UNIVERSAL STATEMENTS | PARTICULAR STATEMENTS | SYMBOLIC NOTATION | TERMS

Forms of Categorical Statements

As we've seen, categorical logic is about sorting individual things into groups and subgroups, and identifying other relationships implied by those groupings.  To do a comprehensive job of this, we need to distinguish between groups that contain every individual with a certain characteristic, and groups that contain just some certain individuals with that characteristic, as well as groups that exclude some or all individuals with that characteristic.  Moreover, we'll want to see how these different forms of statements are interrelated.  Specifically, we want to know whether some kinds of statements either imply or contradict others.  But first things first.

UNIVERSAL STATEMENTS

The statements we've looked at in the sample syllogism about Socrates take the general form All S are P (the capital letters S and P representing the subject and predicate terms, respectively).  This is obviously true in the case of the major premise, All men are mortal.  But if we reword the minor premise to All things that are Socrates are men, and the conclusion to All things that are Socrates are mortal, then we see that the pattern applies to these statements as well.  Note that, although the rewording makes the statements sound very awkward, it doesn't change their meanings in the least.

Now compare such statements to No men are gods.  Whereas All men are mortal makes the claim that everything in the category of men is also in the category of mortal things, No men are gods makes the claim that nothing in the category of men is in the category of gods (or in other words, everything in the category of men is not in the category of gods).  Both statements make claims about all men (or every man), and as such are classified as universal statements.  The difference is that a claim that all S are P is affirmative, while a claim that no S are P is negative.

We could substitute No men are gods as the major premise in our syllogism:

No men are gods.
Socrates is a man.

And if you've already spotted men / man as the middle term, and have figured out that the conclusion to be drawn from these two premises is...

So, Socrates is not a god,

...then you've obviously got the knack of it!

So far, we have two general relationships expressed by categorical statements: universal affirmative (All S are P), and universal negative (No S are P).

Technically, No S are P should be logically equivalent to All S are not P.  However, the meaning of this latter form is ambiguous; in common usage, it could be (and usually is) interpreted as Not all S are P, which is something entirely different from No S are P, as we're about to see.)

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PARTICULAR STATEMENTS

Often, we aren't concerned with all men, or all mortals, or all of anything else.  Sometimes we need to distinguish some men, some mortals, etc., as being in some way different from other men, mortals, or whatever.  Consider the following two premises:

Socrates is a man.
Socrates is a philosopher.
So... what?

What, if anything, can we conclude from these two statements?  All men are philosophers?  No, because Socrates is only one man among many, and other men might be different from Socrates in a number of ways, including whether or not they're philosophers.  How about All philosophers are men?  No, we can't conclude that either, because the information given in these premises doesn't rule out the possibility that non-men (e.g., gods, women, or even other clever species) might be philosophers, too.  However, conclusions we can legitimately draw from these premises include the following:

Some man is a philosopher; or
Some men are philosophers.

In either case, the meaning is this: Something exists in the category of men which is also in the category of philosophers.  Now, this something might be only one thing (Socrates), or it might be many things (Socrates, Plato, Aristotle, Descartes, Leibniz, Hume, Kant, Wollstonecraft, Nietzsche, Sartre, Russell, and hundreds of others).  Indeed, the conclusion opens the door to the possibility that everything in the category of men is also in the category of philosophers, unless some other statement rules this possibility out.

Such a limiting statement might be:

Some man is not a philosopher; or
Some men are not philosophers.

(As we'll note, the second of these is equivalent to the colloquial expression, Not all men are philosophers, which might also be casually (but erroneously) expressed as All men are not philosophers, which could technically be interpreted No men are philosophers—obviously something quite different from the original.  This potential ambiguity arising from imprecise use of language is why logicians scrupulously avoid the All S are not P and Not all S are P constructions, as should we likewise.)

Now, the only difference between Some man is not a philosopher and Some men are not philosophers is that the one is expressed in the singular and the other in the plural; so far as logic is concerned, there's no difference at all, as long as no particular number of men is specified, and as long as the number is greater than zero.  Some men implies at least one man; it can't mean no men.  So, let's arbitrarily pick the plural version, Some men are not philosophers, and pair it with a corresponding but contrary statement, Some men are philosophers.  Again, we have two statements of the same general class—this time particular instead of universal—but again, one statement is negative and the other affirmative.  Thus, we now have four general forms of categorical statements:

Relationship				Structure (Plural)				Structure (Singular)
universal		affirmative		All S are P		or		Every S is P
universal		negative		No S are P		or		No S is P
particular		affirmative		Some S are P		or		At least one S is P
particular		negative		Some S are not P		or		At least one S is not P

Each statement form can be expressed in different ways without changing its overall meaning.  Different ways of expressing the same idea are called stylistic variants.  Some of the most common variants of each form are listed here.

Relationship		General Structure		Typical Stylistic Variants
universal affirmative		All S is / are P.		Each / every S is P.		Any S is / are P.		Only P is / are S.
universal negative		No S is / are P.		Each / every S is not P.		Any S is / are not P.		No P is / are S.
particular affirmative		Some S is / are P.		At least one S is P.		Something is both S and P.		Some P is / are S.
particular negative		Some S is / are not P.		At least one S is not P.		Something is S but not P.		Not all S is / are P.

(Note that our abstract representations of statement forms here are somewhat different from the style we used in the Symbolic Logic section.  In the previous section, we used lower case letters to represent entire statements.  In this case, we're representing individual terms—subject and predicate—within statements, and the capital letters help to emphasize this distinction.)

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UNIVERSAL STATEMENTS | PARTICULAR STATEMENTS | SYMBOLIC NOTATION | TERMS

SYMBOLIC NOTATION

Philosophers are typically delighted whenever matters appear to resolve themselves into neat patterns, such as the one preceding, because this often indicates there are other inferences to be drawn and other points to be made.  This instance is no exception.  Note that, in the copy of the previous table below, another column, labeled Form, has been inserted to the left.  The A, E, I, and O designations in this column are abbreviations for Latin notation used by medieval scholars.  Although the Latin terminology is of no particular interest to us, the letter designations offer us a very handy standard shorthand for these four categorical relationships, so it would be worth our while to commit them to memory, at least for the remainder of this section.

Form		Relationship				Structure (Plural)				Structure (Singular)
A		universal		affirmative		All S are P		or		Every S is P
E		universal		negative		No S are P		or		No S is P
I		particular		affirmative		Some S are P		or		At least one S is P
O		particular		negative		Some S are not P		or		At least one S is not P

If you need a little help remembering which is which, remember that we learned the four relationships in the alphabetical order of the letters that represent them.  Some association of the letter with something in the name of the form may also help, but this is hardly fool-proof.

• A: universal affirmative (All are)

• E: universal negative (Every one is not)

• I: particular affirmative (at least one Is)

• O: particular negative (sOme are not)

Now, there are many things we could do with these four forms within the format of a three-statement syllogism.  But most of these are too complicated, even with the streamlining afforded by this shorthand notation, to be worth our while to study.  However, there are a few very useful and relatively simple patterns that can give us tremendous leverage at understanding and inference-drawing.

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UNIVERSAL STATEMENTS | PARTICULAR STATEMENTS | SYMBOLIC NOTATION | TERMS

TERMS

• form

• universal statement

• universal affirmative

• universal negative

• particular statement

• particular affirmative

• particular negative

• stylistic variant

• A statement

• E statement

• I statement

• O statement

In this lesson, we've considered the four basic forms of categorical statements.  Next, we'll make and illustrate comparisons among these types, drawing broader and more practical insights from the abstract details we've learned here.

=R4=

►  Next: The Square of Opposition

 

UNIVERSAL STATEMENTS | PARTICULAR STATEMENTS | SYMBOLIC NOTATION | TERMS

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