To the extent that vagueness involves semantic indeterminacy, either the object language or metalanguage must fail to obey classical logic. (Varzi 2007, p. 652)
[W]hat determines what the connectives mean are the inferences in which we employ them; the rules of inference implicitly define the connectives. (McGee and McLaughlin 1995, p. 206)
Abstract
Intuitively, vagueness involves some sort of indeterminacy: if Plato is a borderline case of baldness, then there is no fact of the matter about whether or not he’s bald—he’s neither bald nor not bald. The leading formal treatments of such indeterminacy—three valued logic, supervaluationism, etc.—either fail to validate the classical theorems, or require that various classically valid inference rules be restricted. Here we show how a fully classical, yet indeterminist account of vagueness can be given within natural semantics, an alternative semantics for classical proof theory. The key features of the account are: there is a single notion of truth—definite truth—and a single notion of validity; sentences can be true, false, or undetermined; all classical theorems and all classical inference rule are valid; the sorites argument is unsound; ‘definitely’ is treated as a meta-language predicate; higher-order vagueness is handled via semantic ascent.
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Notes
On semantic accounts of vagueness and supervaluationism, see Fine (1975), McGee and McLaughlin (1995, 1998, 2004), and Keefe (2000). Misiuna (2010) provides a semantic theory of vagueness with a non-monotonic, paraconsistent four-valued logic. Barnes (2010) has recently defended a metaphysical account of vagueness, and Barnes and Williams (2011) present a general account of metaphysical indeterminacy. A recent treatment of vaguenss in three-valued logic is Tye (1994); Machina (1976) and Smith (2008) provide degree-theories. There are also contextualist accounts of vagueness—e.g. Shapiro (2006, 2008), Fara (2000), Kamp (1981)—some of which are determinist and some of which are indeterminist.
See McCawley (1993, pp. 107ff.) for a helpful discussion of this result.
For the natural semantics of propositional logic, as well as for the quantified extension presented here, the restriction to persistent sets is unnecessary. From the semantic account of the propositional constants and substitutional quantifiers, it can be straightforwardly proven that, even in its absence, every propositional NS-model and every substitutional NS-model is in fact persistent. The restriction of models to persistent sets is not idle, however, in the full natural semantics of predicate logic (see Garson 2013, Ch. 14): some sets of valuations that satisfy the natural semantic clauses for the quantifiers are not persistent.
This truth condition resembles, but differs from, Beth’s intuitionistic account. See (Garson 2013, Ch. 7) for a detailed discussion.
The propositional NS-models turn out to be isomorphic to models obeying the propositional logic portion of Humberstone’s (1981) interval semantics for the logic of possibility. 〚LL〛 is entailed by Humberstone’s (1981, p. 318) Refinability condition. It is a straightforward matter to prove that classical propositional logic is sound and complete for NS-validity. (See Humberstone 1981, p. 321, or more directly Garson 2013, Sect. 8.7.) A variant of NS was discovered independently and developed by Garson (1990, 2001) in a study of what natural deduction rules express about connective meanings. However, the treatment of disjunction in (Garson 2001, p. 124) appears quite different from Humberstone’s. It has only recently been verified (Garson 2013, Sect. 8.8) that those treatments are equivalent in the classical setting. Humberstone (1981, pp. 316–317) notes a superficial similarity between his interval semantics and supervaluational accounts of vagueness, though he suggests that the imprecision of vague expressions is quite different from the indeterminacy with which he is concerned. Burgess and Humberstone’s (1987) treatment of vagueness, though in some ways related, denies the validity of excluded middle and so is thoroughly non-classical.
Shapiro (2006, 2008) presents a logic for vagueness that is similar to natural semantics (NS) in some respects. His system S, includes a partial precisification relation ≤. S employs Kleene’s three-valued truth-tables for propositional connectives, and then grafts intuitionistic connectives ‘⇒’ and ‘−’ onto that three-valued foundation. Unfortunately, the result is non-classical, and worse, there is no set of rules known to be sound and complete for it (Shapiro 2006, p. 109). However, there is more hope for the intuitionistic subsystem Sint of S that contains only ‘⇒’, ‘−’, ‘&’, and ‘∨’ where the trivalent approach may be, in effect, abandoned. Here the trivalent semantics may be reduced to one where T, F, and U are defined from a bivalent base as in NS.
Shapiro’s semantics for Sint would then be intuitionistic were it not for an important innovation. Shapiro recovers classical logic by modifying the definition of validity so that it preserves forcing rather than truth, where A is forced by v iff for every extension v′ of v, there is an extension v″ of v′ where v″(A) = t.
Note that forcing of A in NS is expressed in the object language by ⌈−−A⌉. (See Glivenko (1929) for why the forcing account of validity becomes classical.) So 〚LL〛, the condition in NS that accepts Double Negation Elimination, amounts to the claim 〚LL*〛 that every forced sentence is true.
- 〚LL*〛:
-
If v forces A, then v(A) = t.
Since the converse of 〚LL*〛 is trivial, NS identifies forcing with truth. Hence, NS undermines the very distinction that Shapiro uses to obtain classical semantics. Instead, it uses a standard definition of validity, with a novel truth condition 〚∨〛, which incorporates a forcing idea.
These facts about NS provide a strong argument against Shapiro’s whole approach. Shapiro must reject 〚LL*〛, since it causes forcing to collapse to truth. However, he himself endorses that very principle: “suppose that ϕ is forced at a partial interpretation N in a frame…. Someone who accepts the truths… in N is thus committed to the truth of ϕ” (Shapiro 2006, p. 80). Since he identifies determined truth with forcing, and determined truth surely entails truth, he cannot avoid 〚LL*〛. So natural semantics provides a much more intuitively acceptable way to deploy forcing in the logic of vagueness.
A replacement of variable v for variable u is proper just in case: if any occurrence of u lies in the scope of a quantifier ⌈∀v ⌉, the quantifier and its bound variables v are replaced with a quantifier ⌈∀w ⌉ and variables w, where w is new to A.
The quantifiers can also be given an objectual semantics; but, in fact, the semantics of the quantifiers generated by their natural deduction rules is neither substitutional nor objectual. (See Garson 1990, pp. 170ff.; 2013, Ch. 14.) In the present account of vagueness, however, the real work is done by the structure imposed by the NS treatment of the propositional connectives, and so we adopt substitutional quantifiers for the sake of simplicity. Nothing hangs on this.
Contrast Asher et al. (2009).
Note also that, since we have only one notion of truth, we have only one notion of validity. We need not choose, as the supervaluationist must, between preservation of truth (local validity) and preservation of supertruth (global validity). There is no consensus about which is the right choice for the supervaluationist. Fine (1975) and Keefe (2000) both assume that truth is supertruth and validity is global validity. Asher et al. (2009) and Varzi (2007) argue that truth is truth at a specification point and validity is local. According to McGee and McLaughlin (1995) our ordinary notion of truth is equivocal; some of its uses are best modeled as truth-at-a-point and some are best modeled as supertruth. Presumably, then, both local and global validity have their places. Though in their (1998) and (2004), McGee and McLaughlin argue that some truth-preserving inferences need to be excluded from within the scope of certain inference rules; this appears to amount to an endorsement of local validity—see especially (2004, p. 135). We have no stake in this internecine fight—and NS allows us to avoid it altogether—but see note 33 below for some worries about localism.
Though Fine (1975) is best known for providing a supervaluational semantics for vagueness, he also introduces a more general semantic framework, of which supervaluations is only one particular implementation. It is possible to define NS within that scheme; however, the novel features of NS appear ad hoc in that setting. Fine’s framework has an effectively trivalent base (T, F, undefined) and defines the extension relation in terms of the preservation of both T and F. From the standpoint of NS, which is essentially bivalent, an embedding into Fine’s scheme is best managed by disallowing the value F, and then grafting on NS truth conditions that appeal to the remaining “side” of Fine’s extension relation. Therefore the things that make NS special, namely that the values T, F and U are defined from a bivalent base, and that its truth conditions are determined from what classical natural deduction rules require to preserve validity, are not well reflected in Fine’s setting.
