## 15-816 Linear Logic |

Martin-Löf popularized the notion of a *judgment* and
*evidence* for a judgment for defining logics and programming
languages. Standard judgment forms are *hypothetical judgments*
and *parametric judgments*. We can use the notion of a judgment
to classify various styles of defining logical systems (restricted
to propositional logic for simplicity).

- If we define the judgment "A is true" without the use of hypothetical judgments we arrive at an axiomatic system in the style of Hilbert.
- If we start from the judgment "A entails B" without the use of hypothetical judgments we arrive at categorical logic.
- If we start from a judgment "A is true" but freely use hypothetical judgments we arrive at natural deduction as devised by Gentzen.
- If we start from two judgments "A is an assumption" and "A is a true conclusion" we arrive at sequent calculus, also due to Gentzen.

We follow natural deduction because of its conceptual simplicity (it requires only one judgment) its proximity to actual mathematical reasoning (Gentzen's original motivation), and because all logical connectives can be defined independently from each other. The last point is particularly salient, since it allows us to easily consider fragments and extensions.

In this lecture we define intuitionistic and classical logic using
natural deduction. The meaning of each logical connective is defined by
introduction and elimination rules. We also introduce the concepts of
*local soundness* and *local completeness* of the
inference rules for a connective, which provide tests whether the
introduction and elimination rules match properly.

- Next: Lecture 03: Linear Natural Deduction
- Previous: Lecture 01: Introduction
- Schedule

Frank Pfenning fp@cs