Paraconsistency and consistency understood as the absence of the

I
JOSÉ M. MÉNDEZ, GEMMA ROBLES, FRANCISCO SALTO, Paraconsistency and
consistency understood as the absence of the negation of any implicative theorem.
Universidad de Salamanca. Campus Unamuno, Edificio FES, 37007, Salamanca, Spain.
E-mail: sefus@usal.es.
URL Address: http://web.usal.es/~sefus.
Dpto. de Hist. y Filosofı́a de la CC, la Ed. y el Lenguaje. Universidad de La Laguna.
Campus de Guajara, Facultad de Filosofı́a, 38071, La Laguna, Tenerife, Spain.
E-mail: gemmarobles@gmail.com.
URL Address: http://webpages.ull.es/users/grobles.
Dpto. de Psicologı́a, Sociologı́a y Filosofı́a, Universidad de Len, Campus Vegazana,
24071, Len.
E-mail: francisco.salto@unileon.es.
URL Address: http://www3unileon.es/personal/wwdfcfsa/web/html.
As is stated in its title, in this paper consistency is understood as the absence of
the negation of any implicative theorem. Then, we define the basic logic adequate to
this concept of consistency in the ternary relational semantics with a set of designated
points, negation being modelled with the Routley Star (see., e.g., [1]). Next, a series
of logics extending this basic one is defined. All logics in this paper are paraconsistent,
but none of them is relevant.
[1] R. Routley et al., Relevant Logics and their Rivals vol. 1, Atascadero,
CA, Ridgeview Publishing Co., 1982.
Acknowledgements. Work supported by research projects FFI2008-05859/FISO and
FFI2008-01205/FISO, financed by the Spanish Ministry of Science and Innovation. G.
Robles is currently a Juan de la Cierva researcher at the University of La Laguna.