UGC NET Study Notes on Logical Reasoning

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In term logic, the square of opposition is a diagram representing the relations between the four basic categorical propositions.
The origin of the square can be traced back to Aristotle making the distinction between two oppositions: contradiction and contrariety.
However, Aristotle did not draw any diagram.
This was done several centuries later by Apuleius and Boethius.

Categorical proposition

In traditional logic, a proposition is a spoken assertion, not the meaning of an assertion, as in modern philosophy of language and logic.
A categorical proposition is a simple proposition containing two terms, subject (S) and predicate (P), in which the predicate is either asserted or denied of the subject
Every categorical proposition can be reduced to one of four logical forms, named A, E, I, and O.
The categorical proposition, in syllogistic or traditional logic, a proposition or statement, in which the predicate is, without qualification, affirmed or denied of all or part of the subject.
Thus, categorical propositions are of four basic forms: “Every S is P,” “No S is P,” “Some S is P,” and “Some S is not P.”

Universal statements are contraries: 'every dog is cat' and 'no dog is cat' cannot be true together, although one may be true and the other false, and also both may be false (if at least one dog is cat, and at least one dog is not cat).

Particular statements are subcontraries. 'Some dog is cat' and 'some dog is not car' cannot be false together.

The particular statement of one quality is the subaltern of the universal statement of that same quality, which is the superaltern of the particular statement because in Aristotelian semantics 'every A is B' implies 'some A is B' and 'no A is B' implies 'some A is not B'.

Contradictory opposition is the relation between two proposition having the same subject but differs in both quality and quantity. The relation between A and O, E and I are called contradictory. In order to refute the truth of the proposition ‘All dogs are cat, it would be enough to show that some dog (or even one dog) are not cat. One exception would disprove the truth of the universal affirmative proposition.