26. Categorical Logic - The Basics

 

Summary

This episode opens the study of categorical logic, the oldest branch of formal logic, reaching back to Aristotle and the ancient Greeks. Like mathematics, deductive logic must be built from the ground up: you cannot test the validity of a categorical syllogism before mastering its building blocks.

Categorical Propositions and the Four Types

Categorical propositions are the building blocks of the syllogism. They are statements about relationships among classes (categories) of things, affirming or denying that one class is wholly or partially included in another. Because a proposition can either affirm or deny (quality) and do so wholly or partially (quantity), there are exactly four subtypes:

• A – universal affirmative ("All dogs are mammals")
• E – universal negative ("No dogs are cats")
• I – particular affirmative ("Some dogs are wolves")
• O – particular negative ("Some dogs are not wolves")

Medieval logicians drew these letters from the Latin words affirmo ("I affirm," giving A and I) and nego ("I deny," giving E and O).

The Four Parts of a Proposition

Propositions themselves break down into four more fundamental parts:

1. The subject term — what the assertion is about (in "All dogs are mammals," dogs).
2. The predicate term — what is asserted about the subject (mammals).
3. The copula — the verb joining subject and predicate, which in logic is always a form of to be (is/are, is not/are not, in any tense). The copula determines quality.
4. The quantifier — the word indicating extent: all or some. It determines quantity. In a universal negative, the word no stands in for both the quantifier all and the negative part of the copula.

Every categorical proposition therefore has both quality (affirmative or negative, the only two options) and quantity (universal, the whole class, or particular, some portion of the class).

Euler Circles

The relationships between subject and predicate can be pictured with Euler circles, named after the 18th-century mathematician Leonhard Euler. There are only five possible diagrams:

• A ("All A is B") uses two diagrams: A and B coinciding completely, or A entirely contained within B.
• I ("Some A is B") can be shown in four ways: the two A-diagrams plus partial overlap (resembling a Venn diagram) and B contained within A.
• E ("No A is B") uses a fifth diagram: two wholly separate circles.
• O ("Some A is not B") is already covered by diagrams three through five.

A key insight: because some can mean any amount up to and including all, the same diagrams that illustrate an A proposition also illustrate an I proposition. Thus if all dogs are mammals, it is also true that some dogs are mammals.

Translating Ordinary Sentences into Standard Form

The rest of the episode addresses translation: rendering everyday sentences into standard categorical form, with all four parts clearly distinguished. There are no rigid rules; the guiding principle is to translate the meaning, not the words, by first asking what the sentence actually states.

Singular subjects. A proper noun names a class of one, so it takes the quantifier all: "Bill went to the store" becomes "All Bill...". The whole class (Bill) is the subject.

Missing copula or extra phrases. Standard form requires the verb to be; other verbs must be rewritten, and prepositional phrases or adjectives need a generic noun of the intended class added. "All Bill went to the store" becomes "All (Bill) is (a person who went to the store)." Di Donato suggests compartmentalizing the four parts, using parentheses, and replacing long terms with single-letter placeholders.

Words that signal quantity. Words such as anyone, everyone, whoever, a, the, or not must be replaced with the only acceptable quantifiers: all, some, and no. "Anyone without hair is bald" becomes "All people without hair are people who are bald."

Judgment calls. "The dolphin is an aquatic mammal" most likely refers to the whole class: "All dolphins are aquatic mammals." But "The morning paper was soaked by the rain" probably refers to a single paper: "Some morning papers were papers soaked by the rain."

The ambiguous "all are not." "All dogs are not black" looks like an E proposition ("No dogs are black"), but the speaker usually means not all dogs are black. So it becomes the particular negative "Some dogs are not black." The construction all S is not P often means some S is not P.

Exclusive propositions (using only or none but) translate into A propositions in two steps: add the quantifier all while dropping the exclusive word, then reverse subject and predicate. "None but animals are veterinary patients" becomes "All veterinary patients are animals." An exception arises when only is preceded by the definite article the: "The only person in the pool is Tom" must preserve the meaning that no one else is present, giving "All persons in the pool are persons identical with Tom." The test is whether the translation is logically equivalent to the original (can one be true while the other is false?).

Negative propositions denying an entire class are E types: "Dogs are not humans" becomes "No dogs are humans," moving the negation into the quantifier position.

Exceptive propositions (signaled by except) contain two claims at once. "All police officers except those wearing gas masks passed out from the gas" splits into a universal affirmative ("All non-gas-mask-wearing officers are officers who passed out from the gas") and a universal negative ("No gas-mask-wearing officers are officers who passed out from the gas"). Di Donato uses the prefix non- to mark the relevant subclasses, previewing term complements for the next episode.

Takeaway

Logic is a language of its own, and translation is an essential skill. When no simple rule applies, always determine what the statement means and translate that meaning. The next episode continues categorical logic with the square of opposition and immediate inference.