Relations between the
classical differential calculus and da Costa’s paraconsistent differential
calculus

**Itala M. Lo****ﬀ****redo 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 C_{1}_{̿}_{ }and the paraconsistent set theory CHU_{1}, 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 C_{1}_{̿}, extended to the language of CHU_{1}, 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 *monomor**ﬁ**sm* 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 di**ﬀ**erential calculus*, 2004. 200p. (Doctorate Thesis). Campinas, State University of Campinas, 2004.