Proof theory · Definition
Consistency
Before you start you need to know
Consistency is a property of a formal system: it must not prove a formula and its negation at the same time.
Definition
A system \(\mathcal{S}\) is consistent if there is no formula \(\varphi\) such that \(\mathcal{S}\) proves both \(\varphi\) and \(\neg\varphi\).
A system is consistent if it does not allow proving a formula and its negation simultaneously. Consistency describes the behavior of the whole system, not the difficulty of a single proof.
In an inconsistent classical theory the principle of explosion appears: from a contradiction any proposition can be derived. That is why a theory that proves everything loses its power to distinguish consequences.
Consistency can be studied by building models or by comparing the theory with another whose consistency is already known.
Think about it
Why is proving \(P\) and \(\neg P\) serious in a classical theory?
Because the contradiction can allow proving any proposition through the principle of explosion.
Exercises 10
- A system is consistent if...
- Which situation signals an inconsistency?
- If a system has contradictory axioms, it may happen that...
- Consistency is a property of...
- To study consistency one asks whether the system can prove...
- A system is inconsistent if it proves...
- In an inconsistent classical theory the principle of... can appear
- Consistency asks whether there is a simultaneous derivation of...
- Which statement about consistency is correct?
- An inconsistent theory is problematic because...
What comes next
References
- Frederic P. Schuller, Lectures on the Geometrical Anatomy of Theoretical Physics · Lecture 1, Introduction / Logic of Propositions and Predicates, YouTube, 2015.