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Arguments and Inference
Human life is full of decisions, including significant choices about what to believe. Although everyone prefers to believe what is true, we often disagree with each other about what that is in particular instances. It may be that some of our most fundamental convictions in life are acquired by haphazard means rather than by the use of reason, but we all recognize that our beliefs about ourselves and the world often hang together in important ways.
If I believe that whales are mammals and that all mammals are fish, then it would also make sense for me to believe that whales are fish. Even someone who (rightly!) disagreed with my understanding of biological taxonomy could appreciate the consistent, reasonable way in which I used my mistaken beliefs as the foundation upon which to establish a new one. On the other hand, if I decide to believe that Hamlet was Danish because I believe that Hamlet was a character in a play by Shaw and that some Danes are Shavian characters, then even someone who shares my belief in the result could point out that I haven't actually provided good reasons for accepting its truth.
In general, we can respect the directness of a path even when we don't accept the points at which it begins and ends. Thus, it is possible to distinguish correct reasoning from incorrect reasoning independently of our agreement on substantive matters. Logic is the discipline that studies this distinction—both by determining the conditions under which the truth of certain beliefs leads naturally to the truth of some other belief, and by drawing attention to the ways in which we may be led to believe something without respect for its truth. This provides no guarantee that we will always arrive at the truth, since the beliefs with which we begin are sometimes in error. But following the principles of correct reasoning does ensure that no additional mistakes creep in during the course of our progress.
In this review of elementary logic, we'll undertake a broad survey of the major varieties of reasoning that have been examined by logicians of the Western philosophical tradition. We'll see how certain patterns of thinking do invariably lead from truth to truth while other patterns do not, and we'll develop the skills of using the former while avoiding the latter. It will be helpful to begin by defining some of the technical terms that describe human reasoning in general.
Our fundamental unit of what may be asserted or denied is the proposition (or statement) that is typically expressed by a declarative sentence. Logicians of earlier centuries often identified propositions with the mental acts of affirming them, often called judgments, but we can evade some interesting but thorny philosophical issues by avoiding this locution.
Propositions are distinct from the sentences that convey them. "Smith loves Jones" expresses exactly the same proposition as "Jones is loved by Smith," while the sentence "Today is my birthday" can be used to convey many different propositions, depending upon who happens to utter it, and on what day. But each proposition is either true or false. Sometimes, of course, we don't know which of these truth-values a particular proposition has ("There is life on the third moon of Jupiter" is presently an example), but we can be sure that it has one or the other.
The chief concern of logic is how the truth of some propositions is connected with the truth of another. Thus, we will usually consider a group of related propositions. An argument is a set of two or more propositions related to each other in such a way that all but one of them (the premises) are supposed to provide support for the remaining one (the conclusion). The transition or movement from premises to conclusion, the logical connection between them, is the inference upon which the argument relies.
Notice that "premise" and "conclusion" are here defined only as they occur in relation to each other within a particular argument. One and the same proposition can (and often does) appear as the conclusion of one line of reasoning but also as one of the premises of another. A number of words and phrases are commonly used in ordinary language to indicate the premises and conclusion of an argument, although their use is never strictly required, since the context can make clear the direction of movement. What distinguishes an argument from a mere collection of propositions is the inference that is supposed to hold between them.
Thus, for example, "The moon is made of green cheese, and strawberries are red. My dog has fleas." is just a collection of unrelated propositions; the truth or falsity of each has no bearing on that of the others. But "Helen is a physician. So Helen went to medical school, since all physicians have gone to medical school." is an argument; the truth of its conclusion, "Helen went to medical school," is inferentially derived from its premises, "Helen is a physician." and "All physicians have gone to medical school."
It's important to be able to identify which proposition is the conclusion of each argument, since that's a necessary step in our evaluation of the inference that is supposed to lead to it. We might even employ a simple diagram to represent the structure of an argument, numbering each of the propositions it comprises and drawing an arrow to indicate the inference that leads from its premise(s) to its conclusion.
Don't worry if this procedure seems rather tentative and uncertain at first. We'll be studying the structural features of logical arguments in much greater detail as we proceed, and you'll soon find it easy to spot instances of the particular patterns we encounter most often. For now, it is enough to tell the difference between an argument and a mere collection of propositions and to identify the intended conclusion of each argument.
