Classical Square of Opposition in Logic: A Study

by Yuvi K - December 16, 2023

Introduction: What is Classical Square of Opposition in Logic?

Classical Square of Opposition is an important concept in logic, which has been studied since antiquity. In the 17th century, logician Ockham developed it into a well-defined logical theory. It is traditionally used for logical operation that involves propositions containing two terms (A and O). These propositions can then be placed into four classes according to the opposition relation that exists between them – A (affirmation), E (exposition), I (infirmation) and O (opposition). The square of opposition summarizes the various properties that the four propositions share, which can then be used by logicians to draw logical conclusions from them.

Definition of Classical Square of Opposition in Logic

The classical square of opposition is a logical device that captures the fundamental logical relations between propositions. It consists of four basic propositions that are arranged in a square, and each proposition stands opposed to its diagonally opposite counterpart. These propositions are “all A is B” (A), “no A is B” (O), “some A are B” (E), and “some A are not B” (I).

For example, if given the proposition “all cats are animals” (A), then the other three propositions ordinate to it in the square would be “no cats are animals” (O), “some cats are animals” (E), and “some cats are not animals” (I).

Properties of Classical Square of Opposition in Logic

The classical square of opposition has certain properties that determine the logical relationship between propositions within it. These properties are as following:

Contradiction

Two propositions are said to be contradictory if their respective truth values are opposite.In the classical square of opposition, this applies to diagonal propositions, such that “all A is B” (A) is contradictory to “no A is B” (O), and “some A are B” (E) is contradictory to “some A are not B” (I).

Contrariety

Contrariety is an opposing relationship between two propositions, such that if either of them is true, the other must be false. In the classical square of opposition, this applies to opposite propositions, such that “all A is B” (A) is contrary to “some A are not B” (I) and “no A is B” (O) is contrary to “some A are B” (E).

Subcontrariety

Subcontrariety is an opposing relationship between two propositions, such that if one of them is false, the other must be true. In the classical square of opposition, this applies to adjacent propositions, such that “all A is B” (A) is subcontrary to “some A is B” (E) and “no A is B” (O) is subcontrary to “some A are not B” (I).

Conclusion:

To sum up, the classical square of opposition is a useful tool in logic that captures the relationship between four basic propositions. Through the use of its properties such as contradiction, contrariety, and subcontrariety, logicians can arrive at logical conclusions based on the propositions contained within it.