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)
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.