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  Aristotelian Logic: The Logic of Categories

[Please note: the description I’m giving here does not reflect Aristotle’s historical motivations and interpretation of the logical system that he developed, which is tied very strongly to his broader philosophical worldview. What I’m describing here, rather, is a standard interpretation of Aristotelian logic as it is presented in modern symbolic logic texts.]

Aristotelian logic is the logic of classes, or categories — hence, it is often called “categorical logic”.

Or rather, it’s the logic of statements that can be represented in terms of classes of things, and relationships between those classes.

For example,the natural language statement “All cows are mammals” would be represented as a relation between the class of cows and the class of mammals (namely, that the class of cows is a subset of the class of mammals, or equivalently, that all members of the class of cows are also members of the class of mammals).

Aristotelian logic gives us tools for representing claims of the following form, which are called “categorical statements”:

  • All S are P
  • Some S are P
  • No S are P
  • Some S are not-P

These reflect the subject-predicate structure of a wide class of statements, but certainly not all statements.

A categorical syllogism is an argument consisting of exactly three categorical statements (two premises and a conclusion) in which there appear a total of exactly three categorical terms, each of which is used exactly twice.

Examples:

1. All humans are mammals.
2. Some builders are human.
Therefore, some mammals are builders.

1. No geese are felines.
2. Some birds are geese.
Therefore, some birds are not felines.

Aristotle examined all the logically distinct types of syllogisms that could be created using the basic categorical statements, and identified which are deductively valid and which are invalid. (There are fourteen valid forms in Aristotle’s system. Medieval logicians gave them all names.)

So, what is the fragment of natural language whose logical structure Aristotelian logic is able to model?

It’s the fragment of natural language that involves statements with a two-place subject-predicate structure, where the subject and the predicate terms can be represented as classes of objects, and where the logical relations between statements are determined by relations of inclusion, exclusion, and overlap among the classes.

That’s it.

One of the more fun activities you do in a symbolic logic class is learn how to translate natural language expressions into the logical syntax of a particular logical system. Informally we call this translation from English (say) to “logicese” (by analogy with “Chinese”, “Japanese”, etc.).

For linguistics majors, the main benefit of working through the exercises like this is that they help to make you aware of grammatical and logical features of language that you wouldn’t otherwise pay attention to.

For example, it’s easy to translate “Some teenagers are McDonald’s employees” because both the subject and predicate terms are plural nouns — each denotes a class of objects. That’s what categorical logic was set up to represent.

Plural nouns can function either as subject or predicate terms in a categorical statement. “Some McDonald’s employees are teenagers” is just as grammatically well-formed as “Some teenagers are McDonald’s employees”.

But now consider this statement: “Some teenagers are disorganized”.

If you switch the subject and predicate terms you get this: “Some disorganized are teenagers”. This is no longer a grammatically well-formed sentence.

The problem is that the word “disorganized” is an adjective, not a noun. Adjectives are used to modify nouns, they can’t normally stand alone.

To write this in the “logicese” of Aristotelian logic, you need to rewrite the adjective as a plural noun, like so:

“Some disorganized people are teenagers.”

Now the subject term denotes a class of people, and not just a bare adjective.

More examples of translation exercises from English into the logical syntax of Aristotelian logic:

  • “Trespassers will be prosecuted.”
    • = “All trespassers are people who will face prosecution.”
  • “Some assembly required.”
    • = “Some parts of this item are parts that need assembling.”
  • “No pain, no gain.”
    • = “No exercise routines without physical pain are exercise routines offering physical gain.”

Fun, right? They can be awkward and robotic, sure, but these translations do offer a certain kind of clarity about what is actually being asserted.

Aristotelian logic has been developed in ways that allow us to translate many kinds of natural language statements into statements about relationships between classes. A full unit on this topic would show you how to translate statements like the following:

  • “Socrates is human.”
    • (The subject term refers to an individual)
  • “Wherever there is smoke, there is fire.”
    • (The subject term employs an adverb phrase)
  • “There are sharks in the local aquarium.”
    • (The subject term has an implied quantifier — some sharks, not all sharks)
  • “Not every novel about romance is interesting.”
    • (Nonstandard quantifier)
  • “Only persons with tickets can enter the arena.”
    • (Exclusive proposition)
  • “Pizza is a healthy meal if it has vegetable toppings.”
    • (Conditional relationships)

Aristotelian logic is powerful and remained the dominant system of logic taught in the universities for 2400 years. But as a theory of deductive reasoning it has important limitations. Aristotle treats some argument forms as valid that we would not today, and it doesn’t include argument forms that we would recognize as valid.

One of the ironies of intellectual history is that while Euclidean geometry is the greatest achievement of Greek deductive science, and Aristotle’s theory of the syllogism was intended as a theory of deductive proof, the theory of the syllogism cannot account for even the simplest proofs in Euclidean geometry.

There are several reasons for this, but a primary reason is that the propositions of geometry are not all of a simple subject-predicate form. In fact, rather few of them are. Instead, geometrical proofs deal with relations among objects, and Aristotle’s system is not designed to model relations. Also, the propositions of geometry do not all have just one quantifier; they may essentially involve repeated uses of “all” and “there exists”.

There are other reasons, but the upshot is that even simple mathematical expressions and mathematical proofs can’t be represented in Aristotelian logic, and this is due to the expressive limitations of the system — it only models a fragment of natural language and natural language reasoning.