Chapter XII: Quantification and Opacity
Ali Akhtar Kazmi
In a 1943 paper, Quine says: "No pronoun (or variable of quantification) within an opaque context can refer back to an antecedent (or quantifier) prior to that context."
The view articulated in these words, which I shall describe as Quine’s Thesis, occupies a central position in many of his later papers, and informs large parts of Chapters IV, V, and VI of Word and Object.
Quine’s thesis should be distinguished from his misgivings about the intelligibility of essentialism, the doctrine that, among the traits of an object some are essential and others are not, to which he thinks quantified modal logic to be committed. Likewise, Quine’s thesis should be distinguished from his more recent doubts about the intelligibility of certain epistemological doctrines to which he thinks the quantified logic of belief to be committed. Quine’s thesis is about quantification in general. It seems to be his view that quantification into opaque constructions faces a purely technical difficulty can be established on the basis of logical and semantic considerations alone. Thus, in "Quantifiers and Propositional Attitudes", after distin-guishing what he calls "the relational senses" of propositional attitudes from their corresponding "notional senses", he notes:
However, the suggested formulations of the relational sense 3/4 viz., ( x) (x is a lion. Eanest strives that Ernest finds x)
( x) (x is a sloop. I wish that I have x)
( x) (Ralph believes that x is a spy)
( x) (Witold wishes that x is president)
all involve quantifying into a propositional attitude idiom from outside. This is a dubious business.
The rest of that paper is an attempt to offer a reconstruction of the relational senses of propositional attitudes which does not involve quantifying into opaque constructions. Similarly, in "Inten-tions Revisited", after noting that quantified modal logic also involves the allegedly illicit quantifying into opaque constructions, Quine offers a reconstruction of quantified modal discourse which is free from this alleged defect.
In Section 1.1 of this paper, I present a general characteri-zation of referential opacity, and contrast it with the notion of a purely referential occurrence of a singular term. In Section 1.2 and 1.3, I examine two lines of argument in defence of Quine’s Thesis, and, argue that they fail to establish it.
1.1
Quine characterizes referential opacity in terms of a principle that h describes as "the principle of substitutivity". Quine formulates this principle in these words: "... given a true statement of identity, one of its two terms may be substituted for the other in any true statement nad the result will be true."
Making allowarances for Quine’s use of the word "statement", this principle may be understood as the claim that:
(A) for all expressions and ß, if, relative to an assignment I, = ß expresses a true proposition, then, for any sentences S andS’, if S contains an occurrence of , and S’ is the result of substituting ß for some occurrence of in S, then, relative to the assignment I, S expresses a true proposition, only if, relative to I, S’ expresses a true proposition.
It should be recognized, as Quine has frequently stressed, that (A) is false. For example, the propositions expressed by (1) ‘Giorgione’ = Barbarelli, and (2) ‘Giorgione’ contains nine letters, are true, whereas, (3) ‘Barbarelli’ contains nine letters, expresses a false proposition. Similarly, (4) Giorgione was so-called because of his size, expresses a true proposition, but (5) Barbarelli was so-called because of his size, does not expresses a true proposition. Counterexamples to (A) are not confined to those cases which involve substitution within contexts of quotation. For instance, though:
(6) 9 = the number of planets,and (7) It is necessary that 9 is odd, both express true propositions, (8) It is necessary that the number of planets is odd, expresses a false proposition.
The temptation to think that (A) is true might arise from a failure to distinguish (A) from the principle that:
(B) The universal closure of every instance of the schema (9) ( x) ( y) (x=y (Fx Fy))
or a notational variant of (9), expresses a true proposition.
But, notice that whereas the proposition that (1) and (4) express true propositions, and (5) fails to expresses a true proposition falsifies (A), it does not falsify (B); and, therefore, (B) does not entail (A).
Quine argues in defense of (B) as follows:
(B) does have the air of a law; one feels that any interpretation of "Fx" violating (B) would be simply a distortion of the manifest intent of `Fx’. Anyway I hope one feels this, for there is good reason to. Since there is no quantifying into an opaque construction, the position of `x’ and `y’ in `Fx’ and `Fy’ must be referential if `x’ and `y’ in those positions are to be bound by the initial `( x)’ and `( y)’ of (9) at all. Since the notation of (9) manifestly intends the quantifiers to bind `x’ and `y’ in all four shown places, any interpretation of `Fx’ violating (B) would be a distortion.
Now, even if one were to disagree with the details of Quine’s argument, his conclusion that nay interpretation of ‘Fx’ violating (B) would be a distortion seems indisputable. One is inclined to say that (B) is false, only if there is a sequence and an expression F, such that there is an assignment under which the element of the sequence assigned to the free occurrences of x and y in Fx and Fy respectively are identical; the sequence satisfies Fy but fails to satisfy Fx. But, if there is an assignment under which a sequence satisfies Fx , then, the element of this sequence assigned to the free occurrences of x in Fy , call it ‘a’, is in the extension of F under that assignment. And, if, under that assignment, the element of the sequence assigned to the free occurrences of y in Fy , call it ‘a’’, is identical with a, a’ is in the extension of F under that assignment; and, therefore, under that assignment, the sequence satisfies Fy. Though one is inclined to say this, it is unlikely that this would satisfy critics of (B). For in the claim that if a is identical with a’, then, if a is in the extension of F under an assignment, the a’ is in the extension of F under that assignment, a critic of (B) will see yet another appeal to (B).
