Questions: Consistency and Inconsistency of Theories

Question 1

A new formal axiomatic theory is shown to be inconsistent. What follows about the theorems provable in that system?

• A. Some theorems are unprovable — the system has gaps
• B. The system is incomplete but may still be useful for some purposes
• C. Every formula is provable in the system, including both a statement and its negation
• D. The system proves no theorems, because the axioms cancel each other out

Answer: (C)

By the principle of explosion (ex falso quodlibet), from any contradiction you can derive any formula whatsoever. An inconsistent system does not prove nothing — it proves everything. Both P and ¬P are theorems. This makes the system useless for distinguishing truth from falsehood. Option A describes incompleteness, which is a different property entirely.

Question 2

Is the set {P → Q, ¬Q, P} consistent?

• A. Yes — no individual formula in the set is itself a contradiction
• B. No — there is no interpretation that satisfies all three formulas simultaneously
• C. Yes — interpretations where P → Q is false satisfy the remaining formulas
• D. It depends on whether P and Q are logically independent

Answer: (B)

If P is true and ¬Q is true (Q false), then P → Q is false (a true antecedent with a false consequent). If P is false, P is not satisfied. No row in the combined truth table makes all three formulas true — the set is inconsistent. Option A is the key misconception: inconsistency is a property of the set as a whole, not of any individual member.

Question 3

A set of formulas can be inconsistent even if none of the individual formulas in the set is itself a contradiction.

• T. True
• F. False

Answer: True

Inconsistency is a collective property: the formulas together impose contradictory demands on an interpretation, even though each individual formula is satisfiable. {P → Q, P, ¬Q} is inconsistent, yet P → Q is satisfiable (make P false), P is satisfiable (make P true), and ¬Q is satisfiable (make Q false). The incompatibility only emerges when you try to satisfy all three simultaneously.

Question 4

An inconsistent theory is useless because it can prove no theorems.

• T. True
• F. False

Answer: False

The opposite is true: an inconsistent theory proves too MANY theorems — in fact, every formula is provable from a contradiction (the principle of explosion). The theory is useless because it proves everything, including contradictions, making it impossible to rely on for sound reasoning. A theory that says everything says nothing meaningful.

Question 5

Why is consistency considered the minimum requirement for a formal theory, rather than merely a desirable property?

Model answer: Because of explosion (ex falso quodlibet): from any inconsistency, every formula is derivable — including both a statement and its negation. A theory that proves everything cannot distinguish true claims from false ones. Consistency is therefore not a bonus feature — it is the precondition for the theory to make any meaningful claims at all. An inconsistent axiom system is not a weak theory; it is no theory.

This is why foundational crises in mathematics (Russell's paradox, Hilbert's program) centered on consistency: if the axioms of arithmetic or set theory were inconsistent, all mathematical reasoning built on them would be worthless. Gödel's incompleteness theorems showed that consistency of a sufficiently powerful system cannot be proved within that system itself — but consistency remains the first requirement, even if unprovable internally.