# 15-816 Linear Logic

## Lecture 02: Natural Deduction

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.

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Frank Pfenning
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