Some critics of (B), on the other hand, are likely to argue that (B) is false, because, for example, (i)
(10) ( x) ( y) (x=y (it is necessary that x is odd it is necessary that y is odd))
is an instance of (9), and, (ii) the proposition expressed by (10) is falsified by the proposition expressed by (9)=the number of planets. It is necessary that 9 is odd. (It is necessary that the number of planets is odd)).
An advocate of (B), then, must reject either (i) or (ii). Consider, for instance:
(11) ( x) ( y) (x=y (‘x’ is the 24th letter of the alphabet ‘y’ is the 24th letter of the alphabet)).
There is, presumably, a way of understanding (11) according to which it exereses a false proposition, a proposition which is also expressed by, for instance:
(12) ( w) ( z) (w=z (‘x’ is the 24th letter of the alphabet ‘y’ is the 24th letter of the alphabet)).
But it is unlikely that it would be thought that the proposition that (11) expresses a false proposition falsifies (B). Instead, one is inclined to say that (11) is not really an instance of (9), that given the interpretation of:
‘x’ is the 24th letter of the alphabet, which is required for (11) to express a false proposition, ‘x’ is the 24th letter of the alphabet, is not an open sentence. Now, some advocates of (B) would be inclined to assimilate the case of (10) to that of (11). They would be inclined to say that (10) is not really an instance of (9) either, because, It is necessary that x is odd, is not an intelligible open sentence. However, the apparent intelligibility of such sentences as:
There is something such that it is necessary that is odd, and The number of planets is such that it is necessary that it is odd, suggests, on the contrary, that It is necessary that x is odd, is an intelligible open sentence, and that, therefore, (10) is an instance of (9). But, if (10) is indeed an instance of (9), then, (B) is true only is (ii) is false.
Quine takes the falsity of (A) as evidence that an occurrence of some singular term in a sentence is not purely referential. For instance, in ‘Reference and Modality’, he writes: ‘Failure of substitutivity reveals 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.’
And elsewhere in the same essay, he notes: the failure of substitutivity shows that the occurrence of the personal name in (4) is not purely referential. These remarks indicate that Quine would endorse the following principle".
(C) For any sentence S, any singular term , and any z, z is a purely referential occurrence of in S, only if, for any sentenceS’, and any singular term ß, if S’ is the result of substituting ß for z, in S, and = ß expresses a true proposition then, relative to an assignment I, then relative to the assignment I, S expresses a true proposition, if and only if, relative to I, S’ expresses a true proposition.
Since, (1) and (4) express true propositions, and (5) does not express a true proposition, if (C) is true, then, the occurrence of ‘Giorgione’ in (4), and, the occurrence of ‘Barbarelli’ in (5) are not purely referential. Similarly, since (6) and (7) express true propositions, and (8) does not express a true proposition, if (C) is true, then, the occurrence of ‘9’ and ‘the number of planets’ in (7) and (8) respectively are both not purely referential. It is worth stressing that (C) is a strong principle. If (C) is true, it is also true that:
(D) For any sentences S and S’, any singular terms , and ß, and any z, if z is an occurrence of in S, and S’ is the result of substituting ß for z in S, then if, relative to an assignment I, = ß expresses a true proposition, but, it is not true that, relative to I, S expresses a true proposition if and only if, relative to I, S’ expresses a true proposition, then, z and the corresponding of occurrence of ß in S’ are both not purely referential.
Though there is evidence that Quine would endorse (C), there is also evidence that he does not intend (C) to be taken as part of a definition of ‘a purely referential occurrence of a singular term’, then, we are owed an account of what this expression means. Quine has, at times, described a purely referential occurrence of a singular term in a sentence as an occurrence of a singular term ‘used in a sentence purely to specify its object’.
Quine’s remark that ‘failure of substitutivity reveals 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’, may appear more helpful.
Presumably, the thought is that the only contribution a purely referential occurrence of a singular term in a sentence makes towards determining the truth-value of that sentence is the specification of the object it refers to. One might, then propose to understand a purely referential occurrence of a singular term in a sentence as follows:
(E’) For any sentence S, any non-vacuous singular term , and any z, z is a purely referential occurrence of in S, if and only if, for any S’, if S’ is the result of substituting, for z in S, a variable which does not occur in S, then, relative to any assignment I, S expresses a true proposition, if and only if, relative to I, whatever z refers to satisfies S’.
(E’) accords with some of the remarks in the literature about the concept of a purely referential occurrence of a singular term. If (E’) is true, then, the first occurrence of ‘Giorgione’ in Giorgione was called ‘Giorgione’ because of his size, is purely referential, since, this sentence exereses a true proposition if and only if, for any variable Giorgione satisfies was called ‘Giorgione’ because of his size.
On the other hand, given (E’), the occurrence of ‘Giorgione’ in (4) Giorgione was so-called because of his size, is presumably, not purely referential. We want to say that for any variable , Giorgione does not satisfy was so-called because of his size.
For any variable , was so-called because of his size, is not a kind of sentence that anything satisfies, and, hence, Giorgione does not satisfy it; but (4) expresses a true proposition if (E’) is true, the occurrence of ‘Giorgione’ in (4) is not purely referential.
However, it should be noted that an advocated of (C) is in no position to endorse (E’). Surely we also want to say that, for any variable , Barbarelli does not satisfy was so-called because of his size.
For any variable , was so-called because of his size, is not a kind of sentence that anything satisfies, and, hence, Barbarelli does not satisfy it; but (5) Barbarelli was so-called because of his size, does not express a true proposition, and, hence, if (E’) is true, the occurrence of ‘Barbarelli’ in (5) is purely referential. But, since unlike (5), (4) and (1) Giorgione = Barbarelli, express true proposition, if (C) is true, the occurrence of ‘Barbarelli’ in (5) is not purely referential; and, hence, if (E’) is true, (C) is not true.
It would seem that our present difficulty arises because (E’) fails to take into account the fact that, for any variable , was so-called because of his size, is not a kind of sentence that anything would either satisfy it or its denial. This suggests that we should revise (E’) as follows:
(E’’) For any sentence S, any non-vacuous singular term , and any z, z is a purely referential occurrence of in S, if and only if, for any S’, if S’ is the result of substituting, for z in S, a variable which does not occur in S, then, S’ is an open sentence, and, relative to any assignment I, S expresses a true proposition, if and only if, relative to I, whatever z refers to satisfies S’.
Unlike (E’), (E’’) is not in conflict with (C). It is not true that if (5) does not express a true proposition, and, for any variable a, Barbarelli does not satisfy was so-called because of his size, then, (E’’) is true only if the occurrence of ‘Barbarelli’ in (5) is purely referential. A further condition needs to he met in order for the occurrence of ‘Barbarelli’ in (5) to be purely referential, i.e. that for any variable , was so-called because of his size, is an open sentence; and, surely that is not the case.
Though (E’’) is not in conflict with (C), it has another consequence which deserves attention. Consider, for instance, the following sentence:
(i) It is possible that the number of planets is odd.
One would be inclined to say that the occurrence of ‘the number of planets’ in (i) is not purely referential. But, if (E") is true, and, the occurrence of ‘the number of planets’ in (i) is not purely referential, then, given that ‘x’ is a variable, (ii) It is possible that x is odd, is not an open sentence. For, suppose that (ii) is an open sentence. Then, surely, 9, the number of planets, satisfies it, and, since, (i) express a true proposition, if (E’’) is true, then, the occurrence of ‘the number of planets’ in (i) is purely referential. Hence, if we are inclined to say that (ii) is an open sentence and that the occurrence of ‘the number of planets’ in (i) is not purely referential, we had better reject (E"). However, even if these are grounds for rejecting (E"), these are not grounds for rejecting the following consequence of (E"):
(E) For any sentence S, any non-vacuous singular term , and any z, z is a purely referential occurrence of in S, only if, for any S’, if S’ is the result of substituting, for z in S, a variable which does not occur in S, then, S’ is an open sentence, and relative to any assignment I, S expresses a true proposition, if and only if, relative to I, whatever z refers to satisfies S’.
It is worth noting that (E) is a weaker principle than (C). Unlike (C), (E) does not guarantee the truth of (D). If (E) is true, then, it is true that (D’) for any sentences S and S’, any singular terms , and ß, and any z, if z is an occurrence of in S, and S’ is the result of substituting ß for z, in S, then, if, relative to an assignment I a = b expresses a true proposition, and, it is not the case that, relative to 1, S expresses a true proposition if and only if, relative to I, S’ expresses a true proposition, then, either z or the corresponding occurrence of ß in S’, is not purely referential.
Consider, for instance, sentences (6), (7), and (8). Since (6) expresses a true proposition, for any variable , 9 satisfies It is necessary that is odd, if and only if, the number of planets, satisfies it. But, since (7) expresses a true proposition, and (8) does not express a true proposition, if (E) is true, then either the occurrence of ‘9’ in (7), or the occurrence of ‘the number of Dlanets’ in (8) is not Durelv referential. Of course. if, for some variable , It is necessary that is odd, is an not open sentence, then, the occurrence of ‘9’ in (7), and, the occurrence of ‘the number of planets’ in (8) fails to be purely referential. But, it does not follow from (E), or from (E) and the fact that (6) and (7) express true propositions and (8) does not, that for some variable , It is necessary that is odd, is not an open sentence.
Now, (E) is in conflict with some of Quine’s remarks about the concept of a purely referential occurrence. Apparently, Quine thinks that not only (C) is true but the following stronger principle (C’) is true as well:
(C’) For any sentence S, any singular term , and any z, z is a purely referential occurrence of in S, if and only if, for any sentence S’, and, any singular term ß,if S’ is the result of substituting ß for z, in S, and relative to an assignment I, = bexpresses a true proposition, then, relative to I, S expresses a true proposition, if and only if, relative to I, S’ expresses a true proposition.
If (C’) is true, then, the occurrence of ‘Giorgione’ in (i) ‘Giorgione’ names a chess player, is purely referential. But, surely, we want to say that, for any variable , ‘ ‘ names a chess player, is not a kind of sentence such that anything would either satisfy it or its denial, that it is not an open sentence. But, if for any variable , ‘ ‘ names a chess player, is not an open sentence, then, (E) is true only if (C’) is not true.
