Proof theory · Definition
Rule of inference
logicproofs
Before you start you need to know
Rules of inference let us move from premises to conclusions in a verifiable way.
Definition
A rule of inference authorizes the formal step from the premises \(\varphi_1,\ldots,\varphi_n\) to the conclusion \(\psi\).
\[\frac{\varphi_1,\ldots,\varphi_n}{\psi}\]
A rule of inference authorizes passing formally from some premises to a conclusion. Its job is to control when the step preserves logical validity.
A true conclusion is not enough for the step to be correct: it must also follow from the premises through an allowed rule.
Rules turn a proof into a verifiable sequence, instead of leaving it as a collection of intuitions or plausible statements.
Think about it
What must justify each step of a proof?
How it is obtained from premises, axioms or earlier steps using an allowed rule.
Exercises 10
- A rule of inference indicates...
- A rule of inference normally relates...
- A conclusion in a formal proof must be accompanied by...
- In a formal proof it is not allowed to...
- Rules of inference help separate...
- A valid rule of inference preserves...
- A rule of inference typically has...
- Applying a rule without its premises holding is...
- Which of these options is a valid rule of inference that obtains \(\psi\) from \(\varphi \to \psi\) and \(\varphi\)?
- A conclusion can be true and yet the step invalid if...
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.