In the following, we present the logic [14], which has dual-interpretation semantics. The - concepts are constructed from atomic concepts, roles, ~ (paraconsistent negation), (classical negation or complement), (intersection), (union), (universal concept quantification) and (existential concept quantification). We use the letters and for atomic concepts, the letter for roles, and the letters and for concepts.

Definition 2.1 Concepts are defined by the following grammar:

Definition 2.2 A paraconsistent interpretation is a structure where 1) is a non-empty set2) is an interpretation function which assigns to every atomic concept a set and to every role a binary relation3) is an interpretation function which assigns to every atomic concept a set and to every role a binary relation4) for any role,.

The interpretation functions are extended to concepts by the following inductive definitions:

.(12)

An expression is defined as. A paraconsistent interpretation is a model of a concept (denoted as) if . A concept is said to be satisfiable in if there exists a paraconsistent interpretation such that.

The interpretation functions and are intended to represent “verification” (or “support of truth”) and “falsification” (or “support of falsity”), respectively. It is noted that includes [3] as a subsystem since in includes in.

Intuitively speaking, is constructed based on the following additional axiom schemes for:

It is noted that the interpretations for ~ and in correspond to the axiom scheme, which means that ~ and are self duals with respect to and ~, respectively. We now give an intuitive example for this axiom. Let stand for the claim that is poor, and let stand for the claim that is rich. Intuitively, is verified (falsified) iff is falsified (verified, respectively). Suppose now that is indeed falsified. This should mean that it is verified that is poor or neither poor or rich. But this is the case iff is not verified, which means that is not falsified.

For each concept, we can take one of the following cases:

1) is verified, i.e., 2) is falsified, i.e., 3) is both verified and falsified4) is neither verified nor falsified.

Thus, may be regarded as a four-valued logic.

In general, a semantic consequence relation ‘is called paraconsistent with respect to a negation connective: if there are formulas and such that does not hold. In the case of, assume a paraconsistent interpretation such that, and not- for a pair of distince atomic concepts and. Then, does not hold, and hence is paraconsistent with respect to:. It is remarked that is not paraconsistent with respect to.

Next, we explain about some differences and similarities between [16] and. In

, the set of multiple interpretation functions were used. These interpretation functions include the following characteristic conditions for negations:

1) for any atomic concept, 2) for any atomic concept, 3) for any atomic concept, 4)5) with6)7)8).

It is remarked that the condition 1 above means that is not paraconsistent with respect to. The subsystem (or special case) (of), which adopts two interpretation functions and, is similar to. The conditions for the constructors and of are almost the same as those of. The main differences are presented as follows:

1) has the “non-paraconsistent” condition: for any atomic concept,

but has no this condition2) adopts the condition:

but has no this condition and adopts the condition:

instead of it.

Some four-valued semantics in [4] were based on, , DL-Lite, etc., and the quasi-classical semantics in [13] was based on. The four-valued semantics in [4] has no classical negation, but has some new inclusion constructors such as strong inclusion. In addition, the quasi-classical semantics in [13] has two kinds of definitions called QC weak semantics and QC strong semantics. The following explanation is based on and QC weak semantics. We use the common language based on, , and/or.

We cannot compare the existing paraconsistent semantics (i.e., the four-valued semantics, the quasi-classical semantics, the single-interpretation semantics and the dual-interpretation semantics) themselves since the underlying DLs are different. Moreover, the motivations of introducing the existing semantics are completely different. For example, in the quasi-classical semantics, the main motivation is to satisfy three important inference rules: modus ponens, modus tollens and disjunctive syllogism. These inference rules are strongly dependent on a specific inclusion constructor and a specific QC entailment. Thus, our comparison without is regarded as not so comprehensive or essential in the sense of the original motivation of the quasi-classical semantics.

The following definition is a slight modification of the definition of [4].

