This paper is a tribute to Delia Graff Fara. It extends her work on failures of meta-rules (conditional proof, RAA, contraposition, disjunction elimination) for validity as truth-preservation under a supervaluationist identification of truth with supertruth. She showed that such failures occur even in languages without special vagueness-related operators, for standards of deductive reasoning as materially rather than purely logically good, depending on a context-dependent background. This paper extends her argument to: quantifier meta-rules like existential elimination; ambiguity; deliberately vague standard mathematical notation. Supervaluationist attempts to qualify the meta-rules impose unreasonable cognitive demands on reasoning and underestimate her challenge.

Este artículo es un tributo a Delia Graff Fara. Extiende su trabajo acerca del colapso de las metareglas (prueba condicional, reductio ad absurdum, contraposición, eliminación de la disyunción) para la validez entendida como preservación de la verdad bajo una identificación de la verdad con superverdad. Graff Fara demostró que dicho colapso se da incluso en lenguajes que no poseen operadores especiales relacionados con la vaguedad, para estándares de razonamiento deductivo tanto material como lógicamente correctos, dependiendo de factores sensibles al contexto. En este artículo se extiende su argumento a: meta-reglas cuantificacionales, como la eliminación existencial; ambigüedad; notación matemática estándar deliberadamente vaga. Los intentos supervaluacionistas de cualificar las meta-reglas imponen exigencias cognitivas poco cabales sobre el razonamiento y subestiman sus retos.

Supervaluationists treat vagueness as a kind of semantic underdetermination. The community’s use of its language fails to determine unique semantic values for its expressions. Many different assignments are in some (under-explained) sense compatible with use—even in a given context (this relativity to context will henceforth be left tacit). Call those assignments compatible with use

Traditional supervaluationism identifies truth with supertruth and falsity with superfalsity, not with truth and falsity respectively on the intended assignment, because there is no intended assignment. Use determines only supertruth and superfalsity. Thus every theorem of classical logic is true, because supertrue, because true on every reasonable assignment. That includes all instances of the law of excluded middle. However, not every sentence of a vague language is either true or false, because borderline sentences are neither supertrue nor superfalse. Consequently, standard disquotational biconditionals for truth are held to fail.

A variant form of supervaluationism identifies truth with disquotational truth rather than supertruth

Speakers may wish to talk in the object-language about borderline status. The natural way to do that is by adding an operator

We can easily turn these ideas into a simple model theory for a formal propositional language with a

What is the appropriate model-theoretic relation of logical consequence in this setting? One proposal is that a valid argument is one that preserves truth at any given point in any given model, which is known as

Г ╞ ‘_{L} α = _{def} for every model <W, V> and

Formally, local validity is a nice classical consequence relation, but it does not suit the spirit of standard supervaluationism. For validity is supposed to be truth-preservation, and standard supervaluationists equate genuine truth with supertruth, not with truth on a given assignment. For them, a genuinely valid argument is one that preserves supertruth in any given model, a different consequence relation known as

Г ‘╞ _{G} α = _{def} for every model <W, V>, if γ is true at

As so often with vagueness, there is a catch: higher-order vagueness. ‘Reasonable’ is itself vague, which makes D vague too, so its semantic value and that of many sentences containing it should themselves vary from one reasonable assignment to another. In the simple semantics just sketched, in any given model, for any given formula

A more natural strategy for the supervaluationist is to follow the example of modal logic, where Kripke showed how to implement model-theoretically its being contingent whether something is necessary, by introducing an accessibility relation R between worlds. Informally, a world

The only formal constraint on the accessibility relation imposed here is reflexivity: every assignment is reasonable according to itself. This validates the plausible principle that whatever is definitely so is indeed so (

Supervaluationist models are now triples <W, R, V>, where W and V are as before and R is the accessibility relation. Since the point of the

Г╞ ‘_{R} α =

The main considerations in this paper are independent of higher-order vagueness. Although we leave room for it by not requiring accessibility to be an equivalence relation, the arguments still work even if that constraint is imposed. Indeed, they work for both regional validity and global validity; for convenience, some will be stated just in terms of regional validity. Although other accounts of validity have been proposed within a supervaluationist setting (

In the fragment of the language without the

Even in the full language with

In a broader sense, however, the logics of global validity and regional validity are non-classical. For they violate some standard

To see how the failure happens in the case of conditional proof, consider the formula _{R} ⊥. Hence, if ‘╞_{R} obeys conditional proof, ╞‘_{R} (_{R}
_{R}

Similar arguments show that global validity and regional validity also violate reductio ad absurdum, argument by cases, and contraposition, since conditional proof is derivable from any one of those meta-rules in this setting.

