Logics of Deontic Inconsistency
Marcelo E. Coniglio
Department of Philosophy, IFCH, and
Centre for Logic, Epistemology and The History of Science (CLE) State University of Campinas (UNICAMP), Campinas, SP, Brazil Security and Quantum Information Group (SQIG), IST / IT, Portugal
coniglio@cle.unicamp.br
Abstract
TheLogics of Formal Inconsistency(LFIs) are paraconsistent logics which internalize the notions ofconsistencyandinconsistencyby means of connectives. Based on that idea, in this paper we propose two deontic systems in which contradictory obligations are allowed, without trivializ- ing the system. Thus, from conflicting obligationsOϕandO¬ϕcontained in (or derived from) an information set, it can be derived that the sen- tence ϕ is deontically inconsistent. This avoids the logic collapse, and, on the other hand, this allows to “repair” or to refine the given informa- tion set. This approach can be used for analyzing paradoxes based on contrary-to-duty obligations.
Keywords: Logics of formal inconsistency, deontic logic, contrary-to-duty oblig- ations.
Introduction
Inconsistencies occur in several contexts, for instance databases and logical para- doxes. Within a classical framework, inconsistencies are identified with contra- dictions. Paraconsistent logics and, in particular, the so-calledLogics of Formal Inconsistency, LFIs (see [2, 4]), deal with contradictions without trivializing the logic system. Thus, contradictions are not necessarily inconsistent (that is, without models or, equivalently, logically trivial orsenseless), and an additional hypothesis ofconsistencyof the contradictory formula is necessary, in order to equate contradictoriness with triviality. The consistency of a formulaϕis made explicit by means of an unary consistencyoperator, ◦, such that ◦ϕ expresses thatϕis ‘consistent’ (or ‘safe’, or ‘well-behaved’, or ‘conclusive’). Equivalently, theinconsistencyof a sentenceϕcan be expressed by a sentence of the form•ϕ where• is an inconsistency operator. Typically, both operators are interdefin- able as expected: •ϕ≡ ¬◦ϕand◦ϕ≡ ¬•ϕ.
The approach ofLFIs to paraconsistency is useful fordetectinginconsisten- cies without falling in a logic collapse. This feature ofLFIs can be applied, for
instance, to remove inconsistencies in databases: a suitable logic tolerating con- tradictions and marking (with an inconsistency operator•) such contradictory sentences could help to “clean up” and to refine the database (see, for instance, [3]).
Based on this idea, we propose two deontic systems, calledLogics of Deon- tic Inconsistency, which are deontic extensions of LFIs and then the negation operator in these logics is paraconsistent. Thus, from contradictory obligations OϕandO¬ϕit can be inferred thatϕisdeontically inconsistent(expressed by a sentence of the formϕ) without trivializing the logic. An adequate Kripke semantics is presented for these systems.
By using logics of deontic inconsistency, deontic paradoxes are not solved but, instead, the information set containing such conflictive sentences (for instance, a database) can be “repaired” and better understood. In fact, the derivation of sentences of the formϕfrom a logically trivial (w.r.t. Standard Deontic Logic SDL) set of sentences helps to detect which sentences are involved in a deontic conflict. Being so, a decision can be taken (for instance, to remove them from the database or to modify them) in order to overcome the conflict. Additionally, this kind of logics could be useful to analyze, for instance, moral dilemmas.
1 Logics of Formal Inconsistency
A basic principle of classical logic is thePrinciple of Non-Contradiction:
¬(ϕ∧ ¬ϕ)
is a theorem, for any formulaϕ. This theorem can be seen (within the framework of classical logic) as a direct consequence of the Ex Contradictione Sequitor Quodlibetproperty:
ϕ,¬ϕ`ψ
for everyϕandψ. The latter can be axiomatized through the schema ϕ⇒(¬ϕ⇒ψ).
In general, the Logics of Formal Inconsistency (LFIs) consider a weaker version of this axiom, say
(bc) ◦ϕ⇒(ϕ⇒(¬ϕ⇒ψ))
The above axiom means that a contradiction (involving a formula ϕ) plus the information thatϕis consistent (represented by the formula◦ϕ) produce a trivial set (from the point of view of logic). In other words, in order to have a trivial (or explosive) set of formulas Γ, it isnotenough to derive a contradiction from Γ:
Γ`ϕ and Γ` ¬ϕ.
In fact, in order to be logically trivial, the set Γ must also derive the consistency of the contradictory formulaϕ:
Γ`ϕ and Γ` ¬ϕ and Γ` ◦ϕ.
The simplestLFIis the logicmbC, introduced in [2] (see also [4]), which is presented below.