Excepting, of course, the degenerate case where v′ = v.
There are also extensions of v in which ‘Plato is bald’ remains unsettled—extenions v′ such that ‘Plato is bald’ is untrue in v′ but true in some further extension of v′.
To handle this formally, we can outfit the object language with numerical terms—say, ‘0’ and the successor functor ‘s’—and a one-place predicate ‘B’ that translates ‘A person with __ hairs is bald’.
The initial valuation described, v, will have a different counterexample to the universal premise, provided by ‘Bn’ and ‘Bn + 1’ for whichever n is the last sentence in the series clearly true in v. (“But wait,” you say; “What about second order vagueness? Is there any such n?” See the discussion of higher order vagueness, Sect. 2.3, below).
Or, at any rate, in each valuation in which there are both bald and not-bald things.
Indeed, we were so tempted. Thanks to Bruno Jacinto for saving us from a howler.
Suppose B0 and ~Bk. For reductio, assume ∀x(Bx→Bx + 1); repeated applications of universal instantiation and modus ponens yield Bk. Thus, ~∀x(Bx→Bx + 1). The latter is classically equivalent to ‘∃x(Bx & ~Bx + 1)’.
Notice, in fact, that nothing in this argument relies on the presence of vagueness or on the existence of a sorites series. For any predicate F—vague or non-vague—and any ordering of objects, a sentence that apparently asserts a sharp borderline—either ‘∃x(Fx & ~Fx + 1)’ or ‘∃x(Fx & ~Fx − 1)’—will be true exactly on those models with both F and non-F things (Here read ‘Fx’ as ‘the xth object in the ordering is F’).
Misiuna (2010) refers to this problem as paradox II. Our solution is to deploy the novel truth conditions of the existential quantifier in NS.
We thank an anonymous referee for reminding us of this point.
Partial precisifications give natural semantics a slight advantage over supervaluational treatments of vagueness that take truth to be supertruth. In such systems—since each supervaluational precisification relevant to the truth of formulas is total—the supervaluationist is committed both to the truth of ‘(∃x)(Bx & ~Bx + 1)’ and to the semantic claim that there is some number i such that v(Bi & ~Bi + 1) = t. Indeed, the supervaluationist must even acknowledge that in each precisification, it is determined which number of hairs marks the border between bald and not-bald. NS allows us to reject the semantic claim.
The reader may generate the right hand side of 〚∃sub〛 by applying the truth conditions 〚~〛 and〚∀sub〛 to ‘~∀x~A’.
Thus it also rules out object-language weak negation, ‘¬’, since this would allow us to express the indeterminacy of A via ⌈¬A & ¬~A⌉. Attempting to force ‘borderline’ or ‘¬’ into the object language collapses U and F, and so imposes bivalence. The resulting operators thus have neither the desired inferential behavior—since all of the classical inference rules for negation will also hold for ‘¬’—nor the desired semantic behavior—since neither ‘¬’ nor ‘borderline’ will be compatible with indeterminacy. See Sect. 3.1 below for an implementation of weak negation as classical negation in the metalanguage.
We thank an anonymous reader for raising the worry that the requirement that ‘definitely’ not appear in the object language is overly restrictive.
See Sect. 4, below, for why the choice of NS is preferred—or perhaps even necessary.
McGee and McLaughlin (1998, p. 225) write, “we can still hold that every classically valid sentence is true, and that every classically valid inference from true premises leads to a true conclusion. By our lights, that’s classical logic.” (See also their 2004, pp. 132–136.) This view has not gained wide favor, and it is commonplace to claim that standard versions of supervaluationism require at least some departure from classicality. Aside from Williamson (1994, pp. 151–152; 2004, p. 120), see, e.g., Fara (2003), Weatherson (2005, pp. 47, 53), Keefe (2000, pp. 179–180), Varzi (2007, pp. 650–653), and Asher et al. (2009, pp. 908–909, 915). The latter three sources each endorse supervaluations; Keefe (2000) argues that the departures from classicality are minor and thus low-cost; Varzi (2007) and Asher et al. (2009) argue for alternative construals of supervaluational validity.
