4454

Autum n 1998

CONTENTS
1.
Teacher’s guide
2.
Student’s guide
3.
Logical form
Form vs content
The distinction between form and content
4.
Truth, validity and soundness
Truth
Validity
Soundness
5.
Fallacies
The study of fallacies
Formal vs informal fallacies
6.
Suggestions for further reading
RMPS: Philosophy Support Materials: Logic I (Higher) 1

RMPS: Philosophy Support Materials: Logic I (Higher) 2

1. TEACHER’S GUIDE
Introduction
An essential part of philosophical activity in any area, be it metaphysics, ethics or epistemology, is the evaluation of arguments. Accordingly, philosophers have more than a passing interest in logic, which can be roughly characterised as the study of argumentation. This unit is designed to introduce students to the basic concepts of formal logic. The purpose is to equip the student with the intellectual tools required for the analysis and evaluation of arguments.
Only the most basic elements of formal logic will be dealt with in this unit. No formal proofs will be required of students, although they will be introduced to the formal apparatus of proofs (variables, constants, argument forms, and some rules of inference). This unit is designed to get students used to thinking about arguments in a more systematic and rigorous manner, and to instil an appreciation of the importance of logical analysis to philosophical activity in general.
The areas of study
•
Logical form
•
Truth, validity and soundness
•
Fallacies.
Delivery strategies
Teachers will obviously have their own preferences concerning the organisation of the course and the allocation of time within it. However, these support notes stress that logic is an accumulative subject - certain concepts must be in place before others can be understood. For example, the notion of logical form must be in place before a student can understand what a valid argument is; and the notions of validity and soundness must be understood before a study of fallacies can be undertaken.
It is difficult to lay down general rules concerning the allocation of time to each concept. Much depends on the students. Some students will find the basic concepts quite straightforward, while others will require more time to master these rather abstract ideas. It is to be expected, however, that a good deal of class time will be allocated to the analysis of real arguments and the study of fallacies.
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Learning and teaching approaches
Again teachers will have their own strategies and preferred ways of doing things. As in the case of mathematics, exercises are an important part of the process of mastering the basic concepts of formal logic.
Students should be encouraged to apply their knowledge of logic to arguments encountered in everyday life, particularly in media. This will serve at least two purposes. First , it will bring home the fact that, despite its abstract nature, logic does apply to worldly affairs. Second , student interest will be greater if they evaluate the arguments of politicians and public figures rather than arguments in textbooks.
Assessment should be continuous, and the opportunity for consolidation of learning ought to take place alongside the more formal assessment required for certification.
Teachers and lecturers should bear in mind that philosophy is also present at Advanced
Higher and they should plan accordingly progression routes for students.
A final note on the emphasis of these support materials: all the basic concepts studied in this unit are more than adequately presented in the many good introductions to formal logic available on the market. Some of these texts have been listed in section 6 on further reading. Rather than repeating this material here, these notes will attempt to clarify some of the more important concepts, and to provide some explanatory material about the nature of logic itself and its importance to philosophy. These ‘meta’- logical comments may help to explain the need for a study of logic in the first place.
Mary Haight’s book is a beginner’s guide to formal logic and, among others, is a useful text to accompany these support materials.
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2.
STUDENT’S GUIDE
While many people are attracted to philosophy because of the inherent interest of the subject matter, very few express an interest in logic. Many are put off because it appears dry and intimidating. Nevertheless, while there is no point in denying that logic is more abstract than, say, ethics or political theory, an understanding of the basic concepts of logic is necessary for work in all areas of philosophy.
Why is logic so important to philosophy? The short answer is that logic is to the philosopher as the experiment is to the scientist. Philosophers and scientists alike need ways of determining whether their theories are worth believing. The scientist tests her theories by making observations (running experiments), and she has a battery of senseextending instruments to help (microscopes, telescopes, etc.). But the scientific method cannot be used to test philosophical theories (if it could the theories would be scientific and not philosophical). The only ‘instrument’ the philosopher can call upon to test her theories is the argument itself. Philosophical theories are only as good as their supporting arguments. Consequently, philosophers must be versed in the basics of logic.
This unit is designed to introduce you to the basic concepts of formal logic. The aim of this study is to equip you with the skills needed to distinguish good arguments from bad ones.
The areas of study are:
•
Logical form
•
Truth, validity and soundness
•
Fallacies.
There are some things we believe because they are obviously true. For example, the proposition ‘2+2=4’ is obviously true to anyone who knows the meanings of ‘2’, ‘4’,
‘+’ and ‘=’. Likewise, if I am standing outside and it is raining, then it will be obvious to me that the proposition ‘It is raining’ is true. I don’t need arguments for these beliefs. (Anyone who asks for one would be thought mad.) But many, if not most of the beliefs we hold are not obviously true in this way. Many of our beliefs are held because we accept some other belief, which we think supports them. This distinction between non-inferential and inferential knowledge is crucial to understanding what logic is about. Logic is wholly concerned with inferential knowledge.
The nature of logic can be clarified by looking at the sorts of questions that interest logicians. For example, if faced with beliefs A, B, C, and D, the logician does not ask,
‘Can I know if beliefs A-D are true?’ Such a question belongs to epistemology. The logician asks a very different question, namely, given beliefs A, B, C, and D, what else can I know? What else must be the case if A, B, C, and D are true? In other words, the logician wants to determine what can safely be inferred from A-D on the assumption that A-D are true. Likewise, when faced with an argument the logician does not ask, Is the conclusion true? The question that concerns the logician is, Does the conclusion really follow from the premises? Indeed logic is best defined as the study of valid inference.
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Perhaps some will wonder if such a study is really necessary. Many are of the opinion that thinking logically and clearly is as natural as breathing, and that no special effort is required. This attitude is quite mistaken, as the study of fallacies will show. Thinking logically is not a natural gift, but a skill that needs to be developed and honed. In the next section we will examine some of the basic concepts and techniques employed by logicians and philosophers to distinguish safe from unsafe inferences.
Arguments fall into two broad categories. Some arguments have conclusions which follow of necessity from the truth of the premises. These are known as deductive arguments. By contrast, some arguments have conclusions which are merely made more probable in virtue of the truth of the premises. These are known as inductive arguments. The certainty of deductive arguments is a product of their logical form, and they are the object of study of formal logic. This unit is wholly concerned with deductive arguments.
By the end of the unit you should be able to:
• identify the logical form of an argument
• distinguish between valid and invalid arguments and between sound and unsound arguments
• explain common fallacies
• evaluate arguments encountered in everyday life.
RMPS: Philosophy Support Materials: Logic I (Higher) 6

3. LOGICAL FORM
The notion of logical form is clearly presented in the textbook accompanying this course. (All of the general introductions to formal logic listed in section 6 provide adequate guidance on this matter.) To avoid unnecessary repetition the following remarks will be limited to two points. First , the distinction between an argument’s content and its form will be made plain; second , the point of drawing this distinction will be explained.
Form vs Content
One of the most difficult aspects of formal logic for beginners is appreciating the distinction between an argument’s logical form and what might be called its content.
The content of an argument is what it is about, i.e., the information contained in the premises and the conclusion. The content of an argument is what usually catches the attention of those new to logic. However, logicians are not particularly interested in the content of arguments. Logicians are interested in the shape or form that an argument takes. The distinction between form and content is best illustrated with examples. Consider the following arguments:
1) If it is raining, then the streets are wet.
2) It is raining.
3) The streets are wet.
and
1) If Dolly is a sheep, then Dolly has a heart.
2) Dolly is a sheep.
3) Dolly has a heart.
(A solid line under a sentence, in this case (2), should be read as ‘therefore’.) The first argument is concerned with raining and the effect rain has on streets, while the second is concerned with the biology of sheep. The content of these arguments could not be more different. Nevertheless, these two arguments share the same logical form or shape. The form can be represented as follows:
1) If something is the case (call it ‘p ‘), then something else is the case
(call it ‘q’).
2) p is the case.
3) Therefore q is the case.
This is an argument form (known formally as ‘modus ponens’). An argument form is obtained by abstracting from an argument’s content in order to focus on the logical relationship between the premises and the conclusion.
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‘P’ and ‘q’ are called variables. Variables are used as place holders in arguments, and they stand for the content of arguments. In the two arguments above, ‘p’ stands for ‘It is raining’ and ‘Dolly is a sheep’, while ‘q’ stands for ‘the streets are wet’ and ‘Dolly has a heart’. Variables are distinguished from constants. The constants in the above argument are the terms ‘if’ and ‘then’. It is the constants that are of interest to logicians because they determine the logical form of arguments. Consider arguments of another form.
1) All cats are mammals.
2) All mammals are warm-blooded
3) All cats are warm-blooded.
and
1) All Scots are British.
2) All Britons are Europeans.
3) All Scots are Europeans.
Again the content of the two arguments is very different, but they share the same logical form, which can be represented as follows:
1) All A are B.
2) All B are C.
3) All A are C.
