deploy both in formal and in informal reasoning practices;
• Talk of the basic rules of inference that correctly abstract from concept-constituting usages is likewise intended not to be restricted to logical practices.
The idea behind the qualifications above is, then, that a correct formulation of those rules should allows us to capture the concept’s inferential role in fully general terms. This leaves it open, of course, how exactly we should articulate the relation between formal and informal reasoning practice – a problem to which I pointed in Chapter II.
interpreted usage in context; and that talk of our general usage of the concept is not reducible to talk of contextually interpreted usages77.
The theoretical counterpart of the data that the considerations above intend to individuate is the following. If the reconstructive project aims at rendering what the concept of universal quantification consists in by means of an appeal to rules of inference, then it should regard the latter as consisting in an inferential instruction that is fully general with respect to specific interpretative contexts, and that should be understood by the subjects who are presented with the rules in such general terms78.
As we shall see in a moment, these observations suggest some natural constraints on an account of the concept of universal quantification.
Preliminary Discussion of ii)
Call a rule, or a set of rules, that correctly abstracts from the constitutive practices for a logical concept C, a concept-defining rule for C.
A widely accepted, although by no means obvious, view is that we should take the concept-defining rules for the universal quantifier to be its standard introduction and elimination rules. To say that the view is by no means obvious is to say the following: that two inferentialist accounts of the logical concepts may agree on what kind of things constitute the concept, on what it is to possess it, and still substantively disagree on which rules correctly display the form of the relevant concept-constituting practice.
Here, however, I want to grant that the accepted view is the most plausible one, at least for the concept of universal quantification. So, the idea that we start with is that the standard introduction and elimination rules for the universal quantifier correctly abstract from the relevant concept-constituting practices.
The first issue that we need to tackle is the following.
77 To say that this is the upshot prima facie is to say that some argumentative and interpretative work is required to establish that this is really what the data show. A discussion of this point will be presented in Chapter IV.
78 In virtue of this, talk of rules in the present discussion has to be understood as talk of inferential instructions, rather than as talk of statements or formulations of the rules in a specific language.
Introduction and elimination rules for a logical concept are normally viewed as the meta-linguistic means to define the correct inferential usage of the concept in the relevant object language.
Consider the standard formulation of ∀-introduction and of ∀-elimination (for
a first order universal quantifier):
∀-introduction: Given a deduction of A from some premises, one may deduce ∀vA(v/t)
from the same premises, where A(v/t) is the result of replacing all occurrences of the individual constant t in the formula A by the individual variable v, provided that no such occurrences of v is bound in A(v/t) and that t occurs in none of the premises.
∀-elimination: From ∀vA one may deduce A(t/v), where A(t/v) is the result of replacing
all free occurrences of the individual variable v in the formula A by the individual constant t79.
Given a formulation such as the one above, one will then provide a semantics in the meta-language. Among the questions that the meta-linguistic interpretation of the rules will try to answer are the following:
a) Should we take A and t in the statements of the rules above to be schematic letters?
b) Should we assume that there are non-trivial semantic constraints on any of the expressions that appear in the rules?
c) In particular, as an example of a semantic constraint, should we make the assumption that the universal quantifier that figures in the statement of the rules ranges over a domain of discourse, and that what is named by t is an object in such a domain?
Etc.
These questions are, then, normally regarded as simply concerning what should be the correct interpretation of the rules in a meta-language, where the correctness of the interpretation will be assessed partly against the purposes and structure of the object language and partly against one’s semantic views and available resources.
The project that constitutes the framework of the present discussion, however, is not simply to determine which interpretation of a rule-based definition of a logical concept will best serve the purposes of an object-language, relative to which the rules are given, or which one will best implement one’s model-theoretic views. In virtue of the status of concept-defining rules in the project that I have labeled as reconstructive, the rule-based definition of the concept of universal quantification has to respond, for its correctness, to the actual features of our concept-constituting practices. More specifically, in the light of the discussion in Chapter II, it has to respond to these practices’ intuitive epistemic features.
The significance, for this project, of questions such as a) to c) above, then, has to be understood in terms of the project’s more ambitious perspective – the answers that one chooses to give to the questions will concern what we take to be the relevant features of our concept-constituting practices. In particular, these will intuitively have to do with what we take to be the scope and epistemic quality of the inferential
commitments that we undertake when we use the concept of universal quantification. To this re-interpretation of questions such as a)-c) I will simply refer, for to ease the exposition, as to what we should take a concept-defining rule (or set of rules) to display.
In the rest of the chapter, talk of the options that we have for rendering what a rule displays, or for how we should understand a rule, is thus to be understood within the framework of such a re-interpretation.
