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 negativeproposition 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 Eand 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 Eis 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) arecontraries. 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 theA 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 Iand 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.