This is another argument form (a type of syllogism). Again, ‘A’, ‘B’ and ‘C’ are variables, playing the role of place holders. (It is a matter of convention that the lower case letters ‘p’, ‘q’, ‘r’ stand for complete sentences, while the upper case letters ‘A’,
‘B’, ‘C’ stand for subject and predicate terms.) The constant in this case is the term
‘all’.
There are many different argument forms. It is the business of the logician to discover these forms, and determine if they are valid or invalid forms of argument. (The technical terms ‘valid’ and ‘invalid’ will be discussed below.)
A further point about argument forms: So far we have discussed argument form as though any particular argument had only one form. Technically this is not so. An argument’s form is discovered, as said above, by abstracting from the content of an argument in order to look at the logical relationship that obtains between the premises and the conclusion. However, one can abstract from the content in more than one way. Consider the argument:
1) All men are mortal.
2) Socrates is a man.
3) Socrates is mortal.
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One way of presenting its form is as follows:
1) All A are B.
2) C is an A.
3) C is B.
But one could also present it as:
1) p.
2) q.
3) r.
(Remember the capital letters A, B and C conventionally stand for words, while the lower case letters p, q, r stand for whole sentences.)
Both of these representations display the form of the argument. But only one of them is interesting from a logical point of view. The first representation shows that the argument is indeed valid; the second fails to show this. The points to emphasise here are, one , an argument can have more than one logical form, and two , an argument is valid if at least one of its forms is valid.
The distinction between form and content
What is the point of drawing the distinction between an argument’s form and its content? The point lies in the fact that an argument can fail for two distinct reasons.
An argument can fail either because one (or more) of its premises is (are) false, or because its form is invalid. Consequently, when appraising an argument one needs to have an eye both to the truth value of its premises (are they true?) and to its form (is it valid or invalid?). If one is not able to draw the distinction between an argument’s form and its content then one will not be in a position to appraise it properly, and then one is more likely to accept bad arguments.
Some further comment and an example or two might help. An argument involves the claim that the truth of the conclusion follows from the truth of the premises. Clearly, if the premises are false then the truth of the conclusion has not been established. So one must look at the truth value of the premises. But a good argument needs more than true premises. The relationship between the premises and the conclusion has to be appropriate as well, and this is a matter of logical form. To see this point consider the following argument:
1) If it is raining, then the streets are wet.
2) The streets are wet.
3) It is raining (or has been raining recently).
Most people will accept (1) because streets do get wet in the rain (unless some extraordinary precaution is taken). Imagine now that you look out the window and you see that the streets are wet, i.e., that (2) is true. In this case both premises would
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be true. But can one safely infer (3)? The short answer is ‘No’, and this is the crucial point about logical form. Streets do indeed get wet in the rain; but they get wet in other circumstances as well (perhaps some students have been raising money for a school trip by washing all the cars in the street). Now the logician can see what is wrong with this argument without recognising that streets can get wet for many different reasons. The problem with this argument is that its underlying form is invalid
(see next section, and the discussion of formal fallacies below). Its form can be represented as follows:
1) if p, then q.
2) q.
3) p.
Any argument that displays this form fails because it is invalid. No argument of this form can be accepted (regardless of the truth of the premises).
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4. TRUTH, VALIDITY, AND SOUNDNESS
Again these concepts are treated fully in the texts listed in section 6. To avoid unnecessary repetition these notes will be limited to a few pertinent remarks, and formal definitions of these concepts.
It is important to recognise that ‘validity’ and ‘soundness’ are ‘terms of art’, i.e., that they are given technical meanings in logic that differ sharply from those found in ordinary parlance. ‘Validity’ and ‘soundness’ in logic are not the same as validity and soundness in ordinary English. Failure to recognise this fact can lead to serious confusion. For example, it is not unusual to say in ordinary parlance, ‘That’s a valid point’ when someone has made a pertinent remark about the topic under discussion.
In this usage ‘validity’ is being attributed to a particular sentence. In logic, however, validity is attributed only to arguments or argument forms, never to particular sentences in isolation. Particular sentences (premises and conclusions) are true or false, not valid or invalid. Again, it is not unusual to say ‘He’s a sound chap, you can trust him’, where ‘soundness’ is being attributed to a person’s character. In logic, however, soundness applies only to arguments. The point to emphasise is that one cannot assume that ordinary English is a reliable guide to logic.
Truth
Truth is a very complex and difficult concept. Thankfully these complexities need not detain us here. Only two points need to be stressed in connection with the basic concepts of logic. First , whatever truth is, it is attributed only to sentences, and not to arguments. Premises and conclusions of arguments are true or false; arguments themselves are never true or false, but only valid or invalid, sound or unsound.
