Da Costa’s Paraconsistent Differential Calculus and a Transference Theorem
Itala M. Loffredo D’Ottaviano1and Tadeu Fernandes de Carvalho2
1 The Group for Applied and Theoretical Logic Centre for Logic, Epistemology and the History of Science
State University of Campinas P.O. Box 6133, 13081-970
Campinas, SP, Brazil [email protected]
2 Mathematics Faculty
Pontific Catholic University of Campinas Campinas, SP, Brazil
Abstract. In this paper, we improve da Costa’s paraconsistent differ- ential calculus, whose underlying set theory and logic are, respectively, da Costa’s paraconsistent set theoryCHU1 and da Costa’s paraconsis- tent predicate calculus with equalityC1=. We present the hyper-ring A and the quasi-ring A∗, that extend the set Rof the real numbers. We introduce some basic definitions and prove extensions of several of the main results of the classical differential calculus. We introduce the con- cept of paraconsistent super-structure, construct special monomorphism between specific super-structures built overRand prove a Transference Theorem from the classical differential calculus into da Costa’s paracon- sistent differential calculus.
Introduction
A logic is paraconsistent if it can be used as the underlying logic to inconsistent but non-trivial theories, which we call paraconsistent theories.
In 1963, da Costa (see da Costa [9,10,12]) introduces his hierarchies of logi- cal calculi for the study of inconsistent but non-trivial theories: the hierarchy of propositional calculiCn,1 ≤n ≤ω, the hierarchy of predicate calculi Cn∗,1 ≤ n≤ω, the hierarchy of predicate calculi with equalityCn=,1≤n≤ω, and the hi- erarchy of calculi of descriptionsDn,1≤n≤ω; he also introduces his hierarchy of inconsistent but apparently non-trivial set theories N Fn,1 ≤ n≤ ω. These systems are well known and related results and topics have been largely studied by several authors (see Arruda [4,5], da Costa and Marconi [15], D’Ottaviano [18,19]).
Motivated by Church’s classical set theoryCHU (see Church [8]), da Costa [11] introduces the new hierarchy of set theories CHUn,1≤n≤ω, also incon- sistent and apparently non-trivial, whose underlying predicate calculi are the corresponding systemsCn=,1≤n≤ω.
In the literature there are paraconsistent reconstructions of the classical dif- ferential calculus, which reflect some of its theoretical and applied aspects.
From this perspective, da Costa proposes a paraconsistent differential cal- culus, as one of the various inconsistent but non-trivial theories that can be developed by using paraconsistent logic and paraconsistent set theory.
After a pre-publication of 1996, da Costa [13] introduces a paraconsistent differential calculus, whose underlying paraconsistent predicate calculus and set theory are, respectively, C1= andCHU1.
Based on the classical set theoryZF, da Costa [13] introduces thering of the hyper-real numbers, denoted byA, and thequasi-ring of the extended hyper-real numbers, denoted by A∗. The algebraic structures A and A∗ are extensions of the fieldRof the standard real numbers; and the elements ofAandA∗are called hyper-real numbers,generalized real-numbers, or simply g-reals.
FromA∗, da Costa proposes the construction of the paraconsistent differen- tial calculus P, whose language is the languageL= of the systemC1= extended to the language ofCHU1, in which we deal with the elements ofA∗.
Carvalho [7] studies and improves the calculus proposed by da Costa. Based on da Costa’s paper, Carvalho introduces definitions for the basic concepts, proves some theorems that generalize important classical results and presents some applications of these results.
In this paper, motivated by Robinson’s non-standard analysis, by Robin- son and Zakon [28] and Stroyan and Luxemburg [30], we introduce the con- cept of paraconsistent super-structure over a non-empty and infinite set X of atoms ofCHU1. The definition ofmonomorphismbetween paraconsistent super- structures is fundamental for the construction and study ofextensionsofsuper- structures.
We prove a Transference Theorem, that “translates” the classical differential calculus into da Costa’s paraconsistent calculus.
As in da Costa’s work, our option by C1= as the underlying logic to our paraconsistent differential calculus is very natural, for it is the underlying logic to the set theoryCHU1, adopted as the underlying set theory to the construction of the differential calculus.
Meanwhile, it would be possible to adopt any other convenient underlying logic, as well as any other underlying set theory, as it is done in Mortensen [25], where different logics are used in non-classical approaches to the differential and integral calculus.
1 Da Costa’s Propositional Paraconsistent Logics C
nThe language L of da Costa’s paraconsistent systems Cn, 1 ≤ n ≤ ω, has as primitive symbols propositional variables, the connectives ¬,∨,∧ and ⊃, and the parentheses (da Costa [9,10,12]).
The notions of formula and theorem, as well as the general conventions and notations, are the standard ones, as in Kleene [24].
LetA andB be formulae. The following operators are added, by definition, to the languageL.
Definition 1. A◦=df ¬(A∧ ¬A).
Definition 2. Ak =df A◦◦...◦ (“◦”k times, for k≥1).
Definition 3. A(k)=df A1∧A2∧. . .∧Ak, for k≥1.
Definition 4. ∼kA=df ¬A∧A(k), for k≥1.
Definition 5. (A↔B) =df (A⊃B)∧(B⊃A).
According to these definitions,A◦ may be read as ‘A is a well-behaved for- mula’ or ‘Ais regular’ and the operator ◦ is usually named ‘ball operator’;Ak may be read as ‘A is ak-times regular formula’; A(k) may be read as ‘A is a well behaved formula of degree k’ or ‘A is a regular formula of degree k’; the symbol ↔corresponds to the usual equivalence; and, for everyCn, 1≤n≤ω, the primitive negation¬is the basic paraconsistent negation of the system, and the connective∼n is called ‘strong negation’.
