Logic: Categorical Propositions and Syllogisms


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By Dr. Garth Kemerling / 11.12.2011
Professor of Philosophy
Capella University

Philosophy Pages

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 system—an 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 class—whether it’s a natural species or merely an arbitrary collection—of 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.”

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.

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 other—that is, either both of them are true or both are false—they 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 Opropositions: 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:

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 Epropositions 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:

term

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.

term

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.

term

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.

proposition

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:

E proposition

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.

I proposition

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.

O proposition

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.

A proposition

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

The Structure of Syllogism

Now, on to the next level, at which we combine more than one categorical proposition to fashion logical arguments. A categorical syllogism 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 minor term of the syllogism as a whole. The major term 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 middle term.

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 major premise of the syllogism. The other premise, which links the middle and minor terms, we call the minor premise.

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.

Standard Form

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 standard form 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 mood and figure. 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.

Since 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.

Form and Validity

This method of differentiating syllogisms is significant because the validity of a categorical syllogism depends solely upon its logical form. Remember our earlier definition: an argument is valid 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.

Diagramming Syllogisms

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:

  1. First draw three overlapping circles and label them to represent the major, minor, and middle terms of the syllogism.
  2. 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.
  3. 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.

 

three circles

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:

major premise

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.)

minor premise

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.

conclusion

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

AAA-1 (valid)

                All M are P.
                All S are M.
     Therefore, All S are P.

AAA-3

AAA-3 (invalid)

                All M are P.
                All M are S.
     Therefore, All S are P.

OAO-3

OAO-3 (valid)

                Some M are not P.
                All M are S.
     Therefore, Some S are not P.

EOO-2

EOO-2 (invalid)

                No P are M.
                Some S are not M.
     Therefore, Some S are not P.

IOO-1

IOO-1 (invalid)

                Some M are P.
                Some S are not M.
     Therefore, Some S are not P.

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 . . .

  1. . . . 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).
  2. . . . 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.
  3. . . . 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.
  4. . . . 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.
  5. . . . 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.
  6. . . . 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.

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