Supervaluationism and Logical Consequence: A Third Way

Pablo Cobreros Supervaluationism and Logical Consequence: A Third Way Abstract. It is often assumed that the supervaluationist theory of vagueness is committed to a global notion of logical consequence, in contrast with the local notion characteristic of modal logics. There are, at least, two problems related to the global notion of consequence. First, it brings some counterexamples to classically valid patterns of inference. Second, it is subject to an objection related to higher-order vagueness. This paper explores a third notion of logical consequence, and discusses its adequacy for the supervaluationist theory. The paper proceeds in two steps. In the first step, the paper provides a deductive notion of consequence for global validity using the tableaux method. In the second step, the paper provides a notion of logical consequence which is an alternative to global validity, and discusses i) whether it is acceptable to the supervaluationist and ii) whether it plays a better role in a theory of vagueness in the face of the problems related to the global notion. Keywords: Supervaluationism, Vagueness, Logical Consequence, Truth. 1. 1.1. Introduction Supervaluationism and vagueness Truth-value gaps help us in characterizing vagueness as a semantic phenomenon. If Tim is a borderline case of the predicate ‘thin’, then the sentence ‘Tim is thin’ has no truth-value. The underlying idea for truth-value gaps is that whatever facts (facts about use, most likely) determine the meaning of the predicate ‘thin’, they leave unsettled the question of whether Tim is thin. The supervaluationist theory provides a way to understand vagueness as a semantic phenomenon. There are several ways in which a vague predicate like ‘thin’ could be made precise in a way consistent with the actual use we make of the expression. For a range of cases any such way of making the predicate precise would be true of them; for example, ‘Gandhi is thin’ (supposing he were still alive) is true in any way of making precise the predicate ‘thin’ which is consistent with the actual use we make of the expression. For a range of cases, the corresponding sentences would be false in any such Special Issue on Vagueness Edited by Rosanna Keefe and Libor Běhounek Studia Logica (2008) 90: 291–312 DOI: 10.1007/s11225-008-9154-1 © Springer 2008 292 P. Cobreros way of making precise the predicate. But there will be a range of cases for which some of these ways of making precise would yield a true statement and some a false one. The supervaluationist theory understands that there is nothing in the world or in the use of a vague predicate (or in any other factor relevant for the meaning of the predicate) that selects a given way of making the vague predicate precise over the others. Thus, any such way of making the vague predicate precise is just as correct as any other. This is the sense of the idea of semantic indecision: nothing in the world, either in the use or in any other factor relevant to the determination of the meaning of a vague predicate, decides which of the ways in which we could make precise the predicate is correct. Just as ‘Tim is thin’ is true for some ways of making ‘thin’ precise but false for others, ‘Tim is thin’ is neither true nor false when we attend to the complete meaning of the predicate. I would like to make two remarks concerning truth and ways of making precise. The previous paragraph implicitly contains the supervaluationist notion of supertruth: for the supervaluationist, a sentence is true (supertrue) just in case it is true in every admissible precisification. A precisification is a truth-value assignment (true or false but not both) to all the sentences of the language. Intuitively, a precisification is what we would reach if we made all the expressions of the language precise (every sentence would receive a truthvalue). But not every precisification is admissible. In the supervaluationist theory there are some constraints on precisifications that play an interesting role in the theory. In the first place, a precisification should respect the clear cases (or clear non-cases): any precisification assigning to the sentence ‘Gandhi is thin’ the truth-value false is not an admissible precisification. In the second place, precisifications should respect semantic connections that might hold even in borderline cases. For example, if Tim is a bit thinner than Tom, any precisification that counts Tom as thin should count Tim as thin; thus, the sentence ‘If Tom is thin then Tim is thin’ is supertrue even if Tim and Tom are borderline-thin. The accommodation of these semantic connections is often invoked as an advantage of supervaluationist semantics over truth-functional theories of vagueness admitting truth-value gaps (such as three-valued semantics). 1.2. Making a difference to truth but not to logic? Supervaluationism reconciles truth-value gaps with classical reasoning, such that the theory ‘makes a difference to truth, but not to logic’ [2, 284]. Fine and Keefe show (in the specification-spaces framework proposed by Fine) that, for the classical propositional language, the classical consequence A Third Way 293 relation coincides with the supervaluationist.1 But supervaluationism allows us to extend the classical propositional language with a definitely operator (usually represented as D) expressing in the object language the nonclassical notion of supertruth; such an operator brings, as it were, the nonclassicalness of the semantics into the object-language. D is susceptible to a possible-worlds semantics which is analogous to that of the modal operator for necessity. An interpretation for this language is an ordered triple