For eachCn, 1≤n < ω, the schemata ofaxiomsand thededuction rule are the following ones.
Axiom 1:A⊃(B⊃A)
Axiom 2:(A⊃B)⊃((A⊃(B ⊃C))⊃(A⊃C)) Axiom 3:A∧B⊃A
Axiom 4:A∧B⊃B Axiom 5:A⊃(B⊃A∧B) Axiom 6:A⊃A∨B Axiom 7:A⊃B∨A
Axiom 8:(A⊃C)⊃((B⊃C)⊃(A∨B⊃C)) Axiom 9:¬¬A⊃A
Axiom 10: A∨ ¬A
Axiom 11n:B(n)⊃((A⊃B)⊃((A⊃ ¬B)⊃ ¬A)) Axiom 12n:A(n)∧B(n)⊃(A∧B)(n)
Axiom 13n:A(n)∧B(n)⊃(A∨B)(n) Axiom 14n:A(n)∧B(n)⊃(A⊃B)(n) Rule of Modus Ponens (MP): A,A⊃BB .
Finally, thesystemCωis defined by Axiom 1toAxiom 10andMP.
The classical propositional logic may be considered as the systemC0 of this hierarchy. This logic is, in fact, given byCωplus reductio ad absurdum, that is:
Axiom 11: (A⊃B)⊃((A⊃ ¬B)⊃ ¬A).
Following, we briefly present some useful known results concerning da Costa’s paraconsistent systemsCn, important for the development of the next sections.
Theorem 1 (da Costa [9]). All the rules and valid schemata of the classical positive propositional calculus are valid in Cn,1≤n < ω.
Theorem 2 (da Costa [9]). In everyCn,1≤n < ω, the strong negation∼n
has all the properties of the classical negation.
Theorem 3 (da Costa [9]).Every system in the hierarchy Cn,1≤n≤ω, is strictly stronger than those which follow it.
Theorem 4 (da Costa [9] and Alves [1]). Every paraconsistent systemCn, 1≤n≤ω, is consistent and non-trivial.
We observe that the Replacement Theorem, although valid inC◦, is not valid in general in Cn, 1≤n≤ω.
As, in every system Cn, 1 ≤ n ≤ ω, the formulae A ⊃ (¬A ⊃ B) and
¬A⊃(A⊃B) are not valid, da Costa’s systems are paraconsistent systemslato sensu, that is, from a contradiction it is not possible in general to deduce any formula.
1.1 The Systems Cn∗,0≤n≤ω
Analogously to the construction of the hierarchy of paraconsistent propositional calculi, we can construct the corresponding hierarchy of predicate calculiCn∗,0≤ n≤ω, in whichC0∗ is the classical predicate calculus andCn∗ is paraconsistent, for everyn,1≤n≤ω.
Thelanguageof the paraconsistent predicate calculi Cn∗,1≤n≤ω, denoted byL∗, is an extension of the languageLintroduced above, by adding, as usually, for everyn, denumerable families ofm-ary predicate symbols andm-ary function symbols, and the universal (∀) and existential (∃) quantifiers.
The usual notions and notations are as in Kleene [24].
Theaxioms and deduction rules of the system C1∗ are the ones of C1, plus the following ones.
Axiom 15. ∀xA(x)→A(t), witht free forxinA(x) Axiom 16. A(t)→ ∃xA(x), witht free forxinA(x) Axiom 17. ∀x(A(x))◦→(∀xA(x))◦
Axiom 18. ∀x(A(x))◦→(∃xA(x))◦
Axiom 19.Either ifAand B are two congruent formulae, or one of them can be obtained from the other by eliminating vacuous quantifiers, thenA↔B Rule 2. A→∀xB(x)A→B(x) , wherexdoes not occur free inA
Rule 3. ∃xA(x)→BA(x)→B , wherexdoes not occur free inA.
We can introduce the calculiCn∗,0≤n≤ω, as in the case of the hierarchy Cn, 1≤n≤ω.
1.2 The Systems Cn=,0≤n≤ω
By adding the binary equality predicate symbol = to the languageL∗of the sys- temsCn∗,0≤n≤ω, we obtain the language L= of the paraconsistent predicate calculi with equality Cn=,0≤n≤ω.
Theaxiomsanddeduction rulesofC1= are the ones ofC1∗, plus the following ones, with the usual restrictions concerning the variables.
Axiom 20. ∀x(x=x)
Axiom 21. x=y→(A(x)↔A(y)), withy free forxin A(x).
The axioms and rules of Cn=,1 < n≤ ω are the ones of Cn∗,1 < n≤ ω, plus Axioms 20 and 21.
2 Da Costa’s Paraconsistent Set Theory CHU
1The paraconsistent set theory CHU1 may be considered an extension of the set theory CHU of Church [8], that corresponds to the theory CHU0 of da Costa’s hierarchy of set theories CHUn,0 ≤ n ≤ ω. Relationships between CHU1and Quine’s set theoryN F (see [?]), and results on da Costa’s hierarchy of paraconsistent set theories and CHU1 can be seen in da Costa [10], Arruda [2], Arruda and da Costa [6], Rosser [29], Forster [23], da Costa, B´eziau and Bueno [14] and Carvalho [7].
The language ofCHU0is the language of Zermelo-Fraenkel systemZF, with the symbol ofdescription ι, that can be introduced either contextualized or as a primitive symbol of the extended language.
Theaxioms of CHU0 are the Axiom of Extensionality, Axiom of the Pair, andAxiom of the Union ofZF, plus the following ones.
Axiom4CHU0.Axiom of the Intersection
∃v(v∈z)→ ∃u∀x(x∈u↔ ∀y(y∈z→x∈y)).
Axiom5CHU0.Axiom of the Infinite
∃v∀x(x∈v↔xis a finite ordinal).
Axiom6CHU0.Choice Axiom x6= Ø→xhas choice function
Axiom7CHU0.Abstraction Axiom (Separation) wf(v)→ ∃u∀x(x∈u↔((x∈v)∧F(x))).
Axiom8CHU0.Substitution Axiom
(∀x∀y∀z(F(x, y)∧F(x, z) → y = z)∧ ∀x∀y∀z(F(x, z)∧F(y, z) → x = y)∧ ∀y(y∈t↔ ∃xF(x, y))∧wf(t))→ ∃v∀x(x∈v↔ ∃yF(x, y)).
Axiom9CHU0.Axiom of the Power Set wf(v)→ ∃u∀x(x∈u↔x⊆v).
Axiom10CHU0. Axiom of the Complement
∃u∀x(x∈u↔x6∈z).
In what follows, we present some basic definitions ofCHU0. Definition 6.
(i)X=def {y:¬(y∈x)}
(ii)ι0x=def {x}
(iii){x1, x2, . . . , xn}=def{x1} ∪ {x2} ∪. . .∪ {xn} (iv)hx1, x2, . . . , xni=def Kuratowski’s orderedn-tuple.
(v)P0
x=def ιu∀y(y∈u↔ ∃z(z∈x∧y∈z));
(vi)Q0
x=defιu∀y(y∈u↔ ∀z(z∈x→y∈z));
(vii)℘(x) =def{y:y⊆x};
(viii) trans(x) =def ∀y(y∈x→y⊆x) (xis a transitive set).
The underlying logic toCHU1 is da Costa’s paraconsistent logic C1=. The axioms of CHU1 are the axioms of CHU0, in which the usual negation ¬ is substituted by the strong negation¬∗ofC1=, plus an axiom that guarantees the
existence of the weak complement and an axiom that guarantees the existence of theRussell’s relationsinCHU1.
Axiom11CHU1. Axiom of the Weak Complement
∃v∀x(x∈v↔x6∈z).
Axiom12CHU1. Paraconsistent Separation Axiom (PS)
∃y∀x1∀x2. . .∀xn(hx1, x2, . . . , xni ∈ y ↔ hx1, x2, . . . , xni 6∈ xi), where 1 ≤ n≤ω.
Therefore, inCHU1 we have two complements, for every setx:
X =def {y :y 6∈x} (complement, relative complement, or weak complement ofx)
X∗ =def {y : y 6∈∗ x} (strong complement of x, with the notation y 6∈∗ x representing¬∗(y∈x)).
InCHU1, we can also define two distinct differences between any two sets:
x−y=def x∩y (difference, relative difference, or weak difference betweenx andy)
x−∗y=def x∩y∗ (strong difference betweenxandy).
The following theorems express fundamental properties ofCHU1.
Theorem 5 (da Costa [11]).CHU1is inconsistent (but apparently non-trivial).
Theorem 6 (da Costa [11]). CHU0 is consistent if, and only if, CHU1 is non-trivial.
3 Da Costa’s Paraconsistent Differential Calculus
Having as underlying set theory the classical theoryZF, da Costa extends the fieldRof the real numbers to the hyper-ringAof the hyper-real numbers, which is extended to the quasi-ring A∗ of the extended hyper-real numbers (see da Costa, B´eziau and Bueno [14]).
According to da Costa, under certain circumstances, an infinitesimal analysis founded in A and A∗ sends us to the ideas of the pioneers of the calculus, as Newton, Leibniz, the Bernoulli’s, de l’Hospital, etc., and recall us the de l’Hospital Principle:
On demande qu’on puisse prendre indiff´eremment l’une pour l’autre deux quan- tit´es qui ne diffrent entr’elles que d’une quantit´e infiniment petite: ou (ce qui est la mˆeme chose) qu’une quantit´e qui n’est augment´ee ou diminue que d’une autre quantit´e infiniment moindre qu’elle, puisse tre consid´er´ee comme demeu- rant la mˆeme...(de L’Hospital [17])
3.1 The Ring A of the Hyper-real Numbers
LetI be a fixed real interval andaa fixed element of the interior ofI.
Definition 7. 3 An infinitesimal variable is a real function f : I ⊆ R → R, such that
x→alimf(x) = 0.
We denote theset of infinitesimal variablesbyV.
Definition 8. The set of thehyper-real numbers, denoted byA, is defined by:
A=def{hr, fi:r∈Randf ∈V}.
Every real numberr, r ∈R, can be identified to the hyper-real of the form hr,0i, which is said astandard real number.
Definition 9. An infinitesimal is a hyper-real number of the form h0, fi, in whichf is an infinitesimal variable.
For everyr∈R, the set of the hyper-reals of the form hr, fiis said to be a monadofr, denoted by [r].
The monad of zero is the set constituted by the infinitesimals.
Definition 10. Theequality oridentityof two hyper-real numbers, denoted by
=, is defined by:
hr, fi=hs, gi if, and only if, r=sandf =g.
Definition 11. Theaddition (+)and themultiplication (×)of hyper-real num- bers are defined by:
(i)hr, fi+hs, gi=def hr+s, f+gi (ii)hr, fi × hs, gi=defhrs, rg+f s+f gi. For any hyper-realhr, fi,hr, fi=hr,0i+h0, fi.
We observe that, in spite of the same notations having been used, the opera- tions + and×between hyper-real numbers are defined from the usual operations of addition and multiplication of real numbers.