(E) purports to give the necessary conditions of a concept which I think, are of interest in discussions of referential opacity. I propose that we accept (E), and that therefore (C’) should be rejected. As for Quine’s remarks about (i), the intuitions which underlie it are captured by another distinction that Quine draws attention to.
Quine writes:
In sentences there are positions where the term is used as a means simply of specifying its object, or purporting to, for the rest of the sentence to say something about, and there are positions where it is not. An example of the latter sort is the position of ‘Tully’ in:
(1) ‘Tully was a Roman’ is trochaic.
When a singular term is used in a sentence to specify its object, and the sentence is true of the object, then certainly the sentence will stay true when any other singular term is substituted that designates the same object. Here we have a criterion for what may be called purely referential position: the position must be subject to the substitutivity of identity. That the position of ‘Tully’ in (1) is not purely referential is reflected in the falsity of what we get by supplanting ‘Tully’ in (1) by ‘Cicero’.
This passage presents a two-fold distinction: one, a distinction among positions occupied by singular terms in a sentence, and, two, a distinction among uses of singular terms in a sentence. Substitutability Salva veritate of coreferential singular terms is offered as a criterion for distinguishing those positions of a singular term in a sentence which are purely referential from those which are not; but, what is apparently given as a justification for this criterion is a claim which involves distinguishing those uses of a singular term in a sentence which are a means simply of specifying its object from those uses which are not. Quine has frequently referred to the latter distinction as a distinction between a purely referential occurrence of a singular term in a sentence and other kinds of occurrence. To avoid confusion between Quine’s distinction among positions, and the associated distinction among occurrences which is partially characterized in (E), let us agree to use the phrase ‘reverentially transparent position’ in place of Quine’s ‘purely referential position’. I shall understand by ‘the position of an occurrence of a singular term in a sentence S’ the result of deleting that occurrence of from S.
Thus, the position of the occurrence of ‘9’ in ‘9 is odd’ is ‘ is odd’, the position of the first occurrence of ‘x ‘ in ‘x = 9. x is odd’ is ‘x = 9. x is odd’, and, the position of the second occurrence of ‘x’ in ‘x = 9. x is odd’ is ‘x = 9. x is odd’. It should be noted that each occurrence of a singular term in a sentence has exactly one position in that sentence; and, that the occurrence of two or more singular terms in different sentences may have the same position in those sentences, as, for instance, ‘3/4 is odd’ is the position of the occurrence of ‘9’ in ‘9 is odd’, and also, the position of ‘The number of planets’ in ‘The number of planets is odd’. Following Quine, I shall define referential transparency of the position of an occurrence of a singular term in a sentence thus:
(F) For any sentence S, any singular term , and any z, if z is the position of an occurrence w, of in S, then z is reverentially transparent, if and only if, for any sentence S’, and, any singular term ß, if S’ is the result of substituting ß for w, in S, and relative to an assignment I, = ß expresses a true proposition, then, relative to I, S expresses a true proposition, if and only if, relative to I, S expresses a true proposition.
And, following Quine, I shall say that the position of an occurrence of a singular term in a sentence is reverentially opaque if and only if it is not reverentially transparent. The position of the occurrence of ‘9’ in ‘9 is odd’ is presumably reverentially transparent, but, the position of the occurrence of ‘9’ in (7) It is necessary that 9 is odd, is referentially opaque. Notice that the position of some occurrence of a singular term in a sentence is reverentially opaque if and only if (A) is false.
I shall say that a one-place sentential operator O is reverentially transparent, if and only if, any position of Z of an occurrence of a singular term in a sentence is reverentially transparent only if OZ is reverentially transparent; and, that a sentential operator is reverentially opaque, if and only if, it is not reverentially transparent. The sentential operators ‘It is true that’, and, ‘It is not the case that’, are reverentially transparent; but ‘It is necessary that’ is reverentially opaque, since ‘is odd’ is reverentially transparent, but ‘It is necessary that is odd’ is not.
The concept of referential transparency of a position, as one would expect, is closely connected with that of purely referential occurrence. Suppose that the position of an occurrence, w, of a singular term in a sentence S, is not reverentially transparent. Given (F), there is, then, a sentence S’, and a singular term ß, such that S’ is the result of substituting ß for w, and, relative to an assignment I, a = b expresses a true proposition, but it is not the case that, relative to I, S expresses a true proposition, if and only if, relative to I, S’ expresses a true proposition. But then, given (D’), either w is not purely referential, or the occurrence of ß in S’ which corresponds to w is not purely referential. Consider, for instance, (7). Since the position of the occurrence of ‘9’ in (7) is not reverentially transparent, given (F) and (D’), there is some singular term , such that, relative to an assignment I, = 9 expresses a true proposition, and the occurrence of a in ‘It is necessary that is odd’ is not purely referential. Given the referential opacity of the position of the occurrence of ‘9’ in (7), and (F) and (D’), it also follows that for any singular term a, and any assignment I, such that = 9 expresses a true proposition relative to I, if the proposition expressed, relative to I, by It is necessary that a is odd differs in truth-value from the proposition expressed, relative to I, by (7), then, either the occurrence of ‘9’ in (7) is not purely referential or the occurrence of a in It is necessary that a is odd is not purely referential. However, it is important to appreciate that it does not follow from the referential opacity of the position of the occurrence of ‘9’ in (7), and (F) and (D’), that the occurrence of ‘9’ in (7) is not Durelv referential.
Quine notes that the existence of reverentially opaque positions shows not only that (A) is false, but that existential generalization is the principle that:
(G) For any sentences S and S’, any singular term , and any variable ß, if ß does not occur in S, and S is the result of substituting ß for some occurrence of in S, then relative to any assignment I, S expresses a true proposition, only if, relative to I, ( ß)S’ expresses a true proposition.
As Quine notes, the existence of vacuous singular terms falsifies (G); ‘There is no such thing as Pagasus’ expresses a true proposition, but, ‘( x) There is no such thing as x’ does not expresses a true proposition. (G) is also falsified by some pairs of sentences consisting of (i) a sentence containing an occurrence of a singular term which is not purely referential, and, (ii) an existential generalization of such an occurrence of a singular term in that sentence. Consider, for instance, (4). (4) expresses a true proposition, but, if (G) is true, then, (4') ( x) x was so-called because of his size, expresses a true proposition as well. But, surely we would say that (4') does not express any proposition, and that, therefore, it does not express a true proposition; and hence, (G) is false. And consider (2). Since (2) expresses a true proposition, if (G) is true, then, (2') ( x) ‘x’ contains nine letters, expresses a true proposition as well. Now, it is not clear what sense is to be made of (2). Perhaps, one is to think of (2) as expressing the same proposition that ‘x’, the 24th letter of alphabet contains nine letters. If so, (G) is false.
From considerations such as these, Quine appears to conclude that ‘if to a reverentially opaque context of a variable we apply a quantifier, with the intention that it govern that variable from outside reverentially opaque context, then what we commonly end up with is unintended sense or nonsense.... In a word, we cannot in general quantify into reverentially opaque contexts’.
Making allowance for Quine’s allusion to unintended sense, Quine’s claim in this passage may be formulated as:
(H) An occurrence of a variable in a sentence may be bound by a quantifier outside of that sentence only if the position of that occurrence of the variable in the sentence is reverentially transparent.
Since the position of the occurrence of ‘x’ in "‘x’ contains nine letters", and, the position of the occurrence of x’ in ‘x was so-called because of his size’ are both reverentially opaque, if (H) is true, the second occurrence of ‘x’ in (2) and, the second occurrence of ‘x’ in (4'), both fail to be bound by the initial quantifiers in (2') and(4 )respectively.
(H) is to be distinguished from the claim that if an occurrence of a singular term in a sentence is not purely referential, then, existential generalization on that occurrence is unwarranted. The latter is suggested by the pairs of sentences (2) and (2'), and (4) and (4'), and Quine, I think, endorses it; but, it is the stronger (H), which articulates Quine’s frequently repeated assertion that there is no quantification into reverentially opaque contexts.
1.2.
Is (H) true?
In a recent paper Kaplan writes:
I have concluded that in 1943 Quine made a mistake. He believed himself to have given a proof of a general theorem regarding the semantical interpretation of any language that combines quantification with opacity. The purported theorem says that in a sentence, if a given position, occupied by a singular term, is not open to substitution by co-designative singular terms salva veritate, then that position cannot be occupied by a variable bound to an initially placed quantifier. The proof offered assumes that quantification receives its standard interpretation. But the attempted proof is fallacious. And what is more, the theorem is false.
Kaplan goes on to reconstruct the alleged proof as follows:
Step 1: A purely designative occurrence of a singular term , in formula is one in which is used solely to designate the object. (This is a definition)
Step 2: If has a purely designative occurrence in , then the truth-value of depends only on what designates, not how designates. (From 1)
Step 3: Variables are devices of pure reference, they cannot have non-purely designative occurrences. (By standard semantics)
Step 4: If and ß designate the same thing, but and differ in truth-value, the occurrences of , in and ß in ß are not purely designative. (From 2)
Now assume (5.1): and ß are co-designative singular terms, and and ß differ in truth-value, and (5.2): is a variable whose value is the object designated by and ß.
Step 6: Either and differ in truth-value or ß and differ in truth-value. (From (5.1) since and ß differ.)
Step 7: The occurrence of in fg is not purely designative. (From 5.2, 6, and 4)
Step 8: is semantically incoherent. (From 7 and 3)
Kaplan notes:
All but one of these steps seem to me to be innocuous. That is step 4 which, of course, does not follow form 2. All that follows from 2 is that at least one of the two occurrences is not purely designative. When 4 is corrected in this way, 7 no longer follows. The error of 4 appears in later writings in a slightly different form. It is represented by an unjustified shift from talk about occurrences to talk about positions. Failure of substitution does show that some occurrence is not purely referential. (Shifting now from ‘designative’ language of ‘Notes on Existence and Necessity’ to the ‘referential’ language of ‘Reference and Modality’). From this it is concluded that the context (read ‘position’) is reverentially opaque. And thus what the context expresses ‘is in general not a trait of the object concerned, but depends on the manner of referring to the object’. Hence, ‘we cannot properly quantify into a reverentially opaque context.’
If we understand the notation of ‘ ‘ in Step 4 as standing for any sentence S which contains one or more occurrences of a singular term , and ‘ ß’ is the result of replacing some occurrence of in S by a singular term ß, then, the proposition expressed in Step 4 is equivalent to (D) of section 1.1. As Kaplan emphasizes, if the proposition expressed in Step 4, or equivalently (D), is true, then, there is a strong argument for (H). Suppose that a sentence contains an occurrence of a singular term. Let us agree to represent S as follows:
S:
Suppose, moreover, that the position of the displayed occurrence of in S is not reverentially transparent. Then, there is a sentence S’, and a singular term ß such that S is the result of replacing the displayed occurrence of in S with ß.
