such as truth-preservation and satisfaction-preservation. Cf, for example, [McGee 2006: 194], in the context of a discussion of the notion of logical consequence for atomic and complex formulas.
102 In this section, I will present Option 3 as it is defended by [Lavine 2006].When turning to a discussion of the option’s plausibility, I shall complicate the discussion by proposing my own variant.
Not only because j) is obviously not well-formed – what do the quantifiers range on? – but because of the two following considerations, concerning respectively 1) our intuitive understanding of a rule of inference and 2) the distinction between the general form of an inference and the semantics of its instances.
1) Our intuitive understanding of the instruction that i) expresses is independent of any consideration about how we specify the range of the quantifiers appearing in the statement of the rule: we understand the rule as concerning what we may infer given the particular formulas φ and ∼φ, not as regarding a relation between all formulas that
we may take as the values of the variable φand all formulas that we may take as the
value of the variable ψ. That is, our intuitive understanding of i) is only dependent
upon is the specification of a purely syntactic constraint on φ and ψ: namely, the
constraint that we should take instances of φ and ψ to be formulas.
The idea is then that a rule of inference consists in an instruction to infer in a certain way independently of any semantic constraint that we may place on the instances of the schematic letters appearing in its statement. Because, Lavine claims, quantificational generality always involves providing such constraints, then the generality of the rule (the extent of our commitments to it) cannot be rendered by any use of the quantifier.
According to Lavine, the appeal to our intuitive understanding of i) can be
strengthened both by a general argument about the assertibility conditions of a schema and by a specific argument about the learnability of inference rules for the universal quantifier that are given in non-schematic form.
The first argument amounts to noting that the assertibility conditions of (claims about) i) and j), and – in general - of (claims about) schematic and quantificational formulations of the generality of an inference rule, are prima facie different. Why?
Well, the view is, in virtue of the fact that the assertibility conditions of a quantified claim will presumably also depend on how the domain of the quantifiers is specified: the claim concerns objects in the domain, and will be assertible if and only if
what is being claimed of those objects holds – where the interpretation that we choose to give of ‘holds’, here, will depend on the specific terms in which the assertibility conditions of a claim are formulated. But in the case of a schema, such conditions seem to have nothing to do with our understanding of what belongs to a domain. When we assert a schema, we make a claim about particular instances of the letters occurring in it, ‘without any need to have a notion in advance of all the suitable instances’ [Lavine 2006: 121].
The second argument goes as follows. Suppose we try to express the generality of (our commitment to) Universal Specification in quantificational terms. The analogue for the universal quantifier of the ill-formed rule of inference j) would then be:
k) ∀(φ)∀(x)∀(τ) ((∀x)φ(x) |- φ(τ))103
Where all the variables appearing in the statement of the rule are quantifiable variables.
One of the problems of this formulation of Universal Specification is the epistemic circularity to which the formalism gives rise. Such circularity concerns both the verification conditions of Universal Specification (what we would have to prove in order to show that the inference from its premises to its conclusion is valid) and what would be required of an epistemic subject to understand the rule when she is first introduced to it. To show that Universal Specification is valid, we would have to apply Universal Specification to it in order to, as it were, knock off each quantifier in the first sequence. For a subject to understand Universal Specification as the instruction spelt out in k), she would already have to grasp (how to apply) Universal Specification in order to grasp the formalism by which the rule is normally introduced. Simply put: a
103 This is what Lavine calls ‘Universal Universal Specification’ [Lavine 2006: 116]. In fact, if the idea is that the generality of the rule is to be rendered in quantificational terms, and quantification always requires the specification of a domain, to achieve the desired generality without making any assumption about the fact that the domain for the quantifiers appearing in the rule is absolutely unrestricted, we would have to state the rule as:
(k*) ∀(D)(∀(φ)∀(x)∀(τ) ((∀x)φ(x) |- φ(τ))) Where D is a first-order variable that ranges over domains of discourse.
rule that is meant to be concept-defining for the universal quantifier requires, both for the proof of its validity and for the definition of what counts as grasping it, using the universal quantifier in the way specified by the rule itself104.
2) While Option 2) takes the concept-defining rules for the universal quantifier to specify a domain for the quantifier, according to Option 3) no such semantic task should be assigned to the rules. Independently of whether one shares the idea that quantificational generality and schematic generality are distinct kinds of generality, there are independent motivations for this choice. In particular, one can reason in the following terms.
We need to distinguish between the status of a general rule of inference and the status of its instances. The general rule is typically formulated in a meta-language, as an instruction that tells us how to use the logical concepts that appear in the object- language relative to which the rule is given. Usage in the object-language will be usage in an interpreted language: it is at this level that (instances of) the rules will be
described as having certain semantic properties (for example, the property of being truth-preserving). Such properties concern fully interpreted formulas, rather than schematic formulations of inference rules [Williamson 2006: 382-3; Rayo & Williamson 2003].
