PROSPECTIVAS DEL MODELO

We have already established the logical deviance of (a), (b) and (c) based on their non-extensionality, non-classicality and incompleteness (among other things) above. In this section I wish to discuss the logical deviance of plural logic, i.e. why it cannot argue on the basis of (2) that it should be included in canonical notation and why it, ultimately, fails as a vindication of _(weak.Q).

We described the business of the logician above as consisting in the reformulation of grammar in order to resolve structural ambiguity and economize on constructions to devise a simple and concise framework for all of science (Quine 2008e, 307). In doing so the logician treated relations, such as “taller than”, as a single morpheme, rather than a predicate ‘tall’ combined with the grammatical particles ‘-er’ and ‘than’ to form “taller than”. By doing so he repudiated the logicality of implications based on the asymmetry and transitivity of the ‘taller than’-relation, for treating it as a whole means that whatever we can infer from ‘something being taller than something else’ depends, to a large part, on the meaning we ascribe to it, rather than the grammatical structure of the sentence itself as provided by the potential particles ‘-er’ and ‘than’ (Quine 2008e, 307).152 The choice of analyzing an expression like ‘taller than’ in this way bore the advantage of simplifying grammar in that we did not have to distinguish between a class or relations to which ‘-er’ and ‘than’ are applicable and a class to which they are not – it enables a unified treatment of predication in our canonical grammar. For similar reasons of simplicity and convenience it was chosen to consider all predicates (except for identity which presents a special case which we will consider in Section 5.3.1) as extra-logical. This bore the advantage of providing a nice definition of logical truth, truth in virtue of grammatical structure, i.e. truth under all lexical substitutions (see above) and a good heuristic as to which truths genuinely belong to a particular science and theory and which are true in virtue of logic. Those that are in full generality statable in the object-language itself belong to the respective science and those that require semantic ascent for their adequate formulation are logical in nature. In order to state logical truths in their full generality one needed to quantify over sentences and therefore advance to a meta-level, since the only kind of generality statable in the object language is in terms of quantification over objects, not, however, over predicates and relations, something necessary to quantify over when talking about lexical substitutions (Quine 1986, 102).153 However, with the introduction oflogical predicates both of these crucial characteristics of logic fail. On the one hand, logical truths become statable in the object language (e.g. (*) ∀x∃yy(x < y)) and the definition of logical truth in terms of lexical substitution fails since sentences like (*) do not remain true under all substitutions of predicates for predicates, i.e. lexical items for lexical items. While this is certainly not sufficient to disqualify< as logical (after all, there is no firm boundary demarcating logic that

152

Cf. (Quine 2008e, 307): “Relative to grammar as thus revised, the asymmetry and transitivity of ‘taller than’ cease to count as logical implication; for they are not reflected in the new grammatical structure.”

153

Chapter 4. Canonical Notation 85

we need to adhere to) it casts doubt on its status and makes us wonder why one should admit it uncritically into logic, which is considered to be a theory of utmost generality. What qualifies <

as belonging to such general theory, i.e. what makes it logical? Similar considerations as the ones pertaining to the logicality of<obviously also pertain to identity,=, which we do want to qualify as logical. We will return to this in Section 5.3.1 where we will present an argument as to why identity should nevertheless be counted to logic and will reconsider the status of<. For now, however, note that no independent argument as to why < should belong to the realm of logic is given and that this, together with the fact that it appears to violate some of the facts about logic grounding its generality leaves it doubtful whether it genuinely belongs into the category of particles rather than the lexicon. Plural logic, on this count, appears to introduce a non-logical and thereby possibly non topic-neutral element into the language of canonical notation.

The by far more significant issue with plural logic concerns the status of its plural variables xx. Quantification, at least on a strict Quinean understanding, is only possible into name position.154

But the semantic values ofxxare supposed to be pluralities, collections of objects that by themselves are not again treated as objects and are thus not in any straightforward sense nameable. Given then that we are not quantifying into name position in plural logic, i.e. not quantifying over objects in the sense that no unique object needs to be assigned as a value to a plural variable and thus no unique determinable reference can be made out, but only somewhat indeterminate reference to objects155 it is not clear whether such quantification should still qualify as genuine quantification in the same sense as our quantification by means of individual variables or rather as something else that compares to our quantification by means of partial analogy, nothing more.

Even if we were able to make sense of this problematic kind of ‘quantification’ which appears to be very different from the singular quantification we are used to, we find ourselves confronted by an even more serious problem: in the formal framework set out for plural logic by Rayo and Boolos it is possible to make do completely without individual variables. We can reduce individual variables to plural variables by means of the following definition (Rayo 2002, 22): let 1(xx) abbreviate

∀yy(yy ≤ xx → xx ≤ yy), where xx ≤ yy stands for ‘they (xx) are some of them (yy)’. We can then define quantification over individual variables as follows: ∃xϕ ≡_def ∃xx(1(xx)∧ϕ) and similarly for the other locutions (Rayo 2002, 22). Individual variables are superfluous in plural logic. But then, given the fundamental status we assigned to the individual variable, what plural logic appears to advocate here is not an emendation of the criterion, but an altogether different understanding of it, not based on reference to individual objects and thereby radically different from our understanding of ontology in terms of reference to to objects. The point becomes clearer when one realizes that, given the reduction of individual variables to plural variables, the restriction of the criterion of ontological commitment to individual variables does not make much sense any more. One can then either modify the criterion to talk about the values of plural variables, by saying that an expression is committed to hose objects that fall under the plural variables/that the plural variables range over,156 or restrict it to certain sentences in every theory (namely those involving 1(xx) above) and say that it is the objects that constitute the pluralities quantified over in the respective sentences which make up the ontology of a theory. In the latter case a proper,

154

Boolos (Boolos 1975) disagrees with this narrow conception of quantification.

155_{Plural quantification still quantifies over objects, of course. The entire point of plural logic was that it quantifies}

over objects, but by means of quantifying over pluralities of them, (potentially) multiple objects in the same quan- tification, at the same time. It might be helpful to think about standard first-order quantification as quantifying over objects in an indefinite sense, indifferent between several objects satisfying the respective condition of the quantifi- cation, but nevertheless referring to one of them, no matter which one, whereas plural quantification quantifies over objects in an indeterminate way, not determining its reference as reference to any particular object at all.

156

non-circular justification would have to be given for why we should restrict ourselves to a certain class of sentences rather than looking for particular expressions when determining the ontological commitment of a theory (what and why is it that makes particular sentences talk about objects and others not), in the former case we are not dealing with a vindication for (weak.Q) anymore, but with a proposal for_(strong.Q). In either case, however, plural logic appears to advance a concept of existence based on different considerations than our original account, thereby breaking with the Quinean grounding of its legitimacy.