Logic

LOGIC
Arguments and Inference
The Discipline of Logic
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 distinctionboth 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.
The Structure of Argument
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."
Recognizing Arguments
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 alltheir conclusions may be false even if their premises are true, and no amount of additional information can help them in the least.
Inductive Inferences
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 informationshort of changing or denying any of the premisescould make the conclusion seem more or less likely; if so, the pattern of reasoning is inductive.
Truth and Validity
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.


Language and Logic
Functions of Language
The formal patterns of correct reasoning can all be conveyed through ordinary language, but then so can a lot of other things. In fact, we use language in many different ways, some of which are irrelevant to any attempt to provide reasons for what we believe. It is helpful to identify at least three distinct uses of language:
The informative use of language involves an effort to communicate some content. When I tell a child, "The fifth of May is Mexican Independence Day," or write to you that "Logic is the study of correct reasoning," or jot a note to myself, "Jennifer555-3769," I am using language informatively. This kind of use presumes that the content of what is being communicated is actually true, so it will be our central focus in the study of logic. An expressive use of language, on the other hand, intends only to vent some feeling, or perhaps to evoke some feeling from other people. When I say, "Friday afternoons are dreary," or yell "Ouch!" I am using language expressively. Although such uses don't convey any information, they do serve an important function in everyday life, since how we feel sometimes matters as much asor more thanwhat we hold to be true. Finally, directive uses of language aim to cause or to prevent some overt action by a human agent. When I say "Shut the door," or write "Read the textbook," or memo myself, "Don't rely so heavily on the passive voice," I am using language directively. The point in each of these cases is to make someone perform (or forswear) a particular action. This is a significant linguistic function, too, but like the expressive use, it doesn't always relate logically to the truth of our beliefs.
Notice that the intended use in a particular instance often depends more on the specific context and tone of voice than it does on the grammatical form or vocabulary of what is said. The simple declarative sentence, "I'm hungry," for example, could be used to report on a physiological condition, or to express a feeling, or implicitly to request that someone feed me. In fact, uses of two or more varieties may be mixed together in a single utterance; "Stop that," for example, usually involves both expressive and directive functions jointly. In many cases, however, it is possible to identify a single use of language that is probably intended to be the primary function of a particular linguistic unit.
British philosopher J. L. Austin developed a similar, though much more detailed and sophisticated, nomenclature for the variety of actions we commonly perform in employing ordinary language. You're welcome to examine his theory of speech acts in association with the discussion in your textbook. While the specifics may vary, some portion of the point remains the same: since we do in fact employ language for many distinct purposes, we can minimize confusion by keeping in mind what we're up to on any particular occasion.
Literal and Emotive Meaning
Even single words or short phrases can exhibit the distinction between purely informative and partially expressive uses of language. Many of the most common words and phrases of any language have both a literal or descriptive meaning that refers to the way things are and an emotive meaning that expresses some (positive or negative) feeling about them. Thus, the choice of which word to use in making a statement can be used in hopes of evoking a particular emotional response.
This is a natural function of ordinary language, of course. We often do wish to convey some portion of our feelings along with information. There is a good deal of poetry in everyday communication, and poetry without emotive meaning is pretty dull. But when we are primarily interested in establishing the truthas we are when assessing the logical merits of an argumentthe use of words laden with emotive meaning can easily distract us from our purpose.
Kinds of Agreement and Disagreement
In fact, an excessive reliance on emotively charged language can create the appearance of disagreement between parties who do not differ on the facts at all, and it can just as easily disguise substantive disputes under a veneer of emotive agreement. Since the degrees of agreement in belief and attitude are independent of each other, there are four possible combinations at work here:
Agreement in belief and agreement in attitude: There aren't any problems in this instance, since both parties hold the same positions and have the same feelings about them. Agreement in belief but disagreement in attitude: This case, if unnoticed, may become the cause of endless (but pointless) shouting between people whose feelings differ sharply about some fact upon which they are in total agreement. Disagreement in belief but agreement in attitude: In this situation, parties may never recognize, much less resolve, their fundamental difference of opinion, since they are lulled by their shared feelings into supposing themselves allied. Disagreement in belief and disagreement in attitude: Here the parties have so little in common that communication between them often breaks down entirely.
It is often valuable, then, to recognize the levels of agreement or disagreement at work in any exchange of views. That won't always resolve the dispute between two parties, of course, but it will ensure that they don't waste their time on an inappropriate method of argument or persuasion.
Emotively Neutral Language
For our purposes in assessing the validity of deductive arguments and the reliability of inductive reasoning, it will be most directly helpful to eliminate emotive meaning entirely whenever we can. Although it isn't always easy to achieve emotively neutral language in every instance, and the result often lacks the colorful character of our usual public discourse, it is worth the trouble and insipidity because it makes it much easier to arrive at a settled understanding of what is true.
In many instances, the informal fallacies we will consider next result from an improper use of emotionally charged language in the effort to persuade someone to accept a proposition at an emotional level, without becoming convinced that there are legitimate grounds for believing it to be true.
Definition and Meaning
Genuine and Verbal Disputes
We've seen that sloppy or misleading use of ordinary language can seriously limit our ability to create and communicate correct reasoning. As philosopher John Locke pointed out three centuries ago, the achievement of human knowledge is often hampered by the use of words without fixed signification. Needless controversy is sometimes produced and perpetuated by an unacknowledged ambiguity in the application of key terms. We can distinguish disputes of three sorts:
Genuine disputes involve disagreement about whether or not some specific proposition is true. Since the people engaged in a genuine dispute agree on the meaning of the words by means of which they convey their respective positions, each of them can propose and assess logical arguments that might eventually lead to a resolution of their differences. Merely verbal disputes, on the other hand, arise entirely from ambiguities in the language used to express the positions of the disputants. A verbal dispute disappears entirely once the people involved arrive at an agreement on the meaning of their terms, since doing so reveals their underlying agreement in belief. Apparently verbal but really genuine disputes can also occur, of course. In cases of this sort, the resolution of every ambiguity only reveals an underlying genuine dispute. Once that's been discovered, it can be addressed fruitfully by appropriate methods of reasoning.
We can save a lot of time, sharpen our reasoning abilities, and communicate with each other more effectively if we watch for disagreements about the meaning of words and try to resolve them whenever we can.
Kinds of Definition
The most common way of preventing or eliminating differences in the use of languages is by agreeing on the definition of our terms. Since these explicit accounts of the meaning of a word or phrase can be offered in distinct contexts and employed in the service of different goals, it's useful to distinguish definitions of several kinds:
A lexical definition simply reports the way in which a term is already used within a language community. The goal here is to inform someone else of the accepted meaning of the term, so the definition is more or less correct depending upon the accuracy with which it captures that usage. In these pages, my definitions of technical terms of logic are lexical because they are intended to inform you about the way in which these terms are actually employed within the discipline of logic.
At the other extreme, a stipulative definition freely assigns meaning to a completely new term, creating a usage that had never previously existed. Since the goal in this case is to propose the adoption of shared use of a novel term, there are no existing standards against which to compare it, and the definition is always correct (though it might fail to win acceptance if it turns out to be inapt or useless). If I now decree that we will henceforth refer to Presidential speeches delivered in French as "glorsherfs," I have made a (probably pointless) stipulative definition.
Combining these two techniques is often an effective way to reduce the vagueness of a word or phrase. These precising definitions begin with the lexical definition of a term but then propose to sharpen it by stipulating more narrow limits on its use. Here, the lexical part must be correct and the stipulative portion should appropriately reduce the troublesome vagueness. If the USPS announces that "proper notification of a change of address" means that an official form containing the relevant information must be received by the local post office no later than four days prior to the effective date of the change, it has offered a (possibly useful) precising definition.
Theoretical definitions are special cases of stipulative or precising definition, distinguished by their attempt to establish the use of this term within the context of a broader intellectual framework. Since the adoption of any theoretical definition commits us to the acceptance of the theory of which it is an integral part, we are rightly cautious in agreeing to it. Newton's definition of the terms "mass" and "inertia" carried with them a commitment to (at least part of) his theories about the conditions in which physical objects move.
Finally, what some logicians call a persuasive definition is an attempt to attach emotive meaning to the use of a term. Since this can only serve to confuse the literal meaning of the term, persuasive definitions have no legitimate use.
Extension and Intension
A rather large and especially useful portion of our active vocabularies is taken up by general terms, words or phrases that stand for whole groups of individual things sharing a common attribute. But there are two distinct ways of thinking about the meaning of any such term.
The extension of a general term is just the collection of individual things to which it is correctly applied. Thus, the extension of the word "chair" includes every chair that is (or ever has been or ever will be) in the world. The intension of a general term, on the other hand, is the set of features which are shared by everything to which it applies. Thus, the intension of the word "chair" is (something like) "a piece of furniture designed to be sat upon by one person at a time."
Clearly, these two kinds of meaning are closely interrelated. We usually suppose that the intension of a concept or term determines its extension, that we decide whether or not each newly-encountered piece of furniture belongs among the chairs by seeing whether or not it has the relevant features. Thus, as the intension of a general term increases, by specifying with greater detail those features that a thing must have in order for it to apply, the term's extension tends to decrease, since fewer items now qualify for its application.
Denotative and Connotative Definitions
With the distinction between extension and intension in mind, it is possible to approach the definition of a general term (on any of the five kinds of definition we discussed last time) in either of two ways:
A denotative definition tries to identify the extension of the term in question. Thus, we could provide a denotative definition of the phrase "this logic class" simply by listing all of our names. Since a complete enumeration of the things to which a general term applies would be cumbersome or inconvenient in many cases, though, we commonly pursue the same goal by listing smaller groups of individuals or by offering a few examples instead. In fact, some philosophers have held that the most primitive denotative definitions in any language involve no more than pointing at a single example to which the term properly applies.
But there seem to be some important terms for which denotative definition is entirely impossible. The phrase "my grandchildren" makes perfect sense, for example, but since it presently has no extension, there is no way to indicate its membership by enumeration, example, or ostension. In order to define terms of this sort at all, and in order more conveniently to define general terms of every variety, we naturally rely upon the second mode of definition.
A connotative definition tries to identify the intension of a term by providing a synonymous linguistic expression or an operational procedure for determining the applicability of the term. Of course, it isn't always easy to come up with an alternative word or phrase that has exactly the same meaning or to specify a concrete test for applicability. But when it does work, connotative definition provides an adequate means for securing the meaning of a term.
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
Informal Fallacies
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 United States. Therefore, Herbert Hoover was the greatest President of the United States.
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
Unwarranted Assumptions
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.
Accident
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.
Converse Accident
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.
False Cause
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.
Complex Question
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
Ambiguous Language
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.
Equivocation
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.
Amphiboly
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.
Accent
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.
Composition
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.
Division
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.
Avoiding Fallacies
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.
Categorical Propositions
