Any comments, critiques or criticisms are much appreciated.

"Necessity Saved"

J. T. Allen

INTRODUCTION

The aim of the present inquiry is not to wash necessity of all controversy; rather, it is to save it from a particular criticism headed by Quine (1953). Saving necessity, thus, lends confirmation, not affirmation.

Quine's complaint over necessity is that the terms (or contexts) in which the putative concept occurs are nonextensional and cannot, therefore, be appended to quantifier logic. Assuredly, if necessity-contexts were nonextensional, this would be a major blow to the logic of necessity-terms (so-called modal logic); for without the auxiliary resources of quantification theory (including identity), putatively permissible sentences, for instance, '(x)(it is possible that x is a person -> it is possible that x is rational)', would not be well-formed formulae in virtue of the fact that quantification rules could not be applied to them. Perhaps this latter charge indisputable or, at any rate, plausible. The issue I wish to raise, however, regards to the former objection that necessity-contexts are nonextensional. In particular, I wish to show that Quine's grounds for this charge are unsound.

QUINE'S ARGUMENT FOR NECESSITY-CONTEXTS' NONEXTENSIONALITY

The underlying principle for extensionality of a context is the following:

(E) A context is extensional if and only if two identical terms (each of which transform the context into a complete, declarative sentence S) can be substituted for each other without changing the truth-value of S.

When a context fails to satisfy (E), it is said to be referentially opaque. When a context, on the other hand, satisfies (E), it is said to be purely referential (or referentially transparent). Quine introduces (E) in order to account for failure of Leibniz's law in certain contexts such as 'has eight letters,' 'is so-called because,' 'believes that,' 'sees that,' etc. Leibniz's law, in its most unsophisticated form, states that if a predicate is true of an object a, then whatever that object is identical to, say, b, the predicate is also true of b.[1]

(LL) Fa

a = b

Thus, Fb

There are contexts (or predicates), however, that, when conjoined with a name for an object such that the atomic sentence is true, and further conjoined with a true identity statement of the name of the object, the resulting sentence, given (LL), is false. Thus, (LL) would appear to be an invalid rule of inference. (E) is meant to sharpen (LL) in order to preserve it against its putative counterexamples. Take, for instance, the true statement

1. Weezer is so-called because of his asthma condition.

Conjoined with the following true identity statement:

2. Weezer = Rivers Cuomo,

the following false statement follows in virtue of (LL):

3. Rivers Cuomo is so-called because of his asthma condition.

Quine's response to such a failure of (LL) regarding the two terms 'Weezer' and 'Rivers Cuomo':

"[R]eveals merely that the occurrence to be supplanted is not purely referential, that is, that the statement depends not only on the object but on the form of the name. For it is clear that whatever can be affirmed about the object remains true when we refer to the object by any other name" (1953).

In other words, (1) is indeed true, but it also depends on the form of the name, not merely on the object in question which is referred to by its name. The name 'Weezer' was so chosen specifically because of a happenstance linguistic feature of the object in question, as (1) says; this gives us information not only about the object, but the object's name as well. In such cases, (LL) is inapplicable. In fact, not only (LL), but other logical rules (or sets thereof) such as quantification theory are equally inapplicable. The blameworthy component is the context, not (LL) or quantification theory. For these rules are to be applied only to statements that ascribe properties to their objects, not their names as well.

Having shown where (E) succeeds in cases that we are fairly content with agreeing with the results, Quine moves on to a more controversial issue: necessity-contexts. Consider the putatively true statement (in modal logic):

4. 9 is necessarily greater than 7.

When conjoined with the following true identity statement:

5. The number of planets = 9,

The following statement follows:

6. The number of planets is necessarily greater than 7,

an outright falsity (in modal logic).[2] Following (E), Quine concludes that necessity-contexts are nonextensional, just as the context 'is so-called because' is.