Second , sentences are usually true if they get the facts right. For example, the sentence ‘Edinburgh is the capital of Scotland’ is true because Edinburgh really is the capital of Scotland. But philosophers and logicians recognise another type of sentence, namely, sentences that are ‘logically true’. A sentence is logically true if it is true in virtue of the meanings of the terms in the sentence.
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Validity
While truth applies to individual sentences, validity applies only to a collection of sentences. More precisely, validity describes a relationship that obtains between premises and conclusion. This relationship is determined by logical form.
•
An argument is valid if, and only if, its form is such that it is impossible for all the premises to be true and the conclusion false.
•
An argument is invalid if, and only if, its form is such that it is possible for all the premises to be true and the conclusion false.
Some examples will help. Take the following valid form:
1) All A are B
2) All B are C
3) All A are C.
This form is valid because it is impossible to substitute terms for the variables A, B and
C that would make the premises true but the conclusion false. This outcome is impossible in virtue of the meanings of the words used in the argument. Any argument of this form is valid because the form guarantees that if the premises are true then the conclusion will be true as well.
Now consider the following invalid form (already encountered above):
1) If p then q.
2) q.
3) p.
This form is not valid because it is possible to substitute sentences for the variables ‘p’ and ‘q’ that would make the premises true but the conclusion false. The following example makes this clear:
1) If Bacon wrote Hamlet, then Bacon was a great writer.
2) Bacon was a great writer.
3) Bacon wrote Hamlet.
The premises are undoubtedly true, but the conclusion is false. This shows that the underlying form of the argument is invalid.
To repeat, an argument is valid if its form is such that it is impossible to move from true premises to a false conclusion. There are some counter-intuitive results of this definition of validity, however. But the impression that something is wrong with this definition is caused by the confusion of ‘validity’ as a term in philosophy, and
‘validity’ as used in ordinary English. The counter-intuitive result is that a valid argument can have false premises and even a false conclusion.
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To see this consider the valid form:
1) All A are B.
2) All B are C.
3) All A are C.
Now for ‘A’ read ‘dogs ‘, for ‘B’ read ‘cats’, and for ‘C’ read ‘birds’.
What we end up with is a lot of nonsense:
1) All dogs are cats.
2) All cats are birds.
3) All dogs are birds.
In this case the premises and conclusion are obviously false. Nevertheless, this argument is formally valid. It is valid because its form is such that it is impossible for the premises to be true and the conclusion false. That is what validity is. A valid form guarantees that if you start with true premises you will end up with a true conclusion.
A valid form guarantees nothing if you start from false premises. The fact that the premises in this case are obviously false is neither here not there as far as the argument’s validity is concerned. An argument is invalid, remember, only if its premises could be true and the conclusion false. (To appreciate the validity of this argument it might help if the reader suspends her robust sense of reality for a moment and asks herself, What would be the case, in an impossible world of course, if all dogs really were cats, and if all cats really were birds? Is it not clear that if these things were true then it would have to be the case that all dogs were birds?)
One final important point. We have already pointed out that an argument can fail for two reasons - at least one of its premises is false, or its form is invalid. This led us to say that it is not enough for an argument to have true premises. The same applies to validity. We cannot accept an argument simply because it is formally valid (the case of dogs cats and birds shows this clearly enough). Validity is a necessary but not a sufficient condition for the acceptability of arguments. An acceptable argument is called a sound argument.
Soundness
Soundness is easy to understand once the notion of validity has been mastered. An argument is sound if, and only if, (a) its premises are true, and (b) its form is valid.
True premises and a valid form are the necessary and sufficient conditions for soundness, or what we might intuitively call ‘good’ arguments. Such arguments are good because if we have true premises and a valid form then we have a guarantee that the conclusion is true.
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5. FALLACIES
The study of fallacies
In ordinary English ‘fallacy’ means little more than ‘erroneous belief’ or ‘false assumption’. In logic the term ‘fallacy’ is reserved for arguments the premises of which do not really imply the conclusion despite the fact that the argument is psychologically persuasive. One of the reasons why the study of logic is necessary is precisely the fact that persuasive arguments are not always logically sound arguments.
A study of the sorts of reasoning errors we are prone to serves several purposes.
First , it makes it less likely that we will commit these errors in the future. Second , because everyone has committed these mistakes at some point in their life, this study brings home the fact that reasoning is a skill and not a natural gift. Finally , by studying how our reasoning can go wrong, we develop a better appreciation of what a sound argument is.
