Completeness Theorems for First-Order Logics of Formal Inconsistency
Rodrigo Podiacki and Walter Carnielli
Centre for Logic, Epistemology and the History of Science – CLE and
Department of Philosophy – UNICAMP P.O. Box 6133, 13083-970
Campinas, SP, Brazil
podiacki@gmail.com, carniell@cle.unicamp.br
Abstract
We investigate the question of characterizing first-orderLFIs (Logics of Formal Inconsistency) by means of two-valued semantics. Although we focus on a particular study-case, the quantified logic QmbC, it will be shown how the method proposed here can be extended to a large fam- ily of quantified paraconsistent logics, supplying a sound and complete semantical interpretation for such logics.
1 First-order Logics of Formal Inconsistency
TheLogics of Formal Inconsistency from now onLFIs are logics able to internalize, in a precise sense, the notions of “consistency” and “inconsistency” in the object-language level, be it by introducing primitive unary connectives, be it by appropriate definitions using the common propositional connectives. Such logics are paraconsistent in the following sense: given a contradiction of the form (ϕ∧ ¬ϕ), in general it is not possible to deduce an arbitrary formulaψfrom it, that is, they do not fall into deductive triviality when exposed to a contradic- tion. This means that the Principle of Explosion, or Principle of Pseudo-Scotus, is not valid in general. However,LFIsmay “explode” ifϕ, besides being contra- dictory, is also consistent (intuitively, this means thatϕbehaves classically, and cannot bear its own contradictory). SoLFIsare submitted to a more restricted principle of explosion, called in [Carnielli, Coniglio and Marcos, 2007] theGentle Principle of Explosion: anLFIexplodes in the presence ofϕ, ¬ϕand◦ϕ, for any arbitraryϕ, where ‘◦ϕ’ expresses the fact thatϕis consistent.
In its beginnings paraconsistent logic was only developed syntactically, that is, it was presented as an axiomatized calculus without any interpretation. In the
70s, there arose the first semantics, known asvaluation semantics,1for the hi- erarchy of propositional paraconsistent logicsCn of da Costa. Nevertheless, the problem concerning convenient interpretation for first-order paraconsistent logic persisted. In 1984, Alves proposed a method which can be calledpre-structural semantics. This method will be adopted in some extent in the present work and will be explained in what follows.
Definition 1. A first-order paraconsistent languageL is composed by:
• a set of individual variables;
• for eachn∈ω, a set of function symbols of arityn;
• for eachn∈ω, a set of predicate symbols of arityn;
• connectives: ¬,◦,∧,∨, →;
• quantifiers: ∃,∀;
• punctuation marks.
The only difference between paraconsistent and classical first-order languages is the presence, in the former, of the unary connective◦, so that the expression
‘paraconsistent language’ is used just to keep the distinction clear. As usual, given a first-order languageL, we assume it has at least one predicate symbol.
Function symbols of arity 0 are individual constants, and predicate symbols of arity 0 are propositional constants. The definitions of term, formula,2 and atomic formula are the usual ones, with the addition of the following clause: if ϕis a formula, so it is◦ϕ.
Definition 2. Letϕandψbe formulas. Ifϕcan be obtained fromψby means of addition or deletion of void quantifiers, or by renaming of bound variables, we say thatϕandψare variants of each other.3
The logicmbCwas introduced in [Carnielli, Coniglio and Marcos, 2007] as a “fundamental”LFI, meaning that its axioms embody a minimum proof capa- bility in order to preserve the positive theorems of classical propositional logic, and at the same time being capable of avoiding trivialization in the presence of contradictions. The axioms of mbCare 1 to 11 bellow. We set the following axioms toQmbC, which is the extension of mbCto first-order logic:
1) α→(β→α)
2) (α→β)→((α→(β→γ))→(α→γ)) 3) α→(β→(α∧β))
4) (α∧β)→α
1[da Costa and Alves 1977].
2We always put ‘formula’ for ‘well-formed formula.’
3See [Avron and Zamansky, 2006].
5) (α∧β)→β 6)α→(α∨β) 7)β →(α∨β)
8) (α→γ)→((β →γ)→((α∨β)→γ)) 9) α∨(α→β)
10)α∨ ¬α
11)◦α→(α→(¬α→β))
12)ϕ[x/t]→ ∃xϕ, iftis a term free forxin ϕ 13)∀xϕ→ϕ[x/t], iftis a term free forxin ϕ 14) Ifαis a variant ofβ, thenα→β is an axiom Rules:
• Modus ponens;
• Introduction of the universal quantifier: α → β / α → ∀xβ, if x isn’t free inα;
• Introduction of the existential quantifier: α →β / ∃xα→ β, if x isn’t free inβ.
Some observations concerning notation. Given a first-order paraconsistent languageL, the algebra of formulas determined by its signature is denoted by
‘F orL◦’, and ‘SL◦’ indicates the set of closed formulas ofL. ‘AF(L)’ and ‘AS(L)’
denote, respectively, the set of atomic formulas and the set of atomic sentences ofL. If ϕ[x1, ..., xn] is a formula with nfree variables, and t1, ..., tn are terms ofL, we put ‘ϕ[x1/t1, ..., xn/tn]’ for the result of substituting each termti for the variable xi, 1 ≤ i ≤ n. The interpretation of a term t (formula ϕ) in a pre-structureAis denoted by ‘A(t)’ (‘A(ϕ)’). If ‘f’ and ‘P’ are, respectively, a function symbol and a predicate symbol ofL, ‘fA’ and ‘PA’ denote, respectively, a function and a predicate in the (pre)structureA.
Definition 3. A pre-structureAforLis a pairh|A|, Iisuch that:
• |A|is a non-empty domain of individuals;
• Iis an interpretation function for each symbol ofL, as usual. That is, for eachn-ary function symbolf ofL,Ipicks out a functionfA:|A|n−→ |A|.
For eachn-ary predicate symbolP of L,I selects an n-ary predicatePA
in the domain, that is, a subset of|A|n.
As it is clear, a pre-structure is exactly what a structure is in first-order classical semantics. The need for this difference in terminology will be explained soon.
Definition 4. LetA=h|A|, Iibe a pre-structure for a languageL. L(A) is the language obtained fromL by addition, as new symbols, of the set of constants {i : i ∈ |A|}. A0 = h|A|, I0i is a pre-structure for L(A), such that I0 is an extension ofIsatisfying: A0(i) =i.
From now on a fixed paraconsistent first-order language L will be presup- posed, as well as a pre-structureA for it. Hence, the extended languageL(A) and the pre-structureA0 forL(A) are determined from the fixedL andA. We shall be always working withA0andL(A). (Sometimes with further extensions.) Definition 5. A QmbC-valuation based on the pre-structureA0 is a function v:SL(A)◦ −→ {0,1}satisfying:
1. v(P(t1, ..., tn) = 1⇔ hA0(t1), ...,A0(tn)i ∈PA0
2. v(α∨β) = 1⇔v(α) = 1 orv(β) = 1 3. v(α∧β) = 1⇔v(α) = 1 andv(β) = 1 4. v(α→β) = 1⇔v(α) = 0 orv(β) = 1 5. v(α) = 0⇒v(¬α) = 1
6. v(◦α) = 1⇒v(α) = 0 orv(¬α) = 0
7. v(∃xϕ) = 1⇔v(ϕ[x/t]) = 1 for some termtin L(A) 8. v(∀xϕ) = 1⇔v(ϕ[x/t]) = 1 for alltinL(A)
9. Ifϕis a variant ofψ, thenv(ϕ) =v(ψ)
In virtue of clauses 5 and 6 above, it is immediate to see that the following problem arises: given L(A) and A0, the mere assignment of values to all the atomic formulas ofL(A), according to clause 1, will not be sufficient per se to determine the values of all its formulas in general. In other words, oncev(ϕ) is assigned for all ϕ ∈ AS(L(A)), there are more than one way of extending that atomic valuation to the setSL(A)◦ . In fact, the next lema shows that there are infinite ways of extending an atomic valuation. Hence, in order to obtain an adequate semantics for the calculus we are investigating, we need a stronger notion than the mere notion of pre-structure.
Definition 6. The complexity of a formula is a functionl :F orL◦ −→N such that: l(ϕ) = 0, ifϕ∈ AF(L); l(ψ#ξ) =l(ψ) + l(ψ) + 1, for # ∈ {∨,∧,→};
l(¬ϕ) =l(ϕ) + 1;l(◦ϕ) = l(ϕ) + 2; andl(Qxϕ) =l(ϕ) + 1, forQ∈ {∀,∃}.
Lema 7. Let v0:AS(L(A)) ∪ {¬ϕ: ϕ∈ AS(L(A))} −→ {0,1} be a function satisfying clauses 1 and 5 of the preceding definition. Then there exists aQmbC- valuationv :SL(A)◦ −→ {0,1} based on A0 which extends v0, that is, such that v(ϕ)= v0(ϕ) for all formulasϕ∈ AS(L(A))∪ {¬ϕ : ϕ∈ AS(L(A))}.
Proof. Just let v(ϕ) = v0(ϕ) for all ϕ∈ AS(L(A)) ∪ {¬ϕ: ϕ ∈ AS(L(A))}, and put the values of formulas of greater complexity according to the clauses of Definition 5.
Definition 8. A QmbC-structure or simply a structure, if no confusion can arise for a first-order language L is a pair E =hA, vi such thatA is a pre-structure for L, and v is a QmbC-valuation or simply a valuation based onA.
Given an structure E forL (based onA), we define a truth-valueE(ϕ) for each closed formulaϕofL(A) in the following manner:
E(ϕ) =v(ϕ),
wherev is the valuation ofE=hA, vi. Ifϕis a formula ofL, anA-instance of ϕis defined in a way similar of that of [Shoenfield, 1967], p. 19: it is a closed formula of the form ϕ[x1/t1, ..., xn/tn] in L(A), such that t1, ..., tn are closed terms ofL(A).
Finally, we can define ‘validity’ according to our semantics.
Definition 9. A formulaϕof a languageLis said to be:
1. valid in the structureE, ifE(ϕ0) = 1 for allA-instanceϕ0 ofϕ;
2. valid in the pre-structureA, ifϕis valid in every structureEbased onA;
3. valid, ifϕis valid in every structureEforL.
As usual, the fact that a formula αis valid in every structure in which all formulas of a set Γ are also valid is indicated by ‘Γ|=QmbCα.’ Given the above definition of ‘validity,’ (in QmbC) it is routine to verify that the proposed semantics is sound: just check that all the axioms ofQmbCare valid and that the rules of inference preserve validity. Next we turn on the definitions which are necessary in order to demonstrate the completeness of our semantics.
Definition 10. Let ∆ ⊆ F or◦L(A) be a set, and E be a QmbC-structure for L(A) based on A0. We say that E is a model of ∆ if and only if E(δ) = 1 for everyδ∈∆. We indicate this fact by ‘E|=QmbC∆.’
Definition 11. A set of formulas ∆⊆F or◦Lis trivial inQmbCif and only if
∆`QmbC α, for every α∈F or◦L; otherwise ∆ is non-trivial.
Definition 12. A set ∆ whose formulas have as its language some extension L04ofLis a Henkin set if and only if (i) for all closed formula∃xϕ∈∆ there is a termt ofL0(A) such that∃xϕ→ϕ[x/t]∈∆, and (ii) ifϕ[x/t]∈∆ for every termt ofL0 free forxinϕ, then∀xϕ∈∆.
We ommit the proof of the following lema.
Lema 13. (α→β)→ ⊥ `QmbCα∧(β → ⊥).
4Typically, an extension by addition of constants. In order to keep generality, we consider a languageLas an extension of itself by adding the set∅of constants
Theorem 14. If ∆ ⊆ F or◦L is a non-trivial set of formulas in QmbC, then there is a setΓ⊇∆ of formulas ofL0, for some extensionL0 of L, such thatΓ is a Henkin set.
Proof. We follow [Henkin, 1949]. Let ∆0be some set satisfying the conditions of the theorem. As the formulas of ∆0are written in some languageL– which, by the way, will be denoted byL(∆0) –, we choose new constantsuακ (α,κarbitrary ordinals) which do not belong to L(∆0). If the language of the set ∆λ, to be constructed in the next paragraph, is already defined, the language of the set
∆λ+1 is built by the addition of the constants uακ. If λ is a limit-ordinal and L(∆σ) if already defined for allσ < λ, thenL(∆λ) = [
σ<λ
L(∆σ). For instance, the language of the set ∆1 is the one obtained fromL(∆0) by the addition of the constantsu1κ, andL(∆ω) = [
n<ω
L(∆n). LetL(∆V) be the language which contains all the constantsuακ of eachL(∆λ), that is,L(∆V) =[
L(∆λ).
Next, we inductively build a chain ∆λ of sets whose formulas have as their languageL(∆λ). Let ϕ0κbe the κ-th formula of ∆0 of the form∃xψ. Then we put ∆1as the set obtained from ∆0by adding (i) the formulaϕ0κ→ψ[x/u1κ], for each formulaϕ0κof the form∃xψof ∆0and all the constants u1κ ofL(∆1), and (ii) ifϕ[x/t]∈∆0for all termstofL(∆0) free forxinϕ, then we put∀xϕ∈∆1. The formula∃xψ →ψ[x/uακ] is called thespecial formula for the constantuακ. In general, if the set ∆λ is already constructed, and ifϕλκ is theκ-th formula of ∆λ of the form∃xψ, then ∆λ+1 is the set obtained from ∆λ by the addition of (i) the formulaϕλκ→ψ[x/uλ+1κ ], for each formulaϕλκ of the form∃xψ of ∆λ and all the constants uλ+1κ of L(∆λ+1), and (ii) if ϕ[x/t] ∈ ∆λ for all terms t of L(∆λ) free for xin ϕ, then ∀xϕ ∈ ∆λ+1. If λ is a limit-ordinal and ∆σ
is already constructed for all σ < λ, then ∆λ = [
σ<λ
∆σ. Finally, we postulate
∆V =[
∆λ. By construction, ∆V is a Henkin set such that ∆0⊆∆V.
It remains to be shown that ∆V is a non-trivial set in QmbC. If that was the case, then ∆V `QmbC ⊥∆0, for some bottom particle inL(∆0), hence also in L(∆λ), for all λ, because F or◦L(∆
0) ⊆F or◦L(∆
λ) ⊆F orL(∆◦
V). As the proof of⊥∆0 from premisses in ∆V involves just a finite number of formulas, and in particular a finite number of special formulas for the constantsuακ, it would be a proof of⊥∆0 from premisses in some ∆λ. Hence it is sufficient to show that all ∆λ are non-trivial, which is done by transfinite induction.
Suppose ∆λis non-trivial, but ∆λ+1`QmbC⊥∆0. By construction, ∆λ+1=
∆λ∪ {∃xψ →ψ[x/uλ+1κ ]}. Then ∆λ∪ {∃xψ →ψ[x/uλ+1κ ]} `QmbC ⊥∆0. By the deduction theorem, ∆λ`QmbC (∃xψ →ψ[x/uλ+1κ ])→ ⊥∆0. By the above lema, ∆λ`QmbC∃xψ∧(ψ[x/uλ+1κ ]→ ⊥∆0). Hence ∆λ `QmbC ψ[x/uλ+1κ ]→
⊥∆0. As the constant uλ+1κ is new, it follows that ∆λ `QmbC ψ[x] → ⊥∆0, and by existential introduction ∆λ`QmbC ∃xψ→ ⊥∆0. But∃xψ∈∆λ, hence
∆λ`QmbC⊥∆0.
If λ is a limit-ordinal, suppose ∆σ is non-trivial for all σ < λ. As ∆λ =
[
σ<λ
∆σ, if ∆λ is trivial, then, for some σ, ∆σ `QmbC ⊥∆0, contradicting the hypothesis.
Definition 15. Let Γ∪ {α} ⊆F orL(A)◦ be a set. Γ is said to be maximal with respect to α in QmbC if Γ6`QmbC α, and, for every formula β /∈Γ, it is the case that Γ, β `QmbC α. In which case Γ is also said to berelatively maximal (with respect to a given formulaα).
Lema 16. Let ∆∪ {α} ⊆F orL(A)◦ such that ∆6`QmbC α. Then there is a set Γ⊇∆ of formulas inL(A)which is maximal with respect toαin QmbC.
Proof. The proof is similar to that of Lindenbaum-Asser’s lema.
Lema 17. Let Lbe a paraconsistent first-order language, and∆∪ {α} a set of formulas inF or◦L such that∆is maximal with respect toαinQmbC. Then ∆ is a closed theory.5
Proof. The proof of this lema, formbC, can be found in [Carnielli, Coniglio and Marcos 2007]. Its adaptation toQmbChas no difficulties. Details are ommited in virtue of space.
Together, lema 16 and theorem 14 guarantee that every set of formulas in L(A), which is maximal with respect to some formula in QmbC(and hence is non-trivial), can be extended to a Henkin set which is maximal with respect to the same formula. The language of formulas contained in this Henkin set, ob- tained fromL(A) according to theorem 14, will be denoted by ‘LH(A)’, and the set of the formulas generated by this language will be designated by ‘F orL◦
H(A)’.
The following lema indicates some important properties of maximal Henkin sets inQmbC.
Lema 18. Let ∆⊂F or◦L
H(A) a Henkin set which is maximal with respect to a formulaα∈F or◦L
H(A) inQmbC. Hence:
(i) (β∧γ)∈∆⇔β∈∆ andγ∈∆ (ii) (β∨γ)∈∆⇔β∈∆ orγ∈∆ (iii)(β →γ)∈∆⇔β /∈∆ or γ∈∆ (iv)β /∈∆⇒ ¬β ∈∆
(v)◦β∈∆⇒β /∈∆or ¬β /∈∆
(vi)∃xϕ∈∆⇒ϕ[x/t]∈∆ for some term t inLH(A) (vii) ∀xϕ∈∆⇒ϕ[x/t]∈∆ for every termtin LH(A) (viii) If ϕis a variant ofψ: ϕ∈∆⇒ψ∈∆
5If Γ is a set of formulas and ∆ is the set of formulas which are consequences of Γ (in some logic), then Γ is a closed theory if and only if Γ = ∆. See [Tarski, 1930], p. 33, where closed theories are called “deductive systems.”
Proof. Item (i) is a consequence of the closure of ∆ (previous lema), axioms 3, 4 and 5, andmodus ponens. Item (ii) is a consequence of the closure of ∆, axioms 6, 7 and 8, andmodus ponens. Item (iii) is a consequence of the closure of ∆, item (ii), axioms 1 and 9, andmodus ponens. Item (iv) is a consequence of the closure of ∆, axiom 10, and modus ponens. For (v), suppose that β ∈ ∆ and
¬β ∈∆; then, for closure, relatively maximality of ∆ and axiom 11, it can be concluded that◦β /∈∆. Items (vi) and (vii) are consequences of the closure of
∆, axioms 12 and 13,modus ponens, and the fact that ∆ is a Henkin set.
Theorem 19. Using the characteristic function of a Henkin set∆⊂F or◦L
H(A), which is relatively maximal inQmbC, it is possible to define a pre-structureB for LH(A) and QmbC-valuation vH based on B such that vH(δ) = 1 if and only ifδ∈∆.
Proof. We employ the method of [Henkin, 1949]. A pre-structureB forLH(A) and aQmbC-valuationvHbased onBare constructed in the following manner.
The domain|B|ofBis composed by the individual constants ofLH(A), which, by construction, has an infinite number of them. For each individual constantc ofLH(A), we defineB(c) =c. For alln-ary function symbolf,n≥1, and all the closed termst1, ..., tn of LH(A), we define fB(B(t1), ...,B(tn)) = f(t1, ..., tn).
For each n-ary predicate symbol P of LH(A), we define the predicate PB as the set of then-tuples ht1, ..., tni such thatP(t1, ..., tn)∈ ∆. It is immediate, for the definition of each PB in B, that ∆ `QmbC P(t1, ..., tn) if and only if P(t1, ..., tn) ∈ ∆ (since ∆ is a closed theory), if and only if ht1, ..., tni ∈ PB (for the definition of the predicate PB). Since, by clause 1 of definition 5, ht1, ..., tni ∈ PB if and only if v(P(t1, ..., tn)) = 1, we conclude that, for all atomic formulaϕ=P(t1, ..., tn)∈∆,vH(ϕ) = 1.
It remains to definevH for all the formulas of ∆. Ifϕhas complexitynand vH(ϕ) is already defined, we putvH(ϕ0) = 1 if and only ifϕ0 ∈ ∆, for all ϕ0 having complexityn+ 1. Since ∆ is a relatively maximal Henkin set, in virtue of lema 18,vH satisfies all the clauses of definition 5.
Corolary 20. The structureH= hB, vHi is a model of∆.
Theorem 21. Completeness. LetΓ∪ {α} be a set inL(A). HenceΓ|=QmbCα implicaΓ`QmbCα.
Proof. Let α∈ F orL(A)◦ such that Γ 6`QmbC α. For theorem 14 and lema 16, it is possible to extend Γ to a Henkin set Γ∗ which is relatively maximal with respect to α in QmbC. As Γ∗ 6`QmbC α, then α /∈ Γ∗. By theorem 19, it is possible to construct a pre-structure B and a QmbC-valuation vH based on B such that the structure H = hB, vHi is a model of Γ∗, and vH(ϕ) = 1 if and only ifϕ∈Γ∗. Hence,H|=QmbC Γ∗, and in particularH|=QmbC Γ; but H6|=QmbCα, which means that Γ6|=QmbCα.
2 Comments and perspectives
QmbCis a simpleLFI. A natural way to reach stronger logics is by addition of axioms concerning e.g. double negations or “consistency propagation”. By means of combinations of such axioms, one obtains a rich variety of paracon- sistent calculi having interesting properties: for example, the definition of some propositional connectives in terms of others.6 Consider the following schemata:
(A) (◦α∧ ◦β)→ ◦(α#β) (B) (◦α∨ ◦β)→ ◦(α#β)
#∈ {∨,∧,→}. If one adds axiom (A) or (B) to the remaining axioms ofQmbC, he obtains two different kinds of “consistency propagation.” In case (A), in order that consistency can “propagate” from simple to more complex formulas, it is sufficient that both of its immediate subformulas are consistent; in case (B), it is enough that one of its subformulas be consistent. To see how the method developed here can be extended in order to obtain a complete semantics to the entire class of first-orderLFIs, letQmbC-A the logic obtained by adding (A) to the axioms of QmbC. By adding the following clause to the definition 5
(A*) v(◦α∧ ◦β) = 1⇒v(◦(α#β)) = 1
one obtains aQmbC-A-valuation. After a suitable definition of ‘structure,’ one reaches aQmbC-A-structure, and then proceeds as in the case of QmbCto establish a completeness theorem for this new calculus.
The possibility of obtaining a semantic characterization to the entire class of first-orderLFIsis an interesting general result, since in this way we open up the way to a model-theoretical treatment to such logics. In principle, there is not much difficulty in variants of Compactness and Lowenheim-Skolem theorems for such logics, and from this ground many results in classical model theory can be regained.
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