So far, I have presented a neo-Fregean notion of the sense of names by treating names as constant quantifiers. I hope that this treatment will be much more satisfactory than any other theory of names. To illustrate this, I shall in the remainder of this paper draw some brief remarks on certain philosophical significance of the quantificational treatment of the sense of names.
22 Nowadays, some logicians prefer to take quantifiers as higher-order predicates (As a matter of fact, this idea also goes back to Frege). For instance, the existential quantifier in a sentence, say ∃xϕ(x), is intended to indicate that the predicate ϕ(x) is not empty. More generally, quantification is to be treated as a function from the given domain to the power set of the given domain. Thus, the universal quantifier is defined as a function which take the domain itself as its value; while the existential quantifier takes some subset of domain as its value. Following this line of thought, we may say that a name, taken as a constant quantifier, will take some singleton as its value.
(a) First of all, our quantificational treatment of names substantially rests upon Frege’s guidelines for his semantic inquiry.
For one thing, under this treatment, the sense of a name in a sentence is construed as a certain constant quantification on the associated predicate(s) that is in turn characterized by virtue of the logical structure of the sentence in which the name occurs. The characterization of the sense of names is thereby a matter of logical investigation, rather than something psychological or subjective.
Moreover, our treatment is in line with the Context Principle. For just like a usual quantifier can be said to have a sense only when it is associated with a variable and then prefixed to an open formula containing some free occurrences of the associated variable, a name can be said to have a sense only when it occurs in a sentence.23 And the quantificational treatment of names explicitly displays that a name always occurs together with some associated predicate(s).24 In addition, one can see that under our treatment, we could never
“lose sight of the distinction between concept and object” with regard to the use of names. At any rate, it is hardly possible for anyone to confuse predicates with quantifiers.
(b) More significantly, the quantificational analysis of names explicitly manifests the logical connection between a name and the associated predicate in a sentence so that it could not only signify what the object is, about which the thought expressed by the given sentence is said to be, but also signifies the very object’s falling
23 Of course, some first-order language may accept formulae/sentences with vacuous quantifiers, i.e., formulae of the form Qxϕ, where no free occurrence of x in ϕ. But a formula of this type says nothing more than what its immediate subformula says.
24 Perhaps, this treatment also provides an answer to the question why sentences, say “Fa,” are different from open formulae such as “Fx.” Bar-Elli raises such a question: why linguistic expressions of the type say “Fa” are to be taken as sentences which can be said to be true or false but why that of the type say “Fx” are not (1998: 179)? For just as we may prefix an x-binding universal/existential quantifier to an open formula with the sole free variable x to turn it into a sentence, the same goes for a name, taken as an x-binding quantifier.
under a concept—the concept that the associated predicate stands for. It seems to me that when Frege remarks that “the sense of a name in a sentence contains the mode of presentation,” the phrase
“the mode of presentation” can be construed as the mode of presentation of a certain object’s falling under a concept.25 It is to this extent that the sense of a name in a sentence can be said to contribute to the sense of that sentence. It strikes me that if the notion of “an object’s falling under a concept” is the primary concern of a thought, perhaps, we should take as our starting-point (or foundation) for semantic investigation the view that predicates will play a central role in the theory of meaning. For following this line of reasoning, it seems likely that all grammatical (logical) subjects can be treated as quantifiers of different types.
(c) Our quantificational account of names opens a promising way to some persisting problems that the use of names in ordinary discourse may give rise to. Noticeably, it provides an explanation of the difference in cognitive value between identity statements “a
= a” and “a = b.” Clearly, under our treatment, they should be
25 It is worth mentioning that by taking a name as a constant quantifier, we can not only take the imported constant quantification as its sense but also fix the intended reference via the specified quantification. The sense of a name understood in this way will be free from Kripke’s criticism of Frege’s notion of sense in his seminal work Naming and Necessity where Kripke argues that
Frege should be criticized for using the term “sense” in two senses.
For he takes the sense of a designator to be its meaning; and he also takes it to be the way its reference is determined. Identifying the two, he supposes that both are given by definite description. Ultimately, I will reject this second supposition too; but even were it right, I reject the first. A description may be used as synonymous with a designator, or it may be used to fix its reference. The two Fregean senses of
“sense” correspond to two senses of “definition” in ordinary parlance.
They should carefully be distinguished. (1980: 59)
One may find that Kripke’s argument could be accepted if one is going to take some associated, or a cluster of, descriptions as the sense of a name.
But, on our approach, Kripke’s argument would miss its target.
reformulated, respectively, as sentences of the form:
(10) axx = x, (11) axbyx = y.
Note that (10) shows that only an x-binding constant quantifier
“a” operating on identity predicate; while (11) shows that two (one x-binding and the other, y-binding) constant quantifiers “a”
and “b,” operating on identity predicate with two distinct free variables x and y. Accordingly, (10) asserts that free occurrences of variables associated with the identity predicate will always take as its value the object associated to the constant quantifier a; while (11) asserts that the two free occurrences of variables, x and y, associated to the identity predicate will take as their values the objects associated to the constant quantifiers a and b, respectively.
From a semantic point of view, granted that the constant quantifier a has a value in a structure, the truth of (10) immediately follows from the identity law: ∀xx = x; but the truth of (11) requires that an object in the given domain is assigned to both a and b as their value in common, which is a matter of semantic stipulation. I think that this should explain the difference in cognitive value between
“a = a” and “a = b.” Moreover, the quantificational treatment of names can prevent some outrageous stipulations of the semantic values of names. For example, Searle (1967) proposed that we may put forth a certain special stipulation of the semantic value of a names so that two distinct occurrences of the same name a in an identity statement “a = a” refer to distinct objects; hence the given identity statement could be viewed as synthetic if it is true.
Surprisingly, it is difficult to find a convincing argument against such an awkward stipulation. But on our account, “a = a” should be rendered as “αxx = x,” and if we take it for granted that on the objectual interpretation of quantification, to each free occurrence of a variable v0 within the scope of a given v0-binding quantifier, the same object will be assigned as its value at a given assignment (or interpretation), then Searle’s proposal can be dismissed.
(d) Meanwhile, on the quantificational account of names, we may accept empty names without commitment to a problematic ontology. As is widely agreed, Russell’s theory of descriptions allows us to comprehend the meaning of a sentence containing empty descriptions by reformulating the involved definite descriptions in terms of some suitable (complex) predicate together with application of the existential quantifier, without presupposition of problematic non-beings. This justifies the expressibility of object-independent propositions in natural language. By the same token, the quantificational treatment of names also enables us to use sentences containing empty names to express object-independent thoughts, namely thoughts that are about non-beings. A sentence with an empty name, understood as a quantifier, at least can be used to signify, in addition to a concept that the given predicate stands for, the falling under relation between the value of the name in use and the specified concept, even though there is no object that the name in use is supposed to signify. After all, as we have already remarked, the primary concern of a thought is with an object’s falling under a concept: to grasp the sense of a simple sentence of the form Pa is no more and no less than to grasp a certain constant quantification on the associated predicate Px. We can thereby grasp the sense of sentences containing empty names without any exceptional ontological commitments. This also justifies Frege’s original view that a name in a sentence may have a sense but no reference, and, a fortiori, the intelligibility of existential statements about non-beings.
We can now speak of the non-existence of Pegasus by stating that Pegasus does not exist without presupposition of the existence of Pegasus. Following this line of thought, we need not take names in existential sentences as concept words, as Frege proposed.26
26 Of course, if this line of reasoning is acceptable, it seems to me that we need not take existence, or the verb “exists,” as a second-order concept/predicate.
But this is a topic beyond the scope of this paper. I have touched on the dispute over whether existence is a first or second-order predicate; for the
(e) A byproduct of the quantificational treatment of names can be found in its application to modal logic. At present, there is an ongoing dispute concerning the distinction between de re and de dicto readings of modal sentences containing names. For example, consider the modal sentence “ φ(a)” where a is a name (or a name letter or an individual constant) and φ(a) is a non-modal sentence. From a semantic point of view, if we intend to ascribe the embodied necessity to the object, taken as the value of a, it is desirable to take the de re reading of “ φ(a).” However, in this case, the rigidity of names has to be presupposed so that we may claim, without any justification, that a name can be used to designate the very same object in all possible worlds. Although a majority of philosophers/logicians prefer to take names as rigid designators, the rigidity of names is purely a matter of semantic stipulation. For syntactically, with regard to the use of a name, say a, a modal sentence of the form φ(a) suggests in no way that names can be used as rigid designators.27 Now, under our
details see Yang (1999).
27 In a first-order modal language with identity, the rigidity of names can be formulated in terms of the following formula:
(*) ⊢ φ(a) ↔ ∀x(x = a → φ(x)).
Or more specifically, as I proposed in Yang (1993):
(**) ⊢ φ(a) ↔ ∀x(x = a → ∀y(y = x →φ(y)).
But when contingent objects are accepted, it would be illegitimate to claim that the semantic value of a, denote |a|, satisfiesφ(x) in a world in which
|a| does not exist. Hence, (*) would fail to hold. Kripke himself seems fully aware of this difficulty so as to propose a weak sense of rigidity of names:
names are rigidity designators in a weak sense when a name refers to the same object in every possible world in which the very object exists. This could be formulated by the following formula:
(†) ⊢ φ(a) ↔ ∃x(x = a ∧ φ(x)).
Or analogously, as I proposed in Yang (1993):
(††) ⊢ φ(a) ↔ ∀x (x = a → ∃y(x = y∧φ(y)), ]
Still, the choice between (*) and (†) remains to be a matter of semantic convention.
treatment, the language in use will no longer contain modal sentences of the form φ(a); and instead, we do have “ axφ(x)”
and ax φ(x). Again, let us take it for granted that all free occurrences of a variable v0 within the scope of the same v0-binding quantifier will take the same object as its value at a given assignment. The same goes for free occurrences of variables within the scope of modal operators. As Bostock (1988: 347) remarks clearly:
A single interpretation will treat every occurrence of the variable that is bound to the same quantifier as designating the same object. Hence, if some relevant occurrence of the variable are in the scope of an embedded modal operator, a particular interpretation of that variable will treat it as designating the same object at all further worlds introduced by the embedded operator.
For clarity, let us call this “Bostock convention.” Clearly, on the basis of Bostock convention with regard to the rigidity of bound variables, a formula of the form “ax φ(x)” will suggest itself that in every possible world the variable x always takes as its semantic value the object associated to the name a, taken as a constant quantifier. The rigidity of names can be thereby expressed in our language. If we insist on the rigidity of names, we may set the following as an axiom:
(12) ⊢ αx φ(x)↔ αxφ(x).
Of course, we may further write “ φ(a)” as an abbreviation of
“ax φ(x).” By contrast, if we reject the rigