BRYSON BROWN

YES, VIRGINIA, THERE REALLY ARE PARACONSISTENTLOGICS

Received 8 January 1998; revised 3 August 1998

ABSTRACT. B. H. Slater has argued that there cannot be any truly paraconsistent logics,because it’s always more plausible to suppose whatever “negation” symbol is used in thelanguage is not a real negation, than to accept the paraconsistent reading. In this paperI neither endorse nor dispute Slater’s argument concerning negation; instead, my aim isto show that as an argument against paraconsistency, it misses (some of) the target. Aimportant class of paraconsistent logics — the preservationist logics — are not subject tothis objection. In addition I show that if we identify logics by means of consequence rela-tions, at least one dialetheic logic can be reinterpreted in preservationist (non-dialetheic)terms. Thus the interest of paraconsistent consequence relations — even those that emergefrom dialetheic approaches — does not depend on the tenability of dialetheism. Of course,if dialetheism is defensible, then paraconsistent logic will be required to cope with it.But the existence (and interest) of paraconsistent logics does not depend on a defenseof dialetheism.

KEY WORDS: aggregation, dialetheism, paraconsistent logic, preservationism

INTRODUCTION

B. H. Slater has argued1 that there cannot be any truly paraconsistent log-ics, because it’s always more plausible to suppose whatever “negation”symbol is used in the language is not a real negation, than to accept theparaconsistent reading. In this paper I neither endorse nor dispute Slater’sargument concerning negation; instead, my aim is to show that as an argu-ment against paraconsistency, it misses (some of) the target. A importantclass of paraconsistent logics — the preservationist logics — are not sub-ject to this objection. So regardless of the outcome of the debate betweenthe dialetheic2 schools of paraconsistency and Quinean reinterpreters likeSlater, paraconsistent logic as a whole cannot be dismissed as oxymoronic.In addition I show that if we identify logics by means of consequencerelations, at least one dialetheic logic can be reinterpreted in preservation-ist (non-dialetheic) terms. Thus the interest of paraconsistent consequencerelations — even those that emerge from dialetheic approaches — does notdepend on the tenability of dialetheism.

Journal of Philosophical Logic28: 489–500, 1999.© 1999Kluwer Academic Publishers. Printed in the Netherlands.

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THE PROBLEM OF DETONATION

Paraconsistent logic arises as an attempt to deal more constructively withinconsistent premise sets than classical logic does. Some of those who takean interest in it are fairly conservative thinkers who are merely sensitiveto our epistemic limits, and prepared to acknowledge the need to reason,from time to time, with premises that cannot all be true, without havingthem explode in a puff of logic.

Of course, the threat of that puff of logic emerges from a very deepseated conception of what logical consequence relations must be like. Weusually think of the semantic consequence relation as truth-preserving:

A: 0 � α iff ∀0′ : 0′ is a maximal satisfiable extension of

0⇒ α ∈ 0′.Similarly, we usually think of the syntactic consequence relation as consis-tency-preserving:

B: 0 ` α iff ∀0′ : 0′ is a maximal consistent extension of

0⇒ α ∈ 0′.Closing0 under either “�” or “`” produces extremely conservative

extensions of0. We can say that the extended set begs no questions0

leaves open, in the sense that no consistent or satisfiable extension of0

is an inconsistent/unsatisfiable extension of the closure of0 under “̀ ”or “�”. With this point in mind, we can repackageA andB in slightlydifferent form:

A ′: 0 � α iff ∀1, if 0 ∪1 is satisfiable, so is0, α ∪1.B′: 0 ` α iff ∀1, if 0 ∪1 is consistent, so is0, α ∪1.

On this account the threat of detonation is inevitable: A logical con-sequence of0 is something whose addition to0 preservesthe satisfiabil-ity/consistency of any satisfiable or consistent extension of0. So any setthat lacks such extensions (as any inconsistent or unsatisfiable set does,since these properties are preserved in all supersets) must have every wffas a consequence.

TWO KINDS OF PARACONSISTENCY

Two possible approaches to paraconsistency emerge from this understand-ing of logical consequence. First, one might simply add to the set of “con-sistent” or “satisfiable” extensions, so that sets which classicaly lack such

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extensions now turn out to have some. This dialetheic approach requiresthat we allow both ‘p’ and ‘¬p’ to be simultaneously true in some cases,which raises prima facie doubts as to whether ‘¬’ is really a negation.Thus those who choose this approach must answer Slater’s argument byproducing reasons to regard their “negation” asreal negation.

But the second approach does not change our usual account of con-sistency or satisfiability at all. Instead, it turns on finding a new propertyor properties which we will require our new consequence relation to pre-serve. Call the propertiesCi, i = 1, n. So long as at least some inconsis-tent/unsatisfiable sets haveCi for somei, there will be a non-empty classof supersets of0 that also haveCi, and we can define a new consequencerelation that does not trivialize for all inconsistent/unsatisfiable sets:

C′: 0 I α iff ∀1, if Ci(0 ∪1), thenCi(0, α ∪1).P. K. Schotch and R. E. Jennings made the first proposal for such a

consequence relation.3 The properties their consequence relation preservesare calledlevels, or degrees of incoherence. They define:

Con(0, ξ) iff there is a family of setsA = φ, a1, . . . , ai for i ≤ ξ suchthat:

(i) a ∈ A⇒ a is consistent.

(ii) ∀γ ∈ 0, ∃a ∈ A : a ` γ.This notion is a generalization of consistency. For a set of tautologiesT ,Con(T , ξ) holds forξ ≥ 0; for a consistent but non-tautologous set,X,Con(X, ξ) holds forξ ≥ 1. Given this notion of consistency with degrees,we can measure thedegree of incoherenceof a set0, i(0):

i(0)def={

minξ | Con(0, ξ), if this limit exists,∞, otherwise.

0’s degree of incoherence is, intuitively, a measure of how drastically wemust divide0’s content up in order to make all the divisions consistent.4

Obviously enough we could tell an exactly parallel story generalizing sat-isfiability instead of consistency. But in either case the upshot is the same:We have here a specific property of premise sets that we might wish to pre-serve when reasoning: Peter Schotch calls any inference relation meetingthis conditionHippocratic, with the motto “primum non nocere”5 in mind.

The new version ofB ′ now reads:

BH: 0[` α iff ∀1i(0 ∪1) = i(0)⇒ i(0, α ∪1) = i(0).So[` preserves the degree-preserving extensions of0 just as` preservesthe consistency-preserving extensions. Another proposal along similar linesis due to Johnston and Jennings.6

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Slater’s dismissal of paraconsistent logics founders here. There is noth-ing for the classical logician to object to at this point; nothing has beensaid about these sets that she cannot see to be true of them. Of course shemight question whether[` is areal consequence relation. But this is a verydifferent issue, and (furthermore) is far from straightforward or obvious.[` differs from` only in how it aggregatespremises, and aggregation isan issue that logicians have generally paid little attention to. The range ofaggregational “power” that goes with different degrees of incoherence inthis logic turns out (for graph-theoretical reasons) to cover the entire rangeof aggregational force a logic might have. Thus, as well as introducing adifferent kind of paraconsistent logic, one which resists Slater’s reinterpre-tive objection, Schotch and Jennings have also pioneered the general logicof aggregation.

REINTERPRETING LOGIC

But there is still more to say. These two classes of paraconsistent logicsare not as sharply distinguished as we might at first think. Logics ini-tially presented as instances of one sort can be reinterpreted as instancesof the other. For example, someone might (perversely) read Schotch andJennings’ logic as rejecting “∧-intro” because sometimes{a, b} can be“satisfied” when{(a ∧ b)} is not.7

My interest here, though, is in a reinterpretation that goes the other way:a preservationist interpretation of a dialetheic paraconsistent logic. Priestand Tanaka present the multi-valued, dialetheic approach to paraconsis-tency in this way:

A many-valued logic will therefore be paraconsistent if it allows both a formula and itsnegation to be designated. The simplest strategy is to use three truth values: true (only) andfalse (only), which function as in classical logic, and both truth and false (which, naturally,is a fixed point for negation). Both varieties of truth are designated. This is the approach ofthe paraconsistent logic LP.8

Both Priest and Routley have frequently used LP as a simple illustrationof a paraconsistent logic. The easiest way to present it is to read the threevalues asT = {T }, F = {F } andB = {T, F } and treat the connectives asfollows:

T ∈ V (p¬αq) iff F ∈ V (α),F ∈ V (p¬αq) iff T ∈ V (α),T ∈ V (p(α ∧ β)q) iff T ∈ V (α) andT ∈ V (β),F ∈ V (p(α ∧ β)q) iff F ∈ V (α) or F ∈ V (β),

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T ∈ V (p(α ∨ β)q) iff T ∈ V (α) or T ∈ V (β),F ∈ V (p(α ∨ β)q) iff F ∈ V (α) andF ∈ V (β).

The resulting consequence relation is arrived at in the usual way:

0 �LP α iff ∀V (V satisfies all members of0 ⊃ V satisfiesα)

whereV ranges over these 3-valued valuations, and bothT andB aredesignated.�LP is clearly paraconsistent; roughly speaking, any inconsis-tency is limited to the variables which must receive “both” to satisfy agiven premise set. Satisfaction under any and all assignments satisfyinga premise set provides a reasonable consequence relation here becauseassignments making classical assignments to all variables aside from thosethat are given non-classical assignments in order to satisfy the premiseset satisfy proper subsets of the sentences satisfied by gratuitously non-classical assignments. So what one might call the “most classical” assign-ments are the ones that determine what follows from a premise set.

Of course, Slater’s objection applies to this logic: the reading of “¬”as a negation needs defense. But I have a different defense of the para-consistency of this logic in mind — one which dialtheists will decline asnot answering to their needs, but one which Slater should consider beforedeclaring this logic is not truly paraconsistent.9

The slogan of the preservationist approach is “Find something you likeabout your premises, and then tailor your consequence relation to preserveit.” We’ve already examined the proposal that we should preserve degreesof incoherence, a generalization of the notion of consistency. But here isanother proposal: Suppose that we regard sentence letters as (potentially)ambiguous, and consider the possibility of arriving a consistent images ofsets of sentences by treating (some) sentence letters in them as ambiguous.To arrive at a consistent image of the set of all wffs, of course, we must treatall the sentence letters as ambiguous. But to arrive at a consistent imageof {p,¬p}, we need only treat ‘p’ as ambiguous. So if we demand preser-vation of some property in this neighborhood, closing under the resultingconsequence relation will not trivialize this set.

That this notion of treating sentence letters as ambiguous is very closelyrelated to the LP approach to paraconsistency is shown by the following:

MAIN THEOREM

GivenV,V ′, γ,R where:

– V is an LP-valuation,

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– V ′ is a classical valuation such that, for every sentence letter:

(i) V (s) = T, F → V ′(s) = T, F ,(ii) V (s) = B → V ′ dispenses withs, but adds new letterssT andsF ,

whereV ′(sT ) = T, V ′(sF ) = F .

– γ is a wff.

Let Rγ be the class of all functionsR ∈ γ → γ ′, whereγ ′ resultsfrom γ by replacing each instance of anys ∈ V −1(B) with one or theother of sT , sF . (EachR thus creates an image ofγ under a possibledisambiguation of the instances of sentence letters inγ assigned “both”byV .)

(i) V (γ ) = T iff ∀R ∈ Rγ, V ′(R(γ )) = T,(ii) V (γ ) = F iff ∀R ∈ Rγ, V ′(R(γ )) = F,

(iii) V (γ ) = B iff ∃R1, R2 ∈ Rγ, V ′(R1(γ )) = T,V ′(R2(γ )) = F.

Proof.By induction on the number of connectives inγ .(1) Supposeγ is a sentence letter,s. Then (i) and (ii) hold trivially given

the definitions ofV,V ′, and (iii) clearly holds as well, since we can satisfythe two clauses by makingR1(s) = sT , andR2(s) = sF .

(2) Assume (i)–(iii) hold for allγ with up ton connectives.(3) Supposeγ containsn+ 1 connectives. Then(a) Supposeγ = ¬α. By our induction hypothesis, (i)–(iii) hold for

α. V ′(R(γ )) = T for everyR ∈ Rγ if and only if V ′(R(α)) = F foreveryR ∈ Rα (i.e. iff every way of disambiguating the sentence lettersin alpha makesα true, and thusγ false). But by our induction hypothesis,V ′(R(α)) = F for everyR ∈ Rα iff V (α) = F , which holds iffV (γ ) =T . Thus (i) holds. A similar argument shows the same for (ii).

For (iii), by our induction hypothesis,V (α) = B iff there areR1 andR2 in Rα such thatV ′(R1(α)) = T and V ′(R2(α)) = F . But this isboth necessary and sufficient for there to be anR3 and anR4 (follow-ing the replacement patterns ofR2 andR1, respectively) inRγ such thatV ′(R3(γ )) = T andV ′(R4(γ )) = F .

(b) Supposeγ = (α ∧ β). ThenV ′(R1(α)) = V ′(R2(β)) = T foreveryR1 ∈ Rα and everyR2 ∈ Rβ iff V (γ ) = T for everyR ∈ Rγ(since everyR ∈ Rγ can be treated as a pair,R1, R2 whereR1 ∈ RαandR2 ∈ Rβ, by makingR replace sentence letters inα asR1 does,and sentence letters inβ asR2 does). But by our induction hypothesis,V ′(R1(α)) = V ′(R2(β)) = T for everyR1 ∈ Rα and everyR2 ∈ Rβ

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iff V (α) = V (β) = T iff V (γ ) = T . Thus (i) holds. Again, a similarargument shows the same for (ii).

For (iii), V (γ ) = B iff V (α) = V (β) = B, orV (α) = B andV (β) =T , or V (α) = T andV (β) = B. By our induction hypothesis,V (α) = Biff ∃R1, R2 ∈ Rα, V ′(R1(α)) = T , V ′(R2(α)) = F , and similarly forβ. And so these three alternative assignments toα andβ are individuallysufficient and disjunctively necessary for there being anR1 ∈ Rγ such thatV ′(R(γ )) = T and anR2 ∈ Rγ such thatV ′(R(γ )) = F .

(c) The argument forγ = (α ∨ β) parallels (b).

COMMENTS

(1) V ′,Rγ allow us to make the same distinctions between wffs that aremade byV , i.e. to duplicate the division of wffs into those assignedT, F andB.

(2) 0 is LP-satisfied byV iff ∃R1 . . . Rn ∈ Rγ1, . . . ,Rγn such thatV ′(Ri(γi)) = T for all γi ∈ 0. Since we can regard such a collectionof Ri as a single function from each instance of ans ∈ V −1(B) in 0to one or another ofsT , sF , we can say that a set0 is LP satisfiable iffone of its disambiguated images is classically satisfiable. Any0 thatcan be LP-satisfied by assigningB to some set of sentence letters has aclassically satisfiable disambiguation which treats only those sentenceletters as ambiguous.

A New Consequence Relation

The close ties between LP and this sort of ambiguity also show up when weconsider consequence relations. For example, a consequence relation veryclosely related to LP results when we choose to preserve the minimumnumber of letters that we must treat as ambiguous in order to produce aconsistent image of some premise set. Since we can treat sentence letters asambiguous even within a single wff, the set{(p∧¬p)}will have non-trivialconsequences for the same reason, so it’s clear that this preservationistlogic is not the same as Schotch and Jennings’ weakly aggregative ap-proach. What logicdo we get if we follow this through? The presentationof Schotch and Jennings’ system above suggests the following definitions:

Ambcon: Ambcon(ξ, 0) iff 0 can be made consistent by treatingξsentence letters as ambiguous.10

1 : The level of ambiguity(“lamb” for short) of a set0,

1(0) = Minξ | Ambcon(ξ, 0).

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And now we can introduce our new consequence relation,F, as follows:

0 F α iff every 1-preserving extension of0

is a1-preserving extension of0, α.

Just as before, nothing is said here that a classical logician will notaccept as true. We do not say that a classically inconsistent0 is reallyconsistent (or satisfiable) after all; we say only that preserving1 insteadof preserving consistency leads to a new consequence relation which doesnot trivialize all inconsistent sets. Whether that consequence relation isinteresting is another question. But we can say at least this about it:

(1) Whenever1(0) = 0, 0 F α iff 0 ` α.

(2) When1(0) > 0, closing0 under` trivializes,

while closing underFmay not.

F does not trivialize all inconsistent sets, and it preserves the classicalconsequence relation for consistent sets. As a consequence relation it is notsubject to Slater’s objection, and, furthermore, it is closely related, thoughnot identical, to LP.

DIFFERENT WAYS TO LOCATE THE AMBIGUITY

Sometimes the ambiguities can be located in different ways. Consider0 ={p, q,¬(p ∧ q)}. Clearly here we can produce a consistent image of0

either by treatingp as ambiguous (turning0 into 0′ = {p, q,¬(p′ ∧ q)})or by treatingq as ambiguous (obtaining0′′ = {p, q,¬(p ∧ q ′)}). Bothtreat a single sentence letter as ambiguous, and so extensions of0 thatmake either “commitment” re. what letter must be treated as ambiguousto produce a consistent image of0 will be 1-preserving. For example,01 = {p, q,¬(p ∧ q),¬p} and02 = {p, q,¬(p ∧ q),¬q} are both1-preserving extensions of0. But, of course,03 = {p, q,¬(p∧q),¬p,¬q}is not1-preserving.

But in some cases different ways of locating the ambiguity will lead todifferent values for1; when this occurs, only extensions which either makeno commitments re. the location of the ambiguity, or which reflect com-mitments which minimize1 will be 1-preserving. Let0 = {p, q, r,¬((p∨q)∧ r)}. To produce a consistent image of this set we can invoke an ambi-guity involving ‘p’ and ‘q’, or one involving ‘r ’ alone. But only extensionsof 0 which locate the ambiguity inr will be 1-preserving. Thus we have:

Min: 0 F ¬r.

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But0 2LP ¬r.11

Thus the logic we have here is not quite LP; it is stronger than LP,because it forces us to choose among the extensions of0 those whichminimize the number of letters that must be treated ambiguous to producea consistent image of0, while LP allows us to consider extensions of0which pursue either way of “satisfying”0, and regards as a consequenceof 0 only what holds in all of these extensions.0 `LP α ⇒ 0 F α holds;but not its converse.

A PRESERVATIONIST TAKE ONLP

However, we can have a preservationist consequence relation that is iden-tical with LP’s. LP andF differ becauseF preserves the minimum numberof letters that must be treated ambiguously. In our last example, therewere two ways of using ambiguity to produce consistent images of0. Wecould treat both ‘p’ and ‘q’ as ambiguous, or ‘r ’ alone. Our counting-based approach forced us to choose the latter, thus giving rise to0 F ¬r.But counting needn’t be the measure of what we want to preserve here.Instead, we can begin with a set of sets,L(0), including every minimal setof sentence letters such that treating those sentence letters as ambiguousallows us to construct a consistent image of0.12 Then we can ask that thisset be preserved by anyadmissible extensionof 0,0E, in the sense that0E

is an admissible extension iffL(0) ⊇ L(0E).Once we have the admissible extensions specified, of course, it’s easy

to apply our general recipe to generate a consequence relation:

0|F α iff α is anL-preserving extension

of everyL-preserving extension of0, or

0|F α iff (0 ⊆ 0′ andL(0) ⊇ L(0′))⇒ L(0) ⊇ L(0′, α).

This new version of our policy of minimizing the amount of ambi-guity required to produce a consistent image of0 brings the resultingconsequence relation,|F, into full agreement with LP:0 �LP α iff ∀V (V satisfies all members of0 ⊃ V satisfiesα). Consider

an assignment to all sentence letters which satisfies0,V0. If V0 assigns“both” to some sentence letters not belonging to a single member ofL(0),the sentences satisfied byV0 will be a proper superset of the sentencesin a more conservative extension, whose satisfaction in LP’s three valuedscheme confines the value “both” to the members of some member ofL(0). Thus theL(0) preserving extensions of0 are the key to0’s LP

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consequences. So long as an assignment preservesL(0), it will be a “min-imal” extension of0, i.e. one that is not trivially a superset of some otherextension of0. The intersection of the extensions determined by theseminimally paradoxical assignment to the sentence letters is the closure of0 under LP’s consequence relation.

On the other hand,0| Fα iff α is aL-preserving extension of everyL-preserving extension of0. But the maximalL-preserving extensions arejust the extensions determined by the minimally paradoxical assignmentswhose intersection determines the LP-consequences of0: A maximal L-preserving extension,0max, must includepγ q, p¬γ q or both, for everywff γ . 0max will include both whenever this is compatible with beingL-preserving. Thus, for some member ofL(0), 1, 0max will include both sand¬s for every sentence letters in 1, and0max will not include bothfor any other sentence letters. The rest of the sentences in0max will beprecisely those satisfied by the LP valuation,V0max that makesT ∈ V (a)for every sentence lettera ∈ 0max andF ∈ V (a) for every sentence lettera such thatp¬aq ∈ 0max.

Proof. (i) {γ | V0max LP satisfiesγ ) ⊆ 0max.Suppose otherwise. Then there is a sentenceγ , satisfied byV0max such

thatγ /∈ 0max. By our main theorem,γ has a partner,R(γ ) which is truein a classical valuation that treats just the sentence letters assignedB byV0max as ambiguous. But these sentence letters are exactly those includedin some member ofL(0). Thusγ is anL-preserving extension of0max,contrary to the maximality of0max.

(ii) 0max⊆ {γ | V0max LP satisfiesγ ).Again, suppose otherwise. Then there is someγ ∈ 0max that is not

satisfied by this valuation. But then, by our main theorem, there is noR ∈Rγ such thatV ′(R(γ )) = T , i.e. 0max is not L-preserving, contrary toassumption.

The closure of0 under|F is the intersection of these maximalL-preserv-ing extensions. Thus

Equivalence: 0|F α iff 0 �LP α.

CONCLUSION

The preservationist approach to paraconsistency eschews dialetheic valu-ations. As a result, it is not subject to Slater’s reinterpretation objection.Further, at least some paraconsitent logics that have been developed usingdialetheic valuations can be reinterpreted in preservationist terms — in

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fact, I conjecture that they all can. Though I do not argue for this here,the philosophical question of the tenability of dialetheism is, in my ownview, open, as is the closely related question of the tenability of dialetheicinterpretations of various paraconsistent logics. I also do not argue herefor or against a particular interpretation of any particular logic. Holdingthese issues aside, my aim has been to show that the existence of thesereinterpretations allows us to separate discussion of paraconsistent logics(in the sense of consequence relations) from debate over the tenabilityof dialetheism. Theseparabilityof these two issues is the principle pointof this paper: Of course, if dialetheism is defensible, then paraconsistentlogic will be required to cope with it. But the existence (and interest) ofparaconsistent logics does not depend on a defense of dialetheism.

NOTES

1 B. H. Slater, “Paraconsistent Logics?”Journal of Philosophical Logic24(1995), 451–454.

2 By ‘dialethec’ I mean logics which achieve paraconsistency by allowing both “A” and“¬A” to receive designated values, for at least some formulae A.

3 See P. K. Schotch and R. E. Jennings, “On Detonating,” in Priest, Routley, Norman(eds.),Paraconsistent Logic: Essays on the Inconsistent(München: Philosophia Verlag,1989).

4 As P. K. Schotch has pointed out (“Paraconsistent Logic: The view from the right,”PSA, 1992, Vol. 2) it’s reasonable to regard a tautologous set as requiring less dividingup than a merely consistent one, since a tautologous set can be freely combined with anyconsistent set salva degree of incoherence, while a consistent set cannot.

5 First, do no harm. See P. K. Schotch, “. . . The view from the right.”6 See “Paradox Tolerant Logic,”Logique et Analyse26 (1983), 291–308.7 The modal logics that are so closely related to this paraconsistent logic would allow

a modal account of satisfaction that divides the satisfaction of inconsistent sets amongsttuples of worlds.

8 Priest, Tanaka “Paraconsistent Logic,” in Stanford’s online philosophy encyclopedia,at http://plato.stanford.edu/entries/logic-paraconsistent/logic-paraconsistent.html

9 Incidentally, in my view at least, the dialethic reading of this logic is not (completely)indefensible. LP’s “¬” really is a lot like classical negation. To cite just a few points,the theorems of the system are the classical tautologies; whenever the variables in A areassigned classical values, A is assigned the appropriate classical value; whenever A isassigned “both” some of its variables are as well, and in particular, contradictions can onlybe designated by receiving the paradoxical value “both”. But whether or not this sort ofdefense finally succeeds, this logic remains a perfectly good paraconsistent logic, as themore conservative reading to follow shows.

10 Note that this process of treating sentence letters as ambiguous can be carried out bysubstituting unique new letters for some but not all instances of the letters to be treated asambiguous. Of course the resulting set of sentences,0′, can be consistent when0 is not.

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11 0’s members can be LP-satisfied by assigning ‘r ’ the valueT , so long as we assign‘p’ and ‘q’ the value both. Thus ‘¬r ’ is not satisfied by every valuation satisfying0.

12 By “minimal” here I mean only that no proper subset of any member ofL is such thattreating its members as ambiguous will allow0 to be rendered consistent.

University of Lethbridge,Department of Philosophy,4401 University Drive,Lethbridge, AB T1K 3M4, Canada