Even that isn't always easy, since arguments embedded in ordinary language can take on a bewildering variety of forms. Again, don't worry too much about this; as we acquire more sophisticated techniques for representing logical arguments, we will deliberately limit ourselves to a very restricted number of distinct patterns and develop standard methods for expressing their structure. Just remember the basic definition of an argument: it includes more than one proposition, and it infers a conclusion from one or more premises. So "If John has already left, then either Jane has arrived or Gail is on the way." can't be an argument, since it is just one big (compound) proposition. But "John has already left, since Jane has arrived." is an argument that proposes an inference from the fact of Jane's arrival to the conclusion, "John has already left." If you find it helpful to draw a diagram, please make good use of that method to your advantage.
Our primary concern is to evaluate the reliability of inferences, the patterns of reasoning that lead from premises to conclusion in a logical argument. We'll devote a lot of attention to what works and what does not. It is vital from the outset to distinguish two kinds of inference, each of which has its own distinctive structure and standard of correctness.
Deductive Inferences
When an argument claims that the truth of its premises guarantees the truth of its conclusion, it is said to involve a deductive inference. Deductive reasoning holds to a very high standard of correctness. A deductive inference succeeds only if its premises provide such absolute and complete support for its conclusion that it would be utterly inconsistent to suppose that the premises are true but the conclusion false.
Notice that each argument either meets this standard or else it does not; there is no middle ground. Some deductive arguments are perfect, and if their premises are in fact true, then it follows that their conclusions must also be true, no matter what else may happen to be the case. All other deductive arguments are no good at all—their conclusions may be false even if their premises are true, and no amount of additional information can help them in the least.
When an argument claims merely that the truth of its premises make it likely or probable that its conclusion is also true, it is said to involve an inductive inference. The standard of correctness for inductive reasoning is much more flexible than that for deduction. An inductive argument succeeds whenever its premises provide some legitimate evidence or support for the truth of its conclusion. Although it is therefore reasonable to accept the truth of that conclusion on these grounds, it would not be completely inconsistent to withhold judgment or even to deny it outright.
Inductive arguments, then, may meet their standard to a greater or to a lesser degree, depending upon the amount of support they supply. No inductive argument is either absolutely perfect or entirely useless, although one may be said to be relatively better or worse than another in the sense that it recommends its conclusion with a higher or lower degree of probability. In such cases, relevant additional information often affects the reliability of an inductive argument by providing other evidence that changes our estimation of the likelihood of the conclusion.
It should be possible to differentiate arguments of these two sorts with some accuracy already. Remember that deductive arguments claim to guarantee their conclusions, while inductive arguments merely recommend theirs. Or ask yourself whether the introduction of any additional information—short of changing or denying any of the premises—could make the conclusion seem more or less likely; if so, the pattern of reasoning is inductive.
Since deductive reasoning requires such a strong relationship between premises and conclusion, we will spend the majority of this survey studying various patterns of deductive inference. It is therefore worthwhile to consider the standard of correctness for deductive arguments in some detail.
A deductive argument is said to be valid when the inference from premises to conclusion is perfect. Here are two equivalent ways of stating that standard:
- If the premises of a valid argument are true, then its conclusion must also be true.
- It is impossible for the conclusion of a valid argument to be false while its premises are true.
(Considering the premises as a set of propositions, we will say that the premises are true only on those occasions when each and every one of those propositions is true.) Any deductive argument that is not valid is invalid: it is possible for its conclusion to be false while its premises are true, so even if the premises are true, the conclusion may turn out to be either true or false.
Notice that the validity of the inference of a deductive argument is independent of the truth of its premises; both conditions must be met in order to be sure of the truth of the conclusion. Of the eight distinct possible combinations of truth and validity, only one is ruled out completely:
| Premises | Inference | Conclusion |
| True | Valid | True |
| XXXX | ||
| Invalid | True | |
| False | ||
| False | Valid | True |
| False | ||
| Invalid | True | |
| False |
The only thing that cannot happen is for a deductive argument to have true premises and a valid inference but a false conclusion.
Some logicians designate the combination of true premises and a valid inference as a sound argument; it is a piece of reasoning whose conclusion must be true. The trouble with every other case is that it gets us nowhere, since either at least one of the premises is false, or the inference is invalid, or both. The conclusions of such arguments may be either true or false, so they are entirely useless in any effort to gain new information.
Definition by Genus and Differentia
Classical logicians developed an especially effective method of constructing connotative definitions for general terms, by stating their genus and differentia. The basic notion is simple: we begin by identifying a familiar, broad category or kind (the genus) to which everything our term signifies (along with things of other sorts) belongs; then we specify the distinctive features (the differentiae) that set them apart from all the other things of this kind. My definition of the word "chair" in the second paragraph of this lesson, for example, identifies "piece of furniture" as the genus to which all chairs belong and then specifies "designed to be sat upon by one person at a time" as the differentia that distinguishes them from couches, desks, etc.
Copi and Cohen list five rules by means of which to evaluate the success of connotative definitions by genus and differentia:
- Focus on essential features. Although the things to which a term applies may share many distinctive properties, not all of them equally indicate its true nature. Thus, for example, a definition of "human beings" as "featherless bipeds" isn't very illuminating, even if does pick out the right individuals. A good definition tries to point out the features that are essential to the designation of things as members of the relevant group.
- Avoid circularity. Since a circular definition uses the term being defined as part of its own definition, it can't provide any useful information; either the audience already understands the meaning of the term, or it cannot understand the explanation that includes that term. Thus, for example, there isn't much point in defining "cordless 'phone" as "a telephone that has no cord."
- Capture the correct extension. A good definition will apply to exactly the same things as the term being defined, no more and no less. There are several ways to go wrong. Consider alternative definitions of "bird":
- "warm-blooded animal" is too broad, since that would include horses, dogs, and aardvarks along with birds.
- "feathered egg-laying animal" is too narrow, since it excludes those birds who happen to be male. and
- "small flying animal" is both too broad and too narrow, since it includes bats (which aren't birds) and excludes ostriches (which are).
Successful intensional definitions must be satisfied by all and only those things that are included in the extension of the term they define.
- Avoid figurative or obscure language. Since the point of a definition is to explain the meaning of a term to someone who is unfamiliar with its proper application, the use of language that doesn't help such a person learn how to apply the term is pointless. Thus, "happiness is a warm puppy" may be a lovely thought, but it is a lousy definition.
- Be affirmative rather than negative. It is always possible in principle to explain the application of a term by identifying literally everything to which it does not apply. In a few instances, this may be the only way to go: a proper definition of the mathematical term "infinite" might well be negative, for example. But in ordinary circumstances, a good definition uses positive designations whenever it is possible to do so. Defining "honest person" as "someone who rarely lies" is a poor definition.
Fallacies of Relevance
Assessing the legitimacy of arguments embedded in ordinary language is rather like diagnosing whether a living human being has any broken bones. Only the internal structure matters, but it is difficult to see through the layers of flesh that cover it. Soon we'll begin to develop methods, like the tools of radiology, that enable us to see the skeletal form of an argument beneath the language that expresses it. But compound fractures are usually evident to the most casual observer, and some logical defects are equally apparent.
The informal fallacies considered here are patterns of reasoning that are obviously incorrect. The fallacies of relevance, for example, clearly fail to provide adequate reason for believing the truth of their conclusions. Although they are often used in attempts to persuade people by non-logical means, only the unwary, the predisposed, and the gullible are apt to be fooled by their illegitimate appeals. Many of them were identified by medieval and renaissance logicians, whose Latin names for them have passed into common use. It's worthwhile to consider the structure, offer an example, and point out the invalidity of each of them in turn.
Appeal to Force (argumentum ad baculum)
In the appeal to force, someone in a position of power threatens to bring down unfortunate consequences upon anyone who dares to disagree with a proffered proposition. Although it is rarely developed so explicitly, a fallacy of this type might propose:
- If you do not agree with my political opinions, you will receive a grade of F for this course.
- I believe that Herbert Hoover was the greatest President of the
- Therefore, Herbert Hoover was the greatest President of the
It should be clear that even if all of the premises were true, the conclusion could neverthelss be false. Since that is possible, arguments of this form are plainly invalid. While this might be an effective way to get you to agree (or at least to pretend to agree) with my position, it offers no grounds for believing it to be true.
Appeal to Pity (argumentum ad misericordiam)
Turning this on its head, an appeal to pity tries to win acceptance by pointing out the unfortunate consequences that will otherwise fall upon the speaker and others, for whom we would then feel sorry.
- I am a single parent, solely responsible for the financial support of my children.
- If you give me this traffic ticket, I will lose my license and be unable to drive to work.
- If I cannot work, my children and I will become homeless and may starve to death.
- Therefore, you should not give me this traffic ticket.
Again, the conclusion may be false (that is, perhaps I should be given the ticket) even if the premises are all true, so the argument is fallacious.
Appeal to Emotion (argumentum ad populum)
In a more general fashion, the appeal to emotion relies upon emotively charged language to arouse strong feelings that may lead an audience to accept its conclusion:
- As all clear-thinking residents of our fine state have already realized, the Governor's plan for financing public education is nothing but the bloody-fanged wolf of socialism cleverly disguised in the harmless sheep's clothing of concern for children.