S’: ß
and, there is an assignment I relative to which = ß expresses a true proposition, but it is not the case that, relative to I, S’ expresses a true proposition. Consider now a sentence S’’ and a variable such that S’’ is the result of replacing the displayed occurrence of in S with ,
S’’:
and suppose that, relative to I, the value of in S’’ is the same as the value of the displayed occurrences of and ß in S and S’ respectively. But, since relative to I, the propositions expressed by S and S’ differ in truth-value, either the propositions expressed, relative to I, by S and S’’ differ in truth-value, or the propositions expressed, relative to I, by S’ and S’’ differ in truth-value. But, then, if (D) is true, occurrence of in S’’ is not purely referential. And. if it is true that (J) an occurrence of a variable in a sentence may be bound by a quantifier outside the sentence only if that occurrence is purely referential, then, the occurrence of in S’’ may not be bound by a quantifier outside of S’’. Hence, if (D) and (J) are true and the position of an occurrence of a variable in a sentence is not reverentially transparent, then, the occurrence of that variable in the sentence may not be bound by a quantifier outside the sentence.
Are (D) and (J) true? To answer this question we need to know what is for an occurrence of a singular term in a sentence to be purely referential. Quine remarks: ‘Failure of substitutivity reveals merely that the occurrence to be supplanted is not purely referential’. I formulated this claim in Section 1.1 as (C). It is easily seen that (C) is true if and only if (D) is true. Perhaps, it would be thought that (C) is one half of a definition of ‘a purely referential occurrence’. It would then be argued that if (C) is a truth of definition, (D) must be true. But, as we have seen, if (D) and (J) are true, (H) is true; and surely, the argument would go on, (J) is a truth of standard semantics; hence, (H) is true.
Now, I think that if (J) is to appear as a premise in any argument for (H), we had better not construe (C) as a truth of definition. Notice that according to (C), an occurrence of a singular term in a sentence is purely referential only if its position in that sentence is reverentially transparent. But, if (C) is a truth of definition, then, it is a truth of definition that if (J) an occurrence of a variable in a sentence may be bound by a quantifier outside the sentence only if that occurrence is purely referential then (J’) an occurrence of a variable in a sentence may be bound by a quantifier outside that sentence only if the position of that occurrence of the variable in that sentence is reverentially transparent.
And (J’) is (H). Hence, if (C) is a truth of definition (J) can appear as a premise in an argument for (H) only on pain of circularity.
In section 1.1, I proposed that we accept (E). I argued that (E) is a weaker principle than (C); that though (D) is a consequence of (C), it is not a consequence of (E). If (E) is true then it is true that (i) if a sentence S contains an occurrence of a singular term a, and (ii) if S’ is the result of substituting ß for an occurrence z of in S, and (iii) relative to some assignment I, = ß expresses a true proposition, but (iv) it is not the case that relative to I, S expresses a true proposition if and only if relative to I, S’ expresses a true proposition, then (v) either z or the corresponding occurrence of ß in S’ is not purely referential.
But it is not a consequence of (E) that given (i) - (iv), both z and the corresponding occurrence of ß in S’ are not purely referential. What Kaplan describes as ‘the error of step 4’ is presumably the error of thinking that (D) is a consequence of (E). But it is not clear from Quine’s writings that he is guilty of this error; Quine endorses (C), and (D) is a consequence of (C). Kaplan writes that the error of step 4 ‘is represented (in later writings) by an unjustified shift from talk about occurrences to talk about positions’. But notice that (C) does in fact license this shift, for (C) states that an occurrence of a singular term in a sentence is purely referential only if its position in that sentence is reverentially transparent. If this shift from talk about occurrences to alk about positions is unjustified, then, (C) is unjustified.
1.3
Quine has observed that if we try to apply existential generalization to (7) It is necessary that 9 is odd, we obtain ( x) It is necessary that x is odd.
But, as he asks rhetorically, what is this object which is necessarily odd? In the light of (7) it is 9, but in the light of (6) 9 = the number of planets, and (13) It is not necessary that the number of planets is odd, it is not. Now, it is not clear why these observations are relevant to (H). Perhaps, as Cartwright says, we should construe Quine as pointing out that a double application of existential generalization to a conjunction of (6) and (7) with (13) yields (14) ( x) ( y) (x = y. It is necessary that x is odd. It is not necessary that y is odd).
But now consider the schema:
(9) ( x) ( y) (x=y (Fx Fy)).
If
(10) ( x) ( y) (x=y (it is necessary that x is odd it is necessary that y is odd))
is an instance of (9), then (14) is in conflict with (B), the claim that the universal closure of every instance of (9) expresses a true proposition. Thus, given that (B) is true, either (10) is not an instance of (9), or (14) does not express a true proposition. Now, presumably the principle of existential generalization whose double application to the conjunction of (6) and (7) with (13) yields (14) is this:
(G) for any sentences S and S’, any non-vacuous singular term , and any variable ß, if ß does not occur in S, and S’ is the result of substituting ß for some occurrence of in S, then relative to any assignment I, S expresses a true proposition, only if, relative to I, ( ß)S’ expresses a true proposition.