1.2 The Three Constraints
Two questions that immediately arise are:
• Which feature or features of our concept-constituting usages of the universal
• Which intuitive constraints should the reconstruction obey?
The two questions partly overlap, as the most intuitive constraint on the
reconstruction is, naturally, that it should render precisely what we take to be relevant features of our concept-constituting practices.
In what follows, I will spell out one such feature, while attempting to provide an answer to the second question.
The following seem to me to be plausible constraints on a reconstruction of the concept-constituting usage of Universal Quantification that takes the quantifier’s introduction and elimination rules as concept-defining80.
The Semantic Contraint: The rules should be understood in such a way as to allow for a semantic treatment of the universal quantifier consistent with the basic inferentialist assumption that what counts as the correct semantics for a concept should be determined by its inferential use.
The Epistemic Constraint (open-endedness): the rules should be understood as displaying the fact that our inferential commitments to the canonical consequences of the concept are open-ended.
The Anti-Relativist Constraint: The way in which the rules are understood should not make the reconstruction, or the very inferential practices that are its subject matter, obviously open to relativist threats81.
80 The constraints have different statuses. The first constraint is motivated by the general
assumptions of an inferentialist account of the logical concepts, and ultimately bears on the account’s internal coherence. The second has to do with the features of our usages of universal quantification that we want the account to render, that is: with the object of the account. The third is a methodological constraint on which I will say more in a moment.
81 The suggestion that we should formulate the ability to counter relativism about the logical concepts as a constraint on the account may raise a suspicion of ad-hocness. I have, however, a good reason for formulating this constraint.
The Semantic Constraint
I take it that the semantic constraint is plausible and basic enough not to need much discussion. It can be expressed in different ways, for example in terms of the idea that a concept’s inferential usage contributes to determine, but is not determined by, the concept’s semantic properties. A way to make this requirement more precise is the following.
On the basis of the assumption that what the logical concepts are constituted by their inferential role, and given that a concept-defining rule of inference displays what such a role consists in in its most general form (that is: independently of any reference to specific interpretations of the concept in a given language), then our understanding of the rule must be consistent with the idea that the semantics of its instances is to be constrained by the commitments that the rule displays82.
It is useful, in this framework, to think of a concept-defining rule of inference as capturing the assertibility conditions of statements in which the relevant logical
concept figures as the main logical operator. Concept-defining rules, then, tell us what counts as a ground for asserting such a statement, and what is the form of the
consequences to which we commit ourselves when asserting the premises of the rule.
The reason is this. Recent philosophical discussions [e.g. McGee 2000; Lavine 2006; Rayo & Uzquiano 2006] have renewed our attention to the fact that the case of the universal quantifier is somewhat special among the logical concepts. This is because standard semantic treatments of the concept seem to generate:
• The well-known semantic and mathematical paradoxes;
• A special case of semantic and epistemic indeterminacy for at least a sub-class of interpreted usages of the concept.
The concern with relativism, in this framework, arises from the second difficulty above – from the recognition, that is, that the threat of relativism appears to affect the concept of universal quantification in a way that is more radical, and potentially more interesting, than for the other logical concepts. This, I hope, will be clearer from the discussion of the constraint itself.
82 The constraint concerns the relation of determination between (the epistemic features of) our inferential commitments and the semantic value of an instance of the concept as this is (contextually) determined in an interpreted language. The idea is then that of a consistency between what we take the rules to display and the direction of the determination.
Because concept-defining rules capture assertibility conditions, they provide us with a criterion for assessing the correctness of the semantics that the relevant statements (that is: interpreted usages of the concepts that the rules define) receive. Simply put, the ascription of truth-conditions (and thus of a truth-value) to such statements must be consistent with our grasp of the circumstances in which they count as established, and of the circumstances in which their consequences do. To say that a concept-defining rule captures the assertibility conditions for a class of statements (i.e. the statements in which the concept defined figures as the main logical operator) is to say that it displays what the general form of these circumstances is.
In what follows, talk of a rule ‘determining’ the semantics of a statement, or of an interpreted instance of the concept that the rule defines, must thus be understood in this sense.
The Epistemic Constraint: Open-Endedness
The implicit idea behind the epistemic constraint is that there should be a match between the epistemic properties of our concept-constituting practices on the one hand, and what we should take the rules of ∀-introduction and ∀-elimination to
display.
The explicit suggestion is that one such epistemic property consists in the open- endedness of our inferential commitments. Intuitively, when we say that a subject’s commitments to inferring X from Y are open-ended, we intend to express the independence of such commitments from actual and potential variations in the linguistic and semantic circumstances in which her commitments are undertaken.
