Wesley C. Salmon, Logic (1984).



When people make statements, they may offer evidence to support them or they may not. A statement that is supported by evidence is the conclusion of an argument, and logic provides tools for the analysis of arguments. Logical analysis is concerned with the relationship between a conclusion and the evidence given to support it.

When people reason, they make inferences. These inferences can be transformed into arguments, and the tools of logic can then be applied to the resulting arguments. In this way, the inferences from which they originate can be evaluated.

Logic deals with arguments and inferences. One of its main purposes is to provide methods for distinguishing those which are logically correct from those which are not.


In one of his celebrated adventures, Sherlock Holmes comes into possession of an old felt hat. Although Holmes is not acquainted with the owner of the hat, he tells Dr. Watson many things about the man -- among them, that he is highly intellectual. This assertion, as it stands, is unsupported. Holmes may have evidence for his statement, but so far he has not given it.

Dr. Watson, as usual, fails to see any basis for Holmes's statement, so he asks for substantiation. "For answer Holmes clapped the hat upon his head. It came right over the forehead and settled upon the bridge of his nose. 'It is a question of cubic capacity,' said he; 'a man with so large a brain must have something in it.' "1 Now, the statement that the owner of the hat is highly intellectual is no longer an unsupported assertion. Holmes has given the evidence, so his statement is supported. It is the conclusion of an argument.

We shall regard assertions as unsupported unless evidence is actually given to support them, whether or not anyone has evidence for them. There is a straightforward reason for making the distinction in this way. Logic is concerned with arguments. An argument consists of more than just a statement; it consists of a conclusion along with supporting evidence. Until the evidence is given, we do not have an argument to examine. It does not matter who gives the evidence. If Watson had cited the size of the hat as evidence for Holmes's conclusion, we would have had an argument to examine. If we, as readers of the story, had been able to cite this evidence, again, there would have been an argument to examine. But by itself, the statement that the owner is highly intellectual is an unsupported assertion. We cannot evaluate an argument unless the evidence, which is an indispensable part of the argument, is given.

To distinguish assertions for which no evidence is given from conclusions of arguments is not to condemn them. The purpose is only to make clear the circumstances in which logic is applicable and those in which it is not. If a statement is made, we may be willing to accept it as it stands. If so, the question of evidence does not arise. If, however, the statement is one we are not ready to accept, then the question of evidence does arise. When evidence has been supplied, the unsupported assertion is transformed into a supported conclusion. An argument is then available, to which logic may be applied.

The term "argument" is a basic one in logic. We must explain its meaning. In ordinary usage, the term "argument" often signifies a dispute. In logic, it does not have this connotation. As we use the term, an argument can be given to justify a conclusion, whether or not anyone openly disagrees. Nevertheless, intelligent disputation -- as opposed to the sort of thing that consists of loud shouting and name-calling -- does involve argument in the logical sense. Disagreement is an occasion for summoning evidence if an intelligent resolution is sought.

Arguments are often designed to convince, and this is one of their important and legitimate functions; however, logic is not concerned with the persuasive power of arguments. Arguments which are logically incorrect often do convince, while logically impeccable arguments often fail to persuade. Logic is concerned with an objective relation between evidence and conclusion. An argument may be logically correct even if nobody recognizes it as such; or it may be logically incorrect even if everyone accepts it.

Roughly speaking, an argument is a conclusion standing in relation to its supporting evidence. More precisely, an argument is a group of statements standing in relation to each other.2 An argument consists of one statement which is the conclusion and one or more statements of supporting evidence. The statements of evidence are called premises. There is no set number of premises which every argument must have, but there must be at least one.

When Watson requested a justification for the statement about the owner of the hat, Holmes gave an indication of an argument. Although he did not spell out his argument in complete detail, he did say enough to show what it would be. We can reconstruct it as follows:
  1. This is a large hat.
  2. Someone is the owner of this hat.
  3. The owners of large hats are people with large heads.
  4. People with large heads have large brains.
  5. People with large brains are highly intellectual.
  6. The owner of this hat is highly intellectual.

This is an argument; it consists of six statements. The first five statements are the premises; the sixth statement is the conclusion.

The premises of an argument are supposed to present evidence for the conclusion. Presenting evidence in premises involves two aspects. First, the premises are statements of fact. Second, these facts are offered as evidence for the conclusion. There are, consequently, two ways in which the premises may fail to present evidence for the conclusion. First, one or more of the premises may be false. In this case, the alleged facts are not facts at all; the alleged evidence does not exist. Under these circumstances, we can hardly be said to have good grounds for accepting the conclusion. Second, even if the premises are all true -- that is, even if the premises do accurately state the facts -- they may not have an appropriate relation to the conclusion. In this case, the facts are as stated in the premises, but these facts are not evidence for the conclusion. In order for facts to be evidence for a conclusion they must be properly relevant to that conclusion. Obviously, it will not do merely to give any true statements to support a conclusion. The statements must have some bearing upon that conclusion.

