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24.7 The “Modern” Square of Opposition

So far you’ve learned Aristotle’s method for categorizing syllogisms.

The beauty of it is that it’s exhaustive: there are only 256 different possible categorical syllogisms, and exactly 15 of them are valid.

So we can evaluate any syllogistic argument by seeing whether or not it’s one of the valid forms.

But there are two important clarifications you need to know about.

Aristotle thought that his system of category logic exhausted all of logic.

First, Aristotle thought that his system of category logic exhausted all of logic.

He thought that any sentence could be paraphrased as one of his forms. And he thought that any valid inference had to be one of his syllogisms (or one of the derivative inference he identified).

For a long time history agreed with Aristotle: for two thousand years everyone thought that Aristotle had started and finished logic. There was nothing else to do but memorize Aristotle’s forms and syllogisms.

Medieval logicians even gave Aristotle’s syllogisms names based on the vowels. For example, “Barbara” was AAA, and “Darii” was AII. And they made up poems to help them remember which syllogisms were valid in which figures.

We now know that not all valid inferences can be modeled in Aristotle’s system.

However, we now know that he was wrong about that: not all valid inferences can be modeled in Aristotle’s system.

For example, the disjunctive syllogism (PvQ and ~P entail Q) depends on truth functional logic. Truth tables will show you that the disjunctive syllogism is valid, but Aristotle’s categorical syllogisms won’t.

That is one reason why you are learning a full first-order logical system (FOL) in this book instead of just focusing on Aristotle’s logic. We now know that FOL is far more powerful than Aristotle’s logic.

Aristotle did not believe in vacuously true generalizations, but modern logicians do.

The second clarification is another point of disagreement between modern logicians and Aristotle.

Recall vacuous generalizations from Section 23.3: “All Ps are Q” is trivially true whenever there are no Ps.

Aristotle disagreed: he thought that “All Ps are Q” entails “Some Ps are Q.”

The Existential Assumption: belief that “All Ps are Q” entails “Some Ps are Q.”

Since “All Ps are Q” entails the existence of Ps, according to Aristotle, this is called the existential assumption.

Aristotle believed the existential assumption.

Because Aristotle believed the existential assumption, he thought that there were more valid syllogisms than we do. Besides the 15 valid syllogisms we mentioned, Aristotle thought there were 9 additional valid ones.

We are only telling you this in case you encounter Aristotle’s logic in other sources and don’t understand why they think there are 24 (15+9) valid syllogisms. You don’t need to learn these additional facets of Aristotle’s logic, because they are not important from a modern point of view.

From the modern perspective, the only entailments that hold between individual forms are the contradictories you learned in Section 24.4.

  • “All Ps are Q” entails the negation of “Some Ps are not Q” (and vice versa).
  • “No Ps are Q” entails the negation of “Some Ps are Q” (and vice versa).

It is common to represent these facts in a square, called the “Modern” Square of Opposition.

Since Aristotle believed in additional entailments, his view is captured by a more complicated diagram often called the “Traditional” Square of Opposition, which you do not need to know. In our first-order logical system, only the “modern” square is relevant.