On the meaning of logical rules II : multiplicatives and additives

El método inicial que imagino era relativamente sencillo. Conocer bien el español, recuperar la fe católica, guerrear contra los moros o contro el turco, olvidar la historia de Europa entre los años de 1602 y de 1918, ser Miguel de Cervantes. Pierre Ménard estudió ese procedimiento (sé que logró un manejo bastante fiel del español del siglo diecisiete), pero lo descartó por fácil. [. . . ] Mi complaciente precursor no rehusó la colaboración del azar : iba componiendo la obra inmortal un poco à la diable, llevado por inercias del lenguaje y de la invención. Yo he contráı el misterioso deber de reconstruir literalmente su obra espontánea. Mi solitario juego está gobernado por dos leyes polares. La primera me permite ensayar variantes de tipo formal o psicológico ; la secunda me obliga a sacrificarlas al texto original y a razonar de un modo irrefutabile esa aniquilación. J.-L. Borges Pierre Ménard autor del Quijote, 1939. The paper expounds the solution to our search for meaning [3] in a particular case : the fragment of logic built from the neutral elements ⊥,>,1,0 by means of the connectives & ,&,⊗,⊕. As explained in the previous paper and summarized in the abstract, the task is to produce a trivial meet between syntax and semantics, but in a non-trivial way. Typically, we cannot content ourselves with a plain game-theoretic paraphrases of logic ; we have to deconstruct familiar syntax and to reconstruct it in another way. But, as in the case of Ménard reconstructing the Quijote, the synthesis is absolutely prior to the analysis. . . in particular the presentation analysis/synthesis that we follow is completely alien to the very spirit of the work. We shall organize a sort of tunnel between syntax and semantics, with a meet in between : the analysis will replace syntax with a game-theoretic variant, and synthesis will elaborate an abstract notion of game, which can be specialized to our fragment and yield exactly the analytic games. Analysis We shall modify the extant sequent calculus into a hypersequentialized version HC, which has internalized the alternation positive/negative to the extent that even negative formulas disappear. . . the usual connectives being replaced with synthetic connectives. The configurations of HC can be fully analyzed by means of various game-