Unit 3: Immediate Inference

Table of Contents

What is Immediate Inference?

An inference is the logical process of drawing a conclusion from premises.
In Aristotelian logic, inferences are divided into two types:

  1. Mediate Inference: An inference where the conclusion is drawn from two or more premises. (e.g., a Categorical Syllogism, covered in Unit 4).
  2. Immediate Inference: An inference where the conclusion is drawn from only one premise.

This unit deals with the three most important types of immediate inference: Conversion, Obversion, and Contraposition. These are "truth-preserving" operations, meaning if the original premise is true, the (validly inferred) conclusion must also be true. They are ways of creating a new, logically equivalent proposition.

Key Concept: Logical Equivalence. Two statements are logically equivalent if they *always* have the same truth value. They mean the same thing, just expressed in a different form. Our goal is to find valid operations that produce equivalent statements.

Conversion

Conversion is an immediate inference that proceeds by swapping the Subject (S) and Predicate (P) terms of the original proposition.

The original proposition is the Convertend. The inferred conclusion is the Converse.

Rule: The quality (Affirmative/Negative) must stay the same. Crucially, no term can be distributed in the converse if it was not distributed in the convertend.

Proposition Convertend (Premise) Distribution Converse (Conclusion) Validity
A All S are P S (D), P (U) All P are S INVALID
(This is called Conversion by Limitation)

Explanation: In "All S are P," P is undistributed. If we swap them to "All P are S," we are now distributing P (Rule: A-prop distributes its subject). This violates the main rule.
Example: "All dogs are mammals" (True) → "All mammals are dogs" (False).
Valid Conversion by Limitation: We can *validly* infer an I proposition: "All S are P" → "Some P are S". (If "All dogs are mammals," then "Some mammals are dogs").
E No S are P S (D), P (D) No P are S VALID (Simple Conversion)

Explanation: Both terms are distributed, so swapping them is perfectly fine.
Example: "No dogs are reptiles" (True) → "No reptiles are dogs" (True).
I Some S are P S (U), P (U) Some P are S VALID (Simple Conversion)

Explanation: No terms are distributed, so no risk of violating the rule.
Example: "Some students are athletes" (True) → "Some athletes are students" (True).
O Some S are not P S (U), P (D) Some P are not S INVALID

Explanation: In "Some S are not P," S is undistributed. When we swap them to "Some P are not S," the S term moves to the predicate position, where it becomes distributed (Rule: O-prop distributes its predicate). This violates the main rule.
Conclusion: The O-proposition has no valid conversion.

Obversion

Obversion is an immediate inference that involves two steps:
  1. Change the Quality (from affirmative to negative, or negative to affirmative).
  2. Replace the Predicate term (P) with its complement (non-P).
The complement of a class is everything outside that class. (e.g., Complement of "dogs" is "non-dogs").

The original proposition is the Obvertend. The inferred conclusion is the Obverse.

Rule: Obversion is valid for all four proposition types (A, E, I, O).

Proposition Obvertend (Premise) Obverse (Conclusion)
A All S are P No S are non-P
Example: "All citizens are voters" → "No citizens are non-voters." (Logically equivalent).
E No S are P All S are non-P
Example: "No dogs are reptiles" → "All dogs are non-reptiles." (Logically equivalent).
I Some S are P Some S are not non-P
Example: "Some students are athletes" → "Some students are not non-athletes." (Logically equivalent).
O Some S are not P Some S are non-P
Example: "Some students are not athletes" → "Some students are non-athletes." (Logically equivalent).

Contraposition

Contraposition is an immediate inference that involves two steps:
  1. Replace the Subject (S) with the complement of the Predicate (non-P).
  2. Replace the Predicate (P) with the complement of the Subject (non-S).

A simpler way to remember this is: First Obvert, then Convert, then Obvert again. (But the rule above is the definition).

The original proposition is the Premise. The inferred conclusion is the Contrapositive.

Proposition Premise Contrapositive Validity
A All S are P All non-P are non-S VALID
Example: "All dogs are mammals" → "All non-mammals are non-dogs." (True → True. Valid).
E No S are P No non-P are non-S INVALID
(This is called Contraposition by Limitation)
Explanation: "No dogs are reptiles" (True) → "No non-reptiles are non-dogs" (False - a "cat" is a non-reptile, but it is also a non-dog).
Valid Contraposition by Limitation: We can validly infer an O proposition: "No S are P" → "Some non-P are not non-S".
I Some S are P Some non-P are non-S INVALID
Explanation: "Some animals are mammals" (True) → "Some non-mammals are non-animals" (False - a "rock" is a non-mammal, but it is also a non-animal).
Conclusion: The I-proposition has no valid contrapositive.
O Some S are not P Some non-P are not non-S VALID
Example: "Some animals are not dogs" (True) → "Some non-dogs are not non-animals" (True - a "cat" is a non-dog, and it is *not* a non-animal... i.e., it is an animal).

Summary Table of Inferences

Exam Tip: Memorize this summary table. It shows all valid immediate inferences.

Proposition Valid Conversion Valid Obversion Valid Contraposition
A: All S are P Some P are S (by Limitation) No S are non-P All non-P are non-S
E: No S are P No P are S All S are non-P Some non-P are not non-S (by Limitation)
I: Some S are P Some P are S Some S are not non-P (None)
O: Some S are not P (None) Some S are non-P Some non-P are not non-S