See van Fraassen (1969, p. 81). Let S* be a supervaluational specification space such that ‘~p’ is super-true (⊨S*~p)—true at all complete specifications in S*—and ‘q’ is undetermined (|S* q)—true at some complete specifications, but false at others. Since |S* q, if ⊨S* q then ⊨S* p; that is, q ⊨S* p. Consider v*, some complete specification in S* in which ‘q’ is true. ⊨S*~p, so ‘p’ is false in v*. Thus, ‘q→p’ is false in v*, and ⊭S* q→p. See also Fara (2003, pp. 214–218). Garson (2013, pp. 140–141) presents similar counterexamples for contraposition, argument by cases, and reductio.
These counterexamples assume that the supervaluationist takes validity to be global validity. If validity is local and specifications are all complete, then supervaluationism does, indeed, validate the classical inferences—trivially so, in fact. Consider the supervaluations of Asher et al. (2009). Simplifying slightly, Asher et al. take a supervaluational model to be any collection of points, each of which is a complete classical model. Truth is, fundamentally, truth at a point, and validity is local: Γ ⊨ l A if and only if for every supervaluational model S and every point v ∈ S, if every member of Γ is true in v, then A is true in v. But this straightforwardly reduces to classical consequence: Γ ⊨ l A if and only if for every classical model M, if every member of Γ is true in M, then A is true in M. So in the system of “debugged” supervaluations, every sentence has its classical truth conditions and validity is, by definition, classical.
On top of this classical framework, Asher et al. introduce two modal operators. Each supervaluational model is outfitted with a binary, reflexive accessibility relation R on S, and a specified subset of S, @. ⌈DA⌉ (read: ‘definitely A’) is true at specification point v if and only if A is true at all v′ such that Rvv′. ⌈ AA⌉ (read ‘actually A’) is true at v if and only if A is true at all v′ ∈ @. Asher et al. take operator ‘A’ to expresses supertruth “properly modally construed” (p. 925). Supertruth is not, then, truth simpliciter, but rather a way of being true—much as is necessary truth.
To us, this system looks like a classical modal logic, rather than a species of supervaluationism. But that’s hardly an objection—maybe, after all, the right logic of vagueness is modal rather than supervaluational. The real trouble with debugged supervaluationism is that, though it is, as advertised, fully classical, it is also fully bivalent—it rules out sentences with indeterminate truth values, and so is incompatible with indeterminism. Remember: truth is truth at a point, and at each point, every sentence is either true or false. To be sure, a true (or false) sentence might have no A-truth value (that is, neither ⌈ AA⌉ nor ⌈ A~A⌉ is true) or no D-truth value (neither ⌈DA⌉ nor ⌈D~A⌉ is true). But this is no more indeterminacy than is contingent truth. Furthermore, the system allows for the following mind-bending situations: Diogenes is bald but not actually bald; Diogenes is definitely bald but not actually bald; Plato is neither actually bald nor actually not bald, and is neither definitely bald nor definitely not bald, but he is, nonetheless, bald.
See Garson (2013, p. 72). We prove here that NS validates conditional proof; proofs for the other rules are equally straightforward. Recall that ⌈A→B⌉ is true at a valuation v just in case for every v′ that extends v, v′(A) = n or v′(B) = t. Let V be any NS model such that A ⊨V B; then for all v ∈ V, either v(A) = n or v(B) = t. (Remember that, since NS has only one notion of truth, global and local validity coincide.) Since every valuation in V meets this condition, every extension of a valuation meets it; thus v(A→B) = t for all v ∈ V. That is: ⊨V A→B.
Williamson is explicitly targeting supervaluationism, but his argument, if successful, would undermine any treatment of vagueness that allows for undetermined truth values. We formulate Williamson’s argument in terms of sentences rather than utterances, as he does, for simplicity’s sake; nothing turns on this.
