Definition:Natural Deduction/Proof Rule

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Definition

A proof rule is a rule in natural deduction which allows one to infer the validity of propositional formulas from other propositional formulas.


Rule of Substitution

Let $S$ be a sequent that has been proved.

Then a proof can be found for any substitution instance of $S$.


Rule of Sequent Introduction

Let the statements $P_1, P_2, \ldots, P_n$ be conclusions in a proof, on various assumptions.

Let $P_1, P_2, \ldots, P_n \vdash Q$ be a substitution instance of a sequent for which we already have a proof.



Then we may introduce, at any stage of a proof (citing SI), either:

The conclusion $Q$ of the sequent already proved

or:

A substitution instance of such a conclusion, together with a reference to the sequent that is being cited.


This conclusion depend upon the pool of assumptions upon which $P_1, P_2, \ldots, P_n \vdash Q$ rests.


Rule of Theorem Introduction

We may introduce, at any stage of a proof (citing TI), a theorem already proved, or a substitution instance of such a theorem, together with a reference to the theorem that is being cited.


Structure of a Proof Rule

A proof rule has a structure, as follows:

  • Definition: This specifies what the proof rule actually does. Note the careful use of can and may in the definition:
  1. Can implies that it is possible to achieve something based on the structure of the system which is being constructed.
  2. May implies that this is what this particular proof rule is allowing you to do.
  • Abbreviation: When deriving a sequent, it is convenient to use a precisely defined shorthand to indicate which rule is being applied at a particular point. While important when used in a medium where space is limited, for example in printed books, on $\mathsf{Pr} \infty \mathsf{fWiki}$ these are not so much relied upon.
  • Deduced from: The truth value of a result which is being deduced by a particular proof rule depends on a specific set of premises, or assumptions made during the course of derivation. This pool of assumptions will vary depending on what the proof rule is and what previously derived result or results the proof rule depends on.
  • Discharged assumptions: This specifies which, if any, assumptions have been discharged, that is, no longer contribute to the truth value of the conclusion being derived.
  • Depends on: This specifies the result from which the proof rule directly derives its result.


Also known as

Other terms which mean the same thing as proof rule when used in the context of logic are:

rule of derivation
rule of proof
transformation rule
inference rule or rule of inference

However, these may have subtly different meanings when used in different contexts, so when discussing logic, the term proof rule is recommended.


Example

An example of a proof rule is:

If $p$ is a theorem, and $p \implies q$ is a theorem, then $q$ is a theorem.

which expresses Modus Ponendo Ponens.


Sources