Axiomatization of Some Basic and Modal Boolean Connexive Logics

Boolean connexive logic is an extension of Boolean logic that is closed under Modus Ponens and contains Aristotle’s and Boethius’ theses. According to these theses (i) a sentence cannot imply its negation and the negation of a sentence cannot imply the sentence; and (ii) if the antecedent implies the consequent, then the antecedent cannot imply the negation of the consequent and if the antecedent implies the negation of the consequent, then the antecedent cannot imply the consequent. Such a logic was first introduced by Jarmużek and Malinowski, by means of so-called relating semantics and tableau systems. Subsequently its modal extension was determined by means of the combination of possible-worlds semantics and relating semantics. In the following article we present axiomatic systems of some basic and modal Boolean connexive logics. Proofs of completeness will be carried out using canonical models defined with respect to maximal consistent sets.

In the case of Boolean connexive logics negation, conjunction and disjunction are two-valued extensional connectives. Implication is defined by the truth-condition for material implication and a condition stating the existence of a relationship (connexion) between the antecedent and the consequent. Only under some conditions introduced in [8] any formula instantiating the schema (A1), (A2), (B1) or (B2) is valid in relating semantics. Such conditions are independent and correspond to connexive laws, if we consider models based on relations that are closed under negation (see [8, pp. 438-439]).
Moreover, Kapsner's conditions of weakly and strongly connexive logics presented in [10] as well as the condition of properly connexive logic (the nonsymmetry of connexive implication) given by Jarmużek and Malinowski in [8] are satisfied.
Some further steps in the analysis of Boolean connexive logics were made in [9] where some modal systems were introduced (cf. [15]). The logics as before were determined semantically, this time by means of possible-worlds semantics combined with relating semantics. The proof-theory was, as in the case of Boolean connexive logics in [8], given by means of tableau systems.
On the basis of the least modal Boolean connexive logic neither (K) nor the law of the duality of the modal operators is valid. Furthermore well-known relational properties of the accessibility relation are not enough to validate modal laws such as (T), (D), (B), (4) and (5). Two methods of making such laws valid were presented in [9]. In this article we are going to focus on the first one and only briefly consider the second one.
The main goal of our article is to axiomatize some of the logics considered in [8,9]. More specifically, we will consider axiomatic systems for the least Boolean connexive logic, its non-modal extension generated by models based on relations that are closed under negation and some modal extensions (generated by models that validate well-known modal schemata). The metalogical analysis will be carried out by adapting the methods applied in the studies of classical and modal logic respectively (cf. [1, pp. 45-64], [2, pp. 111-126], [3, pp. 105-116]). The proofs of completeness for all analysed systems will be delivered by means of canonical models defined with respect to maximal consistent sets.
The structure of the paper is as follows: in Sect. 2, we introduce the languages and notations used in the article; in Sect. 3, we present the semantic analysis of the least Boolean connexive logic and some of its extensions, including modal ones; in Sect. 4, we present an axiomatization of these logics and prove soundness and completeness for them.

Preliminaries: Languages and Notations
In the article we consider two languages. The first is the Boolean connexive language L that consists of propositional variables p 0 , q 0 , r 0 . . .; connectives: ¬ (negation), ∧ (conjunction), ∨ (disjunction); → (implication) and brackets:), (. 2 The second language, a modal Boolean connexive language L , is the modal extension of L. Thus, it is the language L which additionally contains modal operators: (necessity) and ♦ (possibility).
The set of propositional variables is denoted by Var. The set of formulas of the language L (resp. L ) is defined in the standard way and denoted by For (resp. For ). 3 We will omit the outermost brackets in formulas. Obviously For ⊂ For . We use two abbreviations:

Relating Semantics
Let us start our semantic analysis by introducing relating semantics for the language L (cf. [5][6][7]). A model for L (based on a relation R) is an ordered pair v, R such that v is a valuation of Var, i.e. v : Var −→ {1, 0}, and R is a binary relation between formulas, i.e. R ⊆ For × For. 4 The second element of model for L allows us to take under consideration various non-logical connexions. By means of it we can bind such propositions that refer to states of affairs that are related in some way.
The valuation v (resp. the relation R) of a model M for L is denoted by v M (resp. R M ). By R we shall denote the class of all binary relations on For. Let Q ⊆ R. The class of models based on Q is the class of models for L based on any R from Q. Such a class is denoted by MQ. The class of all models for L is MR. 2 In some articles on relating semantics and its connexive applications the symbol → w was used to denote implication (see [8,12]). Such a symbol was introduced in order to distinguish between relating and material implication denoted sometimes as → (cf. [7]). 3 By c(A) we shall denote the complexity of a formula A defined in the standard way, starting with 0 for propositional variables. 4 We write R(x, y), if R ⊆ Σ × Σ holds between elements x, y ∈ Σ and ∼R(x, y), if does not.
Let Σ ⊆ For. We will write M |= Σ instead of for every A ∈ Σ, M |= A.
In the case of negation, conjunction and disjunction, we have the standard extensional interpretation. The truth-condition for implication says that either the antecedent is false or the consequent is true and also the constituent formulas must be related. Hence, the Boolean interpretation of implication is enriched by the non-Boolean requirement of a relationship between propositions.
Let us observe that the notion of truth might be also defined with respect to a relation from R. A formula A ∈ For is true with respect to R ∈ R (in symb.: 5 The notions of valid formulas and semantic consequence are defined in the standard way. Let Q ⊆ R and Σ ∪ {A} ⊆ For. We say that A ∈ For is valid in the class Q ⊆ R (symb.: |= Q A) iff for every R ∈ Q, R |= A.
Let us notice that by [8, p. 435, theorem 5.1]: However, the opposite implications do not hold, i.e. neither Aristotle's laws nor Boethius' laws induce Aristotelian and Boethian relation respectively (see [8, pp. 435-436]). This is not true, however, for relations closed under negation (see [8, p. 438, theorem 6.1]). By closure under negation we mean the following condition: By C (resp. C ¬ ) we denote the class of all connexive relations (resp. the class of all connexive relations closed under negations). In the article we would like to consider the least Boolean connexive logic BC, i.e. the logic determined by the class C and the Boolean connexive logic with the closure under negation BC ¬ , i.e. the logic determined by the class C ¬ .