The meta-rules are defined in terms of the given consequence relation for the language, in this case global or regional validity. Thus whether the meta-rules hold for a language supervenes on the consequence relation for that language; no difference in the former without a difference in the latter. For the

Some confusion arose about the status of the meta-rules for global validity because Kit Fine originally illustrated the failure of conditional proof for global validity with the argument from

As usual, one can complicate the model theory and the models to avoid the inconvenient results, for instance by making the argument from {

Supervaluationists have used their semantic framework to provide semantic accounts of comparative and superlative adjectives and of modifiers such as ‘rather’, ‘in a sense’, and ‘-ish’, starting with work by

Reasoning to a standard of global or regional validity seems to be a tricky business. Some supervaluationists have responded by proposing non-standard variants of the meta-rules that avoid such counterexamples. For instance, Rosanna Keefe suggests formulations in which

Is ordinary deductive reasoning in science—for instance, in drawing consequences from a theory—sensitive to any of the proposed restrictions on the meta-rules? That is far from obvious. Still, supervaluationists might claim that the restrictions are rarely needed, and if we occasionally violate them, that is our fault.

The debate has sometimes focussed on whether the contested meta-rules are ‘part of classical logic’. That has not been very fruitful. They are in a broad sense of ‘classical logic’, but not in a narrower sense. A more interesting question is how much of our deductive practice will be lost if the meta-rules are restricted.

Graff Fara discusses the more interesting question in the latter half of her seminal paper ‘Gap Principles, Penumbral Consequence, and Infinitely Higher-Order Vagueness’

Graff Fara’s examples involve arguments in ordinary English, evaluated according to supervaluationist semantics.

Here is one of Graff Fara’s examples

Therefore,

In some contexts, the supertruth of the premises of
IIB guarantees the supertruth of its conclusion. For, given how vague ‘tall’
is, the borderline zone is definitely more than a millimetre wide (if you doubt
that, Graff Fara invites you to work with a smaller distance). Hence, if

Therefore,

Argument IIA is

Strictly speaking, both arguments involve the comparative operators ‘more’ and ‘-er’, whose semantics may be given in a supervaluationist framework. This is an inessential feature. We could replace the conclusion of IIB and the second premise of IIA with a sentence in mathematical notation about the relative heights of

Could a supervaluationist maintain that since the operative standard of validity is supertruth-preservation, and the

A more discriminating idea is that to appreciate the material regional validity of IIB one must in effect think

In response, the supervaluationist might point out that we rarely accept arguments like IIB, so the problem with contraposing them rarely arises. However, drawing attention to the unnaturalness of IIB is a dangerous move for a supervaluationist to make. For IIB itself feels far less compelling than does the step of contraposition from IIB to IIA, even if one reduces the distance at issue from a millimetre to something much less (but not zero). If one wants to avoid being drawn into the soritical argument IIA, the pre-theoretically natural way to do it is by not accepting IIB in the first place, rather than by accepting IIB and then refusing to contrapose. However materially restricted, regional validity does not come naturally as a standard for judging the material goodness of vague deductive arguments in one’s native language. That is awkward for the supervaluationist who proffers regional validity as the appropriate standard.

Of course, native speakers may be confused about the logic of their own language. Nevertheless, supervaluationists have typically defended their account of the logic of vague languages by arguing that it is not seriously revisionary of inferential practice. But if we avoid reliance on those numerous instances of the meta-rules condemned by supervaluationists only because we are unwilling to rely on numerous object-language arguments (like IIA)

The case of IIA and IIB is far from an isolated example. Similar pairs of arguments can be constructed for any sorites-susceptible term. Graff Fara has presented a deadly serious challenge to the supervaluationist account of deductive validity for vague languages

In this section, I present some variations on Graff Fara’s theme. In the final section, I will comment on the prospects for restricting the meta-rules in the ways supervaluationists have proposed.

In his classic presentation of a supervaluationist treatment of vagueness, Kit Fine describes vagueness as ‘ambiguity on a grand and systematic scale’, and gives it a supervaluationist treatment: ‘an ambiguous sentence is true if true for all disambiguations’

We start with a standard propositional language L, assumed free of vagueness, ambiguity, and additional operators. The semantics is the standard bivalent one, for which classical propositional logic is sound and complete. Let _{R} is not purely logical, since a non-logical constraint on the truth-value of

Some simple observations can now be made.

[a] _{R }

[b] _{R}

[c] _{R}

[d] ¬_{R} ¬p, by an argument like that for [a].

[e] ¬_{R} ¬

[f] ¬_{R}

Consequently, if ‘╞_{R} obeys argument by cases, _{R}
_{R}

The other meta-rules fail similarly. For instance, conditional proof and contraposition fail, since _{R}
_{R}
_{R} ¬_{R} ⊥ but not ¬_{R} ¬

The failure of the meta-rules does not depend on the simplicity of the example. The proof can be adapted to much more complex cases of ambiguity, involving other grammatical categories, more than two readings, pre-existing ambiguity and vagueness in the object-language, and so on. Moreover, as already emphasized, it does not depend on the presence of any special operators in the object-language for dealing with vagueness or ambiguity.

Although the failure of meta-rules for regional validity is usually discussed in the setting of propositional logic, it also affects some meta-rules for quantifiers. An example is a standard elimination rule for the existential quantifier: if from Γ ∪ {α(c)} one can derive β, and the constant _{R} β then Γ ∪ {∃x α(x)} ‘╞_{R} β, subject to the same restrictions.

To assess the rule, we must extend the model theory to handle quantifiers and individual variables. The main choice-point in doing so is whether we allow variation in the quantifier domain from one point of evaluation to another, corresponding to vagueness in how far the quantifiers range. To keep things simple, we assume that there is no such vagueness. In particular, we may specify that the quantifiers range precisely over the natural numbers. If the result is a non-logical consequence relation, that is consonant with Graff Fara’s argument. However, for the time being, we keep the operator

The model has a natural supervaluationist interpretation. For we can read

That example depends on the

CUTOFF Mary is tall and John is not tall and Mary is not more than a millimetre taller than John.

∃CUTOFF For some X, for some Y, X is tall and Y is not tall and X is not more than a millimetre taller than Y.

In the context Graff Fara envisages, CUTOFF is guaranteed not to be supertrue. Thus the argument from CUTOFF to a contradiction is materially regionally valid. Hence, if the existential elimination rule for English preserves material regional validity, then the argument from ∃CUTOFF to a contradiction is also materially regionally valid, by two applications of the rule. But it is not. Given a sufficiently dense distribution of heights in the relevant population between the definitely tall and the definitely not tall, ∃CUTOFF is supertrue. Thus the existential elimination rule fails for material regional validity even in the absence of operators that trade on vagueness.

Contrary to stereotype, some standard mathematical notation is deliberately left vague and context-dependent. For instance, the symbol ≈ is often used for

We can see this by building a toy supervaluationist model of how ≈ works in some simple contexts. Suppose that on each assignment there is a constant δ such that for all real numbers

Now consider the set of formulas {

That is impossible, so the three formulas cannot be supertrue together. Thus the argument from {

Another mathematical symbol deliberately left vague and context-dependent is <<: ‘

Now consider the set of formulas {¬

The evidence from §3 confirms Graff Fara’s claim that the potential for violations of standard meta-rules by material forms of validity is extremely widespread. It extends to ordinary cases of ambiguity, to quantifier rules, and to many mathematical contexts. What morals should we draw from this?

Graff Fara is clearly right that in practice we rarely care whether our deductions are valid by a purely logical standard or a looser, context-dependent one, reliant on background assumptions. We are fine with arguments like ‘She lives in Paris, so she lives in France’, despite their invalidity by a strictly logical standard. Even working mathematicians rarely care whether their proofs are

Suppose that, in a vague and sometimes ambiguous language, our deductive reasoning tends to accord with material regional validity: we tend to accept materially regionally valid deductions and reject materially regionally invalid ones. We must therefore be implicitly avoiding numerous nearby instances of the meta-rules that would take us from materially regionally valid arguments to materially regionally invalid ones. Yet, on the face of it, we freely apply conditional proof, reductio ad absurdum, contraposition, argument by cases, and existential elimination, without specially checking whether the instance in question is problematic for supervaluationist reasons. Did you monitor the purely technical parts of this paper for such fallacies? (Supervaluationists would in fact accept the informal proofs.)

