Optional: W. V. O. Quine, "Reference and Modality", in From a Logical Point of View (New York: Harper and Row, 1953), pp. 139-59 (DjVu)
Quine is interested in the question to what extent we can make proper sense of the notion of necessity, and what philosophical commitments we must make to do so. And this question is tied up, for him, with the question what our logic of necessity should be like.
There are, Quine says, three ways we might treat necessity and contingency logically.
- We may regard them as properties of sentences.
- We may regard them as "statement operators", attaching to statments to make other statments.
- We may regard them as "sentence operators", attaching, like negation, to a formula to make another formula.
The difference between the second and third is that, in the latter case, the formula may contain free variables, which may later be bound by quantifiers.
Quine's overall point may be summarized this way. Step (1) is relatively harmless. We do this in logic anyway when we speak of a sentence as being valid. Step (2) is mostly harmless, and is completely harmless if understood as a notational variant of step (1). But it is dangerous in so far as it encourages one to move to Step (3), which Quine regards as philosophically suspect.
In section I, Quine introduces the notions of purely referential occurrence and of referential opacity. An occurrence of a term is purely referential if one can substitute any other name of the same object without change of truth-value. A 'context' into which one can substitute a sentence is opaque if occurrences of terms in that context are not purely referential. Quine notes that "Quotation is the referentially opaque context par excellence" (p. 159). A context is truth-functional if substitution of sentences with the same truth-value preserves the truth-value of the whole.
On pp. 161-2, Quine presents an argument known as the Slingshot. (It has antecedents in Frege but probably first appears in Alonzo Church.) The argument purports to show that any context that is (i) non-opaque and (ii) permits substitution of logical equivalents, salva veritate, is also truth-functional.
The argument is extremely simple, but Quine's notation may be confusing, so here it is in more modern notation. Suppose that p and q have the same truth-value. Observe that:
- The statement "The x such that (x = Venus & p) or (x = Mars & not-p) = Venus" is logically equivalent to p.
- The statement "The x such that (x = Venus & q) or (x = Mars & not-q) = Venus" is logically equivalent to q.
- The two terms "The x such that (x = Venus & p) or (x = Mars & not-q)" and "The x such that (x = Venus & q) or (x = Mars & not-q)" refer to the same thing: Venus, if p and q are both true; Mars if p and q are both false.
Now suppose that F(p) is a context satisfying assumptions (i) and (ii), Then the following must all have the same truth-value:
- F(the x such that (x = Venus & p) or (x = Mars & not-p) = Venus)
- F(the x such that (x = Venus & q) or (x = Mars & not-q) = Venus)
(i) and (ii) are equivalent by (a); (ii) and (iii) by (c); (iii) and (iv) by (b).
What do you think Russell might say about the Slingshot?
In the context of the paper, the primary lesson of this argument is supposed to be that "quantifying into" referentially opaque contexts is a troublesome matter, to which we shall come shortly.
In section II, Quine notes that we need necessity-like notions in logic (validity, implication), but that these are properly understood as properties of or relations between sentences. He then argues that, with sufficient care taken about quotation, the use of necessity as a statement operator (attaching only to sentences, i.e., formulae with no free variables) can be explained in terms of the `semantical' use: "nec(9 > 5)" just means: Nec('9 > 5').
On the other hand, however, Quine notes that iterated modalities, which are characteristic of modal logic, are not easy to make sense of on this approach. The translation works, but it is not clear what the translated sentences really mean. Quine thinks, then, that, if we explain necessity as logical necessity, that does not really suffice to explain the use of "nec" as a statement operator. In that regard, then, he thinks that even the second grade of modal involvement may not be well-motivated.
In section III, then, Quine turns to the use of "nec" as a sentence operator. With this, we allow such constructs as "(∃x)(nec(x > 5))", meaning: something is necessarily greater than 5.
Quine notes first that, with this move, we make the translation of "nec" to "Nec" impossible, since the variable would have to appear within quotes. A more serious issue, however, is that the context "nec(...)" appears to be referentially opaque, since "nec(9 > 5)" looks true and "nec(the number of planets > 5)" looks false. If so, however, then, as Quine says, "nec(x > 5)" simply isn't true or false of an object at all. (As he tends to ask elsewhere: Which thing is it that is necessarily greater than 5? Is it 9? I.e., the number of planets?) If so, then the value assigned to "x" can't simply be an object.
Articulate this point of Quine's a bit: If we assign "x" the value 9, then is "nec(x > 5)" true or false? Why is the question difficult to answer?
The response Quine envisages is that "nec(the number of planets > 5)" is ambiguous as to the scope of the description.
Explain Quine's point using Russell's notions of primary and secondary occurrence.
Quine notes on p. 173 that, if we are going to quantify into "nec", then we need to be careful about how we apply universal instantiation. Can we formulate the required restriction using Russell's notions of primary and secondary occurrence, again? Why or why not?
Quine concludes by arguing that, if we accept quantification into "nec", then we are committed to "Aristotelian essentialism", i.e., to the doctrine that there is a distinction to be drawn between properties of an object that are essential to it and those that are merely accidental. In particular, Quine notes, it looks as if we are committed to the claim that "being greater than 5" is a property that 9 has essentially, but "being the number of planets" is a property it has only accidentally.
Quine does not say why he regards Aristotelian essentialism as problematic. What sorts of worries do you think he might have?
If we decide to quantify into "nec", then "nec(...)" is a context that is not referentially opaque and, presumably, allows substitution of logical equivalents salva veritate. Why doesn't the Slingshot then imply that "nec(...)" has to be truth-functional? (Hint: Russell again.)