W, R, ν where, W is a (non-empty) set of precisifications, R a binary relation on W and ν a function assigning truth-values to formulas at precisifications. A precisification is a way of making precise every vague expression of the language. The idea of a precisification is formally analogous to that of a possible world in modal semantics: sentences of the language are true or false in each precisification. R is an admissibility relation between precisifications. The informal idea is that the admissibility of a given precisification is itself subject to precisification: a precisification can be admissible for some precisifications but not for others [11, 158]. The relation is required in order to obtain the semantics for systems weaker than S5. A particularity of S5 for the case of vagueness is that it does not permit second-order vagueness [12, 134]. Thus, in order to accommodate the so-called phenomenon of higher-order vagueness, a relation of admissibility is needed. The function ν assigns truth-values to sentences at worlds.2 Classical operators have their usual meaning (relative to worlds); the definition of D is analogous to the definition of the modal operator for necessity: νw (α → β) = 1 iff either νw (α) = 0 or νw (β) = 1 νw (¬α) = 1 iff νw (α) = 0 νw (Dα) = 1 iff ∀w′ such that wRw′ νw′ (α) = 1 1 See [2, 283-4] and [6, 175-6]. The claim, however, needs to be qualified. The classical notion of consequence coincides with that of the supervaluationist in the absence of ‘D’ only if we restrict the consequence relation to single-conclusions. Hyde [4] discusses this issue in depth in comparison with subvaluationism (a weak paraconsistent theory, dual to supervaluationism). 2 As it is the shortest and most standard terminology, I shall keep to modal terminology, talking about worlds instead of precisifications, and saying that w accesses w′ instead of that w′ is admitted by w. I shall assume that the metalanguage behaves classically (some discussions on vagueness allow the metalanguage to behave otherwise). The letters w, w′ , w′′ . . . will be used as variables over possible worlds whereas w0 , w1 . . . are treated as names. In this section and section 2, I shall not assume any restriction on R, for the sake of simplicity. In section 3, restrictions to R play a substantial role in the discussion, as will be pointed out. 294 P. Cobreros It is still a controversial matter as to which of the modal systems is more appropriate for D, but it is usually accepted that the schema Dα → α is a platitude about the logic D (say, if John is definitely tall, then John is tall) and thus, R should be reflexive, thus yielding a semantics for a system at least as strong as the modal T (see for example Williamson’s [12, 130] or the discussion between McGee-McLaughlin [8, 227] and Williamson [13, 119]). 1.3. Global and local validity Once we introduce D, the non-classicality of supervaluationist semantics is reflected in the logic. In particular, it is often assumed that supervaluationism is committed to a global notion of logical consequence3 , in contrast to the local notion characteristic of modal logics. Definition 1 (Local validity). A sentence α is a locally valid consequence of a set of sentences Γ, written Γ
l α, iff ∀ℑW,R,ν ∀w∈W (if ∀γ∈Γ νw (γ) = 1 then νw (α) = 1) Local validity is a natural reading of the idea of necessary preservation of truth in modal logics, because in modal logics, being true (in the intuitive or philosophical sense) is being true-in-a-world (in the technical sense). Instead of preservation of truth-in-a-world, we could consider a notion of logical consequence that preserves the property of being true-at-all-worlds. Definition 2 (Global validity). A sentence α is a globally valid consequence of a set of sentences Γ, written Γ
g α, iff ∀ℑW,R,ν (if ∀γ∈Γ ∀w∈W νw (γ) = 1 then ∀w∈W νw (α) = 1) In the case of arguments without premises, global and local validity coincide (
l α iff
g α), but otherwise things change. Every locally valid argument is globally valid: if Γ
l α then Γ
g α. For if an argument is not globally valid then there is an interpretation in which every premise is true at every world and the conclusion is not true at some; thus there is an interpretation and a world in it such that the premises are true at that world but the conclusion is not. But the converse, if Γ
g α then Γ
l α, is not generally true. In particular, the inference (1) φ
Dφ is globally valid − for any interpretation, if φ is true at all worlds, Dφ is true at all worlds − though it is not locally valid. 3 Fine’s seminal paper implicitly assumes the global notion of consequence [2, 290]. Keefe seems to assume also the global notion in [6, 176] and explicitly assumes it in [5]. 295 A Third Way 1.4. An argument for global validity In her 2003 paper Delia Graff Fara raises an objection related to higher-order vagueness for theories accepting truth-value gaps. The objection uses the rule of D-intro (which mirrors the inference in (1)), (D-intro) φ ⊢ Dφ Fara claims that the rule of D-intro should be valid for the defender of truth-value gaps. The reason is that for truth-value gap theories the Doperator is an object-language expression of the notion of truth; it means something like ‘it is true that’, and in that case ‘it seems impossible for a sentence S to be true while another sentence, “it is true that S”, that says (in effect) that it’s true is not true’ [1, 199–200]. While Fara doesn’t mention global validity, her reasons constitute an indirect argument for this notion, because the inference from φ to Dφ is globally, but not locally, valid. 1.5. Problems of global validity The issue of global validity brings us to the next counterexamples to classically valid patterns of inference, all of them using the inference in (1): (Conditional Proof) (Contraposition) (∨-elimination) (Reductio) φ
g Dφ though
g (φ → Dφ) φ
g Dφ though ¬Dφ
g ¬φ φ
g Dφ ∨ D¬φ and ¬φ
g Dφ ∨ D¬φ though φ ∨ ¬φ
g Dφ ∨ D¬φ φ ∧ ¬Dφ
g Dφ and φ ∧ ¬Dφ
g ¬Dφ though
