Categorical Propositions - Amateur Logician

A, I, E, and O

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Two-Term Logic System: Introducing the Categorical Proposition

– Judgements affirm or deny an attribute of a subject.
– Propositions are verbal or written expressions of a judgement.
– Remember, every such judgement/proposition is either true or false.
– All categorical syllogisms contain categorical propositions.

– Subject: that which is described to have or not have a certain attribute
– Predicate: that which is the attribute
– Copula: the “to be” verb

Categorical Propositions will always be of this form: [subject]—[copula]—[predicate]
Sometimes we can rewrite a given sentence, not of this form, to match this form.
Note also that the copula will always be in the present tense.

Consider the proposition “Socrates is a man.” “Socrates” is the subject we are talking about. Being a “man” is the predicate, since it describes the subject. The copula is “is.”

Or let’s say we have the statement: “Socrates runs to the store.” This is not in proper form. We thus have to change it to “Socrates is running to the store.” (Here the subject is “Socrates.” The predicate is “running to the store.”)

The copula will be of the form “is,” “is not,” “are,” or “are not.” Its logical function is to link together the subject and predicate. That way something can be asserted as a judgement.

Simple Examples. . .

“All gold is a metal.”
Subject: “gold”
Predicate: “metal”
Copula: “is”

“No birds are rational.”
Subject: “birds”
Predicate: “rational”
Copula: “are” (disconnects)

“Some cars are not blue.”
Subject: “cars”
Predicate: “blue”
Copula: “are not”

A subject or predicate doesn’t have to contain a single word; it may have more than one. Terms may be “simple” or “complex.”
— A “simple” term has one word.
— A “complex” term has more than one word.

Hence, this is a perfectly legitimate proposition: “Socrates is running to the store.”

Quantifiers are used. Words like “all” or “some” are quantifiers.

“All gold is a metal” has the quantifier “all.” This means that everything that happens to be gold is being referenced. “Some cars are not blue” has the quantifier “some.” This means that at least one thing that happens to be a car is being referenced.

But also, words or phrases like “any,” “in every case,” “never,” etc. are quantifiers. It’s important, in practice, to recognize synonymous words in order to “translate” or “reduce” a sentence into proper categorical form.

Four Basic Types of Categorical Propositions: A, I, E, and O

Categorical Propositions have Two Basic Features: Quantity and Quality

There’s always a subject and a predicate in a categorical proposition.

Quantity deals with the subject term. Are we referring to all things the subject can refer to? Are we referring to some? Or are we referring to one specific thing?

Quality deals with how the predicate relates to the subject. Is the attribute affirmed of the subject? Or is the attribute not affirmed of the subject?

Universal: predicate is affirmed/denied of all objects denoted by the subject. So, each member of the subject is being referred to.
Particular: predicate is affirmed/denied of some objects denoted by the subject. So, at least one member of the subject is being referred to.
Singular: predicate is affirmed/denied of a specific thing. So, a single or individual thing is being referred to.
Affirmative: predicate really belongs to the subject as an attribute.
Negative: predicate does not belong to the subject as an attribute.

Given that there are three types of quantities and two types of qualities, there are a total of six types of categorical propositions. However, we often can treat singular propositions as if they were universal.^[1] This therefore gives us four types.

Famous Scholastic Logic Mnemonic. . .
“affirmo” = “I affirm” (A, I)
“nego” = “I deny” (E, O)
“Affirmo” and “nego” are Latin for “I affirm” and “I deny.”

Notice we symbolize categorical propositions as A, I, E, or O.

Constructive Use of Categorical Propositions

It’s important to notice key words that may indicate the quantifier type and quality type.

On the one hand, notice various ways that a universal and a particular quantifier may be expressed. Rather than a particular proposition always appearing with the word “some,” it may come in such expressions as “at least one,” “a group of,” or “a few.” Likewise, there are various ways to express a universal quantifier like “all,” “every,” “each,” “any,” etc.

And, on the other hand, the quality can be indicated in various ways. While the “is” or “are” is associated with the affirmative proposition, and the “is not” or “are not” with the negative proposition, there are other ways to express quality. For instance, temporal indication is possible. Hence, the “never” goes with a negative proposition and the “always” with a positive proposition. “Sometimes” and “occasionally,” however, will affect the quantifier.

Back to Our Simple Examples. . .

“Socrates is a man” is singular as it refers to a specific individual. For our purposes, however, we’ll treat singular propositions as universal propositions. And it’s affirmative because the predicate, being a man, is declared to be a real feature of Socrates.

In a nutshell, “Socrates is a man” is an A proposition (affirmative, universal).

Note it does make sense to think of this “singular” proposition as a “universal” proposition. We’re referring to all that is Socrates. He is — totally — a man!

“All gold is a metal” is clearly an affirmative proposition, as it affirms that gold “is a metal.” Nothing is being denied. Further, all things that are gold are being referred to. It refers to “all gold.” The word “all” is a quantifier. So, it’s universal.

“All gold is a metal” is an A proposition (affirmative, universal).

“No birds are rational” is negative, as it denies the attribute of rationality. And it’s universal because it’s not speaking about only some birds but all birds.

“No birds are rational” is an E proposition (negative, universal).

“A few cars are not blue” is negative because it denies the attribute of being blue. It’s not referring to every car. There’s the quantifier “some.” It’s particular.

“A few cars are not blue” is an O proposition (negative, particular).

How about an I proposition (affirmative, particular)?

Here’s an example: “Many philosophers are Aristotelians.”

Symbolizing. . .

Symbolizing categorical propositions is helpful when working with the square of opposition, immediate inference, and with syllogisms.

1) Determine Type of Categorical Proposition (is it A, I, E, or O?)
2) Symbolize the Subject
3) Symbolize the Predicate
4) Write as [Subject]-[type]-[Predicate]

Only use the letters A, I, E, and O for the type.
Pick a letter for the Subject. Pick another letter for the Predicate.

Simple Examples. . .

Proposition: “Socrates is a man.”
Symbolized: SAM
It’s an A type. S is for “Socrates” and M is for “man.”

Proposition: “All gold is a metal.”
Symbolized: GAM
It’s an A type. G is for “gold” and M is for “metal.”

Proposition: “No birds are rational.”
Symbolized: BER
It’s an E type. B is for “birds” and R is for “rational.”

Proposition: “A few cars are not blue.”
Symbolized: COB
It’s an O type. C is for “cars” and B is for “blue.”

Proposition: “Many philosophers are Aristotelians.”
Symbolized: PIT
It’s an I type. P is for “philosophers” and T is for “Aristotelians.”

***
[1] An important exception: singular propositions cannot be treated on the “square of opposition.”