Chapter9Categorical Logic Statements

9.1 Four Kinds of Categorical Statements

There are four kinds of categorical statements, represented by the following standard forms:

Keep this chart handy. It will be important for the rest of this chapter, which explores how each statement works and how to diagram them one by one.

In the chart, under “type” you will see four combinations of the terms “universal,” “particular,” “affirmative,” and “negative.” The quality of a categorical statement is the character of the relationship it affirms between its subject and predicate terms (affirmative or negative). A categorical statement is an affirmative statement 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 statement if it wholly or partially excludes members of the subject class from the predicate class. The quantity of a categorical statement, on the other hand, is a measure of the degree to which the relationship between its subject and predicate terms holds. A categorical statement is universal if it makes an exceptionless claim about the subject and predicate terms. It is particular if it makes claims that hold for one or more members of the subject class.

9.2 Four Parts of Every Categorical Statement

Every categorical statement in standard form has four parts (fig. 9.1).

1. 1. A quantifier—they all start with all, some, or no. In categorical logic, we only make statements about all (every single member), some (which means at least one or many, but at least not none) and no (meaning none or never).
2. 2. A subject term—a word or phrase denoting a class of things serving as the subject of the sentence. This is labelled “subject” because, like a sentence, it is the main thing you are talking about. When we say, “All goats are hungry,” “goats” is the subject term and the subject of the sentence.

1. 3. A predicate term—a word or phrase denoting a class of things serving as the subject complement of the sentence. What does this mean? This is what is said about the subject. The predicate modifies the subject.
2. 4. A copula—a linking verb, a form of the verb “to be,” which connects the subject term with predicate term. The action of a statement (the verb) in categorical logic says something about the being of the subject. We are saying something about what it is.

9.3 Venn Diagrams

John Venn (1834–1923), who was a mathematician at Cambridge University, devised a method of diagramming categorical statements, now called Venn diagrams, which makes representing the relationships between the statements very easy.

Classes of things are represented with a circle. The class of all things that are S is represented on the left lune (think of the moon!). The class of all things that are P is in the right lune. The centre lens represents the intersection between the two, which would be all the things that are both S and P. The outside, labelled “universe of discourse,” is everything else in the world that is not S or P (fig. 9.2).

For example, in figure 9.3, if S stands for the class “Albertans” and P stands for the class “Canadians,” then the lens will represent the class of Albertan Canadians, the left lune will represent the class of non-Canadian Albertans, the right lune will represent non-Albertan Canadians, and the remaining area outside of the two circles, the universe of discourse, will represent all other things in the universe that are not Albertans and not Canadians.

9.4 Universal Affirmative: A

The universal form A is affirmative and takes its name “A” from the medieval Latin word “affirmo” (I affirm). It affirms a rule about all members of a group and is thus universal. A statements make a rule without exception: it is a universal statement if the asserted claim holds for every member of the class designated by its subject term.

A. All S are P: this statement makes a universal declaration about S. It says that all things that are S are also P. Representing this with circles means that you would eliminate the S lune that is not intersecting with P (fig. 9.4).

For example, if our sentence is “All dogs (S) are mammals (P),” we know that there are not any dogs that are not mammals. This seems true, right? What would a dog that is not a mammal be like? Would it still be a dog? No. So we “get rid” of the S that is outside of the P (shade it out). P is a larger set of mammals that would contain all other mammals (humans, cats, etc.). The lens in the middle, then, represents dogs by telling us that they are all the way overlapped by the property of being a mammal.

Sometimes the shading is confusing. What are we colouring in, exactly? In this instance, what you shade is taken out of existence. You are saying this section isn’t allowed to exist—you are saying that it in fact cannot exist. You are demonstrating the rule that the A or E statement makes. This form of mapping is especially important to understand when mapping E statements that also make a rule but make a universal negative.

9.5 Universal Negative: E

The universal form E takes its name from medieval Latin word “nego” (I deny). It is negative because it blocks a possibility. It is also universal because it makes a rule without an exception.

E. No S are P: If No S are P, then there is no possibility of something existing in the overlapping area. For example, if our sentence is “No snakes are poodles,” we know that there is nothing that is both a snake and a poodle, and we indicate this by shading out the overlap area (fig. 9.5).

An E statement makes a rule that there’s no possibility for anything to be both of the terms. In the above, we are saying not only that there are no snake poodles but that snake poodles are impossible. Since the left and right lunes are still open, this diagram demonstrates that being a snake or being a poodle is possible.

Note that when we say no snakes are poodles, it also makes sense to say that no poodles are snakes. This means that in an E statement, the subject and predicate terms can be interchanged. This is called converting the statement: it produces the converse of the original statement. We can see right from the diagram that if no snakes are poodles then no poodles are snakes either. So an E statement is logically equivalent to its converse. (Conversion is a relationship discussed in more depth in Chapter 11.)

9.6 Particular Affirmative: I

An I statement takes its name from the second vowel in “affirmo.” It is affirmative because it says something positive, and it is particular in that it suggests there is at least one member or thing in a class (a “particular” such as a unique thing). This is a different quantity of a categorical statement than a universal (which we have seen in A and E). The quantity of a categorical statement tells us the degree to which the relationship between its subject and predicate terms holds: is it universal, or do we know that the relationship only holds sometimes? If it is a particular statement, then the claim is asserted to hold only for one or more members of the subject class.

I. Some S are P: The statement “some S are P” tells us both that there is at least one thing that is S and that that thing (and possibly others) is also P. We indicate this by putting an x in the overlap of the S and P circles (fig. 9.6). When we put an x somewhere, we are saying there is something there—a particular thing within the terms given.

For example, there are different sizes of dogs; some are miniatures, and some are standards, so if our sentence is “Some standards are poodles,” we know that at least one poodle in the world is that size. The x represents a particular standard poodle.

What an E statement and an I statement share is that they are both logically equivalent to their converse. Think: If some poodles are standards, doesn’t it also tell us that some standards are poodles. This is just to say that if there are poodles that are standards, then there are standards that are poodles.

9.7 Particular Negative: O

The particular negative takes its name from the second vowel in the medieval Latin word “nego” (I deny). A particular negative represents something as existing (particular), but it tells us that the X has a relationship of not being included in the predicate class.

O. Some S are not P: The statement “some S are not P” tells us both that there is at least one thing that is S and that we know of that thing that it is not P. So we put the X in the part of S not overlapping P (fig. 9.7).

Since there are snakes of many kinds and some of them are not puffadders, the statement “Some snakes are not puffadders” is true. This statement directs us to the area of S that isn’t touching P because it tells us specifically that there is something in the S area (the subject term) that is not a P.

We discussed how E statements and I statements are logically equivalent to their converse (you can switch the subject and predicate terms, and they remain true). Note that you cannot do that for O or A statements. This means it is very important that you get these in the proper order when you are mapping. For example, from the fact that some snakes are not puffadders, you cannot infer that some puffadders are not snakes. Just as you cannot infer that all dogs are poodles from the fact that all poodles are dogs.