Standard Categorical Statements

9. Standard Categorical Statements

Introduction

As we begin with the theme of categorical statements, keep in mind that our goal is still the study of argument. Categorical statements are the bones that are held together in the kind of argument known as the categorical syllogism. Syllogisms are three part arguments. Categorical syllogisms are a distinct type of syllogism. This distinctiveness is found in the fact that all the statements of a categorical syllogism state some relationship between categories. Once we have defined and clarified the essence of categorical statements, we can then tackle the categorical argument form in the next chapter. For now there are three things to discuss: definition, standardization, and translation.

1A. Definition

We need a definition of a standard categorical statement. This can be done by working backward from a statement to the qualifying terms "categorical" and "standard."

1B. Statement

A statement is an information giving expression that can be judged to be true or false.

2B. Category

A category is simply a group, a set of things, or a class. The following are categories: roses, cabbages, kings, students. There are many categories within other categories. These are called sub-sets, sets within other sets. "Students who study logic and love it" is a sub-set of students in general (Note: you are in a class all it's own, the "learners and lovers of logic" category). In the infant baptism discussion, the Baptist views infant baptism to be included in the category of things to which administrative discontinuity applies and they then reason that if discontinuity applies to these other areas (spouses, servants) then it may well apply to children (this they claim then shifts the burden of proof to the infant Baptist to show that continuity rather than discontinuity in fact applies to the case of children).

3B. Categorical statement

A categorical statement is an expression that affirms or denies membership in a category or class. Notice how the base definition of a statement in logic has now grown from an information giving expression that is true or false to a true or false information giving expression that affirms or denies membership in a category. The base definition still holds with regard to any argument. This additional aspect applies to arguments that are composed of statements that are all categorical in nature.

4B. Standard categorical statement

There are specific requirements regarding how categorical statements are to be expressed. For purposes of logical analysis, all categorical statements must use the following four things. 1) Quantifiers must be explicitly stated. These are terms that indicate universal or particular scope of reference (specifically, all or some). 2) The copula (the verbal connector, are) must appear in the statement.. 3) There must be a category on both sides of the copula; "nounize" or "thingize" or categorize if needed. 4) For negative statements, not, no, or none must be expressed.

If a statement that is categorical in nature does not specifically use the quantifiers, the copula, categories on both sides, and proper use of the negatives then it must be re-written in standard form fulfilling these requirements. We will see how this is done after we discuss standarization.

2A. Standardization

As we will see, all categorical statements fit into four patterns. For our concerns, there are only four ways that categories are related to each other. There are only four ways of denying and affirming membership of one category in another. These were identified by medieval logicians and have become standardized. We will now consider the four standard forms with respect to their identity, their quality, their quantity, and their terms.

1B. Identity

The four standard forms are simply referred to by means of the vowels A, E, I, and O, which were the abbreviations used by the medieval logicians. The following chart containing these vowels, a symbolic form, and an example reveals the standard form of each statement type:

Abbreviation Symbolic Form Pirates Example

A All x are y All Pirates are dangerous

E No x are y No Pirates are dangerous

I Some x are y Some Pirates are dangerous

O Some x are not y Some Pirates are not dangerous

2B. Quality

To say that a categorical statement has quality is to point out and emphasize the kind of statement it is whether affirmation or denial. If you look at the chart above, you will discover that two of the statements are affirmations and two are denials. The affirmation forms are A and I from the Latin affirmo which means to affirm or to assert. The medieval logicians took the vowels, a and i, from this word to designate the affirmation quality of two of the standard ways of relating categories to each other; hence the A-form and the I-form as names or labels. The negation forms are E and O from the Latin nego, to negate or deny. Hence the E-form and the O-form which designate statements that deny membership of one category in relation to another.

3B. Quantity

Just as all four standard forms have a distinct quality, they also each have quantity. Quantity refers to the fact that every categorical statement is either universal or particular in scope. The medieval logicians selected the vowel names in such a way so they could arrange the forms in alphabetical order and have the universal statement forms together (all..) followed by the particular statement forms (some..). As you move down the alphabet you descend from the universal (A,E) to the particular (I,O).

When we put quality and quantity together we note that the A-form is a universal affirmation (all .. are..), the E-form is a universal denial (all .. are not ..), the I-form is a particular affirmation (some .. are ..), and the O-form is a particular denial (some .. are not ..).

4B. Terms

1C. Number

Every categorical statement has two terms because every categorical statement asserts some kind of relationship between two categories. Terms are simply categories (i.e. the term pirate refers to the category of peg-legged people who roam the high seas).

2C. Identity

The two terms of a categorical statement are the subject term and the predicate term.

Observe the use of term in this connection, where it is a practical synonym for a category. The category identifies the things associated together in the same group. Calling a category a term indicates that a word is being used for its referent (for what it refers to). "Term" has these two uses in this context. It can refer to a word, here, to the word leprechaun (when speaking of a little green person we use the term leprechaun). It can also refer to the group of things, in this example, to the group made up of imaginary little green people. In this second use, term does not refer to a word (the word leprechaun, written or spoken; instead, it refers to the category of little green people. So when we speak of the subject term or predicate term of a categorical statement we speak of words in a sentence but since they represent things they refer to categories (of subject category and predicate category). Similarly, later when we refer to the three terms of a categorical syllogism we refer to the three categories being interrelated in an argument.

3C. Position

When a categorical statement is written in standard form, which for our purposes means proper form (or logical order), the subject term will be cited first and the predicate term will be cited last. What is being asserted is whether or not the subject term (category) has membership in the predicate term (category).

4C. Definition

It is important to note that grammatically the categories being related to each other may be found in any order. Therefore, since it is not always the case that the subject term will be stated first, we need a definition of each term so we can identify them and put them into standard order with the subject term first followed by the predicate term.

1D. The subject term

The subject term is that of which something is being said in a statement. It is that which is being talked about. It is the category that is asserted to have, or not have, membership in the other category of the statement.

2D. The predicate term

The predicate term is that which is said of something. It is the category in which something is asserted to have, or not have, membership. It is the larger circle when relating two things like dogs and animals in the statement "all dogs are animals."

Let's illustrate the interplay of subject and predicate terms by reference to the simple sentence, "Flowers grow." This is a categorical statement but it lacks both the quantifiers and the copula. So, we need to re-write the statement. A rule of thumb for quantifying is the principle that when a class is being referred to in general or as a class, then we will use the universal quantifier. This example will be translated, "All flowers grow."

There could be some contextual reason that forbids the use of "all" as in the statement, "plants bloom." Since all the things that can be classified as plants do not bloom, then this would have to be re-written with the particular quantifier (Some plants bloom).

Back to our "all flowers grow" example, what about the copula, we have no "are" connector in this sentence? It must be supplied. At first, we have the obviously unfinished form:

All flowers are grow.

The way to resolve this problem that results in re-writing is to remember that we have a relation being affirmed between two categories. Therefore, whenever a category is referred to through the use of an adjective or a verb (as here in the verb, grow), we have to categorize or "nounize" the adjective or verb. In this example, what we must do is make "grow" into the noun form: "growing things." Now we have standard form:

All flowers are growing things.

Of course, that of which something is being said is "flowers," the subject term. And that which is said of flowers, that they grow, is the predicate term. It is being affirmed of the subject term, flowers that they have membership in the predicate category of things that grow.

5C. Distribution

The distribution of the terms of a categorical statement is crucially important to argument analysis and the determination of validity. If you grasp this notion clearly now, you will find the evaluation of categorical arguments much easier than might otherwise be the case later.

1D. Definition

Distribution means that all the members of a given class are being referred to. When only some portion of a class of things is being discussed, that class is said to be undistributed. We will refer to a distributed term by means of the capital letter, D and we will refer to an undistributed term by means of the capital letter, D, proceeded by the tilde (~): ~D. One caution is that we must not confuse D with DV.

2D. Contrast

Another caution should be registered. Distribution overlaps with, but is distinct from, quantity. Both of these are scope designators but quantity deals with the scope of the entire statement; whereas, distribution is concerned with the scope of each category within the statement. Quantity describes statements and distribution describes individual terms.

3D. Application

Since there are only four standard forms of the categorical statement, there are only four scenarios of distribution. One way to master distribution is to memorize the four possible patterns that may emerge. Another way is to think through the scenario by applying the principles of quantity and quality to distribution. Quantity applies to the subject term and quality to the predicate.

1E. Distribution of the subject term

The distribution of the subject term is self-defining. That is, the scope of reference of the subject term is defined by the statement quantifier. If it is a universal quantifier (all), then the subject term is distributed (D). If the statement quantifier is particular (some), then the subject term is undistributed (~D).

2E. Distribution of the predicate term

To determine the distribution of the predicate side of a categorical statement, attention has to be focused on the quality of the statement. If the statement is an affirmation, then the predicate term is always undistributed (~D). If the statement is a denial, then the predicate term is always distributed (D). Ultimately, the distribution of the predicate term is also self-defining if we observe the word not following the copula: "not" indicates a distributed predicate term (note that it is implicitly present after the copula in statements saying, "no...are...). Thus, we can work from various cues to the same point of distribution, from noting that the statement is E or O, from picking up the sense that the statement is a denial, and from specifically observing the occurrence of the word not.