If an argument is offered as a justification of its conclusion, two questions arise. First, are the premises true? Second, are the premises properly related to the conclusion? If either question has a negative answer, the justification is unsatisfactory. It is absolutely essential, however, to avoid confusing these two questions. In logic we are concerned with the second question only.3 When an argument is subjected to logical analysis, the question of relevance is at issue. Logic deals with the relation between premises and conclusion, not with the truth of the premises.

One of our basic purposes is to provide methods of distinguishing between logically correct and incorrect arguments. The logical correctness or incorrectness of an argument depends solely upon the relation between premises and conclusion. In a logically correct argument, the premises, whether they are actually true or actually fals, have the following relation to the conclusion: If the premises were true, this fact would constitute good grounds for accepting the conclusion as true. If the facts alleged by the premises of a logically correct argument are, indeed, facts, then they do constitute good evidence for the conclusion. But even if one of the premises are false, the facts alleged by the premises would constitute good evidence for theconclusion if the facts were what the premises claim them to be. That is what we shall mean by saying that the premises of a logically correct argument support the conclusion. The premises of an argument support the conclusion if the truth of the premises would constitute good reason for asserting that the conclusion is true. When we say that the premises of an argument support the conclusion, we are not saying that the premises are true; we are saying that there would be good evidence for the conclusion if the premises were true.

The premises of a logically incorrect argument may seem to support the conclusion but actually they do not. Logically incorrect arguments are called fallacious. Even if the premises of a logically incorrect argument were true, this would not constitute good grounds for accepting the conclusion. The premises of a logically incorrect argument do not have the proper relevance to the conclusion.

Since the logical correctness or incorrectness of an argument depends solely upon the relation between premises and conclusion, logical correctness or incorrectness is completely independent of the truth of the premises. In particular, it is wrong to call an argument fallacious just because it has one or more false premises. Consider the argument concerning the hat in example a. You may already have recognized that there is something wrong with the argument from the size of the hat to the intellectuality of the owner; you might have been inclined to reject it on grounds of faulty logic. It would have been a mistake to do so. The argument is logically correct -- it is not fallacious -- but it does have at least one false premise. As a matter of fact, not everyone who has a large brain is highly intellectual. However, you should be able to see that the conclusion of this argument would have to be true if all of the premises were true. It is not the business of logic to find out whether people with large brains are intellectual; this matter can be decided only by scientific investigation. Logic can determine whether these premises support their conclusion.

As we have just seen, a logically correct argument may have one or more false premises. A logically incorrect or fallacious argument may have true premises; indeed, it may have a true conclusion as well.

b] Premises: All mammals are mortal.
All dogs are mortal.
Conclusion: All dogs are mammals.
This argument is obviously fallacious. The fact that the premises and the conclusion are all true statements does not mean that the premises support the conclusion. They do not. In section 5 we shall prove this argument fallacious by using a general method for treating fallacies. The techniques of section 14 also apply to arguments of this type. For the present, we can indicate the fallacious character of b by pointing out that the premises would still be true even if dogs were reptiles (not mammals). The conclusion would then be false. It happens that the conclusion, "All dogs are mammals," is true, but there is nothing in the premises which provides any basis for it.

Since the logical correctness or incorrectness of an argument depends solely upon the relation between the premises and the conclusion and is completely independent of the truth of the premises, we can analyze arguments without knowing whether the premises are true -- indeed, we can do so even when they are known to be false. This is a desirable feature of the situation. It is often useful to know what conclusions can be drawn from false or doubtful premises. For example, intelligent deliberation involves the consideration of the consequences of various alternatives. We may construct arguments with various premises in order to see what the consequences are.

c] Perhaps you are thinking of buying an expensive foreign sports car. You might suppose to yourself (as a hypothetical premise) that you purchase it. In addition to the cost of the car, there would be other expenses, such as maintenance, license, and insurance. When you add up all of these items, you see that it would create a severe strain on your budget. You recognize, moreover, that one of the main attractions of such a car involves driving at high speeds: in view of the 55-mile-per-hour speed limit, you realize that having it would create a strong temptation to violate the law, thus posing a serious risk of expensive speeding citations and a possible suspension of your driver's license. These considerations might convince you not to purchase such a car at this time.

