Summary: Notes on the simply typed lambda
calculus
Peter Aczel
Manchester University
June 16, 1998
Contents
1 Deduction 1­1
1.1 Inference Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1­1
1.1.1 The Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 1­1
1.1.2 Adding extra axioms . . . . . . . . . . . . . . . . . . . . . . . 1­2
1.1.3 Semantics for Inference Systems . . . . . . . . . . . . . . . . 1­2
1.1.4 Formal Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 1­3
1.1.5 Rules of Inference . . . . . . . . . . . . . . . . . . . . . . . . 1­3
1.2 Intuitionistic Implication . . . . . . . . . . . . . . . . . . . . . . . . . 1­4
1.2.1 A Hilbert­style formal system, H . . . . . . . . . . . . . . . . 1­4
1.2.2 Natural Deduction . . . . . . . . . . . . . . . . . . . . . . . . 1­5
1.2.3 Sequent Formulation, ND, of Natural Deduction . . . . . . . 1­7
1.2.4 Normal ND tree­proofs . . . . . . . . . . . . . . . . . . . . . 1­7
1.2.5 Sequent Calculus SC . . . . . . . . . . . . . . . . . . . . . . . 1­8
1.3 Intuitionistic Propositional Logic . . . . . . . . . . . . . . . . . . . . 1­9

Source: Aczel, Peter - Departments of Mathematics & Computer Science, University of Manchester

Collections: Mathematics