g ¬(φ ∧ ¬Dφ) The objection against the supervaluationist theory is that, once we include the D-operator, its logic deviates too much from classical reasoning [11, 151–2]. In her 2003 paper, Fara presents the following problem for theories which defend truth-value gaps. In a sorites series for a vague predicate like ‘tall’, the first member of the series is tall and the last is not. When confronted with the sorites series we have a no sharp boundary intuition: there seems to be no sharp transition from the objects that are tall to the ones that are not. The truth-value gap defender explains the intuition by saying that there is a gap between the members of the series of which it is true to say that they are tall, and those of which it is false to say that they are tall. Thus, for any member of the series, if it is truly described as tall, it is not the case that its successor in the series is truly described as not tall: 296 P. Cobreros (Gap Principle for T ): DT (x) → ¬D¬T (x′ ) (where ‘T ’ stands for ‘tall’ and x′ is the successor of x in the series) But the no sharp boundary intuition cuts deeper than that, because there seems to be no sharp transition at all; there is no sharp transition from the members of the series truly described as definitely tall to those truly described as not definitely tall either. This idea amounts to a gap principle for DT and generalizes for any iteration of D rendering all the gap-principles of the form: (Gap Principle for D n T ): DD n T (x) → ¬D¬D n T (x′ ) (Equivalent Formulation): D¬D n T (x′ ) → ¬DD n T (x) Thus, in order to fully accommodate the no-sharp boundary intuition, truth-gap theories appear to be committed to the truth of all these gapprinciples. The problem is that for any finite sorites series for a vague predicate like ‘tall’, the truth of all these gap-principles is inconsistent with the rule of D-intro (φ ⊢ Dφ), which Fara argues should be accepted by truth-gap theorists, and is indeed globally valid. The argument proceeds as follows.4 Take a sorites series of m elements for the predicate ‘tall’. The first member of the series is truly described as tall and the last is truly described as not tall. Then, ¬T (m) D¬T (m) ¬DT (m − 1) D¬DT (m − 1) ¬D 2 T (m − 2) D¬D 2 T (m − 2) ¬D 3 T (m − 3)    ¬D m−1 T (1) [premise] [D-intro] [Gap Principle for T ] [D-intro] [Gap Principle for DT ] [D-intro] [Gap Principle for D 2 T ] [Gap Principle for D m−2 T ] On the other hand, since the first member of the series is tall, by m − 1 applications of D-intro we can reach D m−1 T (1), which contradicts the result of the previous argument. Thus, truth-value gap theorists cannot appeal to 4 The argument is based on the same idea as an argument of Wright’s (in [14] and [15]), as Fara herself points out [1, 201]. However, Wright’s argument is defective, as was pointed out by Heck [3]. Fara’s argument is not subject to the problem with Wright’s. A Third Way 297 the truth of all the gap-principles to accommodate the no sharp boundary intuition for any given (finite) sorites series. 2. Checking for global validity: Tableaux As noted, global validity departs from classical consequence, and this is often taken as an objection against it. But why is such a departure necessarily bad? There seem to be at least a couple of reasons. In the first place, we have a good analysis of classical logical consequence, by means of well known deductive systems. In the second place, it is usually claimed that classical consequence is implicit in various kinds of inferential practices. The aim of this section is to provide a deductive notion of consequence for global validity using the tableaux method. This way we directly address the first, more technical, issue of providing a deductive system for global validity. At the same time, we make progress on the second, more philosophical, question: a deductive analysis of global validity will enable us to question whether the sort of inferential moves that differentiate global and and classical consequence are really implicit in practice. The main point of the section (subsection 2.1) is that we can connect global and local validity in a way that enables us to use tableaux to check for global validity. The rest of the section describes the tableaux method given this connection. The appendix indicates out how to give soundness and completeness proofs.5 5 In a 2003 paper P. Kramer and M. Kramer study different supervaluation-based consequence relations. The scope of their study is broader than the one offered here in three senses: they consider first order languages with identity, they also consider multipleconclusions consequence relations, and they include, over the more standard notion of truth-preservation, backwards falsehood preservation (and the logical consequence that is the intersection of these) [7, 227]. The consequence relations of their paper can be divided into two groups: i) those that take truth in a partial model as truth with respect to all its precisifications (named in their paper
t ,
f and
) and those that take truth in a partial model as truth with respect to all its admissible precisifications (named
∗t ,
∗f and
∗ ). The aim of their paper is also different, because they are interested in proving different logical properties of these relations, more than in solving particular philosophical problems that arise from the their application to particular phenomena (such as the case of vague expressions). The only consequence relation that is present in both this paper and theirs is
∗t , that is, the consequence relation of group ii) that comprises the idea of preservation of truth (in this work restricted to single-conclusions); this is global validity. Kramer and Kramer prove that it is closed under substitution (with some restrictions, p. 235–7), and that it is compact and axiomatizable (p. 237–42). The last result provides a deductive notion of consequence, thus partially overlapping with the aim of this section. 298 P. Cobreros 2.1. Connecting global and local validity In order to connect global and local validity we will make use of the following operator and definition: Definition 3 (Operator Du ). For any interpretation