Definition 2.3 (Four-valued semantics) A fourvalued interpretation is defined using a pair of subsets of and the projection functions and. The interpretations are then defined as follows:¹

1) a role is assigned to a relation2) for an atomic concept, where3) if4) if for5) if for⁶⁾

⁷⁾

In the four-valued semantics for [4], different kinds of implications were introduced:

1) (material inclusion)

2) (internal inclusion)

3) (strong inclusion).

The interpretations of, and are respectively presented as follows:

 These implications provide flexible way to model inconsistent ontologies.

The extension of four-valued semantics to the expressive description logic, and the extensions of four-valued semantics to some tractable description logics, Horn-DLs and DL-Lite family were studied in [5].

Next, we discuss about quasi-classical description logic. The following definition is a slight modification of the definition of quasi-classical description logics [12, 13].

Definition 2.4 (Quasi-classical semantics) A quasiclassical weak interpretation is defined using a pair of subsets of without using projection functions. The interpretations are then defined as follows:²

1) a role is assigned to a pair of binary relations2) for an atomic concept, where3)4)5)6)⁷⁾

⁸⁾

The quasi-classical semantics for QC [12] were extended to that of QC [13] to handle inconsistent ontologies. It composes two kinds of semantics, i.e., QC weak semantics and QC strong semantics. QC weak semantics inherits the characteristics of four-valued semantics, and QC strong semantics redefines the interpretation for disjunction and conjunction of concepts to make the three important inference rules (i.e., modus ponens, modus tollens and disjunctive syllogism) hold.

Let be a QC entailment and be a paraconsistent negation connective, which is represented as ~ in the above definition. Then, the following hold:

1) (modus ponense)

2) (modus tollens)

3) (disjunctive syllogism).

Two basic query entailment problems (i.e., instance checking and subsumption checking) were also defined and discussed in [13]. It was also shown that the two basic inference problems can be reduced into the consistency problem.

Finally in this subsection, it is remarked that the pairing functions used in the four-valued and quasiclassical semantics have been used in some algebraic semantics for Nelson’s logics (see e.g., [21] and the references therein). On the other hand, the semantics of is defined using two interpretation functions and instead of the pairing functions. These interpretation functions have been used in some Kripketype semantics for Nelson’s logics (see e.g., [22] and the references therein). It will be shown in the next section that the “horizontal” semantics using paring functions and the “vertical” semantics using two kinds of interpretation functions have thus essentially the same meaning.

Three constructive PDLs, which have single-interpretation semantics, were introduced and studied by Odintsov and Wansing [8]:

1): Constructive version of. It is obtained via a translation into first-order classical logic. A tableau algorithm for was presented in [9].

2): It is obtained via a translation into the quantified N4. The role restrictions and are not dual.

3): It is obtained via an alternative translation into the quantified N4. The role restrictions and are dual. The decidability of was obtained in [8] from a translation into Fischer Servi’s intuitionistic modal logic.

We now give an overview of as follows. has no classical negation connective, but has a paraconsistent negation connective. Also it has no classical implication (or classical inclusion), but has a constructive implication (or constructive inclusion).

uses interpretations where 1) is a non-empty set2) is a reflexive and transitive relation of informational accessibility3) is an interpretation function with some conditions, e.g.a) it maps every atomic concept to a subset ofb) it maps every negated atomic concept to a subset of.

The interpretation function has the following conditions:

1) for an atomic concept, 2) for an atomic concept, 3)4)5)⁶⁾7)8)9)10)11)12)¹³⁾.

It is remarked that the order relation needs some more conditions. For the details, see [8,9].

Similar notions and terminologies for are also used for the new logic. The -concepts are the same as the -concepts. The semantics is defined as a generalization and modification of the quasi-classical weak semantics defined in Definition 2.4. Thus, we use the term “quasi-classical” in the following definition.

Definition 3.1 A quasi-classical interpretation is a structure where 1) is a non-empty set2) is a positive (negative, resp.) polarity function which assigns to every atomic concept a set

(, resp.)3) is an interpretation function which assigns to every atomic concept a pair of sets and to every role a pair of binary relations4) for any role,.

The polarity functions are extended to concepts by the following inductive definitions:

The interpretation function is extended to concepts by:

.

An expression is defined as and. A quasi-classical interpretation

is a model of a concept

(denoted as) if. A concept is said to be satisfiable in if there exists a quasiclassical interpretation such that.

We have the following propositions, which mean that Definition 3.1 is essentially the same definitions as those of the original quasi-classical [12,13] and four-valued [4,5] semantics. See Definitions 2.4 and 2.3.

Proposition 3.2 Let be an interpretation function on a quasi-classical interpretation. Then, the following conditions hold:

Proposition 3.3 Let be an interpretation function on a quasi-classical interpretation. Let and be now represented by P and N, respectively. Also, and for a concept be represented by and, respectively. Define and Then, the following conditions hold:

Next, we show the equivalence between and.

Theorem 3.4 (Equivalence between and) For any concept, is satisfiable in iff is satisfiable in.

Proof.: Suppose that is a quasi-classical interpretation. Then, it is sufficient to construct a paraconsistent interpretation

such that, for any concept,

iff. We define a paraconsistent interpretation by:

1)2) is an interpretation function which assigns to every atomic concept a set and to every role a binary relation3) is an interpretation function which assigns to every atomic concept a set and to every role a binary relation.

Then, we have the fact: for any role,.

It is sufficient to show the following claim which implies the required fact. For any concept,

By (simultaneous) induction on. We show some cases.

Case (is an atomic concept): For 1, we have the following by the definition:. For 2, we have the following by the definition:.

Case: For 1, we have:

(by induction hypothesis for 2). For 2, we have: (by induction hypothesis for 1).

Case: For 1, we have:

(by induction hypothesis for 1). For 2, we have: (by induction hypothesis for 2).

Case: For 1, we have: (by induction hypothesis for 1). For 2, we have: (by induction hypothesis for 2).

Case: For 1, we have:

.

For 2, we have:

,

(by induction hypothesis for 2),

.

: Suppose that is a paraconsistent interpretation. Then, it is sufficient to construct a quasi-classical interpretation such that, for any concept, iff. We define a quasi-classical interpretation by:

1)2) is a positive (negative, resp.) polarity function which assigns to every atomic concept a set (, resp.)3) is an interpretation function which assigns to every atomic concept a pair of sets and to every role a pair of binary relations .

Then, we have the fact: for any role,.

It is sufficient to show the following claim which implies the required fact. For any concept,

Since this claim can be shown in the same way as in the claim of the direction, the proof is omitted here.                                         □

We introduce a new logic, which has a singleinterpretation function. The idea of this formulation is inspired from the paraconsistent semantics for a constructive PDL proposed in [8]. These single-interpretation semantics can also be adapted to Nelson’s paraconsistent logic (see [20]).

Similar notions and terminologies for are also used for. The -concepts are the same as the -concepts.

Definition 3.5 Let be the set of atomic concepts and be the set. A single paraconsistent interpretation is a structure where 1) is a non-empty set2) is an interpretation function which assigns to every atomic (or negated atomic) concept a set and to every role a binary relation.

The interpretation function is extended to concepts by the following inductive definitions:

An expression is defined as. A single paraconsistent interpretation is a model of a concept (denoted as) if. A concept is said to be satisfiable in if there exists a single paraconsistent interpretation such that.

It is remarked that the logic in [8] has the same interpretations for A (atomic concept), (negated atomic concept), and as in. Since is constructive, it has no classical negation, but has constructive inclusion (constructive implication) which is defined by:

.

Next, we show the equivalence between and.

Theorem 3.6 (Equivalence between and) For any concept, is satisfiable in iff is satisfiable in.

Proof. Let be the set of atomic concepts, be the set, and be the set of roles.

: Suppose that is a single paraconsistent interpretation such that has the domain. Then, it is sufficient to construct a paraconsistent interpretation such that, for any concept, iff. We define a paraconsistent interpretation by:

1)2) is an interpretation function which assigns to every atomic concept a set and to every role a binary relation3) is an interpretation function which assigns to every atomic concept a set and to every role a binary relation4) for any role, 5) the following conditions hold:

It is noted that and have the domain.