Since the conjunction of (6) and (7) with (13) expresses a true proposition, given that (B) is true, either (G’) is false or (10) is not an instance of (9). Now, I think that it should be granted that (10) is an instance of (9) if and only if (15) It is necessary that x is odd, is an open sentence. Hence, I think that it should be granted that, given that (B) is true, either (G’) is false or (15) is not an open sentence. However, I do not see why this is any evidence for (H). That (G’) is false is established by the facts that (4) Giorgione was so-called because of his size, expresses a true proposition, but (4') ( x) x was so-called because of his size, does not express any proposition, and hence does not express a true proposition. What is needed to establish (H) is an argument which shows that any apparent counterexample to (G’) involves an attempt to bind an occurrence of a variable which is not in an open sentence.
Cartwright notes:
Perhaps Quine is to be understood rather as follows: It would be counter to astronomy to deny (16) ( y) (y = Phosphorus y = Hesperus), and an application of existential generalization to the conjunction of (16) with (17) astro Hesperus = Phosphorus would yield (18) ( x) ( y) (y = Phosphorus y = x). astro x = Phosphorus).
Again, no one could reasonably deny (19) ( y) (y = Phosphorus y = Phosphorus), and an application of existential generalization to the conjunction of (19) with (20) - astro Hesperus = Phosphorus would yield (21) ( x) ( y) (y = Phosphorus y = x).
- astro Hesperus = Phosphorus).
Consider, then, the thing identical with Phosphorus. Is it a thing such that it is a truth of astronomy that it is identical with Phosphorus? In view of (18) and (21), no answer could be given. There is some one thing identical with Phosphorus. But there is no settling the question whether it satisfies ‘astro x = Phosphorus’. To permit quantification into opaque constructions is thus at odds with the fundamental intent of objectual quantification.
Cartwright sees in this reasoning an argument in defence of (B). Surely the conjunction of (18) and (21), he suggests, is not true; for if it were, the question: ‘Is the thing identical with Phosphorus such that it is a truth of astronomy that it is identical with Phosphorus?’ would be intelligible, but no answer could be given to it. However, seen as an argument for (H), this reasoning, I believe, is invalid. The last sentence, i.e. ‘To permit quantification into opaque constructions is thus at odds with the fundamental intent of objectual quantification’ does not follow from the rest. Consider, for instance, the following argument:
Perhaps Quine is to be understood rather as follows: It would be counter to history to deny (16') ( y) (y = Reagan y = the president of the U.S.), and an application of existential generalization to the conjunction of (16') with (17') It was not the case in 1972 that the president of the U.S. was identical with Reagan, would yield (18') ( x) ( y) (y = Reagan y = x). It was not the case in 1972 that x was identical with Reagan.)
Again, no one could reasonably deny (19') ( y) (y = Reagan y = Reagan), and an application of existential generalization to the conjunction of (19) with (20') It was the case in 1972 that Reagan was identical with Reagan, would yield (21') ( x) (( y) (y = Reagan y = x). It was the case in 1972 that x was identical with Reagan.)
Consider, then, the thing identical with Reagan. Is it a thing such that it was the case in 1972 that it was identical with Reagan? In view of (18) and (21), no answer could be given. There is some one thing identical with Reagan. But there is no settling the question whether it satisfies ‘It was the case in 1972 that x was identical with Reagan’. To permit quantification into opaque constructions is thus at odds with the fundamental intent of objectual quantification.
Surely we must resist the suggestion that no answer could be given to the question ‘Is the thing identical with Reagan such that it was the case in 1972 that it was identical with Reagan?’ The question is intelligible; there is indeed such a thing identical with Reagan; and there is little doubt that this thing is such it was the case in 1972 that it was identical with Reagan. The conjunction (18') and (21') is not unintelligible; it is false.
Now, it ought to be noted, as both Quine and Cartwright would emphasize, that the intelligibility of this question or the intelligibility of the conjunction of (18') and (21') is not guaranteed simply by the intelligibility of quantification and the intelligibility of the role of ‘It was the case in 1972 that’ as an operator on close sentences.
Cartwright notes:
The symbol "°" is sometimes so used that ° count as true if and only if itself is necessary. If that is all there is to go on, we have no option but to count the `°’ construction opaque and hence (i) ( x) ( y) (x = y (°x = x °x = y))
unintelligible. But (ii), (ii) ( x) °(x = x)
and (iii) ( x) ( y) (x = y °x = y)
are witnesses to a contemplated transparent ‘°’ -construction.
Now the intelligibility of such a construction is not guaranteed simply by an antecedent understanding of quantification and of the opaque ‘°’ -construction.
And Quine remarks:
The important point to observe is that granted an understanding of modalities (through uncritical acceptance, for the sake of argument, of the underlying notion of analyticity), and given an understanding of quantification ordinarily so-called, we do not come out automatically with any meaning for quantified modal sentences.
