Consistency

Definition

Consistency means a set of statements does not contain a contradiction.

Basic idea:

No statement P and NOT P are both accepted.

Example consistent set:

P
Q
If P, then Q

Example inconsistent set:

P
NOT P

Consistency means no contradiction.

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Why Consistency Matters

Reasoning depends on stable assumptions.

If a set of premises is inconsistent, it contains conflicting commitments.

Example:

Premise 1: The system is running.
Premise 2: The system is not running.

The premises cannot both be true in the same context.

Inconsistent premises make reasoning unreliable.

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Consistency vs Truth

Consistency does not mean all statements are true.

Example:

The moon is made of cheese.
Cheese is yellow.

This set may be false in content.

But it is not automatically inconsistent unless it contains or implies a contradiction.

Consistency is about compatibility, not factual truth.

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Hidden Inconsistency

Contradictions may not be obvious at first.

Example:

Premise 1: If P, then Q
Premise 2: P
Premise 3: NOT Q

From the first two premises, we infer:

Q

But the third premise says:

NOT Q

So the set is inconsistent.

Inconsistency can be derived, not only stated directly.

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Final Summary

• Consistency means no contradiction.
• A set is inconsistent if it contains or implies P and NOT P.
• Consistency is different from truth.
• A false set can still be consistent.
• Hidden inconsistencies can appear through inference.
• Consistency is required for stable reasoning.