Combination of Relating and Possible-World Semantics
Let us now discuss an interpretation of the modal Boolean connexive language L . For this purpose, we combine a possible-world semantics with a relating one.
A modal frame is an ordered pair W, Q such that W is non-empty set of possible worlds and Q ⊆ W ×W is an accessibility relation. By a relating frame we mean an ordered pair W, {R w } w∈W such that W is non-empty set of possible worlds and {R w } w∈W is a family of binary relations R w ⊆ For × For , where w ∈ W . Notice that a relating frame can be reduced to Q is a modal frame and {R w } w∈W is a relating frame. As we can see, in a combined (resp. relating) frame for every possible world we have a binary relation that makes it possible to consider various non-logical relationships. By F (resp. F r , F) we shall denote the class of all modal (resp. relating, combined) frames. Let A model for L (based on a combined frame W, Q, A set of possible-worlds W (resp. a relation Q, a valuation v) of a model M for L is denoted by W M (resp. Q M , v M ). Let E ⊆ F. The class of models based on E is the class of all models for L based on any frame from E. Such a class of models is denoted by ME. The class of all models for L is Truth in a model for L is defined in the standard way.
As in the case of relating semantics and modal logic, we can define notion of true with respect to various structures. For instance, in [9, p. 220] the following notions were introduced: • A ∈ For is true with respect to a combined frame W, Q, As before, we define valid formulas and semantic consequence in the standard way. Let Σ ∪ {A} ⊆ For and E ⊆ F.
• A is a valid formula in the class ME (in symb.: Obviously, valid formulas might be also defined by means of the notion of truth with respect to classes of combined frames. We get the following fact: Proof. By definitions of valid formulas and formulas true with respect to a given structure. Let {R w } w∈W ∈ F r and w ∈ W . We have the following modal counterparts of relational conditions discussed in the previous Sect. (see [9, pp. 221]): As before, by an Aristotelian (resp. Boethian) frame we mean a relating frame {R w } w∈W such that for every w ∈ W , R w satisfies (a1 ) and (a2 ) (resp. (b0 .), (b1 ), (b2 )), i.e. the family of Aristotelian (resp. Boethian) relations indexed by possible worlds. A connexive frame is a frame that is both Aristotelian and Boethian. By [9, p. 222, theorem 4.2]: • if F ∈ F r is Aristotelian (resp. Boethian) frame, then F |=(A1), F |=(A2) (resp. F |=(B1), F |=(B2)). The opposite implications do not hold, but the proper closure under negation again makes it possible to get equivalence (see [9, pp. 221-222, theorem 4.1]).
By C we denote the class of all combined frames based on the class of all modal frames and the class of all connexive frames for L . The least modal Boolean connexive logic BC is the logic determined by the class C .
Since we want to deal with the modal version of Boolean connexive logics, the question naturally arises of which modal formulas are valid in such logics. Let us consider the following well-known modal schemata: As we know, formulas instantiating the schemata (D 1 ), (D 2 ) and (K) are elements of any normal modal logic, while formulas instantiating the schema (T) are elements of normal modal logics defined by reflexive modal frames, formulas instantiating the schema (D) are are elements of normal modal logics defined by serial modal frames, and so on. 7 None of the above schemata is a schema of valid formulas in logic BC . And even if we consider some extensions of BC defined by means of proper modal frames (for a given modal schema) the considered laws still will not be valid (see [9, pp. 223-224, claim 5.1]). But, it appears that BC is closed under the Necessitation Rule. We have that if A ∈ BC , then A ∈ BC (see [9, pp. 223-224, claim 5.1 (a)]). And obviously BC contains formulas of the following schemata: In [9] two ways were presented in which we can validate the modal laws. The first one assumes some restrictions on modal and relating frames. We consider them below. The second way is similar to the first one, but it refers to a certain mapping called a demodalization. 8 Such a function enables one to simplify constraints that are put on relating frames within the first approach. In this article we focus on the first way.
Let W, Q, {R w } w∈W ∈ F and w ∈ W . In order to validate (K) the following conditions are introduced: We say that a combined frame W, Q, {R w } w∈W satisfies (k1), (k2) iff for every w ∈ W , R w satisfies (k1) and for every w ∈ W , Q and R w satisfies (k2). By [9, pp. 224-225, claim 5.2]: • if F ∈ F satisfies (k1), (k2), then F |=(K). 7 As usual we assume that w, u) and Q(u, t)) ⇒ Q(w, t)); Euclidean iff ∀ w,u,t∈W ((Q(w, u) and Q(w, t)) ⇒ Q(u, t)). We say that a modal frame is xyz iff its accessibility relation is xyz. 8 The notion of demodalization was also introduced in [6] in order to consider a deontic relationship between propositions. Such a relationship does not depend on the modal status of propositions, but only their content. Notice that the demodalization of the kind considered in [9] is a simple eraser transformation defined in [ A, B ∈ For and either A = p 0 ∨ ¬p 0 or B = p 0 ∧ ¬p 0 } (cf. [9]). Let {R w } w∈W ∈ F r and w ∈ W . In order to validate other modal laws we consider the following conditions: We say that a relating frame {R w } w∈W satisfies (xyz) iff for every w ∈ W , R w satisfies (xyz). By [9, pp. 225-226, claim 5.3] we have that: • F ∈ F is based on a reflexive modal frame and a relating frame that satisfies (t) iff F |=(T) • F ∈ F is based on a serial modal frame and a relating frame that satisfies (d) iff F |=(D) • F ∈ F is based on a symmetric modal frame and a relating frame that satisfies (b) iff F |=(B) • F ∈ F is based on a transitive modal frame and a relating frame that satisfies (iv) iff F |=(4) • F ∈ F is based on an Euclidean modal frame and a relating frame that satisfies (v) iff F |= (5).
Taking under consideration the conditions above we introduce the following subclasses of C : • C D 1 is the sub-class of C of combined frames based on the class of all modal frames and the class of relating frames such that (d1) is satisfied • C D 2 is the sub-class of C of combined frames based on on the class of all modal frames and the class of relating frames that satisfies (d2) • C K the sub-class of C of combined frames that satisfies (k1), (k2) • C T is the sub-class of C of combined frames based on reflexive modal frames and relating frames that satisfies (t) • C D is the sub-class of C of combined frames based on serial modal frames and relating frames that satisfies (d) • C B is the sub-class of C of combined frames based on symmetric modal frames and relating frames that satisfies (b) • C 4 is the sub-class of C of combined frames based on transitive modal frames and relating frames that satisfies (iv) • C 5 is the sub-class of C of combined frames based on Euclidean modal frames and relating frames that satisfies (v). By a class C X 1 . . . X n we mean the intersection C X 1 ∩ . . . ∩ C X n . Hence, for instance, C KT5 is the sub-class of C of combined frames that satisfies (k1), (k2) and are based on reflexive, Euclidean modal frames and relating frames that satisfies (t), (v). The logic determined by the class of frames C X 1 . . . X n shall be denoted by BC X 1 . . . X n .

Axiomatic Systems
Let us now present the axiomatization of the logics under consideration here. By PL (resp. PL ) we shall denote the set of all formulas which are instances of classical tautologies in the language L (resp. L ), where ¬, ∧, ∨ are the only classical connectives. By an axiomatic system we mean any set Σ ⊆ For such that PL ⊆ Σ; Σ contains L's schemata (A1), (A2), (B1), (B2) and the following L's schema: 9 The intended axiomatic system for BC is the least axiomatic system and shall be denoted by S. The intended axiomatic system for BC ¬ is the system S ¬ that is the least axiomatic system that contains the following L's schemata: By a modal axiomatic system we mean any set Σ ⊆ For such that PL ⊆ Σ; Σ contains the L 's schemata (Imp), (A1), (A2), (B1), (B2), (Dual ≡ ) and (K ⊃ ); Σ is closed under the introduced modification of Disjunctive Syllogism and the Necessitation Rule, i.e.: if A ∈ Σ then A ∈ Σ. (NR) As in the case of axiomatic systems, any modal axiomatic system is closed under Modus Ponens. The intended axiomatic system for BC is the system S that is the least modal axiomatic system. The intended axiomatic system for BC X 1 . . . X n is the system S X 1 . . . X n that is the least modal axiomatic system that contains L 's schemata (X 1 ), . . . , (X n ).
Let Σ be an axiomatic system and Γ ∪ {A} ⊆ For (resp. Γ ∪ {A} ⊆ For , for modal axiomatic systems). We have the following definition of the notion of a thesis and syntactic consequence: 10 By these definitions we get the following standard fact:

Σ-Consistent and Maximal Σ-Consistent Sets
In this section we recall some well-known metalogical definitions and facts concerning consistent and maximal consistent sets defined with respect to an axiomatic system (cf. [1, pp. 47-49, 53-57]). Let Σ be an axiomatic system (resp. a modal axiomatic system) and Γ ⊆ For (resp. Γ ⊆ For ). Then: The following fact expresses some properties of Σ-consistent sets (cf. [ For modal axiomatic systems we also have the following proposition (cf.
The notion of maximal Σ-consistent set is defined in the standard way. Let Σ be an axiomatic system (resp. a modal axiomatic system) and Γ ⊆ For (resp. Γ ⊆ For ). Γ is maximal Σ-consistent iff the following conditions are satisfied: • Γ is Σ-consistent • for every Δ ⊆ For (Δ ⊆ For , for modal axiomatic systems), if Γ ⊂ Δ then Δ is Σ-inconsistent. The set of all maximal Σ-consistent sets is denoted by Max Σ . Elements of Max Σ are theories in the following sense (cf. Therefore, by means of facts 4.1, 4.4 we get the following corollary: Corollary 4.5. Let Σ be an axiomatic system (resp. a modal axiomatic system) and Γ ∈ Max Σ . Then Σ ⊆ Γ.
By means of fact 4.4 and the definitions of an axiomatic system and syntactic consequence we can prove the next standard fact that characterises connectives ¬, ∧, ∨ on the basis of maximal Σ-consistent sets (cf. [1, p. 53, theorem 2.18 (5)-(7)]). Proposition 4.6. Let Σ be an axiomatic system (resp. a modal axiomatic system) and Γ ∈ Max Σ . Then, for every A, B ∈ For (resp. A, B ∈ For ): 1.
Let Σ be an axiomatic system and Γ ∈ Max Σ . The canonical model for L generated by Γ (for short: Γ-model for L) is a pair v, R such that: We have the following lemma: Notice that the canonical model for L generated by Γ is a model from the class MC, i.e. the class of all models over connexive relations between formulas.  A canonical model for L is the standard canonical model for modal language with a family of relations between formulas indexed by maximal consistent sets. Let Σ be a modal axiomatic system. The canonical model for L generated by Max Σ (for short: • for every w ∈ W , for every A, B ∈ For : Here we have the following standard lemma about canonical models for modal axiomatic systems: Hence, by the definition of Max Σ -model for L , R w (B, C). Since Σ contains L 's schema (Imp) and by fact 4 .6 1 and 3, B ⊃ C ∈ w. Thus, again by the fact 4.6 1 and 3, either B / ∈ w or C ∈ w. By the inductive hypothesis either M, w |= B or M, w |= C. Therefore, since R w (B, C), M, w |= B → C, and so M, w |= A.
Notice that any canonical model for L is a model from the class MC . Proof. We reason similarly as in the proof of fact 4.9.
By fact 4.15 we know that combined frames of canonical models for L are elements of the class C . Now we prove that they are also elements of some proper subclasses of C , if the considered axiomatic systems contain some axiom schemata.  By fact 4.7 and lemma 4.14 we get the following corollary: Corollary 4.18. Let Σ be an axiomatic system and Γ ⊆ For. Then, if Γ is a Σ-consistent set, then there is a w ∈ Max Σ such that M is the Max Σ -model for L and M, w |= Γ.
Proof. Assume all the hypotheses. Suppose Γ is Σ-consistent. By fact 4.7 there is a w ∈ Max Σ , such that Γ ⊆ w. By lemma 4.14, for every A ∈ w, M, w |= A. Hence M, w |= Γ.
"⇐" Suppose A / ∈ Σ. Hence, by fact 4.2, {¬A} is Σ-consistent. By corollary 4.18 there is w ∈ Max Σ such that M is the Max Σ -model for L and M, w |= ¬A. Therefore M, w |= A.

Soundness and Completeness
We first deal with the soundness and completeness theorem for systems S and S ¬ .
Proof. Assume all the hypotheses.