalid; an argument can be valid or invalid, but not true or false.

These two points should be borne in mind, as it is a common mistake to confuse the notions of truth and validity, applying them to the wrong sorts of things.

Here then are two definitions of validity; they are equivalent (they come to the same thing), so you are free to make use of the one you find easier to work with:

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