# 15-816 Linear Logic Lecture 2: Linear Natural Deduction

We discuss the foundation of logic in the tradition of Gentzen, Prawitz, and Martin-L—f by introducing the basic notion of a judgment. The most important judgment is the truth of a proposition, where the meaning of a proposition is given by the rules used to infer it. This leads to a system of natural deduction that characterizes each connective by introduction and elimination rules that must match by satisfying at least local soundness and completeness properties.

In linear logic there is an additional concept, that of a linear hypothetical judgment. This modifies the more traditional notion of a hypothetical judgment by requiring that each linear hypothesis be used exactly once. Linear hypothetical judgments can be characterized by a hypothesis rule and a substitution principle that guide the design of the connective of linear logic.

We then develop the rules for simultaneous conjunction, alternative conjunction, additive truth, multiplicative truth, and linear implication and show that they are locally sound and complete.