User:Tizio/Natural deduction

Box rules

 * one can open a box everywhere
 * first formula is an arbitrary assuption
 * a box can be closed in every point, provided all boxes it contains are closed

Inference rules

 * an inference rule take a number of formulae and boxes and produce a single formula
 * can be applied on formulae and boxes that occur already but not (only) in closed boxes

Introduction and elimination
Most rules are paired: some introduce a connective and some eliminates it. Meaning that some rules take a formula with a connective and produce a related formula without it, some other take some formulae and produce a new one with that connective.

Classical logic may require a non-symmetrical rule.

First-order
Natural deduction is not a refutation procedure. It tries to prove validity directly. In particular, the initial formulae are to be considered as assumptions, so if one wants to prove $$A \rightarrow B$$, placing $$A$$ at the beginning is essentially a start-box assumption that $$A$$ is true.

When a formuila $$B$$ is produced at some point, that means that $$A_1,\ldots,A_n \rightarrow B$$ is valid, where $$A_1,\ldots,A_n$$ are the assumptions that are active at that point. In particular, if $$B$$ contains a constant/variable that is not in the assumptions $$A_1,\ldots,A_n$$, then:


 * since $$A_1,\ldots,A_n \rightarrow B$$ is valid, it holds that "$$\forall c$$"$$. A_1,\ldots,A_n \rightarrow B$$. Since the assumption do not contain $$c$$, that implies $$A_1,\ldots,A_n \rightarrow$$"$$\forall c.$$"$$ B$$, that is $$A_1,\ldots,A_n \rightarrow \forall x . B[x/c]$$
 * in the other way around, a formula with a new constant $$c$$ can only be introduced if the formula is known to be valid for every possible value of the parameter