Now that we've taken notice of many of the difficulties that can be caused by sloppy use of ordinary language in argumentation, we're ready to begin the more precise study of deductive reasoning. Here we'll achieve the greater precision by eliminating ambiguous words and phrases from ordinary language and carefully defining those that remain. The basic strategy is to create a narrowly restricted formal systeman artificial, rigidly structured logical language within which the validity of deductive arguments can be discerned with ease. Only after we've become familiar with this limited range of cases will we consider to what extent our ordinary-language argumentation can be made to conform to its structure.
Our initial effort to pursue this strategy is the ancient but worthy method of categorical logic. This approach was originally developed by Aristotle, codified in greater detail by medieval logicians, and then interpreted mathematically by George Boole and John Venn in the nineteenth century. Respected by many generations of philosophers as the the chief embodiment of deductive reasoning, this logical system continues to be useful in a broad range of ordinary circumstances.
Terms
and Propositions
We'll start very simply, then work our way toward a higher level. The basic unit of meaning or content in our new deductive system is the categorical term. Usually expressed grammatically as a noun or noun phrase, each categorical term designates a class of things. Notice that these are (deliberately) very broad notions: a categorical term may designate any classwhether it's a natural species or merely an arbitrary collectionof things of any variety, real or imaginary. Thus, "cows," "unicorns," "square circles," "philosophical concepts," "things weighing more than fifty kilograms," and "times when the earth is nearer than 75 million miles from the sun," are all categorical terms.
Notice also that each categorical term cleaves the world into exactly two mutually exclusive and jointly exhaustive parts: those things to which the term applies and those things to which it does not apply. For every class designated by a categorical term, there is another class, its complement, that includes everything excluded from the original class, and this complementary class can of course be designated by its own categorical term. Thus, "cows" and "non-cows" are complementary classes, as are "things weighing more than fifty kilograms" and "things weighing fifty kilograms or less." Everything in the world (in fact, everything we can talk or think about) belongs either to the class designated by a categorical term or to its complement; nothing is omitted.
Now let's use these simple building blocks to assemble something more interesting. A categorical proposition joins together exactly two categorical terms and asserts that some relationship holds between the classes they designate. (For our own convenience, we'll call the term that occurs first in each categorical proposition its
subject term
and other its
predicate term
.) Thus, for example, "All cows are mammals" and "Some philosophy teachers are young mothers" are categorical propositions whose subject terms are "cows" and "philosophy teachers" and whose predicate terms are "mammals" and "young mothers" respectively.
Each categorical proposition states that there is some logical relationship that holds between its two terms. In this context, a categorical term is said to be distributed if that proposition provides some information about every member of the class designated by that term. Thus, in our first example above, "cows" is distributed because the proposition in which it occurs affirms that each and every cow is also a mammal, but "mammals" is undistributed because the proposition does not state anything about each and every member of that class. In the second example, neither of the terms is distributed, since this proposition tells us only that the two classes overlap to some (unstated) extent.
Quality and Quantity
Since we can always invent new categorical terms and consider the possible relationship of the classes they designate, there are indefinitely many different individual categorical propositions. But if we disregard the content of these propositions, what classes of things they're about, and concentrate on their form, the general manner in which they conjoin their subject and predicate terms, then we need only four distinct kinds of categorical proposition, distinguished from each other only by their quality and quantity, in order to assert anything we like about the relationship between two classes.
The quality of a categorical proposition indicates the nature of the relationship it affirms between its subject and predicate terms: it is an
affirmative
proposition if it states that the class designated by its subject term is included, either as a whole or only in part, within the class designated by its predicate term, and it is a
negative
proposition if it wholly or partially excludes members of the subject class from the predicate class. Notice that the predicate term is distributed in every negative proposition but undistributed in all affirmative propositions.
The quantity of a categorical proposition, on the other hand, is a measure of the degree to which the relationship between its subject and predicate terms holds: it is a
universal
proposition if the asserted inclusion or exclusion holds for every member of the class designated by its subject term, and it is a
particular
proposition if it merely asserts that the relationship holds for one or more members of the subject class. Thus, you'll see that the subject term is distributed in all universal propositions but undistributed in every particular proposition.
Combining these two distinctions and representing the subject and predicate terms respectively by the letters "S" and "P," we can uniquely identify the four possible forms of categorical proposition:
A universal affirmative proposition (to which, following the practice of medieval logicians, we will refer by the letter "A ") is of the form All S are P. Such a proposition asserts that every member of the class designated by the subject term is also included in the class designated by the predicate term. Thus, it distributes its subject term but not its predicate term. A universal negative proposition (or "E ") is of the form No S are P. This proposition asserts that nothing is a member both of the class designated by the subject term and of the class designated by the predicate terms. Since it reports that every member of each class is excluded from the other, this proposition distributes both its subject term and its predicate term. A particular affirmative proposition ("I ") is of the form Some S are P. A proposition of this form asserts that there is at least one thing which is a member both of the class designated by the subject term and of the class designated by the predicate term. Both terms are undistributed in propositions of this form. Finally, a particular negative proposition ("O ") is of the form Some S are not P. Such a proposition asserts that there is at least one thing which is a member of the class designated by the subject term but not a member of the class designated by the predicate term. Since it affirms that the one or more crucial things that they are distinct from each and every member of the predicate class, a proposition of this form distributes its predicate term but not its subject term. Although the specific content of any actual categorical proposition depends upon the categorical terms which occur as its subject and predicate, the logical form of the categorical proposition must always be one of these four types.
The Square of Opposition
When two categorical propositions are of different forms but share exactly the same subject and predicate terms, their truth is logically interdependent in a variety of interesting ways, all of which are conveniently represented in the traditional "square of opposition."