It is sufficient to show the following claim which implies the required fact. For any concept,

By (simultaneous) induction on. We show some cases.

Case (is an atomic concept): By the definition.

Case: For 1, we have: (by induction hypothesis for 2). For 2, we have: (by induction hypothesis for 1).

Case: For 1, we have: (by induction hypothesis for 1). For 2, we have: (by induction hypothesis for 2).

Case: For 1, we have: (by induction hypothesis for 1). For 2, we have: (by induction hypothesis for 2).

Case: For 1, we have:

.

For 2, we have:

,

(by induction hypothesis for 2),

.

: Suppose that is a paraconsistent interpretation such that and have the domain. Then, it is sufficient to construct a single paraconsistent interpretation such that, for any concept, iff. We define a single paraconsistent interpretation by:

1)2) is an interpretation function which assigns to every atomic (or negated atomic) concept a set and to every role a binary relation3) the following conditions hold:

It is noted that has the domain.

It is sufficient to show the following claim which implies the required fact. For any concept,

Since this claim can be shown in the same way as in the claim of the direction, the proof is omitted here.                                       □

As mentioned before, three constructive PDLs:, and were introduced and studied in [8,9]. By our comparison results of the present paper, we can consider to present the four-valued semantics, the quasi-classical semantics and the dual-interpretation semantics for these constructive PDLs. The notions of constructiveness and paraconsistency are known to be important for logical systems. From the point of view of the truth and falsehood in a logic, the principle of explosion and the excluded middle are the duals of each other. Paraconsistent logics are logics without the principle of explosion, and paracomplete logics are the logics without the excluded middle. Constructive logics are classified as a paracomplete logic. The logics with both the paraconsistency and the paracompleteness are called paranormal (or nonalethic) logics.

Since the precise definitions of the original semantics for and are rather complex, we now present only an outline of the (slightly modified versions of the) semantics of and.

A constructive interpretation is a structure where 1) is a non-empty set2) is a poset3) is a domain function from to (written ad for) such that a) for any, is non-emptyb) for any, if, then.

For each, we interpret an atomic concept and a negated atomic concept ~A as and, respectively. Examples of the interpretations of the composite concepts are presented as follows: For each,

The interpretations of and are rather complex, and hence omitted here. Such interpretations of and imply the differences between the -semantics and the -semantics.

It is remarked that the temporal next-time operator in the temporal description logic [23] is similar to the paraconsistent negation connective in. As mentioned, the connective in is from the paraconsistent negation connective in Nelson’s paraconsistent logic N4 [19,20]. The next-time operator in is from Prior’s tomorrow tense logic [24].

In the following, we explain and the similarities between in and in.

Similar notions and terminologies for are also used for. The symbol is used to represent the set of natural numbers. The -concepts are constructed from the -concepts by adding (next-time operator). An expression is inductively defined by and.

Definition 4.1 - concepts are defined by the following grammar:

Definition 4.2 A temporal interpretation is a structure where 1) is a non-empty set2) each is an interpretation function which assigns to every atomic concept a set and to every role a binary relation3) for any role and any,.

The interpretation function is extended to concepts by the following inductive definitions:

For any, an expression is defined as. A temporal interpretation

is a model of a concept

(denoted as) if. A concept is said to be satisfiable in if there exists a temporal interpretation such that.

The interpretation functions are intended to represent “verification at a time point “.

Intuitively speaking, is constructed based on the following additional axiom schemes for:

where   (2)

where.     (3)

It is noted that in and in are based on some similar axiom schemes. While is regarded as a de Morgan type negation connective, is regarded as a kind of “twisted” de Morgan type connective. By this similarity, we can prove a theorem for embedding into. Such an embedding theorem is similar to a theorem for embedding into. Thus, in an abstract sense, and can be viewed as the same kind of embeddable logics. Indeed, the same embedding-based method can be applied to these logics uniformly.