Now, one may agree that part of what it is to interpret a universally quantified statement is to specify a domain of discourse for the quantifiers. However, on the basis of the considerations just sketched, one may also legitimately ask the question of why such specification should have anything to do either with the rule of inference itself or with our general understanding of what it instructs us to do. Fixing the semantic values of the quantifier is the primary task of a semantic theory of quantificational generality; the object of such theory will be the language in which we carry out the inferences that count as legitimate instances of the concept-defining rules that are
104 A related point is in [Quine 1936: 351-2]. McGee’s understanding of Universal Specification also incurs in an indirect form of the same kind of circularity, as – in the framework of his model-theoretic account of (the semantics for) concept-defining rules - he employs unrestricted second order quantification over classes of structures [McGee 2000: 60, 70].
given in the meta-language. But then, as [Lavine 2006: 117] puts it, “it is not the rules that do the fixing” (of the semantic values of quantified statements), it is rather “the meta-linguistic specification of the permissible instances of the open ended rules” that does105.
3. SCHEMAS AND QUANTIFICATIONAL GENERALITY
How do the three options presented in Section 2 fare with respect to the constraints sketched in Section 1? Recall that these were:
• The Semantic Constraint: the concept-defining rules should be understood in a
way consistent with the idea that what counts as the correct semantics for the universal quantifier is determined by our inferential usage of the concept;
• The Epistemic Constraint, formulated in terms of the open-endedness of our
commitments: the way in which we understand the concept-defining rules should render the open-endedness of the commitments that we undertake when using the concept of universal quantification;
105 Italics mine. I will come back to this consideration in the discussion of the problems of Option 2 (sub-section 3.2).
There is, however, an immediate objection that we should take care of. A rule that is taken to be concept-defining is, in the project that I am defending, a means for reconstructing the general conditions on the correct usage of a logical concept. That is: we look at how competent speakers use the concept, and try to reconstruct what the general form of the inferential commitments that they undertake in this usage is. If usage of a logical concept always takes place in an interpreted language, then intuitively we would like the reconstruction of the general form of such usage to take this into account. In other words, in virtue of the fact that we would like it to render the general features that can be ascribed to our concept-
constituting inferences, it seems natural to require that they also render (the general form of) whichever semantic qualifications characterize usage in an interpreted language.
My simple reply to this objection is that it is based on a wrong assumption. The
assumption is that all the aspects of a speaker’s usage of the concept can or should be captured in inferential terms. This is neither a claim to which the inferentialist would want to subscribe, nor a generally plausible one (what about, for example, the aspects of usage in context that bear on the speakers’ psychological or intentional states?).
• The Anti-relativist and Anti-skeptical Constraint: the reconstruction of our
concept-constituting inferential practices should not be obviously open to the threat of relativism or skepticism.
3.1 On the plausibility of Option 1
Recall that Option 1 has two formulations. According to one formulation, the way in which we should understand Universal Specification is best rendered as the
semantically constrained instruction: m) ∀(D)∀(φ)∀(x ∈ D) φ(x) |- φ(c∈D)
Where all the variables are quantifiable variables. According to the other formulation, the correct understanding of the rule takes φ and, crucially, D, to be schematic letters,
so that we should understand the rule as consisting in the instruction: n) ∀(x) (D(x)→φ(x)) |- D(c) →φ(c)
Here I will only consider the former, and thus focus on m). The reason is the following.
There are two main motivations for giving a schematic reading of the rule.
The first is given by the idea that the schematic generality thus achieved is distinct from the quantificational generality expressed, for example, by m). That is: the idea, which we have already encountered, that the open-endedness of the
commitments that we undertake when inferring with the universal quantifier shouldn’t be rendered in terms of the quantifier itself. But this is the very idea on which Option 3 is also based. The difference between the way in which the schematic formulation of Option 1 and Option 3 render Universal Specification is in the status of the letter c – Option 3 replaces it with a schematic letter s. From the point of view of a proponent of Option 3, then, n) will simply be the result of replacing the schematic letter that should figure in the conclusion of the rule with one of its instances – it will be, that is,
the result of a partial interpretation of the schematic rule. On the other hand, from the point of view of a proponent of (the schematic version of) Option 1, taking the
occurrence of the constant c in the rule to be the result of a partial interpretation is consistent with the general motivation for spelling out the reading of the letter for domains in Universal Specification as schematic: once the open-endedness of (our commitments to) the rule is rendered in this way, the choice of c becomes itself open- ended.
The problem that arises at this point is, then, whether or not the appeal to the notion of schematicity succeeds in capturing the intuitive idea of open-endedness, while respecting the three constraints formulated in sub-section 1.2. This issue will be tackled in the discussion of Option 3.