Note that there is another, technical sense in which NS does preserve bivalence: at each valuation v, every sentence A is such that either v(A) = t or v(A) = n. So every sentence takes exactly one of two semantic values, t and n, at each valuation. But these semantic values aren’t—at least in the first instance—truth values. We’ve identified truth with having a settled valuation of t, and falsity with having a settled valuation of n. From this perspective, since the semantic clauses for the logical constants propagate t and n, rather than T, U, and F, truth and falsity aren’t strictly truth values. Recall the definitions of ‘v(A) = T’, etc.
The conceptual requirement is a constraint on negation, not on falsity. Falsity for object languages without a negation operator is straightforwardly defined in the ordinary way: v(A) = F iff v′ (A) = n for all v′ ∈ V such that v ≤ v′. This is just (DefF), above. The only difference is that such object languages have no way of expressing the falsity of their own sentences. Thanks to an anonymous reviewer for inquiring about this point.
Note that we are not claiming that weak negation is metalinguistic negation in the sense of Horn (1985), widely discussed in the literature. Rather, we claim that the first and second negations in (10) are best interpreted as weak negations, and that weak negation is best formally treated at the metalinguistic level. Shapiro (2006, p. 63) also implements weak negation metalinguistically, though—see note 7 above—his system is non-classical.
Nonetheless, we observe that (10) passes at least one of the traditional tests for metalinguistic negation proposed by Horn (1985). The first negation in (10) doesn’t incorporate:
- (10a):
-
*Plato is non-bald, but he’s also not not bald.
Note that the third ‘not’ does incorporate:
- (10b):
-
Plato isn’t bald, but he’s also not non-bald.
But the second doesn’t:
- (10c):
-
*Plato isn’t bald, but he’s also non-non-bald.
Some nearby natural language examples can’t be given this treatment. Consider the following response to the question “Is Plato bald?”
- (16):
-
He is and he isn’t.
(Thanks to an anonymous referee for this example.) The negation in (16) appears to lack the intonation contour often associated with metalinguistic negation. This isn’t decisive, however, since such stress isn’t a necessary feature of metalinguistic negation. And, as in (10), the negation does seem to resist incorporation:
- (16a):
-
*He’s bald and he’s non-bald.
However, we’re forced to reject (16) as contradictory in any case. If the negation is ordinary object language negation, (16) is a manifest contradiction. If the negation is weak negation, implemented as ordinary negation in the metalanguage, (16) is interpreted as
- (16b):
-
v(A) = T and v(A) ≠ T.
This, too, is a contradiction in the natural semantics framework. If the negation is metalinguistic negation à la Horn (1985), (16) has us the speaker engaging in a sort of performative contradiction, both endorsing and objecting to ‘Plato is bald’. It strikes us that the best thing to say—for everybody but the dialethist—is that (16), though literally false, has the pragmatic effect of conveying that ‘Plato is bald’ is unsettled.
Here, perhaps, we do incur a cost beyond that carried by the usual Tarski hierarchy for a first-order language with classical two-valued semantics: we have a hierarchy of negations in addition to the hierarchy of truth predicates. Such, however, is the price of preserving the classical inference rules in the presence of undetermined truth values. Our hierarchy of negations is similar in spirit—though not in implementation—to those proposed by Fitch (1964) and by Schlenker (2010).
By ‘traditional models’ here, we mean sets of valuations obeying classical truth conditions.
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Acknowledgments
Thanks to Susanne Bobzien, David Etlin, Laurence Goldstein, Bruno Jacinto, Hanti Lin, Diana Raffman, and Meg Viers for helpful comments and discussion. Thanks also to the anonymous reviewers for the paper. A condensed version of this material was presented at the 2012 Eastern APA.
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Brown, J.D.K., Garson, J.W. A New Semantics for Vagueness. Erkenn 82, 65–85 (2017). https://doi.org/10.1007/s10670-016-9806-x
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DOI: https://doi.org/10.1007/s10670-016-9806-x