Definition 1.1 The calculusmbCis defined over the signature{∧,∨,⇒,¬,◦}
as follows:
Axiom schemas:
(Ax1) ϕ⇒(ψ⇒ϕ)
(Ax2) (ϕ⇒ψ)⇒((ϕ⇒(ψ⇒γ))⇒(ϕ⇒γ)) (Ax3) ϕ⇒(ψ⇒(ϕ∧ψ))
(Ax4) (ϕ∧ψ)⇒ϕ (Ax5) (ϕ∧ψ)⇒ψ (Ax6) ϕ⇒(ϕ∨ψ) (Ax7) ψ⇒(ϕ∨ψ)
(Ax8) (ϕ⇒γ)⇒((ψ⇒γ)⇒((ϕ∨ψ)⇒γ)) (Ax9) ϕ∨(ϕ⇒ψ)
(Ax10) ϕ∨ ¬ϕ
(bc) ◦ϕ⇒(ϕ⇒(¬ϕ⇒ψ)) Inference rule:
(MP) ϕ, ϕ⇒ψ
ψ
InmbCit is possible to defineinconsistency(ornon-consistency) from the consistency operator◦as expected: •ϕ=def ¬◦ϕdenotes that the formulaϕis inconsistent.
It is worth noting that, if we substitute axiom (bc) (weak ‘explosion law’) by the ‘explosion law’:
(exp) ϕ⇒(¬ϕ⇒ψ)
(and if we remove the connective◦from the signature) we obtain an axiomati- zation of classical logic.
2 Standard Deontic Logic
Deontic logic is a modal logic designed for dealing with notions such as “it is obligatory that...” or “it is permitted (or permissible) that...”. There exists several formal systems to deal with such operators, but the basic one is the so- called theStandard Deontic Logic(SDL). We briefly recall below a presentation ofSDL. From now on,F orwill denote the algebra of formulas freely generated by the setP of propositional variables and the connectives∨,∧,⇒,¬, O.
Definition 2.1 The calculusSDLis defined over the signature{∨,∧,⇒,¬, O}
as follows:
Axiom schemas:
axiom schemas (Ax1)–(Ax10) frommbC, plus (exp) ϕ⇒(¬ϕ⇒ψ)
(O-K) O(ϕ⇒ψ)⇒(Oϕ⇒Oψ)
(O-E) Ofϕ⇒fϕ wherefϕ=def(ϕ∧ ¬ϕ), forϕ∈F or Inference rules:
(MP) ϕ, ϕ⇒ψ ψ (O-NEC) `ϕ
`Oϕ
It is worth noting that theDeduction Metatheorem(DM) holds in SDL:
(DM) Γ, ϕ`SDLψ if and only if Γ`SDLϕ⇒ψ
Of course, the rule (O-NEC) can only be applied to theorems ofSDL: thus, in generalϕ6`SDL Oϕ. On the other hand, it is possible to perform proof-by- cases(PBC) inSDL:
(PBC) Γ, ϕ`SDLψ and Γ,¬ϕ`SDLψ implies Γ`SDLψ
We adopt from now on the standard notation (ϕ⇔ψ) to denote the formula (ϕ⇒ψ)∧(ψ⇒ϕ). Thus, for everyϕ, ψ∈F orit holds:
(O∧) `SDLO(ϕ∧ψ)⇔(Oϕ∧Oψ) and
(O-exp) Oϕ, O¬ϕ`SDLψ
On the other hand, if P ϕ =def ¬O¬ϕ denotes the permission operator, whereP ϕmeans that “ϕis permissible”, then
(O-D) Oϕ`SDLP ϕ
There exists several paradoxes in the literature concerningSDL. An impor- tant one is described below, which will be also analyzed in the view of the new proposed systems:
Example 2.2 [Contrary-to-duty obligations]
A well-known paradox of deontic logic is due to R. Chisholm (see [5]). The fol- lowing formulation was presented in [1] (see also the analysis in [8]). Consider the following sentences:
(1) It ought to be that John does not impregnate Suzy Mae.
(2) Not-impregnating Suzy Mae commits John to not marrying her.
(3) Impregnating Suzy Mae commits John to marry her.
(4) John impregnates Suzy Mae.
Let jimp and jmarbe propositional constants representing the sentences “John impregnates Suzy Mae” and “John marries Suzy Mae”, respectively. Clearly, (1) and (4) can be formalized as
(i) O¬jimp (iv) jimp
On the other hand, (2) can be formalized either as O(¬jimp ⇒ ¬jmar) or (¬jimp ⇒ O¬jmar). Since the latter can be derived from (iv) (and since the four sentences above are supposed to be logically independent) we adopt the for- mer formalization for (2):
(ii) O(¬jimp⇒ ¬jmar)
With respect to (3), again two alternative formalizations are possible: either O(jimp ⇒ jmar) or (jimp ⇒ Ojmar). Since the former is derived from (i), we adopt the latter:
(iii) (jimp⇒Ojmar)
Despite the set of four sentences (1)-(4) being apparently consistent, its formal- ization (i)-(iv) is inconsistent inSDL, becauseO¬jmaris derived from (i) and (ii) by (O-K) and (MP), whereasOjmaris derived from (iii) and (iv) by (MP).
From this, the set of sentences (i)-(iv) is logically trivial, by (O-exp).
It should be noted that the trivialization of the deontic operator O can be already be obtained in a system weaker thanSDL. Observe that, by removing the axiom schema (O-E) fromSDLit is obtained the modal logicK. Then, by (exp), (O-K) and transitivity of⇒it follows that
`KOϕ⇒(O¬ϕ⇒Oψ)
for every ϕ and ψ. Therefore Γ `K Oψ for every ψ, where Γ is the set of sentences (i)-(iv) of Example 2.2. In other words, the set Γ produces the trivi- alization of the notion of obligationO, even in the weaker systemK.
In order to avoid the logic trivialization from the set Γ, there exist several alternatives. A simple solution is to remove or modify either (exp) or (O-K), in order to block the derivation above. If one wants to keep classical logic as the basic framework, then (exp) cannot be removed, and so a weaker version of axiom (O-K) could be considered. A different solution, which we develop in the following sections, is to keep (O-K) while considering a weaker version of (exp). More specifically, we propose the definition of a deontic logic based on LFIs, which are weaker than classical logic. By using a paraconsistent negation instead of a classical one, (exp) is no longer valid and so the trivialization argument above cannot be reproduced. It is worth noting that (O-E) does not play a crucial role in the trivialization argument above: it just guarantees the validity of (O-D), which is a desirable property of deontic systems, and then it should be maintained. As we shall see, (O-D) admits several formulations within a paraconsistent framework.
3 Logics of Deontic Inconsistency
Based on the idea ofLFIs, we propose from now on deontic calculi weaker than SDL, based onLFIs instead of classical logic. That is, the negation¬can be assumed to be paraconsistent (and so weaker than classical negation), therefore contradictory obligations such as Oϕ and O¬ϕ do not necessarily trivialize the system. Moreover, contradictory obligations do not trivialize the deontic operatorO. This approach is along the same lines than [6].
It is worth noting that the first approach to paraconsistent deontic logics in the literature was given in [7], where a deontic dimension was added to da Costa’s well-known paraconsistent logicC1. This idea was additionally devel- oped in [9]. Our proposal can be considered, in a certain sense, a generalization of [7].
The simplest deontic LFIis obtained frommbC (recall Definition 1.1) by adding a deontic operatorO. LetF or◦be the algebra of formulas freely gener- ated by the setPof propositional variables and the connectives∧,∨,⇒,¬, O,◦.
Definition 3.1 The logicDmbCis defined over the signature{∧,∨,⇒,¬, O,◦}
by adding to the logicmbC(recall Definition 1.1) the following:
(O-K) O(ϕ⇒ψ)⇒(Oϕ⇒Oψ)
(O-E)◦ O⊥ϕ⇒ ⊥ϕ where⊥ϕ=def (ϕ∧ ¬ϕ)∧ ◦ϕ, forϕ∈F or◦ (O-NEC) `ϕ
`Oϕ
It should be noted that fϕ = (ϕ∧ ¬ϕ) does not trivialize DmbC, as in the case of classical logic (as well as in SDL): there existsϕ andψ such that fϕ6`DmbCψ. On the other hand,⊥ϕ`DmbCψfor everyϕ, ψ∈F or◦, by (bc).
ClearlyDmbCenjoys (DM) and (PBC).
The following notion ofdeontic inconsistencyis useful for our purposes:
Definition 3.2 The expressionϕwill stand for the formulaO¬◦ϕ, meaning thatϕis ‘deontically unsafe’, orϕis ‘deontically ill-behaved’, or even thatϕis
‘deontically inconsistent’.
It is immediate to see that contradictory obligations allow to infer that the involved sentence is deontically inconsistent:
Proposition 3.3 In DmbCthe following holds:
`DmbCOϕ⇒(O¬ϕ⇒ϕ) Proof: It is easy to prove that
`DmbCϕ⇒(¬ϕ⇒ ¬◦ϕ)
because it is valid inmbC(see, for instance, [4]). Then, using (O-NEC),
`DmbC O(ϕ⇒(¬ϕ⇒ ¬◦ϕ)) and so, by (O-K), (MP) and transitivity of⇒, we get
`DmbCOϕ⇒(O¬ϕ⇒O¬◦ϕ), that is,
`DmbCOϕ⇒(O¬ϕ⇒ϕ)
as desired.