3 We could use the lateral limits in the definition of infinitesimal variable, observing that the concept of limit here used is the classical one.
Theorem 7. The structure hA,+,×,0,1i is a commutative ring with unity, where0 and1 are the hyper-realsh0,0i andh1,0i, respectively.
The order relation<ofRcan be extended toA.
Definition 12. hr, fi<hs, giif, and only if, either r < s, orr=sandf(x)<
g(x),∀x∈I.
The order relation<is non-linear, inA.
Definition 13 (Carvalho [7]). The orderof an infinitesimal h0, gi, relatively to an infinitesimalh0, fi, is defined by:
a) h0, fi and h0, gi have the same order if lim
x→a
f(x)
g(x) = b, with b a non-zero standard real number;
b) Theorder ofh0, fiissuperior to the order ofh0, giif lim
x→a
f(x) g(x) = 0;
c) h0, fi is of order k relatively to h0, gi if lim
x→a
f(x)
[g(x)]k = b, for b a non-zero standard real number.
Two infinitesimalsh0, fiandh0, giare said to beequivalentif lim
x→a
f(x) g(x) = 1.
Given any functionf, defined in R, it can be extended to a hyper-function f :A→A.
We can naturally express the classical notion of limit in the language of the infinitesimals.
Definition 14. Given a hyper-real function f : B ⊆ A →A,lim
x→rf(x) = b if, and only if, x∈[r]implies f(x)∈[b].
Definition 15. An infinite variable is a function v, v: I ⊆R→ R, such that
x→alimv(x) =∞.
Definition 16. Aninfinite hyper-real number is a pair of the formhv,0i, with v an infinite variable.
Definition 17. The set of the extended hyper-real numbers, denoted byA∗ is defined by:
A∗={a:a∈Aor a is an infinite hyper−real}.
We can extend the operations of addition and multiplication, and the rela- tion of equality ofA to the setA∗, such that the new structurehA∗,+,×,0,1i preserves some of the important properties of the hyper-ringhA,+,×,0,1i.
Meanwhile, as some of the conditions of the definition of ring are not satisfied by this structure, we nameA∗ a quasi-ring.
3.2 The Paraconsistent Differential Calculus
The paraconsistent differential calculus P, that we are going to introduce, is interpreted in the classical structureA∗.
It isnon-trivial- but inconsistent–, if the classical analysis is consistent.
The languageL of Pis the language L= of C1=, extended to the language of CHU1, with functional symbols; special constants to name the individuals of the structure A∗; the predicate <; the operations of A∗; and three types of individual variables, respectively denoting the finite hyper-reals - r, s, . . ., the infinitesimals -δ, ², . . ., and the infinites.
The underlying logic toPis the calculusC1=, and the underlying set theory is da Costa’s paraconsistent set theoryCHU1.
InL, we introduce by definition the predicate ≡, which represents a genera- lized equality predicate, necessary for the comparison of numbers that only differ infinitesimally.
Definition 18. The predicate of generalized equality, or generalized identity, denoted by≡, is defined by:
t1≡t2=def t1−t2=²,
with t1 and t2 terms of the language, ² an infinitesimal, and = the primitive predicate of equality ofL.
We define¬(t1≡t2)by:
t16≡t2=def t16=t2.
A concept of paraconsistent valuation can be introduced for the languageL.
Definition 19. A valuation for L is a function v, from the set of the closed formulae ofL into{0,1}, satisfying:
(1) The conditions of the definition of valuation forC1= of da Costa, B´eziau and Bueno ([14], Chapter 3).
(2) For atomic sentences of the formt1=t2, v(t1=t2) = 1, if t1=t2 is valid in A∗ and
v(t1=t2) = 0, otherwise;
(3) For any sentence P of the considered language, the value v(P) is given by the adequate combination of the previous properties.
According to the above definition, we have that:
(i) For sentences of the formt1< t2, v(t1< t2) = 1, ift1< t2is valid inA∗ and
v(t1< t2) = 0, otherwise;
(ii) For sentences of the formt1≡t2,
v(t1≡t2) = 1, ift1−t2=²is valid inA∗, with ²infinitesimal and
v(t1≡t2) = 0, otherwise;
(iii) For sentences of the formt16≡t2
v(t16≡t2) = 1, ift16=t2(that is, ¬(t1=t2)) is valid inA∗ and
v(t16≡t2) = 0, otherwise.
This definition of valuation, besides being compatible to the de l’Hospital Principle, makes possible verifying the inconsistent character of certain sentences of the paraconsistent calculus, such ast1≡t2, in the case of the joint occurrence ofv(t1≡t2) = 1 andv(¬(t1≡t2)) = 1.
Furthermore, it is fundamental to introduce inPsome equivalent results and extensions of classical results, as illustrated in what follows.
Remark 1. The order relation < is quasi-linear in A and A∗, relatively to the identity≡, what is expressible by the following formula, fort1andt2any terms:
(t1< t2)∨(t2< t1)∨(t1≡t2).
The definition of the limit of a hyper-real function, whenxtends to a standard real number, is the usual one.
Definition 20. Given a hyper-real functionf :B⊆A→A∗ and standard real numbers randb:
x→rlimf(x) = b if, and only if, (∀² > 0)(∃δ > 0)(∀x)(0 < |x−r| < δ →
|f(x)−b|< ²).
Theorem 8 (Carvalho [7]).Given a hyper-real function f :B⊆A→A∗, we have that:
x→rlimf(x) =kif, and only if, (∀x)(x∈B)((x≡r)→(f(x)≡k)).
In the languageL, we can also introduce the concepts of limit of a hyper-real function in the cases in which eitherx, or f(x), tend to the infinite.