The second motivation that one could have for choosing n) over m) is given by the consideration that quantification over all domains, unless suitably constrained or further qualified, is arguably the source of the set-theoretic paradoxes. The strategy that I have followed in this chapter consists in setting aside the issue of paradox, and focusing instead on the project of spelling out concept-defining rules of inference for the universal quantifier. I shall therefore stick to the chosen strategy, and not discuss this further motivation for a schematic reading of Universal Specification.
The formulation of Universal Specification in (m) is subject to the following problems.
The formulation does not allow us to respect the anti-relativist and anti-skeptical constraint.
There are two ways in which the first-order quantifier in (m) ranging on domains can be interpreted. The first is by appeal to the standard model-theoretic definition of the range of a quantifier, according to which a domain is essentially a set [Cartwright 1994]. If we follow this route, then the range of the first quantifier in (m) is the set of all sets. The second consists in interpreting the quantifier plurally, so that
quantification over all domains needn’t require the specification of a set, or a ‘set-like’ object, as the range of the quantifier. To understand the quantifier, we need not undertake any commitment to the existence (or definability) of a set over which it ranges; the possible values of the quantified variables are plural entities [Boolos 1984; 1985].
Consider the first alternative. According to it, quantification over all domains is quantification over the set of all sets. But quantification over the set of all sets is
directly subject to Skolem’s skepticism. That is, it is open to the two-fold objection, which is an upshot of Skolem’s construction, of epistemic indeterminacy and of linguistic inexpressibility. If quantification over the domain of all sets is employed in the very formulation of the concept’s inferential role, the objections endanger, as it were, the determinacy and the expressibility of the concept itself.
In particular, the indeterminacy objection endangers the rationale of the very option under consideration. For what is the point of providing concept-defining rules, which should serve as a suitable reconstruction of the commitments that we undertake in our inferential practices, if there is no determinate fact of the matter as to which concept we deploy in such practices?
The inexpressibility objection endangers our ability to understand the linguistic formulation of the rule as a formulation intended to express its generality. For if such generality is to be understood in terms of quantification over all sets, and the latter is inexpressible as such, then no linguistic formulation of the rule will enable to express the open-endedness of our inferential commitments.
The option is subject to Geach-style relativism. For a consequence of Skolem’s skepticism is that, within any given language, no matter how we strengthen and refine it, there is no fact of the matter as to whether a usage of the quantifiers is
determinately unrestricted or not. As noted above, if this is the case, and if one employs unrestricted quantification over sets to express the concept-defining rule for the very concept of universal quantification, then the determinacy and the
the concept consists in is indeterminate and inexpressible within any given language, there will be no fact of the matter as to whether it will be the same concept across different languages.
Now consider the second alternative, according to which the quantifier has to be interpreted plurally. The alternative is not subject to Skolem’s skepticism: for there is simply no set quantification over which would then be extensionally and
intensionally equivalent to quantification over one of its countable sub-sets. Is it subject to Geach-style relativism? It seems that it is, for consider the following.
Recall that a key-claim of Geach-style relativism is that we cannot hope to define a concept in absolute terms because we have no guarantee that speakers of different languages mean the same when using an expression that is supposed to denote the concept in question.
Now, as [Williamson 2006: 381 ff] argues, a way in which one may start confronting this claim is by considering the concept-defining rules for a logical
concept, and showing that they provide a unique characterization, in the sense that the parallel formulation of the rules with respect to two different languages will define two logically equivalent concepts106.
Let us briefly rehearse Williamson’s argument for the unique characterization of the universal quantifier, before moving on to consider the implications of the argument for the plural reading of the quantifiers in (m).
The argument’s structure is relatively simple. Take the universal quantifier ∀ as
defined, relative to a language L, by the standard ∀-introduction and ∀-elimination
rules (recall the formulation of the rules given in Section 1). Consider also the quantifier ∀*, defined with respect to a language L* by the parallel ∀*-introduction
and ∀*-elimination rules. Assume that the logical vocabulary of the two languages
106 It is not obvious that logical equivalence yields sameness of commitments – for the latter seems to require an additional epistemic component (i.e. our grasp of the logical equivalence). It is, however, certainly a necessary condition – hence the qualification at the beginning of the sentence.
coincides. Assume also that the logical commitments of a speaker S of language L and of a speaker S* of language L* are open-ended: that is, both speakers take the relevant concept-defining rules to hold independently of any expansion of the respective language (in Williamson’s terms, both speakers have a disposition to accept instances of the rules in any extension of the language). Now merge the two languages L and L*, to obtain a new language L+L* whose primitive vocabulary is the union of the primitive vocabularies of the two original languages. Given the pooled commitments of the two speakers, it is possible to show that ∀ and ∀* are logically equivalent. For consider a
formula A of L+L* in which the constant letter t does not occur and no variable except v occurs free (once again, recall the role of t and v in the formulation of ∀-introduction
and ∀-elimination in Section 1). Reasoning in L+L*, from ∀vA one can deduce A(t/v)
by ∀-elimination. As t does not occur in the premise and no occurrence of v becomes