- Therefore, the Governor's plan is bad public policy.
The problem here is that although the flowery language of the premise might arouse strong feelings in many members of its intended audience, the widespread occurrence of those feelings has nothing to do with the truth of the conclusion.
Appeal to Authority (argumentum ad verecundiam)
Each of the next three fallacies involve the mistaken supposition that there is some connection between the truth of a proposition and some feature of the person who asserts or denies it. In an appeal to authority, the opinion of someone famous or accomplished in another area of expertise is supposed to guarantee the truth of a conclusion. Thus, for example:
- Federal Reserve Chair Alan Greenspan believes that spiders are insects.
- Therefore, spiders are insects.
As a pattern of reasoning, this is clearly mistaken: no proposition must be true because some individual (however talented or successful) happens to believe it. Even in areas where they have some special knowledge or skill, expert authorities could be mistaken; we may accept their testimony as inductive evidence but never as deductive proof of the truth of a conclusion. Personality is irrelevant to truth.
Ad Hominem Argument
The mirror-image of the appeal to authority is the ad hominem argument, in which we are encouraged to reject a proposition because it is the stated opinion of someone regarded as disreputable in some way. This can happen in several different ways, but all involve the claim that the proposition must be false because of who believes it to be true:
- Harold maintains that the legal age for drinking beer should be 18 instead of 21.
- But we all know that Harold . . .
- . . . dresses funny and smells bad. or
- . . . is 19 years old and would like to drink legally or
- . . . believes that the legal age for voting should be 21, not 18 or
- . . . doesn't understand the law any better than the rest of us.
- Therefore, the legal age for drinking beer should be 21 instead of 18.
In any of its varieties, the ad hominem fallacy asks us to adopt a position on the truth of a conclusion for no better reason than that someone believes its opposite. But the proposition that person believes can be true (and the intended conclusion false) even if the person is unsavory or has a stake in the issue or holds inconsistent beliefs or shares a common flaw with us. Again, personality is irrelevant to truth.
Appeal to Ignorance (argumentum ad ignoratiam)
An appeal to ignorance proposes that we accept the truth of a proposition unless an opponent can prove otherwise. Thus, for example:
- No one has conclusively proven that there is no intelligent life on the moons of Jupiter.
- Therefore, there is intelligent life on the moons of Jupiter.
But, of course, the absence of evidence against a proposition is not enough to secure its truth. What we don't know could nevertheless be so.
Irrelevant Conclusion (ignoratio elenchi)
Finally, the fallacy of the irrelevant conclusion tries to establish the truth of a proposition by offering an argument that actually provides support for an entirely different conclusion.
- All children should have ample attention from their parents.
- Parents who work full-time cannot give ample attention to their children.
- Therefore, mothers should not work full-time.
Here the premises might support some conclusion about working parents generally, but do not secure the truth of a conclusion focussed on women alone and not on men. Although clearly fallacious, this procedure may succeed in distracting its audience from the point that is really at issue.
Fallacies of Presumption
The fallacies of presumption also fail to provide adequate reason for believing the truth of their conclusions. In these instances, however, the erroneous reasoning results from an implicit supposition of some further proposition whose truth is uncertain or implausible. Again, we'll consider each of them in turn, seeking always to identify the unwarranted assumption upon which it is based.
The fallacy of accident begins with the statement of some principle that is true as a general rule, but then errs by applying this principle to a specific case that is unusual or atypical in some way.
- Women earn less than men earn for doing the same work.
- Oprah Winfrey is a woman.
- Therefore, Oprah Winfrey earns less than male talk-show hosts.
As we'll soon see, a true universal premise would entail the truth of this conclusion; but then, a universal statement that "Every woman earns less than any man." would obviously be false. The truth of a general rule, on the other hand, leaves plenty of room for exceptional cases, and applying it to any of them is fallacious.
The fallacy of converse accident begins with a specific case that is unusual or atypical in some way, and then errs by deriving from this case the truth of a general rule.
- Dennis Rodman wears earrings and is an excellent rebounder.
- Therefore, people who wear earrings are excellent rebounders.
It should be obvious that a single instance is not enough to establish the truth of such a general principle. Since it's easy for this conclusion to be false even though the premise is true, the argument is unreliable.
The fallacy of false cause infers the presence of a causal connectionsimply because events appear to occur in correlation or (in the post hoc, ergo propter hoc variety) temporal succession.
- The moon was full on Thursday evening.
- On Friday morning I overslept.
- Therefore, the full moon caused me to oversleep.
Later we'll consider what sort of evidence adequately supports the conclusion that a causal relationship does exist, but these fallacies clearly are not enough.
