Differences between Derivable rules and Admissible rules?

Question

Can anyone help me to explain and give examples for this question?

"There are two ways to introduce new reference rules: as derivable rules and as admissible rules. What are the differences between them?

Thanks.

Answer 1

The Admissibility and derivability wiki says:-

A derivable rule is one whose conclusion can be derived from its premises using the other rules. An admissible rule is one whose conclusion holds whenever the premises hold. All derivable rules are admissible. To appreciate the difference, consider the following set of rules for defining the natural numbers

You may check out the wiki which says:-

Every derivable rule is admissible, but not vice versa in general. A logic is structurally complete if every admissible rule is derivable, i.e., {\vdash }={\,|!!!\sim }.[5] In logics with a well-behaved conjunction connective (such as superintuitionistic or modal logics), a rule A_{1},\dots ,A_{n}/B is equivalent to A_{1}\land \dots \land A_{n}/B with respect to admissibility and derivability. It is therefore customary to only deal with unary rules A/B.