Definition 21. Given a hyper-real function f : B ⊆A → A∗ and a standard real number b, we have that:
x→hu,0ilim f(x) =bif, and only if,(∃ hv,0i ∈A∗)(∀x)((|x|>|hv,0i|)→(f(x)≡ b)).
We similarly define the other case of infinite limit.
Definition 22 (Carvalho [7]). A hyper-function f :U ⊆A→A∗ iscontinu- ousin a hyper-realhr, gi ∈U if, and only if:
(∀ hs, hi ∈U)(hr, gi ≡ hs, hi →f(hr, gi)≡f(hs, hi).
Otherwise,f is discontinuousinhr, gi.
Definition 23 (Carvalho [7]). A relation defined in a subsetU of A, f :U ⊆ A→A∗, is aquasi-function, or aparafunctionfrom U intoA∗, if it associates a unique element ofA∗ to each element of the domainU, excepting in a discrete set Q={uk}k∈N of points ofU, to whose elements uk the relation f associates more than one image,vk1, vk2, . . . , vknk, nk >1, such thatvik6=vkj butvik≡vkj, for i, j∈ {1,2, . . . , nk} andi6=j.
We have introduced several special concepts, as for instance the concepts of: absolute value of a hyper-real number, quasi-nilpotent infinitesimal, hyper- interval,hyper-sequence(hrn, fni)n∈N),convergenceanduniform-convergence of hyper-sequences, paracontinuity of a hyper-function and paracontinuous quasi- function.
Definition 24. The derivativeof a hyper-real function f :A→A∗, in a stan- dard real number r, is a standard real number, denoted byf0(r), such that:
f(r+²)−f(r) =f0(r)ײ+δ,
where² is an arbitrary infinitesimal, and δ is an infinitesimal that depends on ²and whose order is superior to the order of ².
Remark 2. According to the definition, D is the value of the derivative off in a standard real number r, that is,D=f0(r), if we have that:
f(r+²)−f(r)≡Dײ.
We have that the derivation operations and the main theorems of the usual classical calculus preserve their classical characteristics in the paraconsistent calculus. In Carvalho [7] we proved some of them, as for example theWeierstrass Theorem, Rolle Theorem, Intermediate Value Theoremand theTheorem of the Medium Value.
Besides, in Carvalho [7, p. 127], we present a proposal of a definition for a paraconsistentdefined integral.
We also analyze two applications of the paraconsistent calculus: one of them, of classical nature, based in a problem suggested us by J. Bell through a per- sonal correspondence; the second one, of paraconsistent nature, motivated by the Dirac’s “delta function”.
4 A Transference Theorem
In this section, we present the concept of paraconsistent super-structure and of monomorphism between super-structures, in order to introduce a Transference Theorem from the classical differential calculus into the paraconsistent differen- tial calculus.
For the construction of paraconsistent super-structures over the hyper-ring Aand the quasi-ringA∗, essential in obtaining a Transference Theorem between the classical and the paraconsistent differential calculi, we were motivated by Robinson’s non-standard analysis ([27], reviewed re-edition of the first edition published in 1966), and by Robinson and Zakon [28] and Stroyan and Luxemburg [30].
By adapting the concept of super-structure of Stroyan and Luxemburg [30] to the paraconsistent calculus, we define a paraconsistent super-structure as every super-structure whose underlying set theory is paraconsistent, in our caseCHU1, and that has as its underlying logic a paraconsistent logic, in our caseC1=, whose language is extended by the languageLintroduced in detail in Carvalho [7].
Definition 25. Let X be an infinite set (classically non-empty) of individuals of CHU1. We define the set X inductively:
X0=X, . . .
Xn+1=℘(
[n k=0
Xk), n= 0,1,2, . . .
X = [∞ n=0
Xn
The elements ofX are said to be theentitiesofX and the elements (atoms) of X are the individuals ofX. We indifferently denote the entities by small or capital letters; in particular, the individuals are exclusively denoted by small letters.
Definition 26. The paraconsistent super-structure X over a set X of atoms of CHU1, X classically non-empty and infinite, is the structurehX,=,∈, <,≡i, with X as in the previous definition.
If X = R, we have the super-structure R, whose universe has as basic set the set of the standard real numbers, that are the individuals of Rand whose entities are all of the elements ofR.
IfX =A∗ and ifS∗denotes the universe built fromA∗, whose elements are the individuals ofS∗, then we have the paraconsistent super-structurehS∗,=,∈,
<,≡i.
Definition 27 (Carvalho [7], p. 145).Let X andY be paraconsistent super- structures. An injective function ¨ :X → Y is a monomorphism if it satisfies the following conditions, in which, for simplicity, for every setA of CHU1 the notation ¨Ais used to represent¨(A) (or¨[A]):
i) Ifx=y, then¨x= ¨y (¨preserves the equality relation =);
ii) Ifx≡y, then¨x≡ ¨y (¨ preserves the paraconsistent identity relation
≡);
iii) Ifx < y, then¨x < ¨y (¨preserves the relation<);
iv)¨{X}={¨X}(¨preserves unary sets);
v)¨(A−∗B) = ¨A−∗¨B (¨ preserves the strong difference of sets);
vi)¨(A−B) = ¨A− ¨B (¨preserves the difference of sets);
vii)¨(A×B) = ¨AרB (¨preserves the cartesian product of sets);
viii)¨(τ◦F) = τ(¨F), where τ is a permutation of n elements andF is a n-ary relation (¨commutes with permutation of variables);
ix)¨(F−1) = (¨F)−1 (¨ commutes with inverse relations);
x)¨(Dom(F)) =Dom(¨F)and¨(Im(F)) =Im(¨F), forF a binary rela- tion (¨ preserves the domain and the image of relations);
xi)¨(A∪B) = ¨A∪¨B (¨preserves the union of sets);
xii) GivenA∈ X,¨{(x1, x2, . . . , xn, y) : (x1, x2, . . . , xn)∈y∈A}={(z1, z2, . . . , zn, w) : (z1, z2, . . . , zn)∈w∈ ¨A} (¨ preserves the standard definition of sets);
xiii)¨produces an extension¨Aofσ(A), σ(A)⊆ ¨A, such that σ(A) = ¨A if, and only if, Ais a finite set.
We can verify, by the properties that characterize a monomorphism, that the image¨X of a super-structureX is a super-structure.
The following definition of internal set of a paraconsistent super-structure corresponds to the definition of Robinson and Zakon [28] for the non-standard analysis.
Definition 28. Given a monomorphism¨:X → Y, between the paraconsistent super-structures X andY, the internal elements ofY are the elements of¨X =
∞S
n=0
¨Xn, with X0 the set of individuals of X.
The elements ofY that are not internal elements are said to beexternal ones.
Remark 3. From Definition 27 - xiii, that allows us to define a function σ : EN T(X)→ Y, from the entities ofX into Y, such that, for every entity A of X, σ(A) =def {¨x:x∈A}, we have that:
• An element y of ¨X is an internal element of Y if there is k such that y∈ ¨Xk, that is, if there existsxin X such thaty∈ ¨x.
• In the case thatX0=R, we have thatσ(R) is a (standard) copy ofRin Y, withσ(R)⊆ ¨(R) = ¨R.
• Ifx∈X0=R, then¨x∈ ¨X0 = ¨R⊆ ¨R, and so every individualy of
¨R is a internal element.
• When the super-structureX =R=S∞
n=0Rn is constructed over R0 =R, and the super-structure H is constructed over an extension H0 of R that contains the infinitesimal and infinite numbers, we have that the sets of these infinitesimals and infinite numbers are internal ones.
Similarly to what occurs in Robinson’s non-standard analysis, the internal sets are particularly important for, given a formula α of L interpreted in a paraconsistent super-structure built overR, the quantified variables of its trans- formed ¨(iα), or ¨iα, vary only over internal sets of extensions Hof R, con- structed over extensions ofR. So, it is expected that when the free variables of
¨iαare defined to vary only over internal objects of an extensionHof R, the truth values ofαin Rare the same values of its transformed in¨R ⊆ ¨H.
Definition 29. Aninterpretation functionof the languageLin a paraconsistent super-structureM, built either overRor over any extension ofR, is an injective function i:A⊆C(L)→ M, with A a subset of the setC(L) of the constants of L.
Given an interpretationi, let F be a closed formula of L, whose constants a1, . . . , an belong to the domain ofi:
i) Ift1 andt2 are constants, (t1=t2) is interpreted as “the entity (or indi- vidual)it1is equal (=M, inM) to the entity (or individual)it2”;
ii) If t1 and t2 are constants, (t1 ∈ t2) is interpreted as “the entity (or individual)it1 is a member (∈M, inM) of the entity (or individual)it2”;
iii)Ift1 and t2 are constants, (t1< t2) is interpreted as “the entity (or indi- vidual)it1is smaller (<M, inM) than the entity (or individual)it2”;
iv)Ift1 andt2 are constants, (t1≡t2) is interpreted as “the entity (or indi- vidual)it1is almost identical (≡M, inM) to the entity (or individual)it2.
According to the interpretation given to the symbols =,∈, <and≡ofL, we can say that our interpretations takeLinto the set theoryCHU1.
Definition 30. For every subset B of R, B⊆R, we define the following sets:
IB={(x, y) :x, y∈B∧x=y}
PB={(x, y) :x, y∈B∧x∈y}
OB={(x, y) :x, y∈B∧x < y}
SB={(x, y, z) :x, y, z∈B∧x+y=z}
MB={(x, y, z) :x, y, z∈B∧x.y=z}
Definition 31. Given a δ-incomplete ultrafilterU over a denumerable infinite set J, we define the relations =u, <u,∈u and the operations +u and ×u, be- tween sequences (aj)j∈J of individuals and sequences (Aj)j∈J of entities (non- individuals) ofRJ.
Theorem 9. The relation=u is a congruence relation.
Given an interpretationi: A ⊆C(L) → R, we can establish, by using the properties of the ultrafilterU, an interpretation functioniU :B ⊆C(L)→ RJ, satisfying the following conditions:
• Dom(i)⊆Dom(iU)
• Ifα∈Dom(i) anda= iα, then iU(α) =f ∈ RJ, with f(j) =a, for every j∈J
• TheiUinterpretations of the relations ∈,= and<, and of the operations + and×ofL inRJ are, respectivelly,∈u,=u, <u,+u and×u.
Theorem 10.
RJ,=u,+u,×u
®is a commutative ring with unity.
Theorem 11.
RJ/=u,∗=u,∗+u,∗×u
® is a commutative ring with unity, with the relation and operations of the structure induced from the ones of the previous structure:
[a] ∗=u[b]if, and only if, a=ub
[a] ∗+u[b]∗=u[c] if, and only if,a+ub=uc [a] ∗×u[b]∗=u[c] if, and only if,a×ub=uc.
As R is the basis of R and RJ ⊂ RJ, RJ/=u can be taken as the basis for a super-structure Hthat extends R. In the classical case, this is the super- structure that gives us a non-standard model∗R for the analysis. In our case, we extendHto a paraconsistent model for the analysis, based on the quasi-ring A∗: we can construct a super-structure S∗ = hS∗,=,≡,∈, <i that extends R, such thatRJ⊆ S∗J.
Let us consider aδ-incomplete ultrafilterU over a denumerable infinite setJ. We can define the relations =u,≡u, <u and∈u, and the operations +u and×u
between sequences ofA∗J. As an example, let us introduce one of the definitions.
Given the sequences of terms (rj)j∈J and (s)j∈J ofA∗J, we have that (rj)j∈J ≡u(sj)j∈J if, and only if,{j:rj ≡Sj} ∈U, with≡the generalized identity predicate ofL.
Theorem 12. The relation≡u is a congruence relation inA∗J.
Now, given the operations ∗+u and ∗×u, and the relations ∗ =u,∗≡u,∗∈u
and ∗ <u, by working as in the case ofRJ/=u, we can build a super-structure S over the quotient quasi-ringA∗J/≡u, that extends the super-structureHand, by transitivity, also extends the super-structureR.
By considering the super-structures R over R and S over S∗J/≡u, and the sets I, P, O, Sand M, we finally define the sets ¨I,¨P,¨O,¨S and ¨M, inS, by using the relations and operations obtained for S∗J/≡u.
In the following diagram (Fig. 1), we represent the structures and functions necessary to the proof of the Transference Theorem from a paraconsistent super- structureR, constructed over the setRof the real numbers, into a paraconsistent super-structure S, constructed over an extension of the quasi-ringA∗.
Fig. 1.A monomorphism between the paraconsistent super-structuresRandS
• i, i0, i00, iu, ju - interpretation functions, with domain in subsets A, B, C, D andEof the setC(L) of the constants ofLand rangesR,H,S,RJ andS∗J respectively.
• sR:R → RJ - immersion ofRintoRJ, that identifies every constantr or variablex, ofR, respectively wither to a constant sequence (r, . . . , r, . . .) or to a sequence (x1, x2, . . .) ofRJ.
• Θ:RJ → S∗J - immersion of RJ into S∗J, that associates every sequence of the form (x1, x2, . . . , xn, . . .) to a sequence of the form (hx1,0i,hx2,0i, . . . , hxn,0i, . . .) of finite hyper-real numbers, eventually constants.
• mR :RJ → H - immersion ofRJ into H, that maps sequences (xj) of RJ into classes [(yj)] ofH, with [(yj)] ={(zn)∈ RJ : (zj) =u(yj)}.
• ∗m : F in(S∗J)→ S - function that associates sequences of finite elements ofS∗J, (xj), to classes [(xj)] in S.
• ¨1- monomorphism betweenRandH.
• ¨2 - embedding of H into S, that maps classes [(xj)] of H into classes [(hxj,0i)] ofS.
• ¨- monomorphism between RandS.
• σ1 :C(R)→ H- function, whose domain is the set of the constants of R, that produces a standard copyσ1(R) of the basis RofRinH.
• σ:C(R)→ S - function that determines a copy of RinS such that σR⊂
¨R.
Theorem 13 (Transference Theorem, Carvalho [7], p. 190).Let us take the paraconsistent super-structures RandS, the interpretation functionsi and
i00, and the monomorphism ¨, as just introduced; and let α(x1, x2, . . . , xn) be a formula of the language L, whose free variables are among x1, x2, . . . , xn. In these conditions, α(x1, x2, . . . , xn) is valid in R relatively to vi if, and only if,
¨(α(x1, x2, . . . , xn))is valid inS relatively tovi00.
Proof. By induction on the length of the formulaα. ut The Transference Theorem we have obtained is a fundamental theorem and plays a similar rˆole to the Robinson’s Transference Theorem from the classical differential calculus to the non-standard analysis.
The theorem asserts that a formulaT is a theorem of the classical differential calculus if, and only if, there exists an “interpretation” of T that is a theorem of the paraconsistent differential calculus.
5 Final Remarks
In a series of recent papers, Feitosa and D’Ottaviano have studied inter-relations between logical systems through the analysis of translations between them.
From the definition of the concept of translation between logics proposed by da Silva, D’Ottaviano and Sette [16], Feitosa and D’Ottaviano introduce the concept of conservative translationbetween logics (see Feitosa [21] and Feitosa and D’Ottaviano [22]). D’Ottaviano and Feitosa [20] present conservative trans- lations from the classical propositional logic into the systems Cn,1 ≤ n < ω, propitiating a new approach for the study of the relationships between the sys- temsCn,1≤n < ω, classical logic and other non-classical systems, and between classical and non-classical theories.
The monomorphism¨:R → S built in the proof of the above Transference Theorem constitutes a conservative translation from the classical differential calculus into da Costa’s paraconsistent differential calculus.
We intend to continue the study of the relations between the classical and da Costa’s differential calculi, including from the point of view of the theory of translations developed by Feitosa and D’Ottaviano.
We also intend the extend this work to the paraconsistent integral calculus.
In a future paper, we will present in detail the constructions and proofs of the results of this paper, with the proof of the Transference Theorem.
Acknowledgements
The first author acknowledges support fromFunda¸c˜ao de Amparo `a Pesquisa do Estado de S˜ao Paulo(FAPESP), under the Thematic Project “ConsRel” (Grant Number 2004/14107-2).
References
1. Alves, E.H.L´ogica e inconsistˆencia (Logic and inconsistency). 1976. 137p.
Dissertation (Master) - Philosophy, Letters and Human Sciences Faculty, S˜ao Paulo University, S˜ao Paulo, 1976.
2. Arruda, A.I. Considera¸c˜oes sobre os sistemas formais N Fn (Considera- tions on the formal systemsN Fn). 1964. Thesis. Federal University of Paran´a, 1964.
3. Arruda, A.I. Le sch´ema de la s´eparation dans les syst`emesN Fn.Comptes Rendus de l’Acad´emie de Sciences de Paris, t.280A, p. 1341-1344, 1975.
4. Arruda, A.I. A survey of paraconsistent logic. In: Latin American Symposium on Mathematical Logic, 4., 1978, Santiago, Chile. Mathematical Logic in Latin America: proceedings. Edited by A. I. Arruda, N.C.A. da Costa e R. Chuaqui.
Amsterdam: North Holland, 1980. p. 1-41.
5. Arruda, A.I. Aspects of the historical development of paraconsistent logic. In: G.
Priest, R. Routley and J. Norman(eds). Paraconsistent logic: essays on the inconsistent. Philosophia Verlag, 1989.
6. Arruda, A.I., da Costa, N.C.A. Sur une hi´erarchie de syst`emes formels.Comptes Rendus de l’Acad´emie de Sciences de Paris, t.259, p. 2943-2945, 1964.
7. Carvalho, T.F.Sobre o c´alculo diferencial paraconsistente de da Costa (On da Costa’s paraconsistent differential calculus). 2004. 200 p. Thesis (Doc- torate) - Philosophy and Human Sciences Institute, State University of Campinas, 2004.
8. Church, A. A formulation of the simple theory of types.Journal of Symbolic Logic, v.5, p.56-68, 1974.
9. Da Costa, N.C.A.Sistemas formais inconsistentes (Inconsistent formal sys- tems). 1963. 68 p. Thesis - Federal University of Paran´a, Curitiba, 1963.
10. Da Costa, N.C.A. On the theory of inconsistent formal systems.Notre Dame Jour- nal of Formal Logic, v.15, n.4, p. 497-510, 1974.
11. Da Costa, N.C.A. On paraconsistent set theory.Logique et Analyse, v.115, p. 361- 371, 1986.
12. Da Costa, N.C.A.Sistemas formais inconsistentes (Inconsistent formal sys- tems). Curitiba: Ed. UFPR, 1993.
13. Da Costa, N.C.A. Paraconsistent mathematics. In: I World Congress on Paraconsis- tency, 1998, Ghent, Belgium. Frontiers in paraconsistent logic: proceedings.
Edited by D. Batens, C. Mortensen, G. Priest, J. P. van Bendegen. London: King’s College Publications, 2000. p. 165-179.
14. Da Costa, N.C.A., B´eziau, J.Y., Bueno, O.Elementos de teoria paraconsis- tente de conjuntos (Elements of paraconsistent set theory). Campinas:
Unicamp / CLE, 1998. (Cole¸c˜ao CLE, v.23)
15. Da Costa, N.C.A., Marconi, D. An overview of paraconsistent logic in the 80’s.
The Journal of Non-Classical Logic, v.6, n.1, p. 5-31, 1989.
16. da Silva, J.J., D’Ottaviano, I.M.L, Sette, A.M. Translations between logics. In:
Caicedo, X., Montenegro, C.H. (eds.)Models, algebras and proofs. New York:
M. Dekker, 1999, p. 435-448. (Lecture Notes in Pure and Applied Mathematics, v.
203)
17. De l’Hospital, G.F.A. Analyse des infiniment petits pour l’intelligence des lignes courbes. Paris: Imprimerie Royale, 1696.
18. D’Ottaviano, I.M.L. On the development of paraconsistent logic and da Costa’s work.The Journal of Non-Classical Logic, v.7, p. 89-152, 1990.
19. D’Ottaviano, I.M.L. A l´ogica cl´assica e o surgimento das l´ogicas n˜ao cl´assicas. In:
Evora, F. (ed.)´ S´eculo XIX: o nascimento da ciˆencia contemporˆanea. Campinas:
Unicamp/CLE, 1992. p. 65-93. (Cole¸c˜ao CLE, v. 11).
20. D’Ottaviano, I.M.L, Feitosa, H.A. Paraconsistent logics and translations.Synth`ese, v.125, n. 1/2, p. 77-95, 2000.
21. Feitosa, H.A. Tradu¸c˜oes conservativas (Conservative translations). 1997. 161 p.
Thesis (Doctorate) - Philosophy and Human Sciences Institute - State University of Campinas, 1997.
22. Feitosa, H.A., D’Ottaviano, I.M.L. Conservative translations.Annals of Pure and Applied Logic, v. 108, p. 205-227, 2001.
23. Forster, T.E.Set theory with a universal set. Oxford: Clarendon Press, 1995.
24. Kleene, S.C.Introduction to metamathematics. Amsterdam: North Holland;
New York: Van Nostrand, 1952.
25. Mortensen, C. Inconsistent mathematics. Dortrecht: Kluwer Academic Pub- lishers, 1995. (Mathematics and its Applications, v. 312)
26. Quine, W.V.O. New foundations for mathematical logic. American Mathematical Monthly, v. 44, p. 70-80, 1937.
27. Robinson, A.Non-standard analysis. (Reviwed re-edition of the 1st edition of 1966) Princeton: Princeton University Press, 1996.
28. Robinson, A., Zakon, E. A set theoretical characterization of enlargements. Ap- plications of model theory to algebra, analysis and probability. C.I.T., Holt, Rinehart and Winston, p. 109-122, 1967.
29. Rosser, J.B.Logic for mathematicians. New York: McGraw-Hill, 1953.
30. Stroyan, K.D., Luxemburg, W.A.J. Introduction to the theory of infinitesi- mals. New York: Academic Press (1st ed.), 1976.