In constructing such arguments, we do not pretend that the premises are true; rather, we can examine the arguments without even raising the question of the truth of the premises. Up to this point we have proceeded as if the only function of arguments is to provide justifications for conclusions. We see now that this is only one among several uses for arguments. In general, arguments serve to show the conclusions that can be drawn from given premises, whether these premises are known to be true, known to be false, or are merely doubtful.

For purposes of logical analysis it is convenient to present arguments in standard form. We shall adopt the practice of writing the premises first and identifying the conclusion by a triplet of dots.

d] Everyone who served on the jury was a registered voter.
Jones served on the jury.
Jones was a registered voter.

This argument is logically correct. Outside of logic books, we should not expect to find arguments expressed in this neat form. We must learn to recognize arguments when they occur in ordinary prose, for they are not usually set off in the middle of the page and labeled. Furthermore, we have to identify the premises and the conclusion, for they are not usually explicitly labeled. It is not necessary for the premises to precede the conclusion. Sometimes the conclusion comes last, sometimes first, and sometimes in the middle of the argument. For stylistic reasons arguments may be given in a variety of ways; for example, any of the following variations of d would be quite proper:

e] Everyone who served on the jury was a registered voter and Jones served on the jury; therefore, Jones was a registered voter.
f] Jones was a registered voter because Jones served on the jury, and everyone who served on the jury was a registered voter.
g] Since everyone who served on the jury was a registered voter, Jones must have been a registered voter, for Jones served on the jury.
The fact that an argument is being given is usually conveyed by certain words or phrases which indicate that a statement is functioning as a premise or as a conclusion. Terms like "therefore," "hence," "consequently," "so," and "it follows that" indicate that what comes immediately after is a conclusion. The premises from which it follows should be stated nearby. Also, certain verb forms which suggest necessity, such as "must have been," indicate that the statement in which they occur is a conclusion. They indicate that this statement follows necessarily (i.e., deductively) from stated premises. Other terms indicate that a statement is a premise: "since," "for," and "because" are examples. The statement which follows such a word is a premise. The conclusions based upon this premise should be found nearby. Terms which indicate parts of arguments should be used if, and only if, arguments are being presented. If no argument occurs, it is misleading to use these terms. For instance, if a statement is prefaced by the word "therefore," the reader has every right to expect that it follows from something which has already been said. When arguments are given, it is important to indicate that fact and to indicate exactly which statements are intended as premises and which as conclusions. It is up to the reader to be sure he understands which statements are premises and which are conclusions before he proceeds to subject arguments to analysis.

There is another respect in which arguments encountered in most contexts fail to have the standard logical form. When we subject arguments to logical analysis, all of the premises must be given explicitly. Many arguments, however, involve premises which are so obvious that it would be sheer pedantry to state them in ordinary speech and writing. We have already seen an example of an argument with missing premises. Holmes's argument about the hat was incomplete; we attempted to complete it in example a. Outside a logic book, example d might appear in either of the following forms, depending on which premise is considered more obvious:

h] Jones must have been a registered voter, for he served on the jury.
i] Jones was a registered voter, because everyone who served on the jury was a registered voter.

In neither case would there be any difficulty in finding the missing premise.4

It would be unreasonable to insist that arguments always be presented in complete form without missing premises. The person who puts forth an argument has every right to expect us to try to make the argument as strong as possible. In reconstructing arguments, we should do our best to find plausible premises which will make the argument logically correct. As we shall see, however, in some cases there appears to be no way to make the argument into a strong one. The premises needed to make the argument logically correct do not seem to be true, and those that we are prepared to believe true do not make the argument logicaly correct (see example z of section 7 and example v of section 14).

Although the missing premise is often a statement too obvious to bother making, it can also be a great pitfall. A missing premise can sometimes constitute a crucial hidden assumption. When we attempt to complete the arguments we encounter -- including those we ourselves put forth -- we bring to light the assumptions that would be required to make them logically correct. This step is often the most illuminating aspect of logical analysis. It sometimes turns out that the required premises are extremely dubious or obviously false.

Logical analysis of discourse involves three preliminary steps which we have discussed.

  1. Arguments must be recognized; in particular, unsupported statements must be distinguished from conclusions of arguments.
  2. When an argument has been found, the premises and conclusion must be identified.
  3. If the argument is incomplete, the missing premises must be supplied.
When an argument has been set out in complete and explicit form, logical standards can be applied to determine whether it is logically correct or fallacious.
1 A. Conan Doyle, "The Adventure of the Blue Carbuncle," Adventures of Sherlock Holmes (New York and London: Harper & Row, n.d.), p. 157. Direct quotation and use of literary material from this story by permission of the Estate of Sir Arthur Conan Doyle.