W, R, ν and any world w ∈ W, νw (Du α) = 1 iff ∀w∈W νw (α) = 1 A remark on Du : Du is a modality of universal access in the sense that R plays no role in its definition. As a consequence, any sentence of the form Du φ will be true at some world in an interpretation if and only if it is true at every world in that interpretation. Definition 4 (Set Du (Γ) generated from Γ). Let Γ be the set of sentences {γ1 , . . . , γn }. The set Du (Γ) generated from Γ is the result of applying to each γi in Γ the operator Du ; thus Du (Γ) is {Du γ1 , . . . , Du γn }. The goal of this section is to prove the following proposition: Proposition 1: Γ
g α iff Du (Γ)
l α Proof. Note first that from the definition of global validity (Definition 2), it follows that Γ
g α just in case there is an interpretation such that for every world w in it νw (γ) = 1, for every γ in Γ, and there is some world w in it such that νw (α) = 0. (i) If Γ
g α then Du (Γ)
l α. Suppose, contrapositively, that Du (Γ)
l α; then there is an interpretation and a world w in it such that νw (Du γ) = 1, for every γ in Γ, and νw (α) = 0. Given the definition of Du , for every world w in the interpretation νw (γ) = 1. Thus, there is an interpretation such that for every world w in it, νw (γ) = 1, for every γ in Γ, and there is some world w in it such that νw (α) = 0, that is, Γ
g α. (ii) If Du (Γ)
l α then Γ
g α. Suppose, contrapositively, that Γ
g α; then there is an interpretation such that for every world w in it νw (γ) = 1, for every γ in Γ, and there is a world w in the interpretation such that νw (α) = 0. By the definition of Du and the fact that the set of worlds in the interpretation is non-empty, we have that there is a world w in the interpretation such that νw (Du γ) = 1, for every γ in Γ, and there is a world w in the interpretation such that νw (α) = 0. Name w0 the world at which νw (Du γ) = 1, and w1 the world at which νw (α) = 0. By the remark on Du above, νw0 (Du γ) = 1 iff νw1 (Du γ) = 1. Thus, there is an interpretation and a world in it (namely w1 ) at which νw (Du γ) = 1, for every γ in Γ and νw (α) = 0, that is, Du (Γ)
l α. 299 A Third Way 2.2. Tableaux Definition 5 (Satisfiability). A set of sentences Σ is satisfiable iff ∃ℑW,R,ν ∃w∈W ∀σ∈Σ νw (σ) = 1. As is well known, from the definition of local validity and the definition of satisfiability it follows that: (2) Γ
l α iff Γ ∪ {¬α} is not satisfiable Tableaux tell us how to construct a tree to show whether a given set of sentences is satisfiable. In the case of global validity, we have that Γ
g α if and only if Du (Γ)
l α; thus, the tableaux method for testing whether Γ
g α consists in a tree showing that Du (Γ) ∪ {¬α} is not satisfiable.6 The nodes of a tree for a language containing modal (or similar) operators may have two different forms: either a pair
φ, i, where φ is a sentence and i a natural number, or something of the form irj where i and j are natural numbers. Intuitively, numbers indicate possible worlds and irj means that world i accesses world j. At the beginning of the tree we list the members of Du (Γ) ∪ {¬α} each followed by the number 0. Then we extend the tree with the following rules: Rules for non-modal operators: The rules follow the satisfiability conditions of a given formula. For example, the rules for φ → ψ and ¬(φ → ψ) are, respectively: φ → ψ, i ¬φ, i ψ, i ¬(φ → ψ), i φ, i ¬ψ, i Rules for modal operators: The rules for D and its negation are: Dφ, i irj φ, j (for every such j) ¬Dφ, i irj ¬φ, j (for a new j) Since the rules follow the satisfiability conditions of the sentences already in the tableau, if
Dφ, i is in the tableau, φ should be true in any world 6 A classic text for the tableaux method is Smullyan’s [10], where the method is applied to propositional and first-order languages. I will follow Priest’s text [9], in which the method of tableaux is applied to modal languages. 300 P. Cobreros accessible from i. Likewise, if
¬Dφ, i is in the tableau, there must be at least some world accessible from i in which ¬φ is true. The definition of Du is exactly like the definition of D, except that we omit the use of R. Thus, the rules for Du will be exactly the same as those for D, except that R will not play any role: Du φ, i φ, j (for any j in the tableau) ¬Du φ, i ¬φ, j (for a new j) A branch is closed just in case it contains a node of the form
φ, i and a node of the form
¬φ, i (where i is the same in both cases); otherwise the branch is open. A branch is complete just in case it is either closed, or else a rule has been applied to any formula in the branch that is neither a propositional variable or a negation of a propositional variable. A tableau is closed just in case all its branches are closed. A tableau is complete whenever all its branches are complete. A sentence α is a global deductive consequence of a set of sentences Γ, written Γ ⊢g α, just in case there is a closed tableau for the set Du (Γ)∪{¬α}. Example 1. D(p → q) ⊢g Dp → Dq Du D(p → q), 0 ¬(Dp → Dq), 0 D(p → q), 0 Dp, 0 ¬Dq, 0 0r1 ¬q, 1 p, 1 p → q, 1 ¬p, 1 q, 1 ∗ ∗ 2.3. Example 2. p ⊢g Dp Du p, 0 ¬Dp, 0 0r1 ¬p, 1 p, 1 ∗ Counter-models A complete open tableau tells us how to construct a counter-model for the related inference, following one of its open branches. For any natural number i in the branch we include a world wi in W. For any worlds wi and wj in W, wi Rwj iff irj occurs in the branch. For any propositional variable p, if
p, i appears at some node in the branch then νwi (p) = 1; if
¬p, i appears then νwi (p) = 0; otherwise νwi can assign any value to p. 301 A Third Way Example 3. D(p ∨ q) g Dp ∨ Dq Du D(p ∨ q), 0 ¬(Dp ∨ Dq), 0 ¬Dp, 0 ¬Dq, 0 0r1 ¬p, 1 0r2 ¬q, 2 D(p ∨ q), 0 D(p ∨ q), 1 D(p ∨ q), 2 (p ∨ q), 1 p, 1 q, 1 ∗ (p ∨ q), 2 p, 2 q, 2 ↑ ∗ Let ℑ0 be an interpretation
W, R, ν where: W = {w0 , w1 , w2 }. w0 Rw1 , w0 Rw2 and nothing else is related by R. νw1 (p) = 0, νw1 (q) = 1, νw2 (p) = 1, νw2 (q) = 0 (the rest of the truth-values are assigned arbitrarily). Where W is finite, an interpretation can be depicted diagrammatically: ✘ w1 ✿ ❳③ ❳ w0 ❳ ✘✘✘ ❳ w2 ¬p, q p, ¬q Example 4. g p → Dp ¬(p → Dp), 0 p, 0 ¬Dp, 0 0r1 ¬p, 1 Counter-model: w0 p ✲ w1 ¬p 302 P. Cobreros 3. A third way This section explores a third notion of logical consequence, regional validity,7 which lies between global and local validity; in addition, I give the reasons for why the regional notion rather than the global notion is that to which supervaluationism is committed. The relations between these three notions of consequence are made explicit. The goal of this section is to answer two questions: first, whether the new notion of logical consequence is an option available to the supervaluationist, and second, whether the new notion plays a better role than global validity in a theory of vagueness. The first question is answered by revisiting the argument in subsection 1.4, concerning the commitment of supervaluationism to global validity. The second question is answered by revisiting the problems related to the global notion of consequence in the case of vagueness (subsection 1.5). 3.1. Global, local and regional validity Definition 6 (Regional validity). A sentence α is a regionally valid consequence of a set of sentences Γ, written Γ
r α iff ′ ∀ℑW,R,ν ∀w∈W (if ∀γ∈Γ ∀w∈W wRw′ νw′ (γ) = 1 then ′ ′ ∀w∈W wRw νw′ (α) = 1) The semantics for D includes the idea that each precisification determines both a truth-value assignment and a set of admissible precisifications; thus, for the supervaluationists, (super-)truth is truth in every admissible precisification. But in the case of global validity, what is necessarily preserved is an absolute notion of supertruth, not relative to worlds (or precisifications) as Du bears witness. In contrast, in the case of regional validity, the property necessarily preserved is defined relative to precisifications. Now if the property expressed by D (truth-at-all-accessible-worlds) is the object-language expression of supertruth, it is this property, and not the one preserved by global validity (truth-at-all-worlds), the one that supervaluationist logical consequence should preserve.8 The following diagram graphically explains 7 I am indebted to Elia Zardini for the suggestion for the name. After writing this paper I discovered that Williamson has pointed out that when the accessibility relation fails to be transitive, the supervaluationist theory is committed to something like what I have called regional validity [11, 159, note 32]. Williamson does not develop this idea, because he maintains an epistemicist view on vagueness, in which there is no room for truth-value gaps. Moreover, the importance of the regional notion for the supervaluationist theory emerges in connection to Fara’s argument concerning higher-order vagueness [1]. 8 303 A Third Way the differences between the three notions of consequence, attending to the property preserved in each one. ✎☞ ✎☞ ✘✘ ✿ w1 ✘ ✍✌ ✘❳ w0 ❳ ✎☞ ✍✌ ❳❳ ③ w2 ✎☞ ✍✌ Local validity: ✍✌ The property necessarily preserved is truth-at-each-world w3 ✬ ✿ w1 ✘ w0 ❳ ✘✘✘ ❳❳❳ ③ w2 Global validity: ✫ w3 ✩ % ✎☞ ✿ w1 ✘ ✘ w0 ❳ ✘✘ ❳❳❳ ③ w2 (relative to w ) 0 ✍✌ Regional validity: The property necessarily preserved is truth-at-all-worlds The property necessarily preserved is truth-at-all-accessible-worlds w3 As in the case of global validity, regional validity can be connected to local validity. We will use the following definition: Definition 7 (Set D(Γ) generated from Γ). Let Γ be the set of sentences {γ1 , . . . , γn }. The set D(Γ) generated from Γ is the result of applying to each γi in Γ the operator D; thus D(Γ) is {Dγ1 , . . . , Dγn }. Proposition 2: Γ
r α iff D(Γ)
l Dα Proof. Γ
r α  (definition of regional validity) ′ ∀ℑW,R,ν ∀w∈W (if ∀w∈W wRw′ ∀γ∈Γ νw′ (γ) = 1 ′ ′ then ∀w∈W wRw νw′ (α) = 1)  (definition of D) ∀ℑW,R,ν ∀w∈W (if ∀γ∈Γ νw (Dγ) = 1 then νw (Dα) = 1)  (Definition 7 and definition of local validity), D(Γ)
l Dα Proposition 2 enables us to use the tableaux method to check for regional validity. A sentence α is a regional deductive consequence of a set of sentences Γ, written Γ ⊢r α, iff there is a closed tableau for D(Γ) ∪ {¬Dα}. 304 P. Cobreros Regional validity oscillates between global and local validity, depending on the restrictions on R. If we put no restrictions on the accessibility relation, regional and local validity coincide. However, as noted above, reflexivity is usually taken as a minimal constraint on the semantics of D. If R is just reflexive, regional validity is stronger than local validity but weaker than the global notion. Finally, regional and global validity collapse when R is required to be both reflexive and transitive.9 Concerning the claims contained in this paragraph, I shall show here just that, when R is reflexive, i) regional validity is stronger than local validity, but ii) weaker than global validity and that, when R is both reflexive and transitive, iii) every globally valid argument is also regionally valid. i) It is not the case that if Γ
r α then Γ
l α (when R is reflexive). Proof. In particular, where R is reflexive we have that {γ, ¬Dγ}
r α but {γ, ¬Dγ}
l α. The reflexivity of R is captured in the tableaux method by adding a new rule [9, 40]:  ↓ iri The rule tells us to introduce a new node for any i in the tableau. The tree showing that {γ, ¬Dγ}
r α looks like this, Example 6. {p, ¬Dp} ⊢r q Dp, 0 D¬Dp, 0 ¬Dq, 0 0r0 (refl.) p, 0 ¬Dp, 0 0r1 ¬p, 1 p, 1 ∗ 9 Global and local validity do not coincide tout court; the collapsing result mentioned in the text depends on the expressive resources of the language. Suppose we introduce an operator ‘D∗ ’ such that νw (D∗ α) = 1 iff ∀w′ such that w′ Rw νw′ (α) = 1. With this operator we might find arguments that are globally but not regionally valid in reflexive and transitive interpretations (in particular, φ
g D∗ ¬D∗ ¬φ but φ
r D∗ ¬D∗ ¬φ). I am indebted to an anonymous referee of this journal for pointing this out to me. 305 A Third Way ii) It is not the case that if Γ
g α then Γ
r α. Proof. In particular, φ
g Dφ, as shown in example 2, but φ
r Dφ Example 5. p r Dp Dp, 0 ¬DDp, 0 0r1 ¬Dp, 1 p, 1 1r2 ¬p, 2 Counter-model w0 ✲ w1 p ✲ w2 ¬p The diagram shows that things would have been otherwise in the case where R is required to be transitive. In fact, when R is reflexive and transitive, regional validity collapses with global validity. iii) If Γ
g α then Γ
r α (when R is reflexive and transitive) Proof. Suppose that Γ
r α. In that case, there is an interpretation ℑ =
W, R, ν and a world w0 in it such that, for every γ in Γ νw0 (Dγ) = 1 and νw0 (¬Dα) = 1. Let W ′ be {w|w0 Rw} and R′ , ν ′ the restrictions of R, ν to W ′ . We have to show a) that the interpretation ℑ′ =
W ′ , R′ , ν ′  is still a counter-model for showing that Γ
r α (that is, for any w′ in W ′ , ℑ and ℑ′ assign the same values to sentences at w′ ) and b) that it is in fact a counter-model showing that Γ
g α. To show a), note first that if w′ ∈ W ′ then R′ and R relate w′ exactly to the same worlds, that is, if w′ ∈ W ′ then w′ R′ w iff w′ Rw. For if w′ R′ w then both w ∈ W ′ and w′ Rw. On the other hand, if w′ Rw, as w′ ∈ W ′ , w0 Rw′ and by the transitivity of R, w0 Rw, that is, w ∈ W ′ . Thus, w′ R′ w. a) is proved by induction over the set of wff. The case for propositional variables holds by definition. The case for non-modal operators is straightforward. For φ = Dβ, suppose that w′ ∈ W ′ : ′ (β) = 1 νw′ ′ (Dβ) = 1 iff ∀w∗∈W ′ such that w′ R′ w∗, νw∗ ′ ′ iff ∀w∗∈W ′ such that w R w∗, νw∗ (β) = 1 (by IH) iff ∀w∗∈W such that w′ Rw∗, νw∗ (β) = 1 (by the fact noted above) To prove b) note that w0 has access to every world in W ′ (including itself, given reflexivity). Since for every γ in Γ, νw0 (Dγ) = 1, every member of Γ takes value 1 at every world in W ′ . As νw0 (¬Dα) = 1, there is at least one world in W ′ in which α takes value 0. Thus, ℑ′ shows that Γ
g α. 306 P. Cobreros So far we have three different notions of logical consequence with different extensions, depending on the restrictions placed on the accessibility relation. The local notion of consequence preserves truth at each precisification, and this form of truth does not permit failures of bivalence (each sentence is either true or false at each precisification). Thus, if we want to explain the semantic indeterminacy characteristic of vagueness in terms of truth-value gaps, we cannot be committed to the local notion. Both regional and global validity allow for failures of bivalence, and differ when accessibility is not transitive. So the question arises: which one is better for the supervaluationist theory? The next two subsections intend to address this question. The first subsection argues that the supervaluationist is not committed to the validity of D-intro. This argument intends to show that regional validity, where accessibility is reflexive but not transitive, is an option available to the supervaluationist. The second subsection argues that the regional notion plays a better role than the global notion, and takes into account the problems deriving from the latter. 3.2. The argument for global validity revisited Fara’s remark constitutes an indirect argument for the commitment of supervaluationism to global validity. Fara argues that the inference from φ to Dφ should be valid for the defender of truth-value gaps; the inference is globally valid, but it is not always regionally valid. In particular, it is regionally valid only if the accessibility relation is required to be transitive and (granted reflexivity), as pointed out above, in this case regional validity collapses with global validity. Her challenge was that, given that a sentence φ is true, the sentence stating that φ is true must also be true. I have two comments concerning Fara’s argument. The first is that the particularities of the semantics that cause the failure of the inference from φ to Dφ are well motivated. The second, that regional validity might explain why Fara’s remark seems to be correct. The particular facts of the semantics that cause the failure of the inference are i) the relative-to-precisifications notion of truth at work and ii) the non-transitivity of the admissibility relation (see the counter-model in example 5). Both facts are supported by the idea that the notion of truth at work is vague, which is the intuitive and natural way in which truth-gap theories explain the presence of higher-order vagueness.10 The supervalua10 The appeal here to higher-order vagueness is not an ad hoc maneuver to solve Fara’s objection; the reasons deriving from higher-order vagueness that motivate these facts are independent of the objection, though (of course) they will have an effect on addressing the objection. A Third Way 307 tionist notion of truth (and with it, falsehood and borderline-ness) is itself vague: there are cases that are neither true nor false nor borderline. For the supervaluationist (super-)truth is truth in every admissible valuation; thus, the most likely candidate for being vague in that definition is what counts as admissible. The natural way in which the supervaluationist theory interprets the idea that ‘admissible’ is vague is by saying that ‘admissible’ can be made precise in several ways: whether a precisification is or is not admissible is a relative-to-precisification question.11 Thus, whether a given sentence is (super-)true depends on whether it is true in all precisifications admissible for a given precisification. On the other hand, as McGee and McLaughlin point out in response to Williamson12 , the same reasons that lead the supervaluationist to deny the validity of the schema (F a → ‘F a’ is true) on the grounds of the vagueness of ‘F ’, lead the supervaluationist to deny the validity of the schema (‘F a’ is true → “F a’ is true’ is true) on the grounds of the vagueness of ‘true’. The transitivity of R guarantees the validity of the schema; thus, the failure of the schema requires the failure of R to be transitive. The result of these two facts (that truth is relative to precisifications and that R is not transitive) is a notion of truth that is determinacy implying, in the sense that it allows for failures of bivalence, though this notion is itself vague. Finally, regional validity might explain why Fara’s remark on D as a truth-operator, seems to be correct. It is not the case that φ
r Dφ, but we have that {φ, ¬Dφ}
r α ∧ ¬α. The attraction we feel towards the inference from φ to Dφ can be explained by the fact that anyone holding both φ and ¬Dφ is holding something inconsistent; thus, we tentatively conclude that φ entails Dφ. The conclusion does not follow, however, because the reductio rule employed to reach to the conclusion is not regionally valid. With regional validity we can maintain that there is some psychological compulsion to accept that inference, while the inference itself is nevertheless not regionally correct. 3.3. Problems of global validity revisited One of the problems noted above concerning global validity is that it brings counterexamples to classically valid patterns of inference. The change in validity is a natural consequence of the change caused in the notion of truth; 11 This remark is already present in Williamson’s exposition of the supervaluationist semantics (see [11, 158]). 12 See [8, 224]. It should be said in favour of Williamson that McGee and McLaughlin overlooked [11, footnote 32], as Williamson later pointed out ([13, 119]). 308 P. Cobreros I think that if the supervaluationist accepts the latter, he ought not to be embarrassed by the former. All the counterexamples in the case of global validity make some use of the inference φ
g Dφ; since this inference is not regionally valid, it might be thought that the new notion of logical consequence does not bring counterexamples to classical reasoning. However, as shown above, regional validity does not coincide with local validity when R is reflexive. In particular, where R is reflexive we have that {γ, ¬Dγ}
r α but {γ, ¬Dγ}
l α. As example 6 shows, α plays no role in the branch closure and, thus, it can be anything we like. Thus we have that {γ, ¬Dγ} ⊢r α ∧ ¬α. However, from this it does not follow that {γ} ⊢r ¬¬Dγ (as shown in example 5), nor that {¬Dγ} ⊢r ¬γ. Thus, regional validity brings a counterexample to the classical reductio ad absurdum. Based on the same inference we can encounter another counterexample to conditional proof. While we have that {γ, ¬Dγ} ⊢r α, it is not the case that γ ⊢r ¬Dγ → α.13 How reasonable are these counterexamples? In the case of global validity the inference from φ to Dφ is justified for the reading of D as an objectlanguage expression of the notion of truth, as Fara points out. In the case of regional validity, Fara’s reasons for D-intro are recast, in a weakened form, into the correctness of the inference from {γ, ¬Dγ} to α ∧ ¬α and, in this sense, the inference is justified for the reading of D as an object-language expression of supertruth. In the case of global validity the counterexamples are linked to the inference from φ to Dφ, whereas in the case of regional validity they are linked to the inference from {γ, ¬Dγ} to α ∧ ¬α. I think, therefore, that the problem of counterexamples to classical reasoning is, in the case of regional validity, at worst just as justified as in the case of global validity. The problem related to higher-order vagueness is the one that, I think, tilts the balance towards regional validity. The second problem for global validity is Fara’s argument concerning higher-order vagueness. Fara intends to show that the truth-gap theorist cannot explain, by appealing to gap-principles, the no sharp boundary intuition we have when confronted with any particular (finite) sorites series, because for any such series, there will be principles that are not true. The argument, for the predicate ‘tall’ and a sorites series of m members beginning with a man 2.5 meters tall and ending with one 1.5 meters tall, 13 Williamson has already pointed out that while the supervaluationist is not committed to the inference from φ to Dφ when R is not transitive, he or she is committed to the inference from {γ, ¬Dγ} to α ∧ ¬α, and with it to the counterexamples to contraposition, conditional proof, argument by cases and reductio ad absurdum. See [11, c. 5 note 32] and [13, 119–120]. 309 A Third Way could be depicted in the following way (a diagram of this kind is found in [1, 204]): 2.5 m. ......................................................... 1.5 m. ¬T (m) T (1) ❄ ¬DT (m − 1)  D¬T (m) ❄ ¬D2 T (m − 2)  D¬DT (m − 1) ❄ D¬D2 T (m − 2) [. . .] ❄ n D T (1) n ¬D T (1) Each move downwards is an application of D-intro, while each move leftwards is an application of the relevant gap-principle. As the picture shows, the relevant instance of the relevant gap-principle works in tandem with the inference from φ to Dφ, forcing us to move leftwards to pick out a new element in the series. As the inference from φ to Dφ is not regionally valid, the argument does not work in the case of regional validity.14 I think that Fara’s argument shows, in effect, that with the rule of Dintro we cannot properly accommodate the no-sharp-boundary intuition associated with vagueness. But I take this to be an argument against the rule of D-intro for truth-gap theories in general and against global validity for the supervaluationist theory in particular; once the notion of regional validity is available, it ceases to be an objection to the defender of truth-value gaps (at least if he or she holds a supervaluationist semantics). Conclusion The supervaluationist’s divergence from classical semantics is reflected in the logic once we introduce the definitely operator. But, it is not so clear that the supervaluationist is committed to the global notion of logical consequence. ‘Definitely’ is introduced in the object language in order to express 14 Moreover, it is possible to show the consistency of the gap-principles with finite sorites series given regional validity, though a detailed discussion requires more space than we have here. 310 P. Cobreros the supervaluationist notion of truth, but ‘definitely’ is defined as being relative to precisifications. This ‘relative to precisifications’ property is the one that should be necessarily preserved by logical consequence, and not the absolute notion at work in the case of global validity. I have argued that the proposed notion of logical consequence is an option available to supervaluationism, confronting the argument that asserts that this theory is committed to the global notion. But if the new notion of consequence is an option, then it is a better option than global validity, for while it also brings counterexamples to classically valid patterns of inference, it is not subject to Fara’s objection concerning higher-order vagueness. Appendix: soundness and completeness Soundness and completeness proofs for the tableaux introduced here follow the proofs for standard modal logics.15 I sketch here the proofs for global validity which can be easily adapted for regional validity. Soundness To prove soundness we make use of the following definition and show the lemma. Definition Let ℑ =
W, R, ν be any modal interpretation and let b be any branch of a tableau. We say that ℑ is faithful to b just in case there is a map f from natural numbers to W such that, For every node
φ, i on b, φ is true at world f (i) in ℑ, If irj is on b then f (i)Rf (j) is in ℑ. (We say that f shows ℑ to be faithful to b.) Soundness Lemma For any interpretation ℑ =
W, R, ν and for any branch of a tableau b, if ℑ is faithful to b and a tableau rule is applied to b, then ℑ is faithful to at least one of the branches generated. The lemma is proved by examining the tableau rules case by case. Soundness Theorem For finite Γ, If Γ ⊢g α then Γ
g α 15 See [9, 33–5]. A Third Way 311 Proof. Suppose Γ
g α. Then there is an interpretation ℑ =
W, R, ν and a world w0 in W such that for every Du γ in Du (Γ), νw0 (Du γ) = 1 and νw0 (¬α) = 1. Let f be a function such that f (0) = w0 . Consider now any complete tableau for the inference. f shows ℑ to be faithful to the initial list of the corresponding tableau; by the Soundness Lemma we have that there is at least one complete branch b such that ℑ is faithful to it. If b is closed it contains two nodes
β, i and
¬β, i, in which case (since ℑ is faithful to it) there is a world in ℑ at which both β and ¬β are true. The latter case is not allowed by our definition of interpretation. Thus, if ℑ is faithful to b, then b is not closed. Thus, Γ g α. Completeness To prove completeness we make use of the next definition and prove the lemma. Definition Let b be an open branch of a tableau. The interpretation ℑ =
W, R, ν induced by b is defined as in section 2.3: W = {wi |i is on b}; wi Rwj iff irj is on b; if
p, i occurs on b then νwi (p) = 1, if
¬p, i occurs on b then νwi (p) = 0 and otherwise νwi (p) takes a value arbitrarily. Completeness Lemma Let b be an open complete branch of a tableau. Let ℑ =
W, R, ν the interpretation induced by b. Then, If
β, i is on b, then νwi (β) = 1 If
¬β, i is on b, then νwi (β) = 0 The proof is given by induction over the set of wff. Completeness Theorem For finite Γ, if Γ
g α then Γ ⊢g α. Proof. Suppose that Γ g α. Then, there is a complete tableau for Du (Γ)∪ {¬α} with an open branch, b. The interpretation induced by b is such that for every Du γ in Du (Γ), νw0 (Du γ) = 1 and νw0 (α) = 0. Thus, Γ
g α. Acknowledgements. Thanks to Maria Cerezo, Richard Dietz, Manuel Garcia-Clavel, Patrick Greenough, Rosanna Keefe, Dan Lopez de Sa, Concepcion Martinez, Jose Martinez-Fernandez, Julien Murzi, Erik Norvelle, Paloma Perez-Ilzarbe, Luis Robledo, Stewart Shapiro, Kim Solin, Jose Zalabardo and Elia Zardini for helpful suggestions and comments on earlier 312 P. Cobreros versions of this paper. Many thanks to an anonymous referee of Studia Logica for several very interesting comments. Thanks to the Basque Government for a predoctoral scholarship from 2002 to 2006 (BFI02.35) and a postdoctoral scholarship for the year 2008 (BFI07.235). Thanks to Bancaja (Becas Internacionales Bancaja) for funds to visit the University of Sheffield during the Spring Term of 2007. Thanks to the research projects from the Government of Navarra (ref. 67/2006, de 29 de marzo) and the Government of Spain (ref. HUM2005-05910/FISO). References [1] Fara, D. G., ‘Gap principles, penumbral consequence, and infinitely higher-order vagueness’, Liars and Heaps: New Essays on Paradox, (2003), 195–221. Originally published under the name ‘Delia Graff’. [2] Fine, K., ‘Vagueness, truth and logic’, Synthese, 30 (1975), 265–300. [3] Heck, R., ‘A note on the logic of (higher-order) vagueness’, Analysis, 53 (1993), 201–208. [4] Hyde, D., ‘The prospects of a paraconsistent approach to vagueness’, Unpublished manuscript, (2005). [5] Keefe, R., ‘Supervaluationism and validity’, Philosophical Topics, 28 (2000), 93–106. [6] Keefe, R., Theories of Vagueness, Cambridge University Press, 2000. [7] Kremer, P., and M. Kremer, ‘Some supervaluation-based consequence relations’, Journal of Philosophical Logic, 32 (2003), 225–244. [8] McGee, V., and B. McLaughlin, ‘Review of Vagueness’, Linguistics and Philosophy, 21 (1998), 221–235. [9] Priest, G., An Introduction to Non-Classical Logic, Cambridge Univerisity Press, 2001. [10] Smullyan, R., First Order Logic, Dover, 1995. [11] Williamson, T., Vagueness, Routledge, 1994. [12] Williamson, T., ‘On the structure of higher-order vagueness’, Mind, 108 (1999), 127–143. [13] Williamson, T., ‘Reply to McGee and McLaughlin’, Linguistics and Philosophy, 27 (2004). [14] Wright, C., ‘Further reflections on the sorites paradox’, Philosophical Topics, 15 (1987), 227–290. [15] Wright, C., ‘Is higher-order vagueness coherent?’, Analysis, 52 (1992), 129–139. Pablo Cobreros Department of Philosophy University of Navarra 31080 Pamplona, Spain [email protected]