Moreover, contradictory obligations do not trivialize the operator O and, therefore, do not trivialize the logicDmbC:
Oϕ, O¬ϕ6`DmbC Oψ
for some ϕ and ψ (for instance, for ϕ =p and ψ =q, where p, q are distinct propositional variables). Therefore,
Oϕ, O¬ϕ6`DmbCδ
for someϕandδ. This can be easily proved by using Kripke structures, to be defined in Section 4 below (see Proposition 4.3).
The next example illustrates how the proposed notion of deontic inconsis- tencies appears in deontic paradoxes.
Example 3.4 [Contrary-to-duty obligations, cont.]
Recall Example 2.2 concerning Chisholm’s paradox. Letjimpandjmarbe propo- sitional constants representing the sentences “John impregnates Suzy Mae” and
“John marries Suzy Mae”, and letΓ be the following set of sentences:
(i) O¬jimp
(ii) O(¬jimp⇒ ¬jmar) (iii) (jimp⇒Ojmar)
(iv) jimp
Then, the sentence O¬jmar is derived in DmbC from (i) and (ii), whereas Ojmar is derived in DmbC from (iii) and (iv). Thus, by Proposition 3.3, the sentence jmar is derived from Γ in DmbC, without trivializing the system.
That is, the logicDmbC allows to infer from (i)-(iv) that the sentence “John marries Suzy Mae” is deontically inconsistent, instead of trivializing (as in the
case ofSDL orK).
As mentioned in the Introduction, the derivation in Example 2.2 of deontic inconsistencies concerning the sentences in Γ do not solve Chisholm’s paradox.
However, the fact that sentences of the formϕcan be derived from a logically trivial (w.r.t.SDL) set helps to detect a deontic conflict. From this information, such sentences could be, for instance, removed or suitably modified in order to overcome the conflict.
4 Kripke semantics for DmbC
In this section it is presented a Kripke-style adequate semantics forDmbC. Let 2be the set {0,1}of classical truth values.
Definition 4.1 A Kripke structure for DmbC is a triple hW, R,{vw}w∈Wi, where:
1. W is a non-empty set (of possible-worlds);
2. R⊆W ×W is a relation (ofaccessibility) between possible-worlds which isserial, that is: for everyw∈W there existsw0 ∈W such thatwRw0; 3. {vw}w∈W is a family of mappingsvw:F or◦→2satisfying the following
clauses, for eachw∈W:
(v1) vw(ϕ∧ψ) = 1 iffvw(ϕ) =vw(ψ) = 1;
(v2) vw(ϕ∨ψ) = 0 iffvw(ϕ) =vw(ψ) = 0;
(v3) vw(ϕ⇒ψ) = 0 iffvw(ϕ) = 1 and vw(ψ) = 0;
(v4) ifvw(ϕ) = 0 thenvw(¬ϕ) = 1;
(v5) ifvw(ϕ) =vw(¬ϕ) thenvw(◦ϕ) = 0;
(v6) vw(Oϕ) = 1 iffvw0(ϕ) = 1 for everyw0 ∈W such thatwRw0.
Given a Kripke structure M = hW, R,{vw}w∈Wi for DmbC, a world w in W and a formula ϕ in F or◦, we write M, w ϕ to denote that vw(ϕ) = 1. If Γ ⊆ F or◦ then M, w Γ will stand for M, w ϕ, for every ϕ ∈ Γ.
The notion of semantical consequence within Kripke structures for DmbC is defined as expected. Thus, given a finite set Γ∪ {ϕ} ⊆ F or◦, we say thatϕ follows semantically fromΓinDmbC, written ΓDmbCϕif, for every Kripke structureM forDmbCand everyw∈W: M, wΓ impliesM, wϕ.
Soundness of DmbCwith respect to Kripke structures for DmbCfollows straightforwardly.
Theorem 4.2 (Soundness for DmbC) Let Γ∪ {ϕ} be a set of formulas in F or◦. ThenΓ`DmbCϕimplies that ΓDmbCϕ.
Proof: It is enough to prove that M, w ϕ for every axiom ϕ of DmbC and every Kripke structure M and world w; together with this, it must be proved the soundness of the inference rules. The unique axioms which deserve some attention are (bc) and (O-E)◦ (since the other are clearly valid). The fact that M, w ϕ for every instance ϕ of (bc) is an easy consequence of clauses (v3) and (v5) of Definition 4.1. With repect of (O-E)◦, suppose that vw(O⊥ϕ) = 1. Let w0 ∈ W such that wRw0 (such a w0 exists, because R is serial). Thus vw0(⊥ϕ) = 1 and then vw0(ϕ) = vw0(¬ϕ) = vw0(◦ϕ) = 1, which contradicts clause (v5) of Definition 4.1. Thereforevw(O⊥ϕ) = 0 and so vw(O⊥ϕ ⇒ ⊥ϕ) = 1, by (v3). With respect to (MP), it is immediate to see that, ifM, w ϕ ⇒ψ and M, w ϕ then M, w ψ, by clause (v3). With respect to (O-NEC), if M, w ϕ for every M, w then M, w Oϕ for every
M, w, by clause (v6).
Using soundness of DmbCwith respect to Kripke structures, it is easy to show that contradictory obligations do not trivialize:
Proposition 4.3 Let pandqbe different propositional variables. Then:
(i) Op, O¬p6`DmbCOq;
(ii) Op, O¬p6`DmbC q.
Proof: Letp and q be different propositional variables. Consider the Kripke structureM =hW, R,{vw}w∈WiforDmbCsuch thatW ={w},R={hw, wi}
andvw is the extension toF or◦of the mappingv:P ∪ {¬p : p∈ P} →2such thatv(p) =v(¬p) = 1,v(q) = 0 andv(¬r) = 1 iffv(r) = 0 for everyr∈ Pdiffer- ent fromp. The mappingvwcan be easily obtained inductively fromvby defin- ing: vw(¬ϕ) = 1 iffvw(ϕ) = 0;vw(Oϕ) =vw(ϕ);vw(◦ϕ) = 1 iffvw(ϕ)6=vw(¬ϕ) (provided that the complexity of◦ϕis defined to be greater than the complex- ity of ¬ϕ) and vw(ϕ#ψ) is defined according to clauses (v1)-(v3) of Defini- tion 4.1, for #∈ {∧,∨,⇒}. ThenM, w {Op, O¬p} but M, w 1Oq and so Op, O¬p1DmbCOq. ThusOp, O¬p6`DmbCOq, by Theorem 4.2. Analogously (i.e., using the same Kripke structure) it is proved thatOp, O¬p6`DmbC q.
On the other hand, with additional hypothesis concerning consistency, con- tradictory obligations trivializeDmbC. In order to see this, we introduce the following notion ofdeontic consistency:
Definition 4.4 The expression ϕ will stand for the formula O◦ϕ, meaning thatϕis ‘deontically safe’, orϕis ‘deontically well-behaved’, or even thatϕis
‘deontically consistent’.
Proposition 4.5 In DmbCit holds
Oϕ, O¬ϕ,ϕ`DmbC ⊥ϕ
and then
Oϕ, O¬ϕ,ϕ`DmbC ψ for everyϕ, ψ∈F or◦.
Proof: From (bc) (recall Definition 1.1), (O-K), (MP) and transitivity of⇒, it follows that`DmbC O◦ϕ⇒(Oϕ⇒(O¬ϕ⇒O⊥ϕ)) and so, by (MP),
Oϕ, O¬ϕ, O◦ϕ`DmbCO⊥ϕ.
Then, by (O-E)◦, (MP) and definition ofit follows that Oϕ, O¬ϕ,ϕ`DmbC⊥ϕ. Finally,
Oϕ, O¬ϕ,ϕ`DmbC ψ
since⊥ϕ`DmbCψ, for everyψ.
In order to prove the completeness theorem for DmbC with respect to Kripke structures for DmbC, a canonical model can be constructed. Firstly, the notion ofϕ-saturated set inDmbCis considered.
Definition 4.6 Let ∆∪ {ϕ} ⊆ F or◦ be a set of formulas. We say that ∆ is ϕ-saturated in DmbCif:
(i) ∆6`DmbCϕ;
(ii) ifψ /∈∆ then ∆, ψ`DmbCϕ.
The following properties can be easily proved:
Lemma 4.7 Let ∆ be aϕ-saturated set inDmbC. Then:
(i)∆`DmbCψiff ψ∈∆;
(ii)ψ1∧ψ2∈∆ iffψ1∈∆ andψ2∈∆;
(iii)ψ1∨ψ2∈∆ iffψ1∈∆ orψ2∈∆;
(iv)ψ1⇒ψ2∈∆ iffψ1∈/ ∆or ψ2∈∆;
(v) ifψ /∈∆ then¬ψ∈∆;
(vi) ifϕ,¬ϕ∈∆then ◦ϕ6∈∆.
The following version of Lindenbaum-Asser’s Lemma can be proved, by adapting the classical proof (see, for instance, [4]):
Lemma 4.8 Let ∆∪ {ϕ} ⊆F or◦ be a set of formulas such that ∆6`DmbC ϕ.
Then, there exists aϕ-saturated set ∆0 inDmbCsuch that ∆⊆∆0.
Then following notion will be useful:
Definition 4.9 Let ∆ be a ϕ-saturated set inDmbC. The denecessitationof
∆ is the setDen(∆) =def {ψ∈F or◦ : Oψ∈∆}.
In order to obtain the completeness theorem, one more lemma is needed.
Lemma 4.10 Let∆ be a ϕ-saturated set inDmbC.
(i) The setDen(∆) is a closed theory inDmbC, that is: ifDen(∆)`DmbCψ thenψ∈Den(∆).
(ii) IfOψ /∈∆ thenDen(∆),¬ψ6`DmbC ψ.
Proof: (i) Suppose thatDen(∆) `DmbC ψ. Then, there exists ψ1, . . . , ψn ∈ Den(∆) such that ψ1, . . . , ψn`DmbC ψand so, by (DM), it follows that
`DmbC(ψ1⇒(· · · ⇒(ψn⇒ψ)· · ·)).
By (O-NEC),
`DmbCO(ψ1⇒(· · · ⇒(ψn⇒ψ)· · ·)) and then, by (O-K), (MP) and transitivity of⇒, we get
`DmbC (Oψ1⇒(· · · ⇒(Oψn⇒Oψ)· · ·)).
ButOψ1, . . . , Oψn ∈ ∆, by definition ofDen(∆), therefore ∆`DmbC Oψ, by (MP). ThusOψ∈∆, by property (i) of Lemma 4.7, and thenψ∈Den(∆).
(ii) Suppose that Den(∆),¬ψ `DmbC ψ. Since Den(∆), ψ `DmbC ψ then, using (PBC), it follows thatDen(∆)`DmbCψ. Thus, by item (i),ψ∈Den(∆).
That is,Oψ∈∆.
Definition 4.11 Thecanonical model forDmbCis the triple Mc =hW, R,{v∆}∆∈Wi
such that:
1. W={∆⊆F or◦ : ∆ is aϕ-saturated set inDmbCfor someϕ};
2. R={h∆,∆0i ∈ W × W : Den(∆)⊆∆0};
3. v∆ is the characteristic map of ∆, that is: v∆(ψ) = 1 iffψ∈∆.
Using the lemmas stated above it is easy to prove the following:
Proposition 4.12 The canonical modelMc is a Kripke structure forDmbC.
Proof: We begin by proving that R is serial. Thus, let ∆ be a ϕ-saturated set inDmbC. Then, there must be a formula ψsuch that Den(∆)6`DmbC ψ.
Otherwise, Den(∆) `DmbC ⊥ϕ and then ⊥ϕ ∈ Den(∆), by Lemma 4.10(i).
ThusO⊥ϕ∈∆ and so ∆`DmbC⊥ϕ, by (O-E)◦and (MP). From this it follows that ∆ `DmbC ϕ, a contradiction. Therefore Den(∆) 6`DmbC ψ, for some formulaψ. Then, by Lemma 4.8, there exists a ψ-saturated set ∆0 in DmbC such thatDen(∆)⊆∆0. In other words, there exists ∆0 ∈ W such that ∆R∆0, and thenRis serial.
Let ∆∈ W. By Lemma 4.7(ii)-(vi) it follows that v∆ satisfies clauses (v1)- (v5) of Definition 4.1. It remains to prove that, for every formulaψ inF or◦:
v∆(Oψ) = 1 iff v∆0(ψ) = 1 for every ∆0 such thatDen(∆)⊆∆0.
Given ψ, suppose that v∆(Oψ) = 1 and let ∆0 ∈ W such thatDen(∆)⊆∆0. ThenOψ∈∆ and soψ∈Den(∆), by definition of Den(∆). Thereforeψ∈∆0 and thenv∆0(ψ) = 1.
Conversely, if v∆(Oψ) = 0 then Oψ /∈∆. ThusDen(∆),¬ψ6`DmbC ψ, by Lemma 4.10(ii). Using Lemma 4.8, there exists aψ-saturated set ∆0 inDmbC such that Den(∆)∪ {¬ψ} ⊆∆0. Therefore ∆0 ∈ W such that Den(∆)⊆ ∆0 andv∆0(ψ) = 0. Thus,{v∆}∆∈W satisfies clause (v6) of Definition 4.1 and then
Mc is in fact a Kripke structure for DmbC.
We can finally prove the completeness theorem forDmbC:
Theorem 4.13 [Completeness forDmbC] LetΓ∪ {ϕ}be a set of formulas in F or◦. ThenΓDmbC ϕimplies thatΓ`DmbCϕ.
Proof: Suppose that Γ 6`DmbC ϕ. By Lemma 4.8, we can extend Γ to aϕ- saturated set ∆ in DmbC. Since ∆ 6`DmbC ϕ then ϕ /∈ ∆. Let Mc be the canonical model forDmbC (cf. Definition 4.11). Then, by Proposition 4.12, Mc is a Kripke structure forDmbCand ∆ is a possible-world ofMc such that Mc,∆ Γ (since Γ ⊆ ∆) and Mc,∆ 1 ϕ (since ϕ /∈ ∆). This shows that
Γ1DmbCϕ.
5 Permission and Prohibition
In this section we analyze the definition of the deonticpermissibleandforbidden operators withinDmbC. Additionally, several forms of the law
(∗) Oϕ⇒P ϕ
(whereP stands for the permissible operator) will be analyzed in the context ofDmbC. Observe that (∗) corresponds to (O-D) inSDL.
In order to analize (∗) in the paraconsistent framework, recall the notion of deontic consistency(cf. Definition 4.4).
We begin by observing that, in mbC (and so, in DmbC) it is possible to define a classical negation ∼ as follows: ∼ϕ =def ϕ ⇒ ⊥ϕ. The derived connective∼is considered to be as a classical negation because of the following proposition:
Proposition 5.1 For everyϕ, ψ∈F or◦ it holds:
(i) ϕ,∼ϕ`DmbCψ;
(ii) `DmbC ϕ∨ ∼ϕ;
(iii) ∼∼ϕ`DmbCϕ;
(iv) ϕ`DmbC ∼∼ϕ.
Proof: It follows from the corresponding proof for the negation ∼ defined
analogously inmbC(cf. [4]).
Therefore, the logicDmbChas two negations: a paraconsistent one,¬, and a classical one,∼. Being so, it is possible to define inDmbCfourpermissibility operatorsfrom Oby combining both negations as follows:
(P1) P1ϕ=def ¬O¬ϕ;
(P2) P2ϕ=def ∼O¬ϕ;
(P3) P3ϕ=def ¬O∼ϕ;
(P4) P4ϕ=def ∼O∼ϕ.
The formulaPiϕreads as “ϕisi-permissible”, fori= 1, . . . ,4.
Analogously, it is possible to define inDmbCtwoprohibition operatorsfrom Oby using both negations as follows:
(F1) F1ϕ=def O¬ϕ;
(F2) F2ϕ=def O∼ϕ.
The formulaFiϕreads as “ϕisi-forbidden”, fori= 1,2.
The semantics of these operators works “classically” just in the case of P4 andF2; in the case of P2 just a half of the classical definition of permissibility holds, and in the case of P1, P3 and F1 the semantics is far from classical, because of the paraconsistent characteristics of¬. Thus,
• M, wP4ϕiff there exists somew0∈W such thatwRw0 andM, w0ϕ;
• M, wP2ϕimplies that there exists somew0 ∈W such thatwRw0 and M, w0ϕ;
• M, wF2ϕimplies thatM, w0 1ϕfor every w0∈W such that wRw0. With respect to the law (∗) ruling the derivation of permissibility from oblig- ation, it can be proven the following versions of (O-D) inDmbC:
Proposition 5.2 In DmbCthe following holds, for everyϕ∈F or◦: (i) ϕ`DmbCOϕ⇒P1ϕ
(ii) ϕ`DmbCOϕ⇒P2ϕ (iii) `DmbCOϕ⇒P3ϕ (iv) `DmbCOϕ⇒P4ϕ.
Proof: (i) From Proposition 4.5 it follows that Oϕ,ϕ, O¬ϕ`DmbC P1ϕ.
On the other hand, by definition ofP1,
Oϕ,ϕ,¬O¬ϕ`DmbC P1ϕ
and so, using (PBC),
Oϕ,ϕ`DmbCP1ϕ.
The rest of the proof follows by (DM).
(ii) From Proposition 4.5 it follows that
Oϕ,ϕ, O¬ϕ`DmbC⊥ψ
forψ=O¬ϕ. The rest of the proof follows by (DM) and the definition of P2. (iii) Sinceϕ,∼ϕ`DmbC⊥ϕit follows (as usual) that
Oϕ, O∼ϕ`DmbCO⊥ϕ.
By (O-E)◦and the fact that⊥ϕ`DmbCP3ϕwe get Oϕ, O∼ϕ`DmbCP3ϕ.
On the other hand, by definition ofP3,
Oϕ,¬O∼ϕ`DmbCP3ϕ
and soOϕ`DmbCP3ϕ, by (PBC). The rest of the proof follows by (DM).
(iv) As proved above, Oϕ, O∼ϕ `DmbC ⊥ϕ. Using that ⊥ϕ `DmbC ⊥ψ for ψ= O∼ϕ it follows that Oϕ`DmbC ∼O∼ϕ, by (DM). The rest of the proof
follows by (DM) and the definition ofP4.
With respect to theprohibition operators, the following proposition is easy to prove. We left to the reader the details of the proof.
Proposition 5.3 Let ϕ, ψ ∈ F or◦ and p, q different propositional variables.
Then:
(i) ϕ, Oϕ, F1ϕ`DmbCψ (ii) Op, F1p0DmbCq
(iii) Oϕ, F2ϕ`DmbC ψ.
6 Propagating deontic inconsistency and deon- tic consistency
Finally, we present in this section another deonticLFI, which is stronger than mbC. It is based on the logicLFI1introduced in [3], which allows inconsistency to be propagated. In that paper, a first-order version of LFI1 was used for dealing with databases allowing contradictions.
From now on, the expression •ϕ will stand for the formula¬◦ϕ. Recalling the deontic inconsistency operator introduced in Definition 3.2, it can be written asϕ=O•ϕ.
Definition 6.1 The logic LFI1 is obtained by adding to mbCthe following axiom schemas:
(cef ) ϕ⇔ ¬¬ϕ (ci) •ϕ⇒(ϕ∧ ¬ϕ)
(cj1) •(ϕ∧ψ)⇔((•ϕ∧ψ)∨(•ψ∧ϕ)) (cj2) •(ϕ∨ψ)⇔((•ϕ∧ ¬ψ)∨(•ψ∧ ¬ϕ))
(cj3) •(ϕ⇒ψ)⇔(ϕ∧ •ψ)
Definition 6.2 The logic DLFI1 is defined by adding to the logicLFI1 the following:
(O-K) O(ϕ⇒ψ)⇒(Oϕ⇒Oψ)
(O-E)◦ O⊥ϕ⇒ ⊥ϕ where⊥ϕ=def (ϕ∧ ¬ϕ)∧ ◦ϕ, forϕ∈F or◦ (O-NEC) `ϕ
`Oϕ
Equivalently, DLFI1 can be defined as the logic obtained fromDmbCby adding the axiom schemas of Definition 6.1.
The proof of the following properties of “propagation of deontic inconsis- tency” inDLFI1is straightforward:
Proposition 6.3 The logicDLFI1 satisfies the following:
(i) `DLFI1ϕ⇔(Oϕ∧O¬ϕ) (ii) ϕ, Oψ`DLFI1(ϕ∧ψ) (iii) ψ, Oϕ`DLFI1(ϕ∧ψ) (iv) ϕ, O¬ψ`DLFI1(ϕ∨ψ) (v) ψ, O¬ϕ`DLFI1(ϕ∨ψ)
(vi) `DLFI1(Oϕ∧ψ)⇔(ϕ⇒ψ).
An adequate Kripke semantics forDLFI1can be defined by adding the obvi- ous clauses to the mappingsvwrepresenting the new axioms from Definition 6.1.
We left the details to the reader.
It is interesting to note that, because of the propagation of deontic incon- sistency, the Chisholm paradox produces new sentences which are deontically inconsistent inDLFI1:
Example 6.4 [Contrary-to-duty obligations, cont.]
Recall Example 2.2. Using Example 3.4, from the setΓof sentences (i) O¬jimp
(ii) O(¬jimp⇒ ¬jmar) (iii) (jimp⇒Ojmar)
(iv) jimp
it is derived in the logic DmbC (and then, in DLFI1) the sentence jmar.
SinceΓ`DLFI1(O¬jimp∧jmar)it follows from Proposition 6.3 that Γ`DLFI1(¬jimp∧jmar),
Γ`DLFI1(jimp∨jmar), Γ`DLFI1(¬jimp⇒jmar).
That is, new sentences are proved to be deontically inconsistent fromΓ, by using
a logic system stronger thanDmbC.
7 Final Remarks
This paper proposes a logic context in which conflicting obligations are allowed without trivializing the system. The key is to use a logic basis weaker than classical logic and tolerant to contradictions, that is, a paraconsistent logic.
Being so, the present framework could be a useful tool for analyzing moral dilemmas and deontic paradoxes.
As pointed out above, a similar line of research was introduced in [7] (see also [9]); however, there exist some differences between those proposals and our approach. In [7] and [9] the proposed system, calledC1D, is based on paracon- sistent logicC1in which the consistency operator can be defined in terms of the other connectives as follows: ◦ϕ:=¬(ϕ∧ ¬ϕ). Besides this, consistency “prop- agates” through the connectives (includingO) and so, in particular,◦ϕ⇒ ◦Oϕ is a theorem ofC1D. Moreover,Oϕ⇒ ¬O¬ϕis also a theorem ofC1D which, as we saw in Proposition 5.2(i), is not the case inDmbC: in order to obtainP1ϕ fromOϕ inDmbC it is necessary to add the hypothesis thatϕis deontically consistent, that is,ϕ. Besides this, the logic basis DmbCis strictly weaker thanC1(see [4]) and so we could say that the present approach is slightly more general than that of [7].
Finally, the stronger systemDLFI1allows to deal with more sophisticated notions involving deontic consistency and deontic inconsistency, within a richer language in which deontic and consistency operators interact naturally. Addi- tionally, this system could shed some light on the analysis of deontic inconsis- tencies in the context of databases. The extension ofDLFI1to first-order logic by adapting the semantics presented in [3] is the first step towards this goal.
Acknowledgements: This research was financed by FAPESP (Brazil), The- matic Project ConsRel 2004/1407-2. The author was also supported by an individual research grant from the Brazilian Council of Research (CNPq).
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