2 The term "statement" is used to refer to components of arguments because it is philosophically more neutral than alternatives such as "sentence" or "proposition." No technical definition of "statement" is offered here, because any definition would raise controversies in the philosophy of language which need not trouble the beginner. More sophisticated readers may supply whatever technical definition seems most appropriate to them.

3 There are important exceptions to this statement. They will be discussed in section 12 and 33, but they can safely be ignored until then.

4 Arguments with missing premises have traditionally been called enthymemes -- literally translated, "in the mind" -- but there is no need to memorize that fancy word. It is entirely proper call them simply incomplete arguments. Moreover, in some arguments the conclusion is unstated; the premises are given and we are left to draw our own conclusion. Such arguments have also been called enthymemes, but we shall call them incomplete arguments as well.


In the preceding section we discussed arguments and said that logic can be used to analyze and evaluate them. This is one important function of logic. In the minds of most people, however, logic has another function: it has something to do with thinking and reasoning. Thinking and reasoning consist, in part at least, of making inferences. In this section we shall discuss the application of logic to inferences.

Many of our beliefs and opinions -- indeed, much of our knowledge -- are results of inference. The Sherlock Holmes example provides a simple illustration of this point. Holmes did not see that the owner of the hat was highly intellectual; he saw that the hat was large, and he inferred that the owner was highly intellectual. In the preceding section we discussed Holmes's argument; now let us consider his inference. Holmes had come to the conclusion that the owner of the hat was highly intellectual. This conclusion was a belief or opinion that he held. When Dr. Watson paid him a visit, he stated this conclusion. Holmes had arrived at his conclusion on the basis of evidence that he had. When called upon to do so, he stated his evidence for his conclusion. By stating his evidence and his conclusion, he presented an argument. Before he presented his argument, he had made an inference from his evidence to his conclusion.

There are close parallels between arguments and inferences. Both arguments and inferences involve evidence and conclusions standing in relation to each other. The main difference lies in the fact that an argument is a linguistic entity, a group of statements; an inference is not.

In the first place, the conclusion of an argument is a statement. The conclusion of an inference is an opinion, belief, or some such thing. We began our discussion of logic by dealing with statements, distinguishing between those which are supported and those which are not. A supported statement is the conclusion of an argument. It may have occurred to you that a similar distinction could have been made between beliefs and opinions which are supported and those which are not. We have evidence for some of our beliefs and opinions, while for others we do not. A supported belief or opinion would then be the conclusion of an inference.

If we wish to consider the justification of a belief or opinion, we must examine the evidence for it. In an argument, the evidence is given in statements: the premises. In an inference, the person who makes the inference must have the evidence. To say that a person has evidence is to say that he has knowledge, beliefs, or opinions of a certain kind. For instance, Holmes knew that the hat was large. In addition, he believed that there is a relation between hat size and intellectual ability. This was part of his evidence.

Making an inference is a psychological activity; it consists of drawing a conclusion from evidence, of arriving at certain opinions or beliefs on the basis of others. But logic is not psychology; it does not attempt to describe or explain the mental processes that occur when people infer, think, or reason. Some inferences are, nevertheless, logically correct; others are logically incorrect. Logical standards can be applied to inferences in order to subject them to critical analysis.

In order to evaluate an inference, we must consider the relation between a conclusion and the evidence from which the conclusion is drawn. The conclusion must be stated, and so must the evidence. When the evidence is stated, we have the premises of an argument. When the conclusion is stated, it becomes the conclusion of that argument. The statement of the inference is thus an argument, and it can be subjected to logical analysis and evaluation as we indicated in the preceding section. In the logical analysis of an inference we are not concerned with how the person who made the inference reached his conclusion. We are concerned only with the question of whether his conclusion is supported by the evidence upon which it is based. In order to answer this question, the inference must be stated; when it is stated, it becomes an argument. This is true even if the person who makes the inference states it only to himself.

As we explained in the preceding section, the logical correctness of an argument does not depend upon the truth of its premises. In precisely the same manner, the logical correctness of an inference is independent of the truth of the beliefs or opinions which constitute its evidence. The beliefs or opinions upon which the conclusion of an inference is based may support that conclusion even if these beliefs or opinions are false. Just as we sometimes construct arguments with premises which are dubious or known to be false, so also we often make inferences from suppositions which are dubious or believed to be false. For example, we might make an inference from the suppositions that we will go camping for the next two weeks and that the weather will be rainy during that period. This inference might be an important part of the deliberation which helps us decide how to spend a vacation. In making inference of this kind we need not believe our suppositions; rather, we want to see what the consequences would be if the suppositions were true.

