15816 Linear Logic
Lecture 2: Linear Natural Deduction
We discuss the foundation of logic in the tradition of Gentzen,
Prawitz, and MartinLöf 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.
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Frank Pfenning
