number-theory

76 6 The correspondence with implication6A2 Definition Intuitionist logic (or more precisely, its implicational fragment) hasthe following two rules:Q-+T a(-+E) : (-+I) : [o]io.-+T.Each application of (-+I) is said to discharge (or cancel) some, all, or none of theoccurrences of a above i, and must be accompanied by a discharge label that listsall the occurrences of a it discharges (see below for details). If none are dischargedthe application of (-+I) is called vacuous. Discharged occurrences of a at the topsof branches must be marked by enclosing them in brackets.1Before formally defining deductions and proofs in this system it will be helpfulto look at some examples of how the rules are used. In these examples a positionnotationlike that in 9A1-4 will be included in the deductions to allow the dischargelabels to say precisely which occurrences have been discharged. The position of eachformula will be written in parentheses beside it, and discharge labels will be writtenbetween braces "{ }".6A2.1 Example A proof of (a--+a-->c)-+a-+c.[a-+a--+c] (0011) [a] (0012) (-+E)a--+c (001) [a] (002)c (00)(-+E)(-+I)a-+c (0) {discharging a at 0012, 002}(-+I)(a--+a-+c)--+a.-a-+c (0) {discharging a-+a-+c at 001116A2.2 Example A proof of (a-+a-+c)-+a--+a-+c.[a-+a-+c] (00011) [a] (00012a--+c (0001) [a] (0002)c (000)(-+E)(-+I)a--+c (00)(-+I){discharging a at position 00012}(a--+a--+c) (0) {discharging a at 0002}(-+I)(a-+a-+c)--+a-+a-*c (0) {discharging a-*a--*c at 0001116A3 Discussion (Partial discharging) The proof in 6A2.2 contains an example ofwhat will be called partial discharging. Suppose rule (-+I) in 6A2 is applied to adeduction of -r in which or has n >_ 1 occurrences as assumptions. DefinitionThe system defined above is often called minimal logic. But we shall later be interested in subsystemsof this system so "minimal" seems inappropriate here.