You may still feel uneasy about limiting the application of logic to arguments. Is it true, you might ask, that all inferences can be rendered in language? Are there not, perhaps, some beliefs which cannot be expressed in statements? Is there not, perhaps, some evidence which cannot be stated in words? Is it impossible that someone has a belief which is supported by evidence, and yet either the belief itself or the evidence for it defies formulation in language? This possibility cannot be denied, but caution is required. Suppose the following incident had occurred:

a] Holmes handed the hat to Watson and asked him what he could infer concerning its owner. Watson examined it carefully, and at last he said, "Something about the character of the man is clearly indicated, but it is one of those facets of his spirit which cannot be expressed in words." Hiding his impatience, Holmes asked Watson what evidence he had for this conclusion. Watson replied, "It is an ineffable quality about the hat -- something I couldn't possibly describe."

Such a performance (which never occurred and is completely out of character for Dr. Watson) would surely arouse a reasonable suspicion that no inference had been made, no conclusion drawn, and no evidence found. Whether this suspicion is correct or not, it would be impossible for anyone, including Watson, to subject the alleged inference to logical analysis. Thus, we shall regard the analysis of arguments as the primary function of logic, recognizing that inferences may be dealt with by transforming them into arguments. This means that there is an extremely close relationship between logic and language. We shall see this intimate connection repeatedly as our discussion proceeds.


When a statement has been made, two important questions may be raised: How did it come to be thought of? and What reasons do we have for accepting it as true? These are different questions. It would be a serious error to confuse the questions, and it would be at least as serious to confuse the answers. The first question is one of discovery; circumstances pertaining to this question are within the context of discovery. The second question is one of justification; matters relevant to this question belong to the context of justification.

Whenever the support of a statement is at issue, it is essential to keep clear the distinction between the context of discovery and the context of justification. The justification of a statement consists of an argument. The statement to be justified is the conclusion of the argument. The argument consists of that conclusion and its supporting evidence standing in relation to each other. The discovery of the statement, by contrast, is a psychological process whereby the statement is thought of, entertained, or even accepted.

The distinction between discovery and justification is clearly illustrated by the following example:

a] The Indian mathematical genius Ramanujan (1887-1920) claimed that the goddess of Namakkal visited him in his dreams and gave him mathematical formulas. Upon awaking, he would write them down and verify them.5

There is no reason to doubt that Ramanujan received inspiration in his sleep, whether from the goddess of Namakkal or from more natural sources. These circumstances have nothing to do with the truth of the formulas. The justification relates to their proofs -- mathematical arguments -- which in some cases he supplied after he awakened. The following example illustrates the same point:
b] There is a famous story about Sir Isaac Newton's discovery of the law of gravitation. According to this (probably apocryphal) tale, Newton was sitting in the garden one day and saw an apple fall to the ground. Suddenly he had a brilliant insight: the planets in their courses, objects falling to earth, the tides -- all are governed by the law of universal gravitation.

This is a charming anecdote about the discovery of the theory, but it has no bearing upon its justification. The question of justification can be answered only in terms of observations, experiments, and arguments -- in short, the justification depends upon evidence for the theory and not upon the psychological factors which made the theory occur to Newton in the first place.

Questions of justification are questions about the acceptability of statements. Since a justification of a statement is an argument, justification involves two aspects: the truth of the premises and the logical correctness of the argument. As we have previously emphasized, these two aspects are independent of each other. A justification can fail on account of either one. If we show that the premises are false or dubious, we have shown that the justification is inadequate. Similarly, if we show that the argument is logically incorrect, we have shown that the justification is unsatisfactory. Showing that a justification is inadequate, on either of these grounds, does not show that the conclusion is false. There may be another, adequate justification for the same conclusion. When we have shown that a justification is inadequate, we have merely shown that it does not provide good reason for believing the conclusion to be true. Under these circumstances, we have neither reason to accept the conclusion as true nor reason to reject it as false; there is simply no justification. It will be important to remember this fact as you read the remaining sections of this book. We shall consider many arguments which either have false premises or are logically incorrect. It does not follow that these arguments have false conclusions.

There is, however, such a thing as negative justification. It is sometimes possible to show that a statement is false. As we shall see, the reductio ad absurdum (section 8) and the argument against the person (section 25) are often used in this way. Negative justifications are as much within the context of justification as are positive justifications. Questions concerning the adequacy of a justification are also within the context of justification.