I think that it ought to be conceded that for any reverentially opaque operator O, if all there is to go on about O, is that for any closed sentence, S, OS is true if and only if S is such and such, then we do not thereby gain any understanding of OS , where S’ is an open sentence. The point, I think, is a perfectly general one; one which is independent of any considerations about referential opacity. Indeed, it ought to be conceded that for any operator O, if all there is to go on about O, is that for any closed sentence, S, OS is true if and only if S is such and such, then we do not thereby gain any understanding of OS , where S’ is an open sentence. Consider, for instance, the operator ‘It is not the case that’. If the only available rule for understanding ‘It is not the case that’ is that (i) It is not the case that S is true if and only if S is not true, and quantification is understood, we are not guaranteed any understanding of (ii) ( x) It is not the case that x is odd. For surely, (iii) ( x) ‘x is odd’ is not true does not count as an explanation of (ii). What is obviously needed is an explanation of the role of ‘It is not the case that’ as an operator on an open sentence. But now suppose that (22) It is not the case that x is odd, is specified as an open sentence, and the problem of determining which sequences, if any, satisfy this open sentence is somehow to be settled. It seems to me that it would not be a necessary condition for settling this problem that the position of the occurrence of ‘x’ in (22) be counted as reverentially transparent; for, I am inclined to think that this problem is to be settled independently of any considerations about what singular terms (other than the variables) or what kinds of singular terms (other than the variables) are available. The point is not that there is some doubt about the referential transparency of the position of the occurrence of ‘x’ in (22); it is rather that the referential transparency of this position is not a necessary condition for settling the problem of determining which sequences, if any, satisfy (24). Similarly, suppose that (15) It is necessary that x is odd, and (23) It was the case in 1972 that x was identical with Reagan, are specified as open sentences, and the problem of determining which sequences, if any, satisfy these open sentences is somehow to be settled. It is not a necessary condition for settling this problem that the positions of the occurrences of ‘x’ in (15) and (23) respectively be counted as reverentially transparent. Why is it, then, claimed, as Quine apparently does, that ‘to permit quantification into opaque constructions is thus at odds with the fundamental intent of objectual quantification’.
One cannot help but think that at issue are some principles of instantiation and generalization. Given that (B) is true, if (15) is an open sentence, and the position of the occurrence of ‘x’ in (15) is not reverentially transparent then the following principle of existential generalization is not true:
(L) For any sentences S and S’, any non-vacuous singular term , and any variable ß, if ß does not occur in S, and S’ is an open sentence which is the result of substituting ß for some occurrence of in S, then relative to any assignment I, Sexpresses a true proposition, only if, relative to I, ( ß)S’ expresses a true proposition.
If the position of the occurrences of ‘x’ in (15) is not reverentially transparent then there are singular terms and ß such that = ß. It is necessary that is a odd. It is not necessary that is ß odd expresses a true proposition. But if (15) is an open sentence, then surely x = y. It is necessary that x is odd. It is not necessary that y is odd is an open sentence as well. And if (L) is true, then (14) ( x) ( y) (x = y. It is necessary that x is odd. It is not necessary that y is odd)
expresses a true proposition. But (14) conflicts with (B). Granted that (B) is true, either (L) is not true or (15) is not an open sentence in which the position of the occurrence of ‘x’ is reverentially transparent.
(L) is closely related to the following principle of universal instantiation:
(M) For any sentences S and S’, any non-vacuous singular term , and any variable , if does not occur in S, and S’ is an open sentence which is the result of substituting ß for some occurrence of in S, then relative to any assignment I, ( ß)S’ expresses a true proposition, only if, relative to I, S expresses a true proposition.
Again granted that (B) is true, if (15) is an open sentence in which the position of the occurrence of ‘x’ is not reverentially transparent, then, (M)is false. If the position of the occurrence of ‘x’ in (15) is not reverentially transparent, then there are singular terms and ß such that = ß. It is necessary that is a odd. It is not necessary that is ß odd.
expresses a true proposition. But granted that (B) is true, if (15) is an open sentence, then (10) ( x) ( y) (x = y (it is necessary that x is odd it is necessary that y is odd)
is an instance of (9) which expresses a true proposition. But if (M) is true, and (10) expresses a true proposition, then for any non-vacuous singular terms and ß, = ß (It is necessary that is a odd it is not necessary that is ß odd)
expresses a true proposition. But that conflicts with the claim that there are singular terms and ß, such that = ß. It is necessary that is a odd. It is not necessary that is ß odd expresses a true proposition. Hence, if (B) is true, either (M)is false, or (15) is not an open sentence in which the position of the occurrence of ‘x’ is reverentially transparent. It is readily seen that given that (B) is true, if (M) is true then (H) is true. For given that (B) is true, if (M) is true then the positions of the free occurrences of a variable in any open sentence are reverentially transparent. But, then, (H) is true, since, surely it is only the free occurrences of a variable in an open sentence which may be bound by a quantifier outside of that open sentence.
But is (M) true? It seems to me that (M) is not a principle which is fundamental to the intent of objectual quantification. Objectual quantification is best understood in terms of satisfaction of open sentences, and it appears to me that the problems of determining what it is for a sequence to satisfy an open sentence are to be settled independently of any considerations about what kinds of singular term other than the variable are available. It seems, then, that it is not required for an understanding of objectual quantification that the principle that the position of an occurrence of a free variable in an open sentence is reverentially transparent is true. But since, this principle is true if (M) is true, a defense of (M) is not to be found in any appeal to the fundamental intent of objectual quantification.
Notes
0%
Author: Naeem Ahmad