For the sake of simplicity, in the exposition that follows we may replace talk of concept-constituting commitments with talk of commitments to the concept-defining rules of inference – with the proviso that the object of the commitments in question is given by the inferences whose form is displayed by the rules, rather than by the rules themselves. With this simplification in mind, we can informally express the idea that a commitment to a rule of inference is open-ended as the idea of its independence from:
• The possible or actual expansion of the language via the introduction of non-
logical vocabulary;
• Any change in the meaning of the non-logical vocabulary of the language;
• The acquisition of new theories, or the fact that existing ones are subject to
change.
This idea has often been suggested as a way to characterize our general attitude to (basic) logical principles [Williamson 2003, 2006: 377; McGee 2006: 187; Lavine 2006: 113 ff]. In all the relevant accounts, however, the notion of open-endedness deployed is intended:
• To be an intuitive notion;
• To capture the epistemic quality of a subject’s acceptance of basic logical rules,
rather than of her inferential commitments.
What we need is, then, a clarification of what we want the notion to capture, and a more precise rendering the role that it is supposed to play within the theoretical framework of the current discussion. We need, in other words, the following two things.
1. A clarification of the notion of ‘accepting a rule’. When we say that a subject accepts a basic rule of inference we can mean any of the following:
i) That she believes the rule to be valid (or: understands the rule as a valid rule of inference), that is: truth-preserving under any interpretation of the non-logical vocabulary;
iii) That she explicitly regards the rule as defining a logical concept, and that, as a result of becoming acquainted with the rule, she thereby infers in the way licensed by it (i.e. her inferential usage of the concept consists in performing inferences that are licensed by the rules, in virtue of the fact that she takes the rules to be concept- defining);
iv) That the rule is concept-defining, in the sense that it is the correct
reconstruction of a subject’s inferential commitments, so that it displays the (form of) the subject’s inferential usage of the concept.
In the light of what I take the status of a concept-defining rule to be, in the
framework of this discussion the natural option is the last one. The idea of a subject’s accepting a rule then simply becomes the idea of a match between the commitments that she undertakes when using the concept, on the one hand, and the instruction, in which the rule consists, to infer in a certain way. Simply put: in this view, accepting a rule simply means undertaking, in the relevant circumstances, the inferential
commitments of which the rule displays the general form.
This way of rendering the notion has the independent advantage of not requiring or presupposing any theoretical beliefs or vocabulary on the part of a subject who accepts the rule. Indeed, this seems to match our intuitions about a natural use of the expression ‘subject S accepts rule R’: when S infers, says, that a particular dog barks from the premise that all dogs do, and her usage of the concept ‘all’ suggests that she normally infers to conclusions of the same form from premises of the same form as the ones displayed by the rule, we are naturally disposed to attribute acceptance of the rule to her irrespective of any other consideration about her skills, beliefs or knowledge. 2) A clarification of the view that our commitments to a rule of inference are open- ended, that is: a refinement of our grasp of the notion of ‘any expansion of a language’. What is it, exactly, that we mean when we say that it should display the fact that our commitments will continue to hold in any expansion of the language?
The current philosophical debate on this issue, it seems to me, can be framed in terms of two main theoretical alternatives and one general strategy that underlies them both. The two alternatives consist in saying, respectively:
i) That the concept of ‘any’ is best rendered in terms of non-quantificational generality, and that the concepts denoted by the two expressions ‘any’ and ‘all’ are not reducible to each other;
ii) That the concept denoted by the expression ‘any’ can, with certain provisos, ultimately be rendered in terms of the concept denoted by ‘all’; in particular, that the semantics for a usage of ‘any’ in the object-language can be given, in a meta- language, in terms of quantificational generality.
Proponents of both alternatives normally appeal to notions such as the ones of schematicity or of systematic ambiguity, and use the notions to establish a distinction, between ‘any’ and ‘all’, typically based on the presence or absence of semantic
constraints on the commitments that we undertake when deploying the concepts denoted by the two expressions83.
Proponents of the second alternative do not normally argue for an explicit reduction of one concept to the other; rather, in the attempt to formulate a plausible semantics for the concept of ‘any’, they end up committing themselves to the idea that the concept should ultimately be interpreted in terms that are very similar to, or presuppose, our understanding of the generality expressed by the universal quantifier84. It is this commitment that proponents of the first alternative, such as
[Glanzberg 2000; Lavine 2006] do not share.
83 Examples include: [Glanzberg 2006], [Lavine 2006], [Williamson 2003, 2006], [McGee