THE PROBLEM WITH QUINE'S CRITERION OF EXTENSIONALITY

As suggested in the second footnote of this essay, one could argue with Quine that (6) is true insofar as what we mean by 'the number of planets' is just '9', and nothing more. Quine objects to equating meaning with extension (1951), however; but it nonetheless would satisfy Quine's skepticism over abstract entities meant to serve as 'meanings' of terms, above-and-beyond their extensions. Instead of taking such a route, I will instead pursue an objection to (E), Quine's criterion that, when conjoined with (4)–(6), implies necessity-contexts are nonextensional (and thus, unable to be appended to other logics, including quantification theory). To show this, it is only required that one context independently established as extensional, in conjunction with a true identity statement (indeed, one that Quine explicitly uses to show the nonextensionality of necessity-contexts), implies a false (or meaningless) conclusion.

The counterexample to (E) that I wish to focus on begins with the following sentence, involving a purely referential context:

7. It is true that there are nine planets.

Indeed, such a context in (7) is purely referential. (7) does not depend on the forms of its names for its objects; it depends on the fact of there being a number of planets equal to nine—not on the special linguistic fashion by which the objects in question are referred to. As conspicuous as this seems, (7), conjoined with (LL), and the true identity statement, (5), 'the number of planets = 9', entails the following statement:

8. It is true that there are the number of planets planets,

which is worse off than being false: it is meaningless. Such a sentence as (8) is as grammatically perverse as is the sentence "There are dogs dogs".

The problem that this counterexample unearths is the following dilemma: either (i) (E) is inadequate, in which case Quine's argument against necessity isn't sound; or (ii) (5) is false, in which case, Quine's argument against necessity-contexts isn't sound. Therefore, the counterexample destroys Quine's argument against necessity-contexts, so long as it is affirmed that (7) is a true sentence that is formed out of replacing names for schemata in a purely referential context, and (8) is meaningless. Confessedly, however, this dilemma is deceptive. It is surely less costly for Quine to deny the truth of (5) than it is for him to deal with (E) being an inadequate criterion of extensionality. This is due to the fact that Quine has other arguments against necessity-contexts that involve true identity statements, e.g. 'Morning Star = Evening Star', that do not seem to lead to the same type of counterexamples as above. Quine can merely drop his planets-argument and settle for others. But, this, I fear, lets Quine off too easily. It is still unexplained why (5) and (7) lead to a meaningless conclusion, when (7) appears referentially transparent and (5) appears to be true. Quineans skeptical of necessity need to explain why at least one of these appearances are delusive, thus, preventing (LL) to successfully apply to them.

NOTES:

[1] Here, we can treat "predicate" and "context" as the same. Both predicates and contexts do not express complete sentences, but do so when the schemata or variables are substituted for objects. To avoid confusion, however, (LL) is given in terms of predicates, as it usually is, and not of contexts, though doing otherwise would not commit anyone to any serious logical error.

[2] Note, however, that (6) is only false insofar as 'the number of planets' means something over-and-above its extension. That is to say, if a modal logician ascribes to a theory of meaning that equates the meaning of a term and its referent (or its extension), then (6) is true since both 'the number of planets' and '9' refer to the same abstract mathematical object. If it be objected that it is possible (to the modal logician) that 'the number of planets = 7', then this only implies that 'the number of planets' can have variant meanings in response to observed data (outside the scope of mere behavioral data of language users). Such a logician would say that (6) is true insofar as what is meant by 'the number of planets' is also what is meant by '9'; if the 'the number of planets' becomes '7', then we say 'the number of planets' has changed its meaning, and thus, as expected, the truth-value of (6); something no logician would object to, viz. changing the meaning of a term in a context makes it vulnerable to truth-value change.

REFERENCES:

Carnap, Rudolf. Introduction to Semantics and Formalization of Logic: Two Volumes in One.

1942. N.p.: Harvard University Press, 1943.

Quine, Willard Van Orman. "Two Dogmas of Empiricism." The Philosophical Review 60

(1951): 20-43. Rpt. In From a Logical Point of View. 2nd ed. Cambridge: Harvard University Press, 1980.

---. "Reference and Modality". 1953. In From a Logical Point of View. 2nd ed. Cambridge:

Harvard University Press, 1980.