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Mental Models

Categorical Logic

beginner3 min readlogic · philosophy · mathematics

Aristotle's categorical logic is a system of reasoning based on four types of categorical claims (A, E, I, O) about categories and their members, using syllogisms to draw valid inferences.

THE IDEA

Categorical Logic

In his Prior Analytics, Aristotle (384–322 BCE) laid out a system of logic that would dominate Western and much of Middle Eastern thought for some 2,000 years. Aristotle’s logic concerns categories, their members, and what follows from claims about them. Aristotle was able to capture these relationships in four categorical claims, which have come to be known as:

A Claim: All A are B.
All A are B.
All of the members of Category A are members of Category B.
Examples:
"All cats are mammals."
"All Christians are monotheists."
"All bachelors are unmarried males."

E Claim: No A are B.
No A are B.
None of the members of Category A are members of Category B.
Examples:
"No cats are dogs."
"No Christians are atheists."
"No bachelors are married males."

I Claim: Some A are B.
Some A are B.
At least one member of Category A is a member of Category B.
Examples:
"Some cats are black."
"Some Christians are Catholics."
"Some bachelors are bald."

O Claim: Some A are not B.
Some A are not B.
At least one member of Category A is not a member of Category B.
Examples:
"Some cats are not black."
"Some Christians are not Catholics."
"Some bachelors are not bald."

The fundamental instrument of reasoning in Aristotle's system of logic is the syllogism, which he defines in the Prior Analytics as "discourse in which, certain things being stated, something other than what is stated follows of necessity from their being so." In its simplest form, a syllogism is an argument composed of exactly two premises and one conclusion, the claims of which are A, E, I, and/or O claims. The following are examples of properly formed syllogisms that should be intuitively obvious:

  1. All cats are mammals.

  2. All mammals are warm‑blooded animals.

  3. All cats are warm‑blooded animals.

  4. No Christians are atheists.

  5. All Catholics are Christians.

  6. No atheists are Catholics.

  7. Some cats are black.

  8. All black things are harder to see at night.

  9. Some cats are harder to see at night.

  10. No Republicans are Democrats.

  11. Some Republicans are socially liberal people.

  12. Some socially liberal people are not Democrats.

  13. All men are mortal.

  14. Socrates is a man.

  15. Socrates is mortal.

When to use it

  • Analyzing political arguments that involve categorical statements, such as 'All politicians are corrupt' or 'No conservatives care about the environment.'

  • In mathematics, using set relationships to prove theorems about subsets and complements.

  • Evaluating scientific classifications: checking whether all members of a category share a specific property and testing for counterexamples.

FROM THEORY TO EVERYDAY LIFE

In practice

01

All cats are mammals

  1. All cats are mammals.
  2. All mammals are warm‑blooded animals. Therefore, all cats are warm‑blooded animals.
02

Socrates is mortal

  1. All men are mortal.
  2. Socrates is a man. Therefore, Socrates is mortal.
✳

A MOMENT TO REFLECT

Put it to work.

  1. Can you identify the categorical claim type (A, E, I, or O) in an argument you encounter today?

  2. How might categorical logic help you evaluate the validity of arguments about group membership?

  3. Create your own valid syllogism using at least one A, E, I, or O claim.

References

  1. Prior Analytics — Aristotle (-350)

CONTINUE THE CONVERSATION

Source texts

  • Bad Arguments 100 of the Most Important Fallacies in Western Philosophy (Steven Barbone, Michael Bruce)
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