Begging the Question (petitio principii)
Begging the question is the fallacy of using the conclusion of an argument as one of the premises offered in its own support. Although this often happens in an implicit or disguised fashion, an explicit version would look like this:
- All dogs are mammals.
- All mammals have hair.
- Since animals with hair bear live young, dogs bear live young.
- But all animals that bear live young are mammals.
- Therefore, all dogs are mammals.
Unlike the other fallacies we've considered, begging the question involves an argument (or chain of arguments) that is formally valid: if its premises (including the first) are true, then the conclusion must be true. The problem is that this valid argument doesn't really provide support for the truth its conclusion; we can't use it unless we have already granted that.
The fallacy of complex question presupposes the truth of its own conclusion by including it implicitly in the statement of the issue to be considered:
- Have you tried to stop watching too much television?
- If so, then you admit that you do watch too much television.
- If not, then you must still be watching too much television.
- Therefore, you watch too much television.
In a somewhat more subtle fashion, this involves the same difficulty as the previous fallacy. We would not willingly agree to the first premise unless we already accepted the truth of the conclusion that the argument is supposed to prove.
Fallacies of Ambiguity
In addition to the fallacies of relevance and presumption we examined in our previous lessons, there are several patterns of incorrect reasoning that arise from the imprecise use of language. An ambiguous word, phrase, or sentence is one that has two or more distinct meanings. The inferential relationship between the propositions included in a single argument will be sure to hold only if we are careful to employ exactly the same meaning in each of them. The fallacies of ambiguity all involve a confusion of two or more different senses.
An equivocation trades upon the use of an ambiguous word or phrase in one of its meanings in one of the propositions of an argument but also in another of its meanings in a second proposition.
- Really exciting novels are rare.
- But rare books are expensive.
- Therefore, Really exciting novels are expensive.
Here, the word "rare" is used in different ways in the two premises of the argument, so the link they seem to establish between the terms of the conclusion is spurious. In its more subtle occurrences, this fallacy can undermine the reliability of otherwise valid deductive arguments.
An amphiboly can occur even when every term in an argument is univocal, if the grammatical construction of a sentence creates its own ambiguity.
- A reckless motorist Thursday struck and injured a student who was jogging through the campus in his pickup truck.
- Therefore, it is unsafe to jog in your pickup truck.
In this example, the premise (actually heard on a radio broadcast) could be interpreted in different ways, creating the possibility of a fallacious inference to the conclusion.
The fallacy of accent arises from an ambiguity produced by a shift of spoken or written emphasis. Thus, for example:
- Jorge turned in his assignment on time today.
- Therefore, Jorge usually turns in his assignments late.
Here the premise may be true if read without inflection, but if it is read with heavy stress on the last word seems to imply the truth of the conclusion.
The fallacy of composition involves an inference from the attribution of some feature to every individual member of a class (or part of a greater whole) to the possession of the same feature by the entire class (or whole).
- Every course I took in college was well-organized.
- Therefore, my college education was well-organized.
Even if the premise is true of each and every component of my curriculum, the whole could have been a chaotic mess, so this reasoning is defective.
Notice that this is distinct from the fallacy of converse accident, which improperly generalizes from an unusual specific case (as in "My philosophy course was well-organized; therefore, college courses are well-organized."). For the fallacy of composition, the crucial fact is that even when something can be truly said of each and every individual part, it does not follow that the same can be truly said of the whole class.
Similarly, the fallacy of division involves an inference from the attribution of some feature to an entire class (or whole) to the possession of the same feature by each of its individual members (or parts).
- Ocelots are now dying out.
- Sparky is an ocelot.
- Therefore, Sparky is now dying out.
Although the premise is true of the species as a whole, this unfortunate fact does not reflect poorly upon the health of any of its individual members.
Again, be sure to distinguish this from the fallacy of accident, which mistakenly applies a general rule to an atypical specific case (as in "Ocelots have many health problems, and Sparky is an ocelot; therefore, Sparky is in poor health"). The essential point in the fallacy of division is that even when something can be truly said of a whole class, it does not follow that the same can be truly said of each of its individual parts.
Informal fallacies of all seventeen varieties can seriously interfere with our ability to arrive at the truth. Whether they are committed inadvertently in the course of an individual's own thinking or deliberately employed in an effort to manipulate others, each may persuade without providing legitimate grounds for the truth of its conclusion. But knowing what the fallacies are affords us some protection in either case. If we can identify several of the most common patterns of incorrect reasoning, we are less likely to slip into them ourselves or to be fooled by anyone else.
