Gently Paraconsistent Calculi

In this paper, we consider some paraconsistent calculi in a Hilbert-style formulation with the rule of detachment as the sole rule of interference. Each calculus will be expected to contain all axiom schemas of the positive fragment of classical propositional calculus and respect the principle of gentle explosion.


Introduction
The principle of explosion states that from any set {α, ¬α} of contradictory formulas any other formula β follows. Paraconsistent logic can be described as a logic in which the principle does not hold. The 'definition' is very simple, but it is also very broad. This may lead to some ambiguity and cause interpretive problems, especially if we aim to draw a sharp distinction between paraconsistent and some other nonclassical logics. In this paper, we discuss some possible consequences of the definition. We examine several paraconsistent calculi that respect the so-called principle of gentle explosion, according to which from any set {α, ¬α, ¬¬α} of formulas any other formula β follows. The calculi (of paraconsistent logic) that admit the principle will be called gently paraconsistent.
Let var denote a denumerable set of all propositional variables: p 1 , p 2 , p 3 , etc. The set F of formulas is defined in the standard way using propositional variables from var and the symbols ¬, ∨, ∧ and → for negation, disjunction, conjunction and implication, respectively. The connective of equivalence, α ↔ β, is treated as an abbreviation for (α → β) ∧ (β → α), and hence it will be omitted. We say that a formula α is atomic, if α ∈ var; otherwise α will be called complex. By literals we mean the set LI of all formulas of the form ¬ k p i , where i ∈ N, k ∈ N ∪ {0} and p i ∈ var (if k = 0, then ¬ 0 p i = p i ; if k = 1, then ¬ 1 p i = ¬p i ; etc.). We use lowercase Greek letters for formulas and uppercase Greek letters for subsets of F . In F , we will consider axiomatic propositional calculi in a Hilbert-style formulation with the rule of detachment, (MP) α → β, α / β, as the sole rule of interference. Each calculus C discussed in this paper is expected to have all axiom schemas of the positive fragment of classical propositional calculus (CPC + , for short), that is, all instances of the following schemas: and include the law of gentle explosion: (DS 2 ) α → (¬α → (¬¬α → β)).
Definition 1. For C, any α ∈ F and any Γ ⊆ F , we say that α is provable from Γ within C (in symbols: Γ C α) if, and only if there is a finite sequence of formulas, β 1 , β 2 , . . . , β n such that β n = α and for each i n, either β i ∈ Γ, or β i is an axiom of C, or for some j, k i, we have β k = β j → β i . A formula α is a thesis of C (in symbols: ∅ C α) iff α is provable from ∅ within C (henceforth, we will use iff as shorthand for if, and only if).
Notice that each calculus C may be identified with a triple F , Ax C , C , but it is determined by its set of axioms Ax C which is included in F . Moreover, it can be verified that C is a finitary consequence relation satisfying the so-called Tarskian properties (viz. reflexivity, monotonicity and transitivity).
The deduction theorem holds for any calculus having (MP) as the sole rule of inference, and (A1), (A2) as its axiom schemas. Thus we have Theorem 1. For any Γ ⊆ F and α, β ∈ F : It follows from (A9), the deduction theorem and (MP) that the following lemma holds as well: Lemma 2. For any Γ, ∆ ⊆ F and α, β, γ ∈ F : if Γ ∪ {α} C γ and Γ ∪ {β} C γ, then Γ ∪ {α ∨ β} C γ. Remark 1. The following formulas are provable in CPC + : The formulas will be useful for proving the results presented below.

Gently Paraconsistent Calculi
The set of axiom schemas of C enriched with (ExM) α ∨ ¬α and (NN2) α → ¬¬α yields the axiom system of classical propositional calculus (in short: CPC). From the viewpoint of paraconsistency, neither (ExM) nor (NN2) seems to be controversial, and therefore they could be generally accepted. There is a problem, however, in admitting (DS 2 ) and (NN2) simultaneously. This is because the pair of formulas is equivalent, on the grounds of CPC, to (DS) α → (¬α → β). The latter, being viewed as a highly contentious logical law, should be rejected. Not surprisingly then, (NN2) cannot be universally accepted either. On the other hand, the formula (NN1) ¬¬α → α appears to be more applicable than (NN2), in the sense that its application does not need to be limited, for example, to certain complex formulas (cf. Sections 2.2 and 2.5).

The Calculus A1
The basic gently paraconsistent calculus is A1. The calculus is defined by the axioms of CPC + , (DS 2 ) and (MP) as the sole rule of inference. It is a proper subsystem of the paracomplete logic CLaN. The CLaN, as proposed in [3], is axiomatized by CPC + , (DS) and (MP). The calculus A1 may be seen as an example of a paranormal calculus (in Miró Quesada's terminology), that is, a calculus which is both paraconsistent and paracomplete. Some other examples of the paranormal calculi are given in [4,5].

Definition 2.
An A1-valuation is any function v : F −→ {1, 0} that satisfies, for any α, β ∈ F , the following conditions: The proof of soundness proceeds by induction on the length of a derivation in A1. To prove the completeness, we apply the method which is based on the notion of maximal nontrivial sets of formulas (see [6,7]). To begin with, let us recall some important definitions and results. Definition 4. Let C = F , Ax C , C be a calculus (satisfying Tarskian properties) and ∆ ⊆ F . We say that ∆ is a closed theory of C iff for any β ∈ F , we have ∆ C β iff β ∈ ∆. We say that ∆ is maximal nontrivial with respect to α ∈ F in C iff (i) ∆ C α and (ii) for every β ∈ F , if β ∈ ∆ then ∆ ∪ {β} C α. Lemma 3 ([6], Lemma 2.2.5.). Every maximal nontrivial set with respect to some formula is a closed theory.
Proof. We need to prove that the mapping v is an A1-valuation. The proof splits into a number of cases. The case (∧) follows directly from the definition of ( ), the axioms (A4)-(A6) and Lemma 1; the case (∨) from ( ), (A7)-(A9) and Lemma 1.
Notice that the so-called Lindenbaum-Łoś theorem holds,for any finitary calculus C = F , Ax C , C .
Proof. Assume that Γ A1 α and let ∆ be a maximal nontrivial set with respect to α in A1 such that Γ ⊆ ∆. Then, α ∈ ∆. Because Lemma 4 holds, there is an A1-valuation v such that v(α) = 0 and, for any Though the calculus A1 is very weak and does not provide any adequate grounds for practical inference, it offers a good starting point for further research. In the subsequent paragraphs, we will discuss various gently paraconsistent extensions of A1.
The proof of soundness is by induction on the structure of proofs in E1. The completeness proof strategy is exactly the same as that of the proof of Theorem 3. The key point is to show that the following lemma holds: Lemma 6. For any maximal nontrivial set ∆ with respect to α in E1, any δ ∈ F , the mapping v : F −→ {1, 0} defined as ( ): v(δ) = 1 iff δ ∈ ∆, is an E1-valuation.
Another example is the paranormal logic I 1 P 1 . The logic was considered in [4,[8][9][10]  An I 1 P 1 -valuation is any function v : F −→ {1, 2, 3, 0} compatible with the above truth tables. An I 1 P 1 -tautology is a formula which under every valuation v takes on the designated values {1, 2}.

The Calculus B1
The calculus B1 is obtained from A1 by adding the formula (ExM) as a new axiom schema, which indicates that B1 is axiomatizable by CPC + , (DS 2 ), (ExM) and (MP). Since the law of excluded middle is unprovable in A1, we obviously have that A1 B1. The calculus B1 was considered in [12,13] as the strongest in the hierarchy of B n -calculi (n ∈ N). Definition 8. A B1-valuation is any function v : F −→ {1, 0} that satisfies, for any α, β ∈ F , the following conditions:
BE1 and BE1 CB1. There exists, however, some paraconsistent calculi in which (DS ¬ ) does not fail (see [15], for discussion on the topic). The example of such a calculus is P1.
Sette's calculus is maximal with respect to CPC (see [16], pp. 179-180). Consequently, it is the top extension of all gently paraconsistent calculi discussed in this paper.

Final Remarks
We considered several paraconsistent calculi that admitted the principle of gentle explosion, namely  It is noteworthy that some well-known (not-gently) paraconsistent logics can be obtained by eliminating (DS 2 ) from the axiom schemas. For instance, dropping (DS 2 ) from B1 results in obtaining the logic CLuN (see [3], for details); dropping (DS 2 ) from CB1 results in obtaining in the logic Cmin(see [23]) . The calculi form together the lattice structure shown in Figure 2.

Conflicts of Interest:
The authors declare no conflict of interest.