"All S are P." (A )- - - - - - -(E ) "No S are P."
* *
* *
* *
*
* *
* *
* *
"Some S are P." (I )--- --- ---(O ) "Some S are not P."
Propositions that appear diagonally across from each other in this diagram (A and O on the one hand and E and I on the other) are contradictories. No matter what their subject and predicate terms happen to be (so long as they are the same in both) and no matter how the classes they designate happen to be related to each other in fact, one of the propositions in each contradictory pair must be true and the other false. Thus, for example, "No squirrels are predators" and "Some squirrels are predators" are contradictories because either the classes designated by the terms "squirrel" and "predator" have at least one common member (in which case the I proposition is true and the E proposition is false) or they do not (in which case the E is true and the I is false). In exactly the same sense, the A and O propositions, "All senators are politicians" and "Some senators are not politicians" are also contradictories. The universal propositions that appear across from each other at the top of the square (A and E ) are contraries. Assuming that there is at least one member of the class designated by their shared subject term, it is impossible for both of these propositions to be true, although both could be false. Thus, for example, "All flowers are colorful objects" and "No flowers are colorful objects" are contraries: if there are any flowers, then either all of them are colorful (making the A true and the E false) or none of them are (making the E true and the A false) or some of them are colorful and some are not (making both the A and the E false). Particular propositions across from each other at the bottom of the square (I and O ), on the other hand, are the subcontraries. Again assuming that the class designated by their subject term has at least one member, it is impossible for both of these propositions to be false, but possible for both to be true. "Some logicians are professors" and "Some logicians are not professors" are subcontraries, for example, since if there any logicians, then either at least one of them is a professor (making the I proposition true) or at least one is not a professor (making the O true) or some are and some are not professors (making both the I and the O true). Finally, the universal and particular propositions on either side of the square of opposition (A and I on the one left and E and O on the right) exhibit a relationship known as subalternation. Provided that there is at least one member of the class designated by the subject term they have in common, it is impossible for the universal proposition of either quality to be true while the particular proposition of the same quality is false. Thus, for example, if it is universally true that "All sheep are ruminants", then it must also hold for each particular case, so that "Some sheep are ruminants" is true, and if "Some sheep are ruminants" is false, then "All sheep are ruminants" must also be false, always on the assumption that there is at least one sheep. The same relationships hold for corresponding E and O propositions. 1997-2002 Garth Kemerling. Last modified 27 October 2001. Questions, comments, and suggestions may be sent to: the Contact Page.
Immediate Inferences
If we expand the scope of our investigation to include shared terms and their complements, we can identify logical relationships of three additional varieties. Since each of these new cases involves a pair of categorical propositions that are logically equivalent to each otherthat is, either both of them are true or both are falsethey enable us to draw an immediate inference from the truth (or falsity) of either member of the pair to the truth (or falsity) the other.
Conversion
The converse of any categorical proposition is the new categorical proposition that results from putting the predicate term of the original proposition in the subject place of the new proposition and the subject term of the original in the predicate place of the new. Thus, for example, the converse of "No dogs are felines" is "No felines are dogs," and the converse of "Some snakes are poisonous animals" is "Some poisonous animals are snakes."
Conversion grounds an immediate inference for both E and I propositions That is, the converse of any E or I proposition is true if and only if the original proposition was true. Thus, in each of the pairs noted as examples in the previous paragraph, either both propositions are true or both are false. In addition, if we first perform a subalternation and then convert our result, then the truth of an A proposition may be said, in "conversion by limitation," to entail the truth of an I proposition with subject and predicate terms reversed: If "All singers are performers" then "Some performers are singers." But this will work only if there really is at least one singer. Generally speaking, however, conversion doesn't hold for A and O propositions: it is entirely possible for "All dogs are mammals" to be true while "All mammals are dogs" is false, for example, and for "Some females are not mothers" to be true while "Some mothers are not females" is false. Thus, conversion does not warrant a reliable immediate inference with respect to A and O propositions.
Obversion
In order to form the obverse of a categorical proposition, we replace the predicate term of the proposition with its complement and reverse the quality of the proposition, either from affirmative to negative or from negative to affirmative. Thus, for example, the obverse of "All ants are insects" is "No ants are non-insects"; the obverse of "No fish are mammals" is "All fish are non-mammals"; the obverse of "Some musicians are males" is "Some musicians are not non-males"; and the obverse of "Some cars are not sedans" is "Some cars are non-sedans."
Obversion is the only immediate inference that is valid for categorical propositions of every form. In each of the instances cited above, the original proposition and its obverse must have exactly the same truth-value, whether it turns out to be true or false.
Contraposition
The contrapositive of any categorical proposition is the new categorical proposition that results from putting the complement of the predicate term of the original proposition in the subject place of the new proposition and the complement of the subject term of the original in the predicate place of the new. Thus, for example, the contrapositive of "All crows are birds" is "All non-birds are non-crows," and the contrapositive of "Some carnivores are not mammals" is "Some non-mammals are not non-carnivores."
Contraposition is a reliable immediate inference for both A and O propositions; that is, the contrapositive of any A or O proposition is true if and only if the original proposition was true. Thus, in each of the pairs in the paragraph above, both propositions have exactly the same truth-value. In addition, if we form the contrapositive of our result after performing subalternation, then an E proposition, in "contraposition by limitation," entails the truth of a related O proposition: If "No bandits are biologists" then "Some non-biologists are not non-bandits," provided that there is at least one member of the class designated by "bandits." In general, however, contraposition is not valid for E and I propositions: "No birds are plants" and "No non-plants are non-birds" need not have the same truth-value, nor do "Some spiders are insects" and "Some non-insects are non-spiders." Thus, contraposition does not hold as an immediate inference for E and I propositions. Omitting the troublesome cases of conversion and contraposition "by limitation," then, there are exactly two reliable operations that can be performed on a categorical proposition of any form:

A
proposition:
All S are P.


Obverse
No S are non-P.


Contrapositive
All non-P are non-S.




E
proposition:
No S are P.



Converse
No P are S.


Obverse
All S are non-P.




I
proposition:
Some S are P.



Converse
Some P are S.


Obverse
Some S are not non-P.




O
proposition:
Some S are not P.



Obverse
Some S are non-P.


Contrapositive
Some non-P are not non-S.




Existential Import
It is time to express more explicitly an important qualification regarding the logical relationships among categorical propositions. You may have noticed that at several points in these two lessons we declared that there must be some things a certain kind. This special assumption, that the class designated by the subject term of a universal proposition has at least one member, is called existential import. Classical logicians typically presupposed that universal propositions do have existential import.
But modern logicians have pointed that the system of categorical logic is more useful if we deny the existential import of universal propositions while granting, of course, that particular propositions do presuppose the existence of at least one member of their subject classes. It is sometimes very handy, even for non-philosophers, to make a general statement about things that don't exist. A sign that reads, "All shoplifters are prosecuted to the full extent of the law," for example, is presumably intended to make sure that the class designated by its subject term remains entirely empty. In the remainder of our discussion of categorical logic, we will exclusively employ this modern interpretation of universal propositions.
Although it has many advantages, the denial of existential import does undermine the reliability of some of the truth-relations we've considered so far. In the traditional square of opposition, only the contradictories survive intact; the relationships of the contraries, the subcontraries, and subalternation no longer hold when we do not suppose that the classes designated by the subject terms of A and E propositions have members. (And since conversion and contraposition "by limitation" derive from subalternation, they too must be forsworn.) From now on, therefore, we will rely only upon the immediate inferences in the table at the end of the previous section of this lesson and suppose that A and O propositions and E and I propositions are genuinely contradictory. Diagramming Propositions The modern interepretation of categorical logic also permits a more convenient way of assessing the truth-conditions of categorical propositions, by drawing Venn diagrams, topological representations of the logical relationships among the classes designated by categorical terms. The basic idea is fairly straightforward:
[if gte vml 1]> [if gte vml 1]> [if gte vml 1]> [if gte vml 1]> [if gte vml 1]> [if gte vml 1]>
Each categorical term is represented by a labelled circle. The area inside the circle represents the extension of the categorical term, and the area outside the circle its complement. Thus, members of the class designated by the categorical term would be located within the circle, and everything else in the world would be located outside it.
We indicate that there is at least one member of a specific class by placing an inside the circle; an outside the circle would indicate that there is at least one member of the complementary class.
To show that there are no members of a specific class, we shade the entire area inside the circle; shading everything outside the circle would indicate that there are no members of the complementary class.
Notice that diagrams of these two sorts are incompatible: no area of a Venn diagram can both be shaded and contain an ; either there is at least one member of the represented class, or there are none.
In order to represent a categorical proposition, we must draw two overlapping circles, creating four distinct areas corresponding to four kinds of things: those that are members of the class designated by the subject term but not of that designated by the predicate term; those that are members of both classes; those that are members of the class designated by the predicate term but not of that designated by the subject term; and those that are not members of either class.
Categorical propositions of each of the four varieties may then be diagrammed by shading or placing an in the appropriate area: The universal negative (E ) proposition asserts that nothing is a member of both classes designated by its terms, so its diagram shades the area in which the two circles overlap.
[if gte vml 1]> The particular affirmative (I ) proposition asserts that there is at least one thing that is a member of both classes, so its diagram places an in the area where the two circles overlap. Notice that the incompatibility of these two diagrams models the contradictory relationship between E and I propositions; one of them must be true and the other false, since either there is at least one member that the two classes have in common or there are none. The particular negative (O ) proposition asserts that there is at least one thing that is a member of the class designated by its subject term but not of the class designated by its predicate term, so its diagram places an in the area inside the circle that represents the subject term but outside the circle that represents the predicate term. Finally, the universal affirmative (A ) proposition asserts that every member of the subject class is also a member of the predicate class. Since this entails that there is nothing that is a member of the subject class that is not a member of the predicate class, an A proposition can be diagrammed by shading the area inside the subject circle but outside the predicate circle. Again, the incompatibility of the diagrams for A and O propositions represents the fact that they are logically contradictory; one of them must be true and the other false.
Categorical Syllogisms
Now, on to the next level, at which we combine more than one categorical proposition to fashion logical arguments. A is an argument consisting of exactly three categorical propositions (two premises and a conclusion) in which there appear a total of exactly three categorical terms, each of which is used exactly twice.
One of those terms must be used as the subject term of the conclusion of the syllogism, and we call it the name="/dy/m7.htm#mint" of the syllogism as a whole. The name="/dy/m.htm#majt" of the syllogism is whatever is employed as the predicate term of its conclusion. The third term in the syllogism doesn''t occur in the conclusion at all, but must be employed in somewhere in each of its premises; hence, we call it the name="/dy/m7.htm#midt". Since one of the premises of the syllogism must be a categorical proposition that affirms some relation between its middle and major terms, we call that the name="/dy/m.htm#majp" of the syllogism. The other premise, which links the middle and minor terms, we call the name="/dy/m7.htm#minp". Consider, for example, the categorical syllogism: No geese are felines. Some birds are geese. Therefore, Some birds are not felines. Clearly, "Some birds are not felines" is the conclusion of this syllogism. The major term of the syllogism is "felines" (the predicate term of its conclusion), so "No geese are felines" (the premise in which "felines" appears) is its major premise. Simlarly, the minor term of the syllogism is "birds," and "Some birds are geese" is its minor premise. "geese" is the middle term of the syllogism. name=mood In order to make obvious the similarities of structure shared by different syllogisms, we will always present each of them in the same fashion. A categorical syllogism in name="/dy/s7.htm#strd" always begins with the premises, major first and then minor, and then finishes with the conclusion. Thus, the example above is already in standard form. Although arguments in ordinary language may be offered in a different arrangement, it is never difficult to restate them in standard form. Once we''ve identified the conclusion which is to be placed in the final position, whichever premise contains its predicate term must be the major premise that should be stated first.
