Similarly to the case of traditional syllogistic, which was interpreted within classical monadic predicate calculus, it is possible to develop a paraconsistent syllogistic. It is based on, for instance, the monadic calculus corresponding to the paraconsistent predicate logic C?1. In order to reach that, it suffices that one translate the propo-sitionsA, I, E andO intoC1?: the translations are as follows, which were based on the classical setting:
Aab ∀x(a(x)→b(x)) Iab ∃x(a(x)∧b(x)) Eab ∀x(a(x)→ ¬b(x)) Oab ∃x(a(x)∧ ¬b(x))
There are two brief remarks to be made within this context. (1) The valid positive deductions in C0?, the classical predicate calculus, are also valid inC1?; that is, when no explicit negation is involved, the positive deductions of C?0 andC1∗ are the same. (2) InC1?one can find paraconsistent predicates, such that, for instance, there are elements that satisfy the predicate and, at the same time, do not satisfy it; i.e., for some predicate pthe following holds:
∃x(p(x)∧ ¬p(x)).
Thus, based on arguments rather similar to the ones found in the classical case, it is possible to verify the validity of inferences, and one changes accordingly the theories of opposition, conversion, immediate inferences and syllogism. (Each pred-icate within the universe of discourse has three parts: of the elements that satisfy it, of those that do not satisfy it, and of those that simultaneously satisfy it and do not satisfy it. Simple graphics supply then evidence for the validity, or for the invalidity, of certain inferences and conversions.)
Based on this approach, one can prove the following result. In the paraconsis-tent logic C1?, all modes of the first and of the third figures of the syllogism are valid; none of the second is valid; and of the fourth, just Bramantip and Dimaris modes are valid. It is worth mentioning that since C1? has a strong negation, of a classical trend, and if such negation is adopted in the interpretation of syllogistic reasoning, then the classical theory is obtained. As is known, Lukasiewicz has ax-iomatised the theory of categorical syllogism, based on the classical propositional calculus and admitting as specific axioms certain categorical propositions, as well as some appropriate definitions. Based on the paraconsistent propositional calcu-lus, for instance the calculus C1, it is also possible to formulate an axiomatics for paraconsistent syllogistic, articulated in parallel lines to the theory just outlined.
Moreover, we should note that there are further extensions or modifications of the Aristotelian syllogistic that also admit paraconsistent versions, such as Hamilton’s, De Morgan’s and Gergone’s.
3 Paraconsistent set theory
Cantor’s naive theory was based mainly on two fundamental principles: the pos-tulate of extensionality (if the sets xandy have the same elements, then they are equal), and the postulate of separation or comprehension (every property deter-mines a set, composed of the objects that have this property). The latter postulate, in the standard (first-order) language of set theory, becomes the following formula (or scheme of formulas):
∃y∀x(x∈y↔F(x)). (7) Now, it is enough that one replaces the formula F(x), in (7), by x /∈ x in order to derive Russell’s paradox. That is, the principle of separation (7) entails inconsistency. Thus, if one adds (7) to first-order logic, conceived as the logic of a set theoretic language, a trivial theory is obtained. There are also other paradoxes, such as those of Currys and Moh Schaw-Kwei, that indicate that (7) is trivial or, more precisely, trivialises set theory, if its underlying logic is classical, even ignoring negation.10 In other words, classical positive logic is incompatible with (7); the same holds also for several other logics, such as the intuitionistic one. Classical set theories are distinguished by the restrictions that are imposed on (7), to the effect of avoiding paradoxes. In order that the theory thus obtained does not become too weak, some further axioms, besides extensionality and separation (with due restrictions), are added, depending on the particular case in question. Thus, for instance, in Zermelo-Fraenkel (ZF), separation is formulated in the following way:
∀z∃y∀x(x∈z↔(x∈ ∧F(x))), (8)
10Let us exemplify this fact with Curry’s paradox. In (7), substitute for F(x) the expression x∈x→α, whereαis a formula whatever in whichydoes not appear free. Then, by (7), we get
∃y∀x(x∈y↔(x∈x→α)). Let us callc, in honour of Curry, this sety; so,∀x(x∈c↔(x∈x→ α)). But thenc∈c↔(c∈c→α), hence (i)c∈c→(c∈c→α) and (ii) (c∈c→α)→c∈c).
But the law of contraction (γ→(γ→β))→(γ→β) holds, so (i) entails (iii)c∈c→α. From (ii) and (iii), we get c∈c. Finally, from this last sentence and from (iii), by Modus Pones, we obtainα. This shows that (7) entails triviality even without negation.
where the variables are subject to the usual conditions. In ZF, then, F(x) deter-mines the subset of the elements of the setz that have the propertyF (or satisfy the formula F(x)). In the Kelly-Morse system, on the other hand, separation is formulated as follows:
∃y∀x(x∈y↔(F(x)∧ ∃z(x∈z)), (9) while, in Quine’s system NF, the notion of stratification is employed, and the scheme of separation is written like (7), provided that F(x) is stratified (besides the stan-dard conditions regarding the variables).
However, we can ask whether it would be possible to examine the problem from a distinct viewpoint: what is needed in order to maintain the scheme (7) without restrictions (with no regard to the conditions on the variables)? The answer is immediate: one should change the underlying logic, so that (7) does not inevitably lead to trivialisation. The separation scheme, without strong restrictions, leads to contradictions. Hence, such a logic has to be a paraconsistent one. It was slowly verified that there are infinitely many ways to weaken the classical restrictions imposed on the separation scheme, each of them corresponding to distinct categories of paraconsistent logic. Furthermore, extremely weak logics have been formulated to be the underlying logics of set theories in which (7) is used without any restrictions (not related to the choice of variables).
An important point is that several paraconsistent set theories contain the classi-cal one, in Zermelo-Fraenkel’s, Kelly-Morse’s or Quine’s formulations. Hence, para-consistency goes beyond the classical domain, and allows, among other things, the reconstruction of traditional mathematics (see [108], [76], [109], [184]). It is quite fair then to claim that paraconsistent theories extend the classical ones, just as Poncelet’s imaginary geometry comprises the standard Euclidean geometry. More-over, we should stress a difficulty found in the very foundations of logic. Classical elementary logic (it would, in fact, be enough to consider only part of its positive logic) and the separation postulate seen to be both evident; we may even to claim that they are equally evident or intuitive. However, they are mutually incompatible, and constitute thus a case of incompatible evidences (this fact generates a difficulty from the viewpoint of classical logic). Without presenting detailed philosophical analyses, we shall just note that classical theories adopt a particular line of ap-proach, and paraconsistent theories, another one. Such an exploration contributes for a better comprehension of the classical position itself, a clearer understanding of negation, and a more realistic perception of the possibility of discourse, even if one partially puts aside the principle of non-contradiction.
3.1 The systems N F
n, 1 ≤ n ≤ ω
Here we begin by describingN F1.11The underlying logic ofN F1is the calculusC1=, that is, we assume the language of this calculus plus the propositional postulates
→1to¬4of page 8, the postulates (I)-(VII) for the predicate calculus (see page 17), and the postulates for equality (I’) and (II’) (page 21). The specific postulates of N F1 are:
(NF1) (Extensionality) ∀x∀y(∀z(z∈x↔z∈y)→x=y)
(NF2) (Separation) ∃y∀x(x∈y ↔α(x)), wherexand y are distinct variables, y does not occur free inα(x) and this formula is either stratified or is of the form x /∈x.12
11Caiero and de Souza have developed a paraconsistent version of theM Lsystem in [55].
12We recall that a formula is stratified if it is possible to replace each variable occurring in it by a numeral in the following manner: we replace everywhere the same variable by the same numeral
If we add toC1= the scheme¬(α∧ ¬α), we get the classical first-order predicate calculus with identity. The system NF of Quine is obtained by adjoining to this calculus the postulates (NF1) and (NF2) above, provided that this second postulate is subjected to the sole restriction that α(x) must be stratified; we shall denote Quine’s system by N F0.
In order to introduce the set theoriesN Fn, 1≤n < ω, we employ the calculus Cn=, 1≤n < ωas their underlying logics plus the specific postulates above. Then we have:
Theorem 3.1.1 N Fn,1≤n < ω contains N F0.
Proof: It follows from the fact that ifαis a theorem ofN F0, and if we replace all occurrences of¬in this formula by¬(n), obtainingα0, thenα0 is provable inN Fn, 1≤n < ω, where¬(n)αis the formula¬α∧α(n).
Theorem 3.1.2 If N Fn,1≤n < ω is non-trivial, then N F0 is consistent.
Proof: Let us suppose that N Fn, 1 ≤ n < ω is non-trivial and that N F0 is inconsistent. Letα1, . . . , αn a proof of a contradiction inN F0, whereαn isβ∧ ¬β.
Then, if α01, . . . , α0n are formulas obtained from the αi as explained in the proof of the preceding theorem, then this last sequence would be a derivation ofβ0∧ ¬(n)β0 in N Fn. But in this system (γ∧ ¬(n)γ)→δis a valid scheme, and soN Fn would be trivial.
We can prove the following Theorem 3.1.3
1. In the hierarchy N Fn, 1 ≤n < ω, every system is stronger that those which follow them.
2. If N F1 is non-trivial, then allN Fn, 1< n < ω are also non-trivial.
3. N Fn, 1≤n < ω is inconsistent.
Let us make some comments on item 3. InN Fn, Russell’s set, that is, the set R =def {x:x /∈x}, does exist (see the next subsection), that is, in these systems we have ` ∃y∀x(x∈ y ↔ x /∈x). As we shall see below, it is easy to prove that R∈R∧R /∈R; so, the systemsN Fn are inconsistent. The other items are proved without difficulty (see [76]).
The next step is to show that if N F0 is consistent, then N F1 is non-trivial.
Therefore, due to item 2 of the preceding theorem, the consistency of N F0entails the non-triviality of N Fn, 1≤n < ω. Let us define a systemN F∗1 as follows: we keep with the propositional postulates (see page 8)→1−− →3,∧1−−∧3,∨1−−∨3 only and add the following new postulates: (→∗4) (Peirce’s Law) ((α→β)→α)→α and (→∗5) (¬α→β)→((¬α→ ¬β)→α), whereβ is not atomic. This new set of postulates of course provides an axiomatization for the classical propositional logic.
The remaining postulates of N F∗1 are those ofN F1, except for those which turn to be redundant. For instance, since αo is provable inN F∗1 when αis not atomic, it results that postulates (V) and (VI) of page 17 are provable. So, N F1 is weaker thanN F∗1.
Lemma 3.1.1 The consistency of N F0 entails the non-triviality ofN F∗1.
so that, for each occurrence of∈, the numeral immediately following∈is the immediate successor of the numeral immediately preceding∈[140, p. 213].
Proof: Let f be a map whose domain is the set of formulas of N F∗1 and whose range is the set of formulas of N F0, defined as follows, where V =def {x:x=x}:
1.f(x=y) =def x=y 2.f(x∈y) =def x∈y
3.f(x∈y) =def x∈y 4.f(x /∈y) =def x∈ V ∧y∈ V 5.f(∀xα) =def ∀xf(α) 6.f(∃xα) =def ∃xf(α) 7.f(α∧β) =def f(α)∧f(β) 8.f(α∨β) =def f(α)∨f(β) 9.f(α→β) =def f(α)→f(β)
.
Then, using the preceding results, we can see that if αis a theorem ofN F0, then f(α) is a theorem ofN F∗1. Since the rules of inference of N F∗1 are valid inN F0, any theoremαofN F∗1 induces a theoremf(α) ofN F0. Therefore, supposing that N F0 is consistent, N F∗1 cannot be trivial; for instance, ∅ ∈ ∅is not a theorem of N F∗1, sincef(∅ ∈ ∅) =∅ ∈ ∅ is not provable inN F0.
Theorem 3.1.4 If N F0 is consistent, thenN F1 is non-trivial.
Proof: The consistency of N F0 implies the non-triviality of N F1 because this system is weaker thanN F∗1.
Theorem 3.1.5 IfN F0 is consistent, then all the inconsistent systemsN Fn,1≤ n < ω are non-trivial.
Proof: It suffices to note that the theoriesN Fn, 1≤n < ωare weaker thanN F∗1. Changing a little bit the proof of Theorem 3.1.4, we can prove the following result: if N F0 is consistent, then the system obtained from N Fn, 1≤n < ω by adjoining axioms guaranteeing the existence of the sets of all non-k-circular sets (k = 1,2, . . .) is non-trivial. For example, the set of all non-3-circular sets is the following set: {x:¬∃y1∃y2∃y3(x∈y1∧y1∈y2∧y2∈y3∧y3∈x)}.
N Fωis like N F1, except that its underlying logic isCω= instead ofC=1. N Fωis weaker thanN F1, so ifN F0is consistent, it is non-trivial.