In review of distribution determination, we can think through the following questions. First, what is the distribution of the predicate term in the statement "All x are y"? What is the distribution of the subject term in the statement "no x are y"? How about the predicate term in "some x are not y"?

3A. Translation/Re-writing

Because of the difference between the order and style of expression of English grammar and logical order and style, ordinary English statements have to be translated into the standard AEIO forms with the correct quantifiers, the copula, and proper "nounization" (According to Eric Chang, "What we need is more nouniness"). Admittedly, the translations into logical form will often sound stilted and artificial. They will also often be very poor English expressions. What they have going for them is that they reflect good logic for purposes of precise analysis.

1B. A helpful phrase

A key phrase in translating the relationship that exists between terms in a categorical statement is "included in the class of." For example, that which is intended by the A-form, "all x are y" is brought out somewhat when we translate it, "all x are included in the class of y."

2B. Difficulties

Normal English quantification and negation present many non-standard forms of expression. We want to highlight some basic difficulties in this regard by means of the devices we can use to handle them.

1C. The identical with device

Consider the citation of individuals as in the Socrates example. "Socrates is a man" becomes the A-form, "All persons identical with Socrates are men." Sometimes the statement is stated in such a way that it is hard to pick out the terms. Ralph's market is an example: Ralph's market does not sell banoranges.

Where are the terms? How can we pin them down? The re-writing can be done in steps. First, we can insert the copula: "Ralph's market [are] not sell banoranges." This will not suffice for obvious reasons, so we need another step. Second, we can "nounize" the predicate side to "not sellers of banoranges." But how about the subject side? Here we have an individual market that is a non-seller of banoranges. How can we quantify it? "Some of Ralph's market is a not seller" is unacceptable since there are no banoranges anywhere in the store. What we need is some way to say "All Ralph's market are not sellers" in a way that at least approaches some semblance of meaningful communication. Two ways will work here. We could say that "All markets that belong to Ralph (and we know he only has one) are not sellers of banoranges." Or, we use the "identical with" device for translation. In this example, we would then have the following E-form: All market's identical with Ralph's are not sellers of banoranges (or are not market's that sell banoranges).

2C. Times/places/cases device

For the translation of references to times, places, and cases, we can use the time/place/case device. Thus, "when it rains, it pours" becomes "all times of raining are times of pouring." And "dishonesty is immoral" becomes "all cases of dishonesty are cases of immorality."

3C. The repeat the subject term device

Sometimes the best way to re-write into standard form is to repeat the subject term adding to it the descriptions found on the predicate side of the statement ("all students love logic" becomes "all students are students who love logic."). To avoid redundancy, the phrase "people who" could follow the copula and then be followed by all the descriptions even preserving the tense of the verb: "all students will go to the play at midnight if there is one" becomes "all students are people who will go to the play at midnight if there is one."

By adding these descriptions, the subject term is made into a larger category that can serve as the predicate term. For example, "all cowboys ride horses" can be "nounized" by repeating the subject term: "all cowboys are cowboys who ride horses" (of course, you could use "all cowboys are people who ride horses").

4C. A negated subject term for unless

Thus, "all may go unless a student" becomes "all non-students are people who may go." Notice that "unless" is literally, "if not" (all "if not students" may go) and that "unless" is ambiguous by itself. Also, note that this does not necessarily mean that "no students may go" because we need not take being a non-student as required in order to be a "goer." If being a non-student is sufficient but not necessary to being a goer then students are not excluded. We simply have the case that all who are not students may go without saying what pertains to students (cf. a context where an affirmative answer is given to the question: can students go as well?). If we had more information that showed that what is meant is that if and only if one is a non-student then they are goers, then we would know that students are excluded (for this we need more information in the context). We will come back to the use of unless when we cover conditionals in truth functional language.

5C. Reversing

Only (and alone) is a word of peculiar character. Three steps are needed to translate it: 1) it translates as a universal, 2) it attaches to the predicate term, and 3) the subject and predicate must be reversed. Thus, "Only doctors are allowed to prescribe drugs" becomes "All who are allowed to prescribe drugs are doctors." Likewise, "students alone may attend" becomes "All who may attend are students."

6C. "Some are not" is used for "not all are"

7C. "Some" translates most and few.

Categorical Statements WS1

For more examples see Churchill: p148-149 (A, 1-10; B, 1-5) p154 (B, 1-10) pp180-181 (B, 1-5; C, 2, 5)

A. Define

statement