# Logic and Computation Seminar

## Next session

### Nelson’s logic S

Besides the better-known Nelson logic (N3) and paraconsistent Nelson logic (N4), in 1959 David Nelson introduced, with motivations of arithmetic and constructibility, a logic that he called S. The logic S was originally introduced by means of a calculus (crucially lacking the contraction rule) with infinitely many rule schemata and no semantics (other than the intended interpretation into arithmetic). We look here at the propositional fragment of S, showing that it is algebraizable (in fact, implicative) in the sense of Blok and Pigozzi with respect to a variety of three-potent involutive residuated lattices. We thus introduce the first known algebraic semantics for S as well as a finite Hilbert-style calculus equivalent to Nelson’s presentation; this also allows us to settle the relation between S and the other two above-mentioned Nelson logics.

Note the unusual weekday.

Funded under the scope of UID/EEA/50008/2013.