Unit 2: Immediate Inference and Square of Opposition

Introduction to Immediate Inference
Conversion
Obversion
Contraposition
Traditional Square of Opposition
Aristotelian and Boolean Squares
Exam Focus: Tips and FAQs

Introduction to Immediate Inference

In logic, inference is the process by which one proposition (the conclusion) is arrived at on the basis of one or more other propositions (premises). Immediate inference is a type of deductive inference where the conclusion is drawn directly from a single premise. Unlike mediate inference (syllogism), there is no middle term used to link concepts.

Conversion

Conversion is an immediate inference produced by interchanging the subject and predicate terms of a categorical proposition.

A-Proposition (All S is P): Converts to Some P is S. This is called "Conversion by Limitation" because we cannot infer "All P is S" without committing a fallacy.
E-Proposition (No S is P): Converts to No P is S. This is a simple conversion.
I-Proposition (Some S is P): Converts to Some P is S. Also a simple conversion.
O-Proposition (Some S is not P): Cannot be validly converted.

Common Pitfall: Never try to convert an O proposition. For example, "Some dogs are not pets" does not mean "Some pets are not dogs."

Obversion

Obversion involves changing the quality of the proposition (affirmative to negative or vice versa) and replacing the predicate term with its contradictory term (complement).

Original Proposition	Obverse Form
A: All S is P	E: No S is non-P
E: No S is P	A: All S is non-P
I: Some S is P	O: Some S is not non-P
O: Some S is not P	I: Some S is non-P

Rule: The truth value of the original and the obverse remains exactly the same.

Contraposition

Contraposition is a complex immediate inference where the subject is replaced by the complement of the predicate, and the predicate is replaced by the complement of the subject.

A: Converts to All non-P is non-S.
O: Converts to Some non-P is not non-S.
E: Converts to Some non-P is not non-S (by limitation).
I: Invalid. No valid contrapositive for I-propositions.

Traditional Square of Opposition

The Square of Opposition describes the logical relationships between the four categorical propositions (A, E, I, O) when they share the same subject and predicate.

Contradictories (A & O, E & I): They have opposite truth values. If one is true, the other must be false.
Contraries (A & E): They cannot both be true, but they can both be false.
Subcontraries (I & O): They cannot both be false, but they can both be true.
Subalternation (A to I, E to O): If the universal is true, the particular must be true.

Aristotelian and Boolean Squares

There is a fundamental difference in how traditional and modern logic interpret the "existence" of the subject.

1. Traditional (Aristotelian) Square

Assumes Existential Import: Universal propositions (A, E) imply that the subject actually exists in the real world. All four relationships (contrary, subcontrary, subaltern, contradictory) are considered valid.

2. Modern (Boolean) Square

Rejects Existential Import for universals: Universal propositions are seen as hypothetical (If S exists, then P). Because of this, only the Contradictory relationship remains valid. Contraries, subcontraries, and subalternation are rejected in the Boolean interpretation.

Exam Focus: Tips and FAQs

Exam Tip: In conversion, remember E and I are simple. A requires limitation. O is impossible.

Common Mistakes

Mistaking contraries for contradictories. Contraries (A/E) can both be false!
Applying Aristotelian subalternation in a Boolean context. In modern logic, "All S is P" does NOT prove "Some S is P".

Frequently Asked Questions

Q: Why can't O-propositions be converted?
A: Because conversion would change the distribution of the terms. In "Some S is not P", P is distributed. If converted to "Some P is not S", S would become distributed, which wasn't the case in the premise.

Mnemonics

Use "C-O-C" to remember the types of immediate inference: Conversion, Obversion, Contraposition.