Standard Form Categorical Propositions: Quantity, Quality, and Distribution

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Standard Form Categorical Propositions: Quantity, Quality, and Distribution

Abstract: The most important properties of standard form categorical propositions are explained and illustrated. I. What is a categorical proposition?
	A. The long range goal of our study of traditional logic[1] is to give a theory of deduction, i.e., to explain the relationship between the premisses and conclusion of a valid argument and provide techniques for the appraisal of deductive arguments. Hence, we will be distinguishing between valid and invalid arguments.
		1. A deductive argument is defined as one whose premisses are claimed to provide conclusive evidence for the truth of its conclusion.
		2. A valid deductive argument is one in which it is impossible for the premisses to be true without the conclusion being true also.
	B. Our study of deduction, for the present, will be about arguments stated in categorical propositions.[2] Here is an example of this kind of argument:
		No honest people are persons who embroider the truth. Some politicians are persons who embroider the truth. Some politicians are not honest people.
		1. A categorical proposition is defined as any proposition that can be interpreted as asserting a relation of inclusion or exclusion, complete or partial, between two classes.[3]
		For example, the proposition or statement that “All human beings are mortal” implies anything that is a human being is included into the class of mortal things.
		2. A class is defined as a collection of all objects which have some specified characteristic in common. Understanding this definiion is no more complicated than observing that the class of “lightbulbs” all have the common characteristic of “being a lightbulb.”
			Thus, we can have four possible class relations in the various kinds of categorical propositions:
			Utilizing the classes, “people” and “good beings” we can illustrate these possibilities:
			a. complete inclusion ⟶ “All people are good beings.”
			b. complete exclusion ⟶ “No people are good beings.”
			c. partial inclusion ⟶ “Some people are good beings.”
			d. partial exclusion ⟶ “Some people are not good beings.”
			We can also describe these four kind of statements respectively as
			a. universal affirmative
			b. universal negative
			c. particular affirmative
			d. particular negative
		3. Often, it is convenient to look at the general form of the statements given above. To these forms, special names are given: A, E, I, and O.
			A: All S is P. [4]
			E: No S is P. [5]
			I: Some S is P.
			O: Some S is not P.
			… where S and P stand for the logical subject and the logical predicate of the statement respectively.
			4. An oft-used mnemonic device for remembering the quality of statements is to remember the red underlined letters in the Latin words: Affirmo and Nego. In this fashion, A and I statements are seen to be affirmative, E and O are remembered as negative.
		5. Note, in categorical propositions, the logical subject S and the logical predicate P represent something different from the grammatical subject and the grammatical predicate as taught in English grammar classes.
			For example, in the pretense statement, “All (unfledged floithoisters) are (things apt to become unflaggled)” … the logical subject is composed of all of the words between the quantifier “All” and the copula ”are,” and the logical predicate is composed of all of the words after the copula “are” until the end of the statement. So the logical subject of this statement is “unfledged floithoisters” and the logical predicate is “things apt to become unflaggled.”
		6. Also note that the word “Some” in the particular affirmative O statement is taken to mean “at least one.” This meaning differs somewhat from its ordinary language meaning. For example, the statement “Some women are physicists at CERN is considered true if there is at least one woman physicist at CERN. (Additionally this statement would be true even if it turned out that all except one person were women physicists at CERN.)
		7. A model statement, then, can be represented in general as
			Quantifier [subject term] copula [predicate term].
II. Analysis of the Categorical Proposition: Quality, Quantity, and Distribution
	A. The quantity of a categorical proposition is determined by whether or not it refers to all members of its subject class (That is, the statement is considered either universal or particular in quantity.) To question “How many members of the subject class are being discussed?” asks for quantity.
	B. The quality of a categorical proposition is determined by whether the asserted class relation is one of inclusion or exclusion (That is, the statement or proposition is considered either affirmative or negative in quality.)
	C. Indicators of “how many” are called quantity indicators (i.e., quantifiers) and specifically are “All,” “No,” and “Some.”
	D. Indicators of affirmative and negative are quality indicators (i.e., qualifiers) and specifically are “are,” “are not,” “is,” “is not,” and “No.”
		Note that “No” is both a quantifier and a qualifier.
	E. In sum, thorough knowledge of the following table is absolutely essential to do well in categorical logic:

Name	Form				Quantity	Quality	Distribution
							Subject	Predicate
A	All S is P				universal	affirmative	distributed	undistributed
E	No S is P				universal	negative	distributed	distributed
I	Some S is P				particular	affirmative	undistributed	undistributed
O	Some S is not P				particular	negative	undistributed	distributed

	F. Distribution of a term.
		1. A distributed term is a term of a categorical proposition that is used with reference to every member of a class. If the term is not being used to refer to each and every member of the class, it is said to be undistributed.
		2. Consider the following propositions:
			A: All birds are winged creatures.
			E: No birds are wingless creatures.
			I: Some birds are black things.
			O: Some birds are not black things.
The kinds of distribution of subject and predicate terms appearing in the above listed statements are explained here:
	Subject					Predicate
	A: refers to all birds					does not refer to every member, e.g., bats, flying fish.
	E: refers to all birds by indicating that they are not part of the predicate class					refers to all wingless creatures by indicating that they are not part of the subject class
	I: refers only to some birds					refers only to some black things, viz., those which are birds
	O: refers only to some birds, not all of them					refers to all members of the class! Viz., not one of them is in the class referred to by "some birds"
		3. For the predicate of the O proposition, consider the following analogy. If we know that there is a book not in a bookcase, then we know something about each and every shelf in that bookcase—namely, we know that the book is not on that shelf.
		4. There are three ways to remember the distribution status of subject and predicate for standard form categorical propositions:
			a. Memorize it. (Undoubtedly, this is most recommended way.)
			b. Figure it out from an example (as was done above).
			c. Remember the following rule:
				The quantity of a standard form categorical proposition determines the distribution of the subject (such that if the quantity is universal, the subject is distributed and if the quantity is particular, the subject is undistributed), and …
				the quality of a standard form categorical proposition determines the distribution status of the predicate (such that if the quality is affirmative, the predicate is undistributed, and if the quality is negative, the predicate is distributed).

Notes

1. Traditional logic and the logic of the syllogism descended from Aristotelian logic. Bocheński writes, “[Aristotle] exercised a decisive influence on the history of logic for more than two thousand years, and even today much of the doctrine is traceable back to him.” I. M. Bocheński, A History of Formal Logic, (New York: Chelsea,1961), 40.↩

2. Categorical propositions or statements are simple propositions. Other kinds of statements include compound statements such as the hypothetical (If… then …), the disjunctive (Either … or …), and the conjunctive (Both … and …) statements.↩

3. P.T. Geach explains that the use of class logic here to explain the traditional logic of terms leads to confusion:

‘[I]n [the] use of schematic letters like ‘S’ and ‘P’ you find, for example, in one and the same context the phrase ‘every S’, which requires that ‘S’ be read as a general term like ‘man’ and the phrase ‘the whole of S’, which requires that ‘S’ be a singular designation of a class taken collectively, like ‘the class of men’ obviously ‘man’ and ‘the class of men’ are wholly different sorts of expression.↩

4. P.T. Geach points out:

“[T]he extreme grammatical oddity” of “All S is P” being translated from “Omne S est P” of the historical Latin texts which ought be, according to him, “Every S is P,” so that the whole class of the subject term is universally quantified and no distribution error occurs.

[P.T. Geach, Logic Matters (University of California Press, 1980), 69.]

However, we follow the usual practice of using “All S is P” for the A statement, recognizing that the statement form need not be necessarily understood as following English grammar prose rules for sentences and can be understood as “All Ss.” ↩

5. Note that “No S is P” does not mean

“All S is not P”

because the latter statement form is ambiguous and can mean either

“No S is P”

“Some S is not P.”

For example, “All lodestones are not non-magnetic ore” in standard form means

“No lodestones are non-magnetic ore”

because lodestones (i.e., magnetite) are naturally occurring magnets — all of them, by definition, are magnetic.

But the statement “All swans are not white” in standard form means

“Some swans are not white”

because in nature swans are observed to exist in several other colors also.

To take another example, the statement “All that glitters is not gold” would be conservatively translated into standard form as

“Some things that glitter are not gold.”

not “No things that glitter are gold” because gold things can glitter.

Consequently, statements of the form “All S is not P” are not always the same thing as the E standard form proposition “No S is P.”. To translate properly, one must be aware of the meaning of the terms in the statements.

If specific information about the subject class of a statement is not known, the usual translation of statements of the form “All S is not P” is

“Some S is not P.”

For example, if nothing is known about the hunting habits of honey badgers, the statement

“Not all honey badgers are solitary hunters”

could be safely translated as

“Some honey badgers are not solitary hunters.”

(and one would not know whether or not it was also true that “Some honey badgers are not solitary hunters”)↩

After studying these notes, try the Chart Quiz where you can practice listing the quantity, quality, distribution of standard from categorical propositions.

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