The error of treating items in the context of discovery as if they belonged to the context of justification is called the genetic fallacy. It is the fallacy of considering factors in the discovery or genesis of a statement relevant, ipso facto, to the truth or falsity of it. For example,
c] The Nazis condemned the theory of relativity because Einstein, its originator, was a Jew.
This is a flagrant case of the genetic fallacy. The national or religious background of the discoverer of a theory is certainly relevant only to the context of discovery. The Nazis used such a fact as if it belonged to the context of justification.

The genetic fallacy must, however, be treated with care. As we shall see (sections 24 and 25), both the argument from authority and the argument against the man have fallacious forms, often cases of the genetic fallacy. Both arguments, however, also have correct forms, which must not be confused with the genetic fallacy. The difference is this. Items within the context of discovery can sometimes be correctly incorporated into the context of justification by showing that there is an objective connection between that aspect of the discovery and the truth or falsity of the conclusion. The argument then requires a premise stating this objective connection. The genetic fallacy, in contrast, consists of citing a feature of the discovery without providing any such connection with justification.

The distinction between discovery and justification is closely related to the distinction between inference and argument. The psychological activity of making an inference is a process of discovery. The person who makes an inference must think of the conclusion, but this is not the entire problem of discovery. He must discover the evidence, and he must discover the relationship between the evidence and the conclusion. The inference is sometimes described as the transition from the evidence to the conclusion. If this is taken to mean that thinking, reasoning, and inferring consist in starting from evidence which is somehow given and proceeding by neat logical steps to a conclusion, then it is certainly inaccurate. In the first place, the evidence is not always given before the conclusion. Sometimes a conclusion occurs to you first; then you have to try to find evidence which will support it or show it to be false. Sometimes you have a little evidence, then you think of a conclusion, and finally you have to discover more evidence before you have a completed inference. Even if you start with some evidence and simply proceed from it to a conclusion, in most cases thinking does not proceed in logical steps. Our mind wander, we daydream, reveries intrude, irrelevant free association occur, and blind alleys are followed. No matter how it happens, however, the inference is sometimes completed, and we end up with evidence and conclusion standing in relation to each other. All of this pertains to discovery. When the process of discovery is finished, the inference can be transformed into an argument, as we explained in the preceding section, and the argument can be examined for logical correctness. The resulting argument is not by any means a description of the thought processes which led to the conclusion.

It should be evident that logic does not attempt to describe the ways in which people actually think. You may wonder, however, whether it is the business of logic to lay down rules to determine how we ought to think. Does logic provide a set of rules to guide us in reasoning, in problem solving, in drawing conclusions? Does logic prescribe the steps we should follow in making inferences? This conception is a common one. People who reason effectively are often said to have a "logical mind," and they are said to reason "logically."

Sherlock Holmes is a prime example of a man with superb reasoning powers. He is extremely skillful at making inferences and drawing conclusions. When we examine this ability we see, however, that it does not lie in utilizing a set of rules to guide his thinking. For one thing, Holmes is vastly superior to his friend Watson in making inferences. Holmes is willing to teach Watson his methods, and Watson is an intelligent man. Unfortunately, there are no rules Holmes can communicate to Watson which will make Watson capable of Holmes's feats of reasoning. Holmes's abilities consist of factors such as his keen curiosity, his high native intelligence, his fertile imagination, his acute powers of percepton, his wealth of general information, and his extreme ingenuity. No set of rules can provide a substitute for such abilities.

If there were a set of rules for making inferences, these rules would constitute rules for discovery. Actually, effective thinking requires free play of thought and imagination. To be bound by rigid methods or rules would tend only to hamper thought. The most fruitful ideas are often precisely those which rules could not produce. Of course, people can improve their reasoning abilities by education, practice, and training, but all this is a far cry from learning and adopting a set of rules of thought. In any case, when we discuss the specific rules of logic we shall see that they could not begin to be adequate as methods of thinking. The rules of logic, if imposed as limitations on ways of thinking, would become a straitjacket.

What we have said about logic may be disappointing. We have placed a good deal of emphasis upon the negative side by saying what logic cannot do. Logic cannot provide a description of actual thought processes -- such problems are in the domain of psychology. Logic cannot provide rules for making inferences -- such matters pertain to discovery. What, then, is logic good for? Logic provides critical tools with which we can make sound evaluations of inferences. This is the sense in which logic does tell us how we ought to think. Once an inference has been made, it can be transformed into an argument, and logic can be applied to determine whether it is correct. Logic does not tell us how to make inferences, but it does tell us which ones we ought to accept. A person who accepts incorrect inferences is being illogical.

To appreciate the value of logical tools, it is important to have realistic expectations about their use. If you expect a hammer to do the job of a screwdriver you are bound to be disappointed, but if you understand its function you can see its usefulness. Logic deals with justification, not with discovery. Logic provides tools for the analysis of discourse; such analysis is indispensable to intelligent expression of our own views and the clear understanding of the claims of others.

4 G. H. Hardy, P. V. Seshu Aiyar, and B. M. Wilson, eds., Collected Papers of Srinivasa Ramanujan (Cambridge: University Press, 1927), p. xii.


What we have said so far applies to all types of arguments. The time has come to distinguish two major types: deductive and inductive. There are logically correct and incorrect forms of each. Here are correct examples.
a] Deductive: Every mammal has a heart.
All horses are mammals.
Every horse has a heart.
b] Inductive: Every horse that has ever been observed has had a heart.
Every horse has a heart.

There are certain fundamental characteristics which distinguish deductive and inductive arguments. We will mention two primary ones.
  1. If all of the premises are true, the conclusion must be true.
  1. If all of the premises are true, the conclusion is probably true but not necessarily true.
  1. All of the information or factual content in the conclusion was already contained, at least implicitly, in the premises.
  1. The conclusion contains information not present, even implicitly, in the premises.

It is not difficult to see that the two examples satisfy these conditions.

Characteristic I. The only way in which the conclusion of a could be false -- that is, the only possible circumstance under which it could fail to be true that every horse has a heart -- is that either not all horses are mammals or not all mammals have hearts. In other words, for the conclusion of a to be false, one or both of the premises must be false. If both premises are true, the conclusion must be true. On the other hand, in b, it is quite possible for the premise to be true and the conclusion false. This would happen if at some future time, a horse is observed which does not have a heart. The fact that no horse without a heart has yet been observed is some evidence that none ever will be. In this argument, the premise does not necessitate the conclusion, but it does lend some weight to it.

Characteristic II. When the conclusion of a says that all horses have hearts, it says something which has already been said, in effect, by the premises. The first premise says that all mammals have hearts, and that includes all horses according to the second premise. This argument, like all other correct deductive arguments, states explicitly or reformulates information already given in the premises. It is for this reason that deductive arguments also have characteristic I. The conclusion must be true if the premises are true, because the conclusion says nothing which was not already stated by the premises. However, the premise of our inductive argument b refers only to horses that have been observed up to the present, whereas the conclusion refers to horses that have not yet been observed. Thus, the conclusion makes a statement that goes beyond the information given in the premise. It is because the conclusion says something not given in the premise that the conclusion might be false even though the premise is true. The additional content of the conclusion might be false, rendering the conclusion as a whole false. Deductive and inductive arguments fulfill different functions. The deductive argument is designed to make explicit the content of the premises; the inductive argument is designed to extend the range of our knowledge.

We may summarize by saying that the inductive argument expands the content of premises by sacrificing necessity, whereas the deductive argument achieves necessity by sacrificing any expansion of content.

It follows immediately from these characteristics that deductive correctness (known as validity -- see section 5) is an all-or-nothing affair. An argument either qualifies fully as a correct deduction or fails completely; there are no degrees of deductive validity. The premises either completely necessitate the conclusion or fail entirely to do so. Correct inductive arguments, in contrast, admit of degrees of strength, depending upon the amount of support the premises furnish for the conclusion. There are degrees of probability that the premises of an inductive argument can supply to a conclusion, but the logical necessity that relates premises to conclusion in a deductive argument is never a matter of degree.

Examples a and b illustrate clearly the basic properties of inductive and deductive arguments and the important differences between them. These same characteristics can be used to classify far less trivial arguments.
c] The relation between a scientific generalization and its supporting observational evidence is inductive. For example, according to Kepler's first law, the orbit of Mars is an ellipse. The observational evidence for this law consists of a number of isolated observations of the position of Mars. The law itself refers to the position of the planet, whether it is observed or not. In particular, the law states that the future motion of the planet will be elliptical, that its motion was elliptical before any person observed it, and that its motion is elliptical when clouds obscure the sky. Clearly, this law (the conclusion) has far more content than the statements describing the observed positions of Mars (the premises).6

d] Mathematical argument is deductive. The most familiar example is Euclidean plane geometry, which almost everyone studies in high school. In geometry, theorems are proved on the basis of axioms and postulates. The method of proof is to deduce the theorems (conclusions) from the axioms and postulates (premises). The method of deduction assures us that the theorems must be true if the axioms and postulates are true. This example shows, incidentally, that deductive arguments are not always trivial. Although the content of the theorem is given in the axioms and postulates, this content is by no means completely obvious. Making the implicit content of the axioms and postulates explicit is genuinely illuminating.

One of the main reasons for emphasizing the basic characteristics of inductive and deductive arguments is to make clear what can be expected of them. In particular, deduction has a severe limitation. The content of the conclusion of a correct deductive argument is present in the premises. An argument, therefore, cannot be a logically correct deduction if it has a conclusion whose content exceeds that of its premises. Let us apply this consideration to two extremely important examples.

e] Rene Descartes (1596-1650), often considered the founder of modern philosophy, was deeply disturbed by the error and uncertainty of what passed for knowledge. To remedy the situation, he attempted to found his philosophical system upon indubitable truths and derive a variety of far-reaching consequences from them. In his Meditations he takes great pains to show that the proposition "I think, therefore, I exist" cannot be doubted. He then proceeds to elaborate a comprehensive philosophical system in which, eventually, he arrives at conclusions concerning the existence of God, the nature and existence of material objects, and the basic dualism of the physical and the mental. It is easy to form the impression that Descartes is attempting to deduce these consequences from the preceding premise alone, but no such deduction could be logically correct, for the content of the conclusions far exceeds that of the premise. Unless Descartes was guilty of gross logical blunders, he must have used additional premises, nondeductive arguments, or both. These simple considerations should serve as a warning that the initial interpretation of Descartes's argument is probably mistaken and that a closer examination is required.
f] One of the most profound and vexing problems in moral philosophy is that of providing a justification for value judgments. The British philosopher David Hume (1711-1776) saw clearly that value judgments cannot be justified by deducing them from statements of fact alone He writes, "In every system of morality which I have hitherto met with, I have always remarked that the author proceeds for some time in the ordinary way of reasoning, and establishes the being of a god, or makes observations concerning human affairs; when of a sudden I am surprised to find that instead of the usual copulations of proposition is and is not, I meet with no proposition that is not connected with an ought or an ought not. This change is imperceptible, but is, however, of the last consequence. For as this ought or ought not expresses some new relation or affirmation, it is necessary that it should be observed and explained; and at the same time that a reason should be given for what seems altogether inconceivable, how this new relation can be a deduction from others which are entirely different from it."7

The characteristics just described serve to distinguish correct deductive from correct inductive arguments. At this point you might be wondering how to identify incorrect deductions and inductions. We shall, after all, encounter some well-known invalid forms of deductive argument (e.g., in sections 7 and 14), as well as some inductive fallacies (e.g., in sections 21 Mand 22). Nevertheless, strictly speaking there are no incorrect deductive or inductive arguments; there are valid deductions, correct inductions, and assorted fallacious arguments. There are, of course, invalid arguments that look something like valid deductions; these are loosely called "invalid deductions" or "deductive fallacies." They are identified as deductive fallacies because they can easily be confused with correct deductions. Similar remarks could be made about the mistakes that plague inductive reasoning. There are no precise logical characteristics that delineate incorrect deductions and incorrect inductions; basically it is a psychological matter. An incorrect argument that might be offered as a valid deduction, either because the author of the argument is making a logical error or because he or she hopes to fool someone else, would normally be considered a deductive fallacy. Incorrect induction could be described analogously.

You may have noticed, quite correctly, that any inductive argument can be transformed into a deductive argument by the addition of one or more premises. It might, therefore, be tempting to regard inductive arguments as incomplete deductive arguments rather than as an important and distinct type. This would be a mistake. Even though every inductive argument can be made into a deductive argument by the addition of premises, the required premises are often statements whose truth is very doubtful. If we are trying to justify conclusions, it will not do to introduce highly dubious premises. Actually, the kind of argument which extends our knowledge is indispensable. For instance, if no such mode of argument were available, it would be impossible to establish any conclusions about the future on the basis of our experience of the past and present. Without some type of inductive reasoning, we would have no grounds for predicting that night will continue to follow day, that the seasons will continue to occur in their customary sequence, or that sugar will continue to taste sweet. All such knowledge of the future, and much else as well, depends upon the power of inductive arguments to support conclusions that go beyond the data presented in their premises.

6 The fact that Kepler's first law is a consequence of Newton's laws does not affect the example, for Newton's laws, themselves, have content that goes far beyond the observational evidence on which they rest. In either case, there is an inductive relation between the observational evidence and the laws.

7 David Hume, A Treatise of Human Nature, Bk. III, Pt. I, Sec. I.