Relations between the
classical differential calculus and da Costa’s paraconsistent differential
calculus
Itala M. Loffredo D’Ottaviano
Centre for Logic,
Epistemology and the History of Science-CLE
Department of Philosophy
State University of
Campinas-UNICAMP)
itala@cle.unicamp.br
In 2000, da Costa
introduced a paraconsistent differential calculus, whose underlying
paraconsistent predicate calculus and set theory are, respectively, his known
paraconsistent predicate calculus with equality C1̿ and the paraconsistent set theory CHU1, introduced by da
Costa in 1986. Two special algebraic structures, the hyper-ring A and the quasi-ring A*, which extend the field
R of the standard real numbers, are constructed: the elements of A and A*
are called hyper-real numbers, generalized real-numbers, or simply g-reals.
From A*, da Costa proposes the
construction of a paraconsistent differential calculus ℙ, whose language is the
language L of C1̿, extended to the language of CHU1, in which we deal with the
elements of A*.
Carvalho (2004)
studies and improves the calculus proposed by da Costa, presenting the definitions of hyper-real function, derivative and integral
of a hyper-real function. Extensions of several fundamental theorems of the
classical differential calculus are obtained and some applications of these
results are also presented, by using the paraconsistent apparatus.
In this paper,
motivated by works by Robinson, Zakon, Stroyan and Luxembourg, we introduce the
concepts of paraconsistent super-structure and monomorfism between paraconsistent super-structures. We prove a Transference Theorem, which “translates” the
classical differential calculus into da Costa’s paraconsistent calculus.
CARVALHO,
T.F. On da Costa’s paraconsistent differential calculus, 2004. 200p. (Doctorate Thesis). Campinas, State University of Campinas, 2004.