A Note on Stahl's Opposite System

How to Make a New Logic

“…We remark that Goodman's anti-intuitionistic propositional calculus and my systems SP and SC, in the notion of [11], which are thus the Stahlization of classical propositional and predicate logics, since the former is a consistent system 12 in which not all but some contradictions are derived, and the later is the extreme case of paraconsistent logic, namely the logic in which all theses are inconsitent. 7 Better to say, in the area of the foundation of mathematics, we do not mean however that category theory is the foundation of mathematics.…”

“…The real novelty of Stahl's formulation from the standpoint of axiomatic rejection consists in the axiomatization of all the contradictions free from the notion of provability 9 . In our use of words, Inoué [11] observed that the idea of Stahl is nothing but the Stahlization of classical propositional logic in Gentzen-style. His observation is based on the inversion principle after P. Lorenzen with respect to negation in Gentzen's sequent calculus LK 10 .…”

“…His observation is based on the inversion principle after P. Lorenzen with respect to negation in Gentzen's sequent calculus LK 10 . He could extend Stahl's logic to the first-order predicate level on the basis of the Stahlization (see [11] ). It should be mentioned that I, the author of [11] , also was stimulated by category theory to get the idea 11 .…”

“…He could extend Stahl's logic to the first-order predicate level on the basis of the Stahlization (see [11] ). It should be mentioned that I, the author of [11] , also was stimulated by category theory to get the idea 11 .…”

“…We can find the example of Stahlization in two papers, i.e. Goodman[9] and Inoué[11]. Though[9] is ornamented by many considerations about his4 Rather technical examples are e.g.…”