The set of axiom schemas of
enriched with (
)
and (
)
yields the axiom system of classical propositional calculus (in short:
). From the viewpoint of paraconsistency, neither (
) nor (
) seems to be controversial, and therefore they could be generally accepted. There is a problem, however, in admitting (
) and (
) simultaneously. This is because the pair of formulas is equivalent, on the grounds of
, to (
)
. The latter, being viewed as a highly contentious logical law, should be rejected. Not surprisingly then, (
) cannot be universally accepted either. On the other hand, the formula (
)
appears to be more applicable than (
), in the sense that its application does not need to be limited, for example, to certain complex formulas (cf.
Section 2.2 and
Section 2.5).
2.1. The Calculus
The basic
gently paraconsistent calculus is
. The calculus is defined by the axioms of
, (
) and (MP) as the sole rule of inference. It is a proper subsystem of the paracomplete logic
. The
, as proposed in [
3], is axiomatized by
, (
) and (MP). The calculus
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 -valuation is any function that satisfies, for any , the following conditions:
(∧) = 1 iff = 1 and = 1
(∨) = 1 iff = 1 or = 1
(→) = 1 iff = 0 or = 1
() if = = 1, then = 0.
Definition 3. A formula α is an -tautology iff for every -valuation v, = 1. For any and , α is a semantic consequence of (in symbols: ) iff for any -valuation v: if = 1 for any , then = 1.
Theorem 2. For every and : if then .
The proof of soundness proceeds by induction on the length of a derivation in
. 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 be a calculus (satisfying Tarskian properties) and . We say that is a closed theory of iff for any , we have iff . We say that is maximal nontrivial with respect to in iff (i) and (ii) for every , if then .
Lemma 3 ([
6], Lemma 2.2.5)
. Every maximal nontrivial set with respect to some formula is a closed theory. Observe that the lemma holds for . Additionally, we have
Lemma 4. For any maximal nontrivial set with respect to α in , any , the mapping defined as (): = 1 iff , is an -valuation.
Proof. We need to prove that the mapping v is an -valuation. The proof splits into a number of cases. The case (∧) follows directly from the definition of (), the axioms – and Lemma 1; the case (∨) from (), – and Lemma 1.
Case (→): (if–then) Assume, for a contradiction, that = 1, ) = 1 and ) = 0. Then, by (), we have that , and . Now, by Lemma 1, we get , , that is, . The formula () is a thesis of and the deduction theorem holds, so . Since the relation is transitive (see Lemma 1), then , which means that . However, . This entails a contradiction.
(then–if) There are two subcases to consider. Subcase (i): Suppose, for a contradiction, that ) = 0 and = 0. This implies that and , by (). Since is a maximal nontrivial set with respect to , then and . Hence, , , by the deduction theorem, and consequently, . Observe that () is a thesis of , so , by the deduction theorem. The relation is transitive, and therefore . Since is deductively closed, then . However, , by the main assumption. This yields a contradiction. Subcase (ii): Suppose that ) = 1. Then, by (), we get . This implies, by Lemma 1, that . Since is an axiom schema of , then, by the deduction theorem, we have . The relation is transitive, and hence . If , then , which means that = 1.
Case (): Assume, for a contradiction, that = 1, = 1 and = 1. Then by (), we have , and . By Lemma 1, we obtain , and . This implies that . Since () is an axiom schema of , then , by the deduction theorem. The relation is transitive, so . Observe that is deductively closed, then . However, by the main assumption, . This entails a contradiction. □
Notice that the so-called Lindenbaum–oś theorem holds, for any finitary calculus = .
Lemma 5 ([
2], Theorem 3.31; [
6], Theorem 2.2.6)
. For any and such that , there is a maximal nontrivial set with respect to α in such that . Thus, the completeness of follows
Theorem 3. For all and : if , then .
Proof. Assume that and let be a maximal nontrivial set with respect to in such that . Then, . Because Lemma 4 holds, there is an -valuation v such that = 0 and, for any , = 1. Hence, . □
Though the calculus 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 paraconsistent extensions of .
2.2. The Calculus
The calculus
is defined by
, (
), (
)
, where
, and (MP). There are only few paraconsistent calculi in which (
) is provable. One of them is Sette’s calculus
. Anticipating what comes next in
Section 2.5, Sette’s calculus will be the
top paraconsistent extension of the calculi admitting (
) and (
), simultaneously.
Definition 5. An -valuation is any function that, for any , satisfies all the conditions of -valuation and, additionally: () if = 1, then = 0, where .
Definition 6. A formula α is an -tautology iff for every -valuation v, = 1. For any and , α is a semantic consequence of (in symbols: ) iff for any -valuation v: if = 1 for any , then = 1.
Theorem 4. For every and : iff .
The proof of soundness is by induction on the structure of proofs in . 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 , any , the mapping defined as (): = 1 iff , is an -valuation.
Proof. Case (∧), (∨), (→) and (): The proof proceeds analogously to that of Lemma 4. Case (): Assume, for a contradiction, that = 1 and = 1, where . Then by (), we have and . It follows from Lemma 1 that and , which results in . By () and the deduction theorem, we easily show that . The relation is transitive, so . Since is deductively closed, then . However, (the main assumption). This entails a contradiction. □
Definition 7. Let be the set of all theses of . For any calculi and in , we say that is an extension of iff . We say that is a proper subsystem of (in symbols: ) iff and .
Remark 2. .
There is an alternative way to extend so that the resulting calculus preserves () as provable. Let be the calculus defined by , (), () , where , and (MP). It then follows from the deduction theorem, (), () and (MP) that () is a thesis of . Note that of the form () is not an -tautology. So, by completeness, it is not provable in , either. This suggests that the new calculus is strictly stronger than , i.e., .
Another example is the paranormal logic
. The logic was considered in [
4,
8,
9,
10]. It is characterized by the four-valued matrix
where
and
are the sets of logical values and designated values, respectively; the connectives
are defined in the following way:
| → | 1 | 2 | 3 | 0 | 1 | 1 | 1 | 0 | 0 | 2 | 1 | 1 | 0 | 0 | 3 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 |
| ∧ | 1 | 2 | 3 | 0 | 1 | 1 | 1 | 0 | 0 | 2 | 1 | 1 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| ∨ | 1 | 2 | 3 | 0 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 3 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
|
An -valuation is any function compatible with the above truth tables. An -tautology is a formula which under every valuation v takes on the designated values .
All axioms of are -tautologies, and (MP) preserves tautologicality. However, the formula , being an -tautology, is unprovable in . This yields that .
A slightly different example is with
. The logic, as proposed in [
5,
11], is axiomatizable by (
), (
), (
)–(
), (
)
, where
, and (MP). The formula
of the form (
) is unprovable in
, and neither is
of the form (
) provable in
, then we have that
and
.
2.3. The Calculus
The calculus
is obtained from
by adding the formula (
) as a new axiom schema, which indicates that
is axiomatizable by
, (
), (
) and (MP). Since the law of excluded middle is unprovable in
, we obviously have that
. The calculus
was considered in [
12,
13] as the strongest in the hierarchy of
-calculi (
).
Definition 8. A -valuation is any function that satisfies, for any , the following conditions:
(∧) = 1 iff = 1 and = 1
(∨) = 1 iff = 1 or = 1
(→) = 1 iff = 0 or = 1
() if = 0, then = 1
() if = 1, then = 0 or = 0.
Definition 9. A formula α is a -tautology iff for every -valuation v, = 1. For any and , α is a semantic consequence of (in symbols: ) iff for any -valuation v: if = 1 for any , then = 1.
Theorem 5. For every and : iff .
The completeness proof is carried out similarly as for the calculus
. In addition to Lemmas 3 and 5 given in
Section 2.1, the following lemma is of particular importance:
Lemma 7. For any maximal nontrivial set with respect to α in , any , the mapping defined as (): = 1 iff , is a -valuation.
Proof. Case (∧), (∨), (→) and (): We proceed analogously to the proof of Lemma 4. Case (): Assume, for a contradiction, that = 0 and = 0. Then by (), we obtain and . Since is a maximal nontrivial set with respect to , then and . By Lemma 2, we get . Since () is a thesis of and Lemma 1 holds, then . Recall that is a closed theory, so . However, . This entails a contradiction.
Case (): Suppose, for a contradiction, that = 1, = 1 and = 1. The remaining part of the proof is similar to the case () of Lemma 4 and thus omitted. □
As
of the form (
) is not a thesis of
and
of the form (
) is not provable in
, it follows that
and
.
2.4. The Calculi and
The calculus comprises the axioms of , (), (), () and (MP), which clearly yields that both and . The is an example of calculus which is paraconsistent only at the level of literals: a pair of the formulas and yields any iff is not a propositional variable nor is its iterated negation.
Definition 10. A -valuation is any function that satisfies, for any , all the conditions of -valuation and additionally: () if = 1, then = 0, where .
Definition 11. A formula α is a -tautology iff for every -valuation v, = 1. For any and , α is a semantic consequence of (in symbols: ) iff for any -valuation v: if = 1 for any , then = 1.
Theorem 6. For every and : iff .
Proof. The proof proceeds as in Theorems 2–4. □
The calculus
was introduced in [
14]. It arose as a result of the extension of
with the law of double negation (
)
, which suggests that
. Moreover, we have
Remark 3. The calculus is axiomatizable by , (), () and (MP).
Proof. See op. cit., p. 227, for details. □
Definition 12. A -valuation is any function that satisfies, for any , the following conditions:
(∧) = 1 iff = 1 and = 1
(∨) = 1 iff = 1 or = 1
(→) = 1 iff = 0 or = 1
() if = 0, then = 1.
() if = 1, then = 0.
Definition 13. A formula α is a -tautology iff for every -valuation v, = 1. For any and , α is a semantic consequence of (in symbols: ) iff for any -valuation v: if = 1 for any , then = 1.
Theorem 7. For every and : iff .
Proof. We refer the reader to op. cit., pp. 230–231, for details. □
Since
of the form (
) is not a thesis of
and
of the form (
) is not provable in
, it follows that
and
. There exists, however, some paraconsistent calculi in which (
) does not fail (see [
15], for discussion on the topic). The example of such a calculus is
.
2.5. Sette’s Calculus
The calculus
, proposed in [
16], is defined in the language with negation and implication as primitives by (
), (
), (
)
, (
)
, (
)
. The sole rule of inference is (MP). Some alternative axiomatizations of
have been developed since then (see e.g., [
9,
17,
18,
19,
20,
21,
22]).
Sette’s calculus is sound and complete with respect to the matrix
, where
and
are the sets of logical and designated values, respectively. The connectives of → and ¬ are determined by the following truth tables (cit. per [
16], p. 176):
A
-valuation is any function
v from the set of formulas to the set of logical values, i.e.,
, compatible with the above truth tables. A
-tautology is a formula which under every valuation
v takes on the designated values
. Conjunction and disjunction are definable connectives:
;
(cit. per [
9,
20]).
Remark 4. The calculus is axiomatizable by , (), (), (), where , and (MP).
Proof. The proof splits into two steps. To show that the axioms (
)–(
), (
), (
) and (
) are
-tautologies, and (MP) preserves tautologicality, it suffices to apply the three-valued semantics for
(plus the definitions of ‘missing’ connectives). Next we need to demonstrate that (
), (
) and (
) are provable in the proposed axiomatization. This in turn follows from the results of [
17], pp. 270–272, and [
19], pp. 1111–1113. □
Sette’s calculus is maximal with respect to
(see [
16], pp. 179–180). Consequently, it is the top extension of all gently paraconsistent calculi discussed in this paper.