Obviously, in practice we rarely insert ‘definitely’ into the input or output of the meta-rules to ensure (material) regional validity. Do we instead apply qualified versions of the meta-rules with extra conditions on the status of the input argument, such as McGee and McLaughlin and others impose? Such proposals typically require the input argument to meet a more demanding standard of validity than regional validity. Thus, if ordinary reasoners pay attention to regional validity at all, they must keep track of

One hypothesis is that we accept regionally valid arguments, and make free but illicit use of the unqualified meta-rules, because we are unaware of the fallacies. On that hypothesis, our reasoning should get us into serious trouble. Does it? A first thought is that we do get into serious trouble: slippery slopes and sorites paradoxes. However, it is not clear that our problems there depend on illicit use of the meta-rules. For if they do, why are the problems so limited, when their cause is almost ubiquitous? Why do mathematicians not get into trouble with their use of symbols like ≈ and <<? After all, mathematicians are the people who give our patterns of deductive reasoning their most systematic and severe stress-testing.

Consider ambiguity as a toy model of vagueness, as supervaluationists such as Fine have done. Of course, ambiguity

That is a bank.

Therefore, that is both a financial bank and a river bank.

BANK is untempting, to say the least. However one disambiguates the premise, the conclusion does not follow, even materially. Yet BANK is materially regionally valid, on a supervaluationist treatment of ambiguity: if the premise is true on both disambiguations, the conclusion is true (compare §3.2, [a] and [b]). Of course, committed supervaluationists may be able to talk themselves into accepting the validity of BANK. But it does not come naturally to reason according to BANK.

The same goes for vagueness proper. For instance, when I think as a mathematician, I feel no temptation to regard {

I will not labour the case. Graff Fara’s point goes too deep for any technical fix. Supervaluationists need to undertake a far more searching investigation of what our deductive practice would really be like if properly tailored to context-dependent material regional validity. In this, as in so much else, she has left us a rewarding challenge for the future.

Delia characterizes herself as an epistemicist
at

For simplicity, the object-language is assumed to contain only declarative sentences.

One would not expect ‘

The distinction between global and regional
validity is also involved in the debate between

Proof: Suppose that Γ ∪ {α} entails β. Hence Γ ∪ {α ∧ ¬β} entails ⊥. Hence, by reductio ad absurdum, Γ entails ¬(α ∧ ¬β); equally, by contraposition, Γ ∪ {¬⊥} entails ¬(α ∧ ¬β); in both cases, Γ entails α → β. Also, from the original supposition, Γ ∪ {α} entails α → β. Moreover, {¬α} entails α → β. Hence, by argument by cases, Γ entails α → β. The proofs assume that any set of premises entails its classical truth-functional consequences and that entailment obeys the standard structural rules for logical consequence, in particular the cut rule; these assumptions hold for both global and regional validity.

One can also consider an alternative
consequence relation that holds between premises and conclusion if and only if
(i) whenever the conjunction of the premises is supertrue, the conclusion is
superfalse and (ii) whenever the conclusion is superfalse then the conjunction
of the premises is superfalse. However, since

Unclarity on which object-language the
meta-rules concern vitiates the objection in

See

An example is
Williams 2008, where each model has both an accessibility relation over all
delineations (points) and an independently specified subset of delineations,
whose members are called sharpenings. Supertruth in a model is defined as truth on all

Her model-theoretic framework differs slightly from the present one, but not in ways that matter for present purposes.

To generate a sorites paradox from multiple instances of IIA, one needs the structural rule of Cut: if Г entails α and Δ ∪ {α} entails β then Г ∪ Δ entails β. Cut holds even for restricted versions of regional validity, for if all members of Г ∪ Δ are supertrue at a point

In

Pedantically speaking, the three object-language variables

Thanks to Volker Halbach, Ming Xiong, other
participants in a class at Oxford, and two referees for