Unfortunately, because we err in so many different ways, it is difficult to study fallacies systematically (there is no theory of fallacies). Because there is no one general way in which we go wrong, logic textbooks tend simply to list different types of fallacies, often classifying them in different ways. There is no profit in trotting out these fallacies here because they are more than adequately presented in the texts listed in section 6. I would recommend Chapters 4 and 5 of The Snake and the Fox and
Thinking about Thinking for beginners, Chapter 3 of Copi’s Introduction to Logic for a more formal account, and Hamblin’s Fallacies for those with a special interest in the subject. The rest of this section will be restricted to a single end, namely the clarification of the distinction between formal and informal fallacies.
Formal vs informal fallacies
As mentioned above, some arguments fail because their form is invalid. When the invalid form in question is particularly deceptive it is called a formal fallacy. We have already had occasion to examine a formal fallacy in one of the examples used above.
Recall the argument form:
1) if p, then q.
2) q.
3) p.
This formal fallacy is called ‘affirming the consequent’. We know that this form is invalid because we have been able to find sentences to replace ‘p’ and ‘q’ which make the premises true and the conclusion false. (For ‘p’ read ‘Bacon wrote Hamlet’, and for ‘q’ read ‘Bacon was a great writer’.) We were alerted to this error of reasoning because it is common knowledge that Bacon did not write Hamlet. But the form is very deceptive, particularly when we do not know whether the conclusion is true or false, and even more so when we already believe the conclusion to be true.
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Consider the following argument:
1) If Newtonian mechanics is correct, then Saturn will be visible at a particular location in the sky next Tuesday.
1) Saturn was visible at the predicted location on Tuesday.
2) Newtonian mechanics is correct.
It is very easy to think that the conclusion follows; but inspection shows that it has the same form as the former, clearly invalid, argument. Failure to recognise this fallacy is at the root of the common but erroneous belief that scientific theories can be proved true.
But some fallacies are informal, that is to say, the error they contain is not one of logical form. Consider the following argument:
1) If Aristotle said p, then p must be true.
2) Aristotle said p.
3) p must be true.
This is an ‘argument from authority’, a traditional informal fallacy. It is an informal fallacy because, while unacceptable, it actually has a valid form (modus ponens). The problem with this argument is not its form, but its questionable first premise. Human beings are easily impressed by the standing of others, especially when we (often correctly) appreciate that others know a lot more than we do. But no matter how impressive a figure we appeal to, a conclusion is not true because that figure says so.
There are many more fallacies to study, both formal and informal. It is to be expected that this study will prove both entertaining and useful, despite the fact that logic is usually quite dry and abstract. The entertainment comes from the humour of recognition. Comedians can always count on a laugh when they put their finger on something we all know to be true, particularly when it is embarrassing. Identifying fallacies has something of this recognition factor because you will find that you have committed these fallacies yourself at some point or other. The usefulness of such a study is that you will be able to recognise many bad arguments almost immediately once you are familiar with the common fallacies.
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7. SUGGESTIONS FOR FURTHER READING
There are many good introductory textbooks to formal logic on the market which cover the material dealt with in this unit. The following titles are recommended:
Basic introductions
Haight, M. The Snake and the Fox: An Introduction to Logic . London:
Routledge, forthcoming. (A beginner’s guide to formal logic.)
Flew, A. Thinking about Thinking (Or, Do I sincerely want to be right?
). Glasgow:
Fontana Press, 1975. (Contains a very good opening chapter on the key notions of formal logic, and a very readable account of the most common fallacies.)
More advanced works
Copi, I.M. Introduction to Logic . New York: Macmillan. Any edition. (A classic to be recommended. Contains clear accounts of all the key notions, an excellent chapter on fallacies, and exercises.)
Salmon, W. Logic . Second edition. New Jersey: Prentice-Hall, 1973.
(Contains clear accounts of key concepts.)
Cohen and Nagel. An Introduction to Logic . London: Routledge and Kegan
Paul, 1966. (This is part of a larger work entitled An Introduction to
Logic and Scientific Method . A classic account of the nature of logic is provided in chapter one.)
For a specialist study of fallacies see:
Hamblin, C.L. Fallacies . Newport News: Vale Press, 1970. (A study for those with a special interest in fallacies.)
There is growing number of textbooks devoted to informal logic and critical thinking.
These may be of some use to teachers and lecturers who want more background in the nature of argumentation. The following titles may be of some use.
Thomson, A. Critical Thinking: A Practical Introduction . London:
Routledge, 1996.
Phelan and Reynolds. Argument and Evidence: Critical analysis for the
Social Sciences . London: Routledge, 1996.
Fisher, M. The Logic of Real Arguments . Cambridge: CUP, 1988.
Fogelin, R.J. Understanding Arguments: An Introduction to Informal Logic .
New York: Harcourt, 1978.
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