Medieval logicians devised a simple way of labelling the various forms in which a categorical syllogism may occur by stating its name="/dy/m9.htm#mood". The mood of a syllogism is simply a statement of which categorical propositions (A, E, I, or O) it comprises, listed in the order in which they appear in standard form. Thus, a syllogism with a mood of OAO has an O proposition as its major premise, an A proposition as its minor premise, and another O proposition as its conclusion; and EIO syllogism has an E major premise, and I minor premise, and an O conclusion; etc. name=fig there are four distinct versions of each syllogistic mood, however, we need to supplement this labelling system with a statement of the figure of each, which is solely determined by the position in which its middle term appears in the two premises: in a first-figure syllogism, the middle term is the subject term of the major premise and the predicate term of the minor premise; in second figure, the middle term is the predicate term of both premises; in third, the subject term of both premises; and in fourth figure, the middle term appears as the predicate term of the major premise and the subject term of the minor premise. (The four figures may be easier to remember as a simple chart showing the position of the terms in each of the premises: M P P M M P P M 1 \ 2 3 4 / S M S M M S M S All told, there are exactly 256 distinct forms of categorical syllogism: four kinds of major premise multiplied by four kinds of minor premise multiplied by four kinds of conclusion multiplied by four relative positions of the middle term. Used together, mood and figure provide a unique way of describing the logical structure of each of them. Thus, for example, the argument "Some merchants are pirates, and All merchants are swimmers, so Some swimmers are pirates" is an IAI-3 syllogism, and any AEE-4 syllogism must exhibit the form "All P are M, and No M are S, so No S are P." name=form This method of differentiating syllogisms is significant because the validity of a categorical syllogism depends solely upon its name="/dy/l5.htm#logf". Remember our earlier definition: an argument is name="/lg/e01.htm#val" when, if its premises were true, then its conclusion would also have to be true. The application of this definition in no way depends upon the content of a specific categorical syllogism; it makes no difference whether the categorical terms it employs are "mammals," "terriers," and "dogs" or "sheep," "commuters," and "sandwiches." If a syllogism is valid, it is impossible for its premises to be true while its conclusion is false, and that can be the case only if there is something faulty in its general form. Thus, the specific syllogisms that share any one of the 256 distinct syllogistic forms must either all be valid or all be invalid, no matter what their content happens to be. Every syllogism of the form AAA-1 is valid, for example, while all syllogisms of the form OEE-3 are invalid. This suggests a fairly straightforward method of demonstrating the invalidity of any syllogism by "logical analogy." If we can think of another syllogism which has the same mood and figure but whose terms obviously make both premises true and the conclusion false, then it is evident that all syllogisms of this form, including the one with which we began, must be invalid. Thus, for example, it may be difficult at first glance to assess the validity of the argument: All philosophers are professors. All philosophers are logicians. Therefore, All logicians are professors. But since this is a categorical syllogism whose mood and figure are AAA-3, and since all syllogisms of the same form are equally valid or invalid, its reliability must be the same as that of the AAA-3 syllogism: All terriers are dogs. All terriers are mammals. Therefore, All mammals are dogs. Both premises of this syllogism are true, while its conclusion is false, so it is clearly invalid. But then all syllogisms of the AAA-3 form, including the one about logicians and professors, must also be invalid. This method of demonstrating the invalidity of categorical syllogisms is useful in many contexts; even those who have not had the benefit of specialized training in formal logic will often acknowledge the force of a logical analogy. The only problem is that the success of the method depends upon our ability to invent appropriate cases, syllogisms of the same form that obviously have true premises and a false conclusion. If I have tried for an hour to discover such a case, then either there can be no such case because the syllogism is valid or I simply haven''t looked hard enough yet. name=vdsyll The modern interpretation offers a more efficient method of evaluating the validity of categorical syllogisms. By combining the drawings of individual propositions, we can use Venn diagrams to assess the validity of categorical syllogisms by following a simple three-step procedure: First draw three overlapping circles and label them to represent the major, minor, and middle terms of the syllogism. Next, on this framework, draw the diagrams of both of the syllogism''s premises. Always begin with a universal proposition, no matter whether it is the major or the minor premise. Remember that in each case you will be using only two of the circles in each case; ignore the third circle by making sure that your drawing (shading or ) straddles it. Finally, without drawing anything else, look for the drawing of the conclusion. If the syllogism is valid, then that drawing will already be done. Since it perfectly models the relationships between classes that are at work in categorical logic, this procedure always provides a demonstration of the validity or invalidity of any categorical syllogism. Consider, for example, how it could be applied, step by step, to an evaluation of a syllogism of the EIO-3 mood and figure, No M are P. Some M are S. Therefore, Some S are not P. First, we draw and label the three overlapping circles needed to represent all three terms included in the categorical syllogism: Second, we diagram each of the premises: Since the major premise is a universal proposition, we may begin with it. The diagram for "No M are P" must shade in the entire area in which the M and P circles overlap. (Notice that we ignore the S circle by shading on both sides of it.) Now we add the minor premise to our drawing. The diagram for "Some M are S" puts an inside the area where the M and S circles overlap. But part of that area (the portion also inside the P circle) has already been shaded, so our must be placed in the remaining portion. Third, we stop drawing and merely look at our result. Ignoring the M circle entirely, we need only ask whether the drawing of the conclusion "Some S are not P" has already been drawn. Remember, that drawing would be like the one at left, in which there is an in the area inside the S circle but outside the P circle. Does that already appear in the diagram on the right above? Yes, if the premises have been drawn, then the conclusion is already drawn. But this models a significant logical feature of the syllogism itself: if its premises are true, then its conclusion must also be true. Any categorical syllogism of this form is valid. Here are the diagrams of several other syllogistic forms. In each case, both of the premises have already been drawn in the appropriate way, so if the drawing of the conclusion is already drawn, the syllogism must be valid, and if it is not, the syllogism must be invalid. AAA-1 (valid) All M are P. All S are M. Therefore, All S are P. AAA-3 (invalid) All M are P. All M are S. Therefore, All S are P. OAO-3 (valid) Some M are not P. All M are S. Therefore, Some S are not P. EOO-2 (invalid) No P are M. Some S are not M. Therefore, Some S are not P. IOO-1 (invalid) Some M are P. Some S are not M. Therefore, Some S are not P. Practice your skills in using Venn Diagrams to test the validity of Categorical Syllogisms by using Ron Blatt''s excellent name="venndiagram.html" target=new.
Establishing Validity
Rules and Fallacies
Since the validity of a categorical syllogism depends solely upon its logical form, it is relatively simple to state the conditions under which the premises of syllogisms succeed in guaranteeing the truth of their conclusions. Relying heavily upon the medieval tradition, Copi & Cohen provide a list of six rules, each of which states a necessary condition for the validity of any categorical syllogism. Violating any of these rules involves committing one of the formal fallacies, errors in reasoning that result from reliance on an invalid logical form.
In every valid standard-form categorical syllogism . . .
. . . there must be exactly three unambiguous categorical terms.
The use of exactly three categorical terms is part of the definition of a categorical syllogism, and we saw earlier that the use of an ambiguous term in more than one of its senses amounts to the use of two distinct terms. In categorical syllogisms, using more than three terms commits the fallacy of four terms (quaternio terminorum).
. . . the middle term must be distributed in at least one premise.
In order to effectively establish the presence of a genuine connection between the major and minor terms, the premises of a syllogism must provide some information about the entire class designated by the middle term. If the middle term were undistributed in both premises, then the two portions of the designated class of which they speak might be completely unrelated to each other. Syllogisms that violate this rule are said to commit the fallacy of the undistributed middle.
. . . any term distributed in the conclusion must also be distributed in its premise.
A premise that refers only to some members of the class designated by the major or minor term of a syllogism cannot be used to support a conclusion that claims to tell us about every menber of that class. Depending which of the terms is misused in this way, syllogisms in violation commit either the fallacy of the illicit major or the fallacy of the illicit minor.
. . . at least one premise must be affirmative.
Since the exclusion of the class designated by the middle term from each of the classes designated by the major and minor terms entails nothing about the relationship between those two classes, nothing follows from two negative premises. The fallacy of exclusive premises violates this rule.
. . . if either premise is negative, the conclusion must also be negative.
For similar reasons, no affirmative conclusion about class inclusion can follow if either premise is a negative proposition about class exclusion. A violation results in the fallacy of drawing an affirmative conclusion from negative premises.
. . . if both premises are universal, then the conclusion must also be universal.
Because we do not assume the existential import of universal propositions, they cannot be used as premises to establish the existential import that is part of any particular proposition. The existential fallacy violates this rule.
Although it is possible to identify additional features shared by all valid categorical syllogisms (none of them, for example, have two particular premises), these six rules are jointly sufficient to distinguish between valid and invalid syllogisms.
Names for the Valid Syllogisms
A careful application of these rules to the 256 possible forms of categorical syllogism (assuming the denial of existential import) leaves only 15 that are valid. Medieval students of logic, relying on syllogistic reasoning in their public disputations, found it convenient to assign a unique name to each valid syllogism. These names are full of clever reminders of the appropriate standard form: their initial letters divide the valid cases into four major groups, the vowels in order state the mood of the syllogism, and its figure is indicated by (complicated) use of m, r, and s. Although the modern interpretation of categorical logic provides an easier method for determining the validity of categorical syllogisms, it may be worthwhile to note the fifteen valid cases by name:
The most common and useful syllogistic form is "Barbara", whose mood and figure is
AAA-1
:
All M are P.
All S are M.
Therefore, All S are P.
Instances of this form are especially powerful, since they are the only valid syllogisms whose conclusions are universal affirmative propositions.
A syllogism of the form
AOO-2
was called "Baroco":
All P are M.
Some S are not M.
Therefore, Some S are not P.
The valid form
OAO-3
("Bocardo") is:
Some M are not P.
All M are S.
Therefore, Some S are not P.
Four of the fifteen valid argument forms use universal premises (only one of which is affirmative) to derive a universal negative conclusion:
One of them is "Camenes" (
AEE-4
All P are M.
No M are S.
Therefore, No S are P.
Converting its minor premise leads to "Camestres" (
AEE-2
All P are M.
No S are M.
Therefore, No S are P.
Another pair begins with "Celarent" (
EAE-1
No M are P.
All S are M.
Therefore, No S are P.
Converting the major premise in this case yields "Cesare" (
EAE-2
No P are M.
All S are M.
Therefore, No S are P.
Syllogisms of another important set of forms use affirmative premises (only one of which is universal) to derive a particular affirmative conclusion:
The first in this group is
AII-1
("Darii"
All M are P.
Some S are M.
Therefore, Some S are P.
Converting the minor premise produces another valid form,
AII-3
("Datisi"
All M are P.
Some M are S.
Therefore, Some S are P.
The second pair begins with "Disamis" (
IAI-3
Some M are P.
All M are S.
Therefore, Some S are P.
Converting the major premise in this case yields "Dimaris" (
IAI-4
Some P are M.
All M are S.
Therefore, Some S are P.
Only one of the 64 distinct moods for syllogistic form is valid in all four figures, since both of its premises permit legitimate conversions:
Begin with
EIO-1
("Ferio"
No M are P.
Some S are M.
Therefore, Some S are not P.
Converting the major premise produces
EIO-2
("Festino"
No P are M.
Some S are M.
Therefore, Some S are not P.
Next, converting the minor premise of this result yields
EIO-4
("Fresison"
No P are M.
Some M are S.
Therefore, Some S are not P.
Finally, converting the major again leads to
EIO-3
("Ferison"
No M are P.
Some M are S.
Therefore, Some S are not P.
Notice that converting the minor of this syllogistic form will return us back to "Ferio."
Arguments in Ordinary Language
People reasoning in ordinary language rarely express their arguments in the restricted patterns allowed in categorical logic. But with just a little revision, it is often possible to show that those arguments are in fact equivalent to one of the standard-form categorical syllogisms whose validity we can so easily determine. Let's consider a few of the methods by means of which we can "translate" ordinary-language arguments into the forms studied by categorical logic.
Translation into Standard Form
In the simplest case, we may need only to re-arrange the propositions of the argument in order to translate it into a standard-form categorical syllogism. Thus, for example, "Some birds are geese, so some birds are not felines, since no geese are felines" is just a categorical syllogism stated in the non-standard order minor premise, conclusion, major premise; all we need to do is put the propositions in the right order, and we have the standard-form syllogism:
No geese are felines.
Some birds are geese.
Therefore, Some birds are not felines.
Reducing Categorical Terms
In slightly more complicated instances, an ordinary argument may deal with more than three terms, but it may still be possible to restate it as a categorical syllogism. Two kinds of tools will be helpful in making such a transformation:
First, it is always legitimate to replace one expression with another that means the same thing. Of course, we need to be perfectly certain in each case that the expressions are genuinely synonymous. But in many contexts, this is possible: in ordinary language, "husbands" and "married males" almost always mean the same thing.
Second, if two of the terms of the argument are complementary, then appropriate application of the immediate inferences to one of the propositions in which they occur will enable us to reduce the two to a single term. Consider, for example, "No dogs are non-mammals, and some non-canines are not non-pets, so some non-mammals are pets." Replacing the first proposition with its (logically equivalent) obverse, substituting "dogs" for the synonymous "canines" and taking the contrapositive of the second, and applying first conversion and then obversion to the conclusion, we get the equivalent standard-form categorical syllogism:
All dogs are mammals.
Some pets are not dogs.
Therefore, Some pets are not mammals.
The invalidity of this syllogism is more readily apparent than that of the argument from which it was derived.
Recognizing Categorical Propositions
Of course, the premises and conclusion of an ordinary-language argument may not be categorical propositions at all; even in this case, it may be possible to translate the argument into categorical logic. For each of the propositions of which the argument consists, we must discover some categorical proposition that will make the same assertion.
One especially common but troublesome instance is the occurrence of singular propositions, such as "Spinoza is a philosopher." Here the subject clearly refers to a single individual, so if it is to be used as the subject term of a categorical proposition, we must suppose that it designates a class of things which happens to have exactly one member. But then the categorical proposition that links Spinoza with the class designated by the term "philosopher" could be interpreted as an
A
proposition (All S are P) or as an
I
proposition (Some S are P) or as both of these together. In such cases, we should generally interpret the proposition in whichever way is most likely to transform the argument in which it occurs into a valid syllogism, although that may sometimes make it less likely that the proposition is true.
Other cases are easier to handle. If the predicate is adjectival, we simply substantize it as a noun phrase in order to make a categorical proposition: "All computers are electronic" thus becomes "Some computers are electronic things," for example. If the main verb is not copulative, we simply use its participle or incorporate it into our predicate term: "Some snakes bite" becomes "Some snakes are animals that bite." If the elements of the categorical proposition have been scrambled, we restore each to its proper position: "Bankers? Friendly people, all" becomes "All bankers are friendly people." And, in a variety of cases your texbook discusses in detail, the statements of ordinary language often contain significant clues to their most likely translations as categorical propositions.
Remember that in each case, our goal is fairly to represent what is being asserted as a categorical proposition. To do so, we need only identify the two categorical terms that designate the classes between which it asserts some relation and then figure out which of the four possible relationships (A , E , I , or O ) best captures the intended meaning. It is always a good policy to give the proponent the benefit of any doubt, whenever possible interpreting each proposition both in a way that recommends it as likely to be true and in a way that tends to make the argument in which it occurs a valid one.
Occasionally these methods are not enough to provide for the translation of ordinary-language arguments into standard-form categorical syllogisms. Next, we examine a few special instances that require a more significant transformation.
Introducing Parameters
In order to achieve the uniform translation of all three propositions contained in a categorical syllogism, it is sometimes useful to modify each of the terms employed in an ordinary-language argument by stating it in terms of a general domain or parameter. The goal here, as always, is faithfully to represent the intended meaning of each of the offered propositions, while at the same time bringing it into conformity with the others, making it possible to restate the whole as a standard-form syllogism.
The key to the procedure is to think of an approriate parameter by relation to which each of the three categorical terms can be defined. Thus, for example, in the argument, "The attic must be on fire, since it's full of smoke, and where there's smoke, there's fire," the crucial parameter is location or place. If we suppose the terms of this argument to be "places where fire is," "places where smoke is," and "places that are the attic," then by applying our other techniques of restatement and re-arrangement, we can arrive at the syllogism:
All places where smoke is are places where fire is.
All places that are the attic are places where smoke is.
Therefore, All places that are the attic are places where fire is.
This standard-form categorical syllogism of the form
AAA-1
is clearly valid.
Enthymemes
Another special case occurs when one or more of the propositions in a categorical syllogism is left unstated. Incomplete arguments of this sort, called enthymemes are said to be "first-," "second-," or "third-order," depending upon whether they are missing their major premise, minor premise, or conclusion respectively. In order to show that an enthymeme corresponds to a valid categorical syllogism, we need only supply the missing premise in each case.
Thus, for example, "Since some hawks have sharp beaks, some birds have sharp beaks" is a second-order enthymeme, and once a plausible substitute is provided for its missing minor premise ("All hawks are birds"), it will become the valid
IAI-3
syllogism:
Some hawks are sharp-beaked animals.
All hawks are birds.
Therefore, Some birds are sharp-beaked animals.
Sorites
Finally, the pattern of ordinary-language argumentation known as sorites involves several categorical syllogisms linked together. The conclusion of one syllogism serves as one of the premises for another syllogism, whose conclusion may serve as one of the premises for another, and so on. In any such case, of course, the whole procedure will comprise a valid inference so long as each of the connected syllogisms is itself valid.
Sorites most commonly occur in enthymematic form, with the doubly-used proposition left entirely unstated. In order to reconstruct an argument of this form, we need to identify the premises of an initial syllogism, fill in as its missing conclusion a categorical proposition that legitimately follows from those premises, and then apply it as a premise in another syllogism. When all of the underlying structure has been revealed, we can test each of the syllogisms involved to determine the validity of the whole.
Understanding how these common patterns of reasoning can be re-interpreted as categorical syllogisms may help you to see why generations of logicians regarded categorical logic as a fairly complete treatment of valid inference. Modern logicians, however, developed a much more powerful symbolic system, capable of representing everything that categorical logic covers and much more in addition.
Logical Symbols
Although traditional categorical logic can be used to represent and assess many of our most common patterns of reasoning, modern logicians have developed much more comprehensive and powerful systems for expressing rational thought. These newer logical languages are often called "symbolic logic," since they employ special symbols to represent clearly even highly complex logical relationships. We'll begin our study of symbolic logic with the propositional calculus, a formal system that effectively captures the ways in which individual statements can be combined with each other in interesting ways. The first step, of course, is to define precisely all of the special, new symbols we will use.
Compound Statements
The propositional calculus is not concerned with any features within a simple proposition. Its most basic units are whole propositions or statements, each of which is either true or false (though, of course, we don't always know which). In ordinary language, we convey statements by complete declarative sentences, such as "Alan bears an uncanny resemblance to Jonathan," "Betty enjoys watching John cook," or "Chris and Lloyd are an unbeatable team." But when we're thinking about the logical relationships that hold among two or three or more such statements, it would be awfully clumsy to write out the entire sentence at every occurrence of each of them. Instead, we represent specific individual statements by using capital letters of the alphabet as statement constants. Thus, for example, we could use A , B , and C to represent the statements mentioned aboveletting A stand for "Alan bears an uncanny resemblance to Jonathan," B stand for "Betty enjoys watching John cook," and C stand for "Chris and Lloyd are an unbeatable team." Within the context of this discussion, each statement constant designates one and only one statement.
When we want to deal with statements more generally, we will use lower-case letters of the alphabet (beginning with "p") as statement variables. Thus, for example, we might say, "Consider any statement, p , . . ." or "Suppose that some pair of statements, p and q , are both true . . . ." Statement variables can stand for any statements whatsoever, but within the scope of a specific context, each statement variable always designates the same statement. Once we've begun substituting A for p , we must do so consistently; that is, every occurrence of p must be taken to refer to A . But if another variable, q , occurs in the same context, it can stand for any statement whatsoever B , or C , or even A .
Next we introduce five special symbols, the statement connectives or operators:
~
(also symbolized as & or
Ù )
Ú
É (also symbolized as )
º (also symbolized as « )
The syntax of using statement connectives to form new, compound statements can be stated as a simple rule:
For any statements, p and q , ~ p

p

q

p

Ú q

p

É q and

p

º q

are all legitimate compound statements.
This rule is recursive in the sense that it can be applied to its own results in order to form compounds of compounds of compounds . . . , etc. As these compound statements become more complex, we'll use parentheses and brackets, just as we do in algebra, in order to keep track of the order of operations. Thus, since A , B , and C are all statements, so are all of the following compound statements: