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44Логические исследования 2019. Т. 25. № 2. С. 46-60 УДК 510.644
Logical Investigations 2019, Vol. 25, No. 2, pp. 46-60 DOI: 10.21146/2074-1472-2019-25-2-46-60
JANUSZ ClUOIURA
Paraconsistency and Paracompleteness
Janusz Ciuciura
University of Lodz, Faculty of Philosophy and History, Institute of Philosophy, 3/5 Lindleya Str., 90-131 Lodz, Poland. E-mail: [email protected]
Abstract: A logic (L, Kp) is said to be paraconsistent if, and only if {a, —a} 3, for some formulas a, /. In other words, the necessary and sufficient (the latter is problematic) condition for a logic to be paraconsistent is that its consequence relation is not explosive. The definition is very simple but also very broad, and this may create a risk that some logics, which have not too much in common with the paraconsistency, are considered to be so. Nevertheless, the definition may still serve as a reasonable starting point for more thorough research.
Paracomplete logic can be defined in many different ways among which the following one may be of some interest: A logic (L, hq) is said to be paracomplete if, and only if {/ ^ a, ^ a} a, for some formulas a, /3. But again, just as in the case of paraconsistent logic, the definition is very general and may be seen to overlap with the logics that have nothing in common with the paracompleteness.
In the paper, we define some calculi of paraconsistent and paracomplete logics arranged in the form of hierarchies, determined by several criteria. We put central emphasis on logical axioms admitting only the rule of detachment as the sole rule of inference and on the so-called bi-valuation semantics. The hierarchies (no matter which one) are expected to shed some light on the aforementioned issue.
Keywords: Paraconsistent logic, paraconsistency, paracomplete logic, paracompleteness
For citation: Ciuciura J. "Paraconsistency and Paracompleteness", Logicheskie Issledo-vaniya / Logical Investigations, 2019, Vol. 25, No. 2, pp. 46-60. DOI: 10.21146/20741472-2019-25-2-46-60
1. Introduction
Let var denote a (non-empty) denumerable set of all propositional variables. The set of formulas F is inductively defined (in Backus-Naur form) as follows:
f ::= p | —a | a V a | a A a | a ^ a,
where p e var, a e F and the symbols —, V, A, ^ denote negation, disjunction, conjunction and implication, respectively.1
xThe connective of equivalence, a o 3, is treated as an abbreviation for (a ^ 3)A(3 ^ a).
© Ciuciura J.
A logic (L, hp) is said to be paraconsistent if, and only if {a, —a} £, for some formulas a, £. In other words, the necessary and sufficient condition for a logic to be paraconsistent is that its consequence relation is not explosive. Despite this definition being very broad and open to criticisms on a number of fronts, it may still serve as a good basis for further discussion. In what follows, however, we will need some additional criteria to be considered valid, i.e.
1. the principles of explosion: a ^ (—a ^ £), (a A —a) ^ £
2. the law of contradiction: —(a A —a)
3. laws of double negation: ——a ^ a, a ^ ——a
must not be provable in any calculus of the hierarchy.2 2. Paraconsistent Calculus B1
The paraconsistent calculus B1 is defined, in Hilbert-style formalization, by the following axiom schemas:
(A1) a ^ (£ ^ a)
(A2) (a ^ (£ ^ 7)) ^ ((a ^ £) ^ (a ^ 7))
(A3) ((a ^ £) ^ a) ^ a
(A4) (a A £) ^ a
(A5) (a A £) ^ £
(A6) a ^ (£ ^ (a A £))
(A7) a ^ (a V £)
(A8) £ ^ (a V £)
(A9) (a ^ y) ^ ((£ ^ y) ^ (a V £ ^ 7)) (ExM) a V —a
(DS2) a ^ (—a ^ (——a ^ £)).
The sole rule of inference is Detachment Rule, (MP) a ^ £, a / £.3 It is easily seen that B1 contains the positive fragment of Classical Propositional Calculus. Besides, it is an extension of PI.4
2For modern discussion on the topic, see e.g. [Avron et all, 2018] (Chapter 2) and [Carnielli, Coniglio, 2016] (Chapter 1).
3The calculus B1 originally appeared in [Ciuciura, 2014] as mbC1. The abbreviation mbC1 was chosen to emphasise that the calculus mbC proposed by Carnielli, Coniglio and Marcos (see [Carnielli et all, 2007, pp. 37-41]) was a logical inspiration for mbC 1/B1. We should stress, however, that mbC 1/B1 is not equivalent to mbC. See [Ciuciura, 2018, p. 140], for details.
4In literature, the logic PI is also known as CLuN. See [Batens, 1980, pp. 204-205], and [Batens, De Clercq, 2004, p. 229].
Definition 1. Let a G F and r CJ .A formula a is provable from r within B1 (r hbi a, in symbols) iff there is a finite sequence of formulas, £1, £2,..., £n such that £n = a and, for each i < n, at least one of the following is true:
1. £i G r
2. £i is an axiom of B1
3. £i is obtained from some of the previous £j by application of the rule of (MP).
Definition 2. A formula a is a thesis of B1 iff 0 h#i a.
Before going any further, we should demonstrate that B1 meets the criteria specified in the clauses 1, 2 and 3 of Introduction. With that in mind, let us consider the matrix
M1 = <{1, 2, 0}, {1, 2}, -, A, V, ^ ),
where {1,2,0} is the set of logical values, {1,2} is the set of designated values and the connectives -, A, V, ^ are defined by the tables5:
1 2 0 —1
1 1 1 0 1 2
2 1 1 0 2 0
0 1 1 1 0 1
A 1 2 0 V 1 2 0
1 1 1 0 1 1 1 1
2 1 1 0 2 1 1 1
0 0 0 0 0 1 1 0
Each axiom schema of B1 is valid in the matrix M1 and (MP) preserves validity. The principles of explosion take the non-designated value 0: 1 ^ (-1 ^ 0) = 1 ^ (2 ^ 0) = 1 ^ 0 = 0, (1 A -1) ^ 0 = (1 A 2) ^ 0 = 1 ^
0 = 0; so do the laws of double negation: --0 ^ 0 = -1 ^ 0 = 2 ^ 0 = 0,
1 ^ --1 = 1 ^ -2 = 1 ^ 0 = 0 and the law of (non-)contradiction: -(1 A —1) = -(1 A 2) = -1=0.
Lemma 1. For every r, A CJ and a, £, 7 G F, we have:
1. if a G r, then r bbi a
2. if r C A and r K^i a, then A K^i a
5The truth tables for the binary connectives were of interest to many logicians, e.g. da Costa [da Costa, 1974, p. 499] and Sette [Sette, 1973, pp. 176, 179]. The three-valued table for negation was presented in [Post, 1921, pp. 180-181] (the so-called cyclic negation) and [Slupecki, 1939, p. 112].
3. if A l-£i a and, for every ft e A such that r -bi ft, then r -bi a
4. if r U {a} -bi 7 and r U {ft} -bi 7, then r U {a V ft} -bi 7
5. if r U {a} -bi 7 and A -bi a, then r U A -bi 7
(in particular, if r U {a} -bi Y and 0 -bi a, then r -bi Y)
6. r -Bi a iff for some finite A C r, A -gi a.
Proof. The proof is similar to that of the classical case. We refer the reader to [Wojcicki, 1988] and [Pogorzelski, Wojtylak, 2008] for details. ■
Theorem 1. Deduction theorem holds for B1.
Proof. It suffices to observe that B1 includes (A1), (A2) and the sole rule of inference in B1 is (MP). ■
Theorem 2. Some (weaker) variants of the indirect deduction theorem hold for B1, i.e.
1. if r U {a} -bi {ft, —ft, ——ft}, then r -bi -a
2. if r U {—a} -bi {ft, —ft, ——ft}, then r -bi a.
Proof. First, notice that (a ^ ft) ^ ((a ^ —ft) ^ ((a ^ ——ft) ^ —a)) and (—a ^ ft) ^ ((—a ^ —ft) ^ ((—a ^ ——ft) ^ a)) are provable in B1. Now assume that r U {a} -bi {ft, —ft, ——ft}. Then, by the deduction theorem, we obtain that r -bi {a ^ ft, a ^ —ft, a ^ ——ft}. Since 0 -bi (a ^ ft) ^ ((a ^ —ft) ^ ((a ^ ——ft) ^ —a)), so {a ^ ft, a ^ —ft, a ^ ——ft} -bi —a. The relation -bi is transitive (by Lemma 1), therefore r -bi —a.
The second item can be proved analogously. ■
Remark 1. The formulas
1. —(a A —a A ——a)
2. a ^ (—a ^ ———a)
3. ——a ^ —(a A —a)
4. (a ^ ft) ^ (—a V ft) are provable in B1.
Remark 2. B1 is not maximal with respect to P1.6
Proof. Let CB1 denote the calculus obtained from B1 by adding a new axiom schema, that is, ——a ^ a. Note that the axiom is a theorem of P1 [Sette, 1973, pp. 174-175] and so are all axioms of B1 [Ciuciura, 2018, pp. 116-120].
6Sette used — and ^ as primitive connectives. Other connectives, viz. conjunction, disjunction and equivalence, were introduced through definitions. See [Sette, 1973, pp. 178-179].
We show now that the formula (a ^ ß) ^ --(a ^ ß)7 is unprovable in CB1. To demonstrate this fact, employ the matrix
M2 = <{1, 2, 0}, {1, 2}, -, A, V, ^ ),
where {1, 2, 0} consists of the logical values, {1,2} contains the designated values and the connectives -, A, V, ^ are defined by the truth tables:
120
1 1 2 0 1 0
2 1 2 0 2 1
0 1 1 1 0 1
A 1 2 0 V 1 2 0
1 1 1 0 1 1 1 1
2 1 1 0 2 1 1 1
0 0 0 0 0 1 1 0
All axiom schemas of CB1 are valid in the matrix M2 and (MP) preserves validity. Observe that the formula (a ^ £) ^ --(a ^ £) is not valid in M2 since (1 ^ 2) ^ --(1 ^ 2) = 2 ^ --2 = 2 ^ -1 = 2 ^ 0 = 0. Hence, there exists a calculus that is both an extension of B1 and a proper subsystem of P1. This calculus is CB1. ■
Below we will propose a semantics for B1, but first we recall a few simple facts about the calculus. Since they are quite obvious and do not require detailed comments, their proofs will be omitted.
Remark 3. Enriching the set of axiom schemas of B1 with the formula a ^ --a, results in obtaining the axiom system of Classical Propositional Calculus.
Remark 4. Enriching the set of axiom schemas of B1 with the formulas --a ^ a and (a * £) ^ --(a * £), where * G {A, V results in obtaining P1 of Sette.
As a direct consequence of the above remarks, we have the following:
Remark 5. B1 is a proper subsystem of Sette's calculus P1.
Remark 6. The calculus B1 is not maximal with respect to Classical Propositional Calculus.
7The fifth axiom of P1. See ibid., p. 173.
Now we present a bi-valuational semantics for B1.
Definition 3. A B1-valuation is a function v : F —> {1,0} defined in the following way:
(V) v(a V ft) = 1 iff v(a) = 1 or v(ft) = 1 (a) v(a A ft)= 1 iff v(a) = 1 and v(ft) = 1 v(a ^ ft) = 1 iff v(a) = 0 or v(ft) = 1 (—) if v(—a) = 0, then v(a) = 1 (——) if v(——a) = 1, then (v(a) = 0 or v(—a) = 0).
Definition 4. A formula a is a B1-tautology iff for every B1-valuation v,
v(a) = 1.
Definition 5. For any a e F and r C F, a is a semantic consequence of r (r =bi a, in symbols) iff for any B1 -valuation v: if v(ft) = 1 for any ft e r, then v(a) = 1.
A doubt might arise as to how some formulas should be interpreted. Let us give an example to illustrate the point. Suppose that we work with the formula ———a. Then the question appears: How should the formula be interpreted, as —(——a), or maybe rather as ——(—a)? The answer is quite obvious: it depends on the logical value we assign to the formula via B1 -valuation. To be more precise, if v(———a) = 0, then v(——a) = 1, and consequently v(a) = 0 or v(—a) = 0; if v(———a) = 1, then v(—a) =0 or v(——a) = 0, etc.
Theorem 3. For every r C F and a e F, r -bi a iff r |=bi a.
Proof. The proof of soundness can be done in the standard way. The proof of completeness is by contraposition: Assume that r —bi a. The method applied in this paper is based on the notion of maximal non-trivial sets of formulas. We use the technique described in [Carnielli, Coniglio, 2016] (see Section 2.2). To start with, let us recall some important definitions and results.
Definition 6. Let A C F and a e F, we say that a set A is a closed theory of (L, -p) if, and only if the following holds: A -p a iff a e A.
Definition 7. Let A C F and a e F, we say that a set A is maximal nontrivial with respect to a in (L, -p) iff
1. A —p a, and
2. for every ft e F, if ft e A then A U {ft} -p a.
By Lemma 1, it follows that the consequence relation -bi satisfies the so-called Tarskian properties (reflexivity, transitivity and monotonicity). Then, the next lemma holds for -bi as well:
Lemma 2. Let (L, lp) be a logic that satisfies the Tarskian properties, then, any maximal non-trivial with respect to a in (L, lp) set of formulas is a closed theory.8
Consequently, we may formulate the so-called Lindenbaum-Los' theorem:
Theorem 4. Let (L, lp) be a Tarskian and finitary logic over the language L. Let r U {a} C L be such that r |p a. Then there exists a set A such that r C A C L with A being maximal non-trivial with respect to a in L.
Proof. We refer the interested reader to [Carnielli, Coniglio, 2016] (see the proof of Theorem 2.2.6) and [Pogorzelski, Wojtylak, 2008] (see Theorem 3.31), for details. ■
Now, we need to prove the following lemma:
Lemma 3. Let A C F and a G F, where A is a maximal non-trivial set with respect to a in B1. The mapping v : F —> {1, 0} defined as
(*) v(0) = 1 if and only if 0 G A,
for every 0 G F, is a B1 -valuation.
Proof. We limit ourselves to proving that the lemma holds for (-) and (--). The rest of the proof proceeds in the same way as in Theorem 2.2.7 of [Carnielli, Coniglio, 2016].
(-). Let 0 = -£. Assume that v(-£) = 0 and, by contradiction, v(£) = 0. Then by (*), we have -£ G A and £ G A. Observe that A is a maximal non-trivial set with respect to a, so A U {£} lbi a and A U {-£} lbi a. From Lemma 1(4), we obtain A U {£ V -£} lbi a. Since the law of excluded middle, £ V-£, is a thesis of B1, then, by Lemma 1(5), we have A lbi a. Note that A is a closed theory. This entails that a G A. But a G A by the main assumption, i.e. r |bi a, and Theorem 4. A contradiction. This implies that if v(-£) = 0, then v(£) = 1. As a result, the mapping v satisfies the clause (-) of Definition 3.
(--). Let 0 = --£. Suppose that v(--£) = 1 and, by contradiction, v(£) = 1 and v(-£) = 1. Then by (*), we obtain that --£ G A, £ G A and -£ G A; and consequently, A lbi --£, A lbi £ and A lbi -£, by Lemma 1(1). This means that A lbi {£, -£, --£}. Notice that £ ^ (-£ ^ (--£ ^ y)) is a thesis of B1. Thus {£, -£, --£} lbi 7, by the deduction theorem. The relation lbi is transitive (by Lemma 1), so A lbi 7. Observe
8 Cit. per [Carnielli, Coniglio, 2016], Lemma 2.2.5.
that A is deductively closed, then 7 e A. Since 7 is any formula of B1, so in particular 7 = a and a e A. But a e A. A contradiction. This implies that if v(——ft) = 1 then v(ft) = 0 or v(—ft) = 0. Therefore, the mapping v satisfies the clause (——) of Definition 3. ■
Now, recall that r —bi a. Let A be a maximal non-trivial set with respect to a in B1 extending r. Since Lemma 3 holds for B1, there is a valuation function v such that: v(ft) = 1 for any ft e r, and v(a) = 0. But if v(ft) = 1 for any ft e r, and v(a) = 0, then r |=bi a. It means that if r —bi a, then r |=bi a, and finally that r |=bi a implies r -bi a. ■
We have already noticed that B1 fulfils the basic criteria to be regarded as a paraconsistent calculus: neither the consequence relation -bi is explosive nor the law of non-contradiction is provable in B1. On the other hand, it can be easily proved that {a, —a, ——a} -bi ft, for any formulas a, ft and so is the formula —(a A —a A ——a) a thesis of B1. This could be greeted with some scepticism and one might raise the question whether the calculus with a ^ (—a ^ (——a ^ ft)) being a thesis was paraconsistent at all. If we answer negatively, the next question arrives: What about the calculi with a ^ (—a ^ (——a ^ (———a ^ ft))) or a ^ (—a ^ (——a ^ (———a ^ (————a ^ ft)))) being provable? Are they paraconsistent or not? This turns the issue into a sorites-style argument. To put it metaphorically, the point here is that we are always made to specify how many destructive formulas (i.e. a, —a, ——a, ———a, etc.) is needed to accept any ft.
3. Hierarchy of Bn-calculi (n e N)
There are several hierarchies of paraconsistent logic among which Newton da Costa's hierarchy of C-systems is probably the most famous one. The hierarchy introduced in this section differs in many respects from the hierarchy C-systems. One of them is that the law of double negation, ——a ^ a, does not hold in any Bn-calculus; the other is simplicity.
For each n e N, let Bn result from (A1)-(A9) and (MP) by adding to them the axiom schemas:
(ExM) a V —a
(DSk) a ^ (—a ^ (—2a ^ (... ^ (—ka ^ ft)...),
where k = n + 1 and —ka is an abbreviation for ——a.9
k
9If n = 0, then (DS1) a ^ (—a ^ 3) is an axiom schema of B0 (and B0 is Classical Propositional Calculus).
Consequently, the semantics needs to be modified as follows:
Definition 8. Let n G N. A Bn-valuation is a function v : F —> {1, 0} inductively defined in the following way: (V) v(a V £) = 1 iff v(a) = 1 or v(£) = 1 (a) v(a A £) = 1 iff v(a) = 1 and v(£) = 1 v(a ^ £) = 1 iff v(a) = 0 or v(£) = 1 (-) if v(-a) = 0 then v(a) = 1
(-■k) if v(-■ka) = 1 then (v(a) = 0 or v(-a) = 0 or ... or v(-k-1a) = 0), where k ^ 2.
Definition 9. A formula a is a Bn-tautology iff for every Bn-valuation v,
v(a) = 1.
Definition 10. For any a G F and r C F, a is a semantic consequence of r (r |=Bn a, in symbols) iff for any Bn-valuation v: if v(£) = 1 for any £ G r, then v(a) = 1.
Theorem 5. For every r C F and a G F, r \~b™ a iff r \=sn a.
Proof. The proof is analogous to that of Theorem 3. ■
Observe that each calculus in the hierarchy B1, B2, ..., Bn is weaker than the preceding one(s); obviously, except for B1. To put it more accurately:
Remark 7. For any m, n G N such that m > n, Bm is a proper subset of Bn.
Proof. Note that the only difference between the axiomatizations Bm and Bn is due to the axiom schema (DSm+1)/(DSn+1), respectively. Hence, it suffices to demonstrate that, if m > n, then (DSm+1) is a thesis of Bn, whereas (DSn+1) is not provable in any Bm-calculus. But this can be easily proved with the help of the completeness theorem and semantics for the calculi. ■
Remark 8. Enriching the set of axiom schemas of any Bn-calculus (n G N) with the formula a ^ --a, results in obtaining the axiom system of Classical Propositional Calculus.
From the philosophical viewpoint, each member of the hierarchy brings an answer to the question that has arisen in the section 2: 'How many destructive formulas do we need to accept any £?'
4. Paracomplete calculus Q1
Paracomplete logic can be defined in many different ways among which the following one may be of some interest: A logic (L, -q} is said to be paracomplete iff {ft ^ a, —ft ^ a} a, for some formulas a, ft. But, once again, just as in the case of paraconsistent logic (see Introduction), the definition is very general and may be seen to overlap with the logics that do not have too much in common with the paracompleteness. As a result, some additional criteria should be established to introduce a paracomplete calculus. On the basis of these generalizations, it seems noteworthy to mention at least some the most important and frequently used ones.
Definition 11. A logic (L, -q} is said to be paracomplete iff it cumulatively meets the following conditions:
1. {ft ^ a, —ft ^ a} a, for some formulas a, ft
2. {a} {ft, —ft}, for some formulas a, ft
3. 0 a V —a, for a formula a
4. 0 (—a ^ a) ^ a, for a formula a
5. 0 (a ^ —a) ^ —a, for a formula a.
In what follows, we will additionally postulate that each paracomplete calculus must satisfy two extra requirements, viz.
6. 0 a ^ ——a, for a formula a
7. 0 ——a ^ a, for a formula a.10
Now we can introduce the first calculus of paracomplete logic. The calculus, denoted as Q1, is presented in Hilbert-style formalization:
(A1) a ^ (ft ^ a)
(A2) (a ^ (ft ^ 7)) ^ ((a ^ ft) ^ (a ^ 7))
(A3) ((a ^ ft) ^ a) ^ a
(A4) (a A ft) ^ a
(A5) (a A ft) ^ ft
(A6) a ^ (ft ^ (a A ft))
(A7) a ^ (a V ft)
(A8) ft ^ (a V ft)
(A9) (a ^ y) ^ ((ft ^ y) ^ (a V ft ^ y))
10 A list of the alternative definitions of paracomplete logic is given, e.g., in [Petrukhin, 2018, pp. 425-426]. Some examples of the paracomplete calculi can be found in [Ciuciura, 2015b], [Karpenko, Tomova, 2017], [Loparic, da Costa, 1984], [Popov, 2002] and [Sette, Carnielli, 1995].
(ExM2) a V -a V --a (dS) a ^ (-a ^ ß).
The sole rule of inference is Detachment Rule, i.e. (MP) a ^ ß, a / ß. The definitions of a formal proof (deduction), a syntactic consequence within Q1 and a thesis of Q1 are analogous to those given in Section 2.
Remark 9. Q1 can be viewed as a dual counterpart of the paraconsistent calculus B1 (in a sense given in [Loparic, da Costa, 1984, pp. 119-120]).
To demonstrate that the calculus Q1 satisfies the conditions specified in the above, it suffices to apply the matrix
M3 = <{1, 2, 0}, {1}, -, A, V, ^ ),
where 1 is the only designated truth value in M3, the connectives -, A, V, ^ are defined as follows:
1 2 0 —1
1 1 0 0 1 0
2 1 1 1 2 1
0 1 1 1 0 2
A 1 2 0 V 1 2 0
1 1 0 0 1 1 1 1
2 0 0 0 2 1 0 0
0 0 0 0 0 1 0 0
Notice that that all axiom schemas of Q1 are valid in and the rule of detachment preserves validity. To show that Q1 satisfies the criteria, it is enough to assign 0 to a in the formulas: a V -a, (-a ^ a) ^ a, (a ^ -a) ^ -a and --a ^ a; and 1 to a in a ^ --a. To prove that, for some a, £ e F, neither {£ ^ a, -£ ^ a} a nor {a} {£, -£} holds in Q1, assign the value 0 (or 2) to a and 0 to £ in the former; and 1 to a and 0 to £ in the latter.
Remark 10. Lemma 1 and the deduction theorem hold for Q1.
Remark 11. The formulas
1. (£ ^ a) ^ ((-£ ^ a) ^ ((--£ ^ a) ^ a))
2. ((-a ^ a) ^ a) o (a V -a)
3. -(a V -a) ^ --a
4. (-a V £) ^ (a ^ £) are provable in Q1.
A bi-valuational semantics for Q1 coincides, to some extent, with the semantics presented in Section 2. The essential difference lies in the way the connective of negation is defined.
Definition 12. A Q1 -valuation is a function v : F —> {1,0} defined as follows:
(V) v(a V ft) = 1 iff v(a) = 1 or v(ft) = 1 (a) v(a A ft) = 1 iff v(a) = 1 and v(ft) = 1 v(a ^ ft) = 1 iff v(a) = 0 or v(ft) = 1 (—) if v(—a) = 1 then v(a) = 0 (——) if v(——a) = 0 then (v(a) = 1 or v(—a) = 1).
Definition 13. A formula a is a Q1-tautology iff for every Q1-valuation v,
v(a) = 1.
Definition 14. For any a e F and r C F, a is a semantic consequence of r (r |=qi a, in symbols) iff for any Q1-valuation v: if v(ft) = 1 for any ft e r, then v(a) = 1.
Theorem 6. For every r C F and a e F, r -qi a iff r |=qi a.
Proof. The proof of soundness is standard (by induction), and we omit it here. The proof of completeness is more involved. The proof method presented in this section is based on the notion of relatively maximal sets of formulas. To begin with, assume that r |=qi a and r -qi a. If r -qi a, then there exists A cF such that 1. a e A; 2. A is deductively closed, that is, A -qi ft iff ft e A; 3. A is relatively maximal with respect to the formula a, i.e A -qi a and, for every ft e F, if ft £ A then A U {ft} -Qi a; 4. r c A.
Now we ought to prove that the Lindenbaum-Asser theorem holds. The proof is similar to the one found in the proof of the completeness of mbC (see [Carnielli et all, 2007], Section 3.3), so we omit it here. Before introducing the definition of the canonical model for -qi , we need to prove an auxiliary lemma.
Lemma 4. Let A C F and ft, y e F, then 1. if —ft e A, then ft e A
if ——ft e A, then ft e A or —ft e A ft V y e A iff ft e A or y e A ft A y e A iff ft e A and y e A ft ^ y e A iff ft e A or y e A.
Proof. We only prove the cases 1 and 2. The proof of the other cases is similar to the ones in Classical Propositional Calculus.
1. Suppose that -£ G A and, by contradiction, £ G A. Then A -£ and A hgi £. It means that A {£, -£}. Observe that 0 hgi £ ^ (-£ ^ a) (DS). Since {£, -£} hgi a (by the deduction theorem), then A Kqi a (by the transitivity of hgi ). A is deductively closed, so a G A. But a G A. A contradiction.
2. Assume that --£ G A and, by contradiction, £ G A, -£ G A. It results in the following A U {--£} hqi a, A U {-£} hqi a and A U {£} hqi a. From Lemma 1(4), we receive A U{£ V-£ V--£} hgi a. Since (ExM2) is a thesis of Q1, then, by Lemma 1(5), we get A Kqi a. Note that A is deductively closed, so a G A. But a G A. A contradiction. ■
Let us define the canonical Q1-model for Kqi as Mc = (A, vc), where
All we have to do, at this point, is to show that the canonical valuation vc satisfies the conditions (-), (--), (A), (V) and (^), which is obvious, because Lemma 4 holds. Now, since A /qi a then, for every £ e A, v(£) = 1 and v(a) = 0. This implies that A =qi a. Since r c A, so r =qi a. But
5. Hierarchy of Qn-calculi (n e N)
The Q1 is not the only paracomplete calculus that satisfies the criteria discussed in the previous section. There is a whole family of such calculi: Q1, Q2, ..., Qn, having properties analogous to those of Q1. We put them into the form of a hierarchy.11 Each member of the hierarchy is expected to fulfil the following condition: Qm is a proper subset of Qn, for any m, n e N such that m > n. To achieve the goal, we need to replace the axiom (ExM2) with a more general schema, that is
(ExM k) a V —a V —i—a V ... V —i a,
where k = n + 1, n e N. This entails replacing the evaluation condition for (-.-I) with a more general one:
(-) if v(-ka) = 0 then (v(a) = 1 or v(-a) = 1 or ... or v(-k-1a) = 1),
where k ^ 2. Roughly speaking, the more negated formulas appear as the disjuncts in (ExMk), the weaker calculus is.
nThe hierarchy proposed in this paper is not the only one that appeared in logical literature. There are some interesting hierarchies in da Costa-style presentation, e.g. da Costa and Marconi's hierarchy of paracomplete calculi Pn (see [da Costa, Marconi, 1986]) or Arruda and Alves' logic of vagueness (see [Arruda, Alves, 1979a] and [Arruda, Alves, 1979b]).
r =qi a. A contradiction.
Theorem 7. For every r C F and a e F, r -Qn a iff r |=Qn a, n e N.
Proof. The theorem can be proved in the same manner as in the case where n = 1. The only essential difference is that we have to make some changes in the auxiliary lemma. To be precise, we need to replace
2. if ——ft e A then ft e A or —ft e A with
2k if —kft e A then ft e A or — ft e A or ... or —k 1ft e A, where k ^ 2.
Remark 12. For any m, n e N such that m > n, we have Qm is a proper subset of Qn.
Proof. The proof is analogous to that of Remark 7. ■
Remark 13. Enriching the set of axiom schemas of any Qn-calculus (n e N) with the formula ——a ^ a, results in obtaining the axiom system of Classical Propositional Calculus.
In conclusion, let us stress that every member of the hierarchy has the prerequisites for being a good paracomplete calculus. The idea behind it is much simpler than that given in [Arruda, Alves, 1979a], [Arruda, Alves, 1979b] or [da Costa, Marconi, 1986], and this is supposed to be an advantage.
References
Arruda, Alves, 1979a - Arruda, A.I., Alves, E.H. "Some remarks on the logic of vagueness", Bulletin of the Section of Logic, 1979, Vol. 8, No. 3, pp. 133-138. Arruda, Alves, 1979b - Arruda, A.I., Alves, E.H. "A semantical study of some systems of vagueness logic", Bulletin of the Section of Logic, 1979, Vol. 8, No. 3, pp. 139144.
Avron et all, 2018 - Avron, A., Arieli, O., Zamansky, A. "Theory of Effective Pro-positional Paraconsistent Logics", in: Studies in Logic, Mathematical Logic and Foundations, Vol. 75. College Publications, 2018. Batens, 1980 - Batens, D. "Paraconsistent extensional propositional logics", Logique
et Analys, 1980, Vol. 23, No. 90-91, pp. 195-234. Batens, De Clercq, 2004 - Batens D., De Clercq, K. "A Rich Paraconsistent Extension of Full Positive Logic", Logique et Analys, 2004, Vol. 47, No. 185-188, pp. 227-257. Carnielli, Coniglio, 2016 - Carnielli, W., Coniglio, M.E. "Paraconsistent Logic: Consistency, Contradiction and Negation", in: Logic, Epistemology, and the Unity of Science, Vol. 40, Springer International Publishing, 2016. 398 pp. Carnielli et all, 2007 - Carnielli, W., Coniglio, M.E., Marcos, J. "Logics of Formal Inconsistency", in D.M. Gabbay, F. Guenthner (eds.) Handbook of Philosophical Logic, 2nd edition, Vol. 14. Springer, 2007, pp. 1-93.
Ciuciura, 2018 - Ciuciura, J. Hierarchies of the paraconsistent calculi, Wydawnictwo Uniwersytetu Lodzkiego, Lodz, 2018. (In Polish)
Ciuciura, 2014 - Ciuciura, J. "Paraconsistent heap. A Hierarchy of mbCn-systems", Bulletin of the Section of Logic, 2014, Vol. 43, No. 3-4, pp. 173-182.
Ciuciura, 2015a - Ciuciura, J. "Paraconsistency and Sette's calculus P1", Logic and Logical Philosophy, 2015, Vol. 24, No. 2, pp. 265-273.
Ciuciura, 2015b - Ciuciura, J. "A Weakly-Intuitionistic Logic I1", Logical Investigations, 2015, Vol. 21, No. 2, pp. 53-60.
da Costa, 1974 - da Costa, N.C.A. "On the theory of inconsistent formal systems", Notre Dame Journal of Formal Logic, 1974, Vol. 15, No. 4, pp. 497-510.
da Costa, Marconi, 1986 - da Costa, N.C.A., Marconi, D. "A note on paracomplete logic", Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Mate-matiche e Naturali. 1986. Rendiconti, Serie 8, Vol. 80, No. 7-12, pp. 504-509.
Karpenko, Tomova, 2017 - Karpenko, A., Tomova, N. "Bochvar's three-valued logic and literal paralogics: Their lattice and functional equivalence", Logic and Logical Philosophy, 2017, Vol. 26, No. 2, pp. 207-235.
Loparic, da Costa, 1984 - Loparic, A., da Costa, N.C.A. "Paraconsistency, Paracom-pleteness, and Valuations", Logique et Analyse, 1984, Vol. 27, No. 106, pp. 119-131.
Petrukhin, 2018 - Petrukhin, Y. "Generalized Correspondence Analysis for Three-Valued Logics", Logica Universalis, 2018, Vol. 12, No. 3-4, pp. 423-460.
Pogorzelski, Wojtylak, 2008 - Pogorzelski, W.A., Wojtylak, P. Completeness Theory for Propositional Logics, Studies in Universal Logic. Basel: Birkhauser, 2008. 178 pp.
Popov, 2002 - Popov, V.M. "On a three-valued paracomplete logic", Logical Investigations, 2002, Vol. 9, pp. 175-178. (In Russian)
Post, 1921 - Post, E.L. "Introduction to a general theory of elementary propositions", American Journal of Mathematics, 1921, Vol. 43, No. 3, pp. 163-185.
Sette, 1973 - Sette, A.M. "On the propositional calculus P1", Mathematica Japonicae, 1973, Vol. 18, No. 3, pp. 173-180.
Sette, Carnielli, 1995 - Sette, A.M., Carnielli, W.A. "Maximal weakly-intuitionistic logics", Studia Logica, 1995, Vol. 55, No. 1, pp. 181-203.
Slupecki, 1939 - Slupecki, J. Dowod aksjomatyzowalnosci pelnych systemow wielowar-tosciowych rachunku zdas, Sprawozdania z pos. Tow. Nauk. Warsz, 32, wydz. III, 1939. (In Polish)
Wojcicki, 1988 - Wojcicki, R. Theory of Logical Calculi. Basic Theory of Consequence Operations, Synthese Library vol. 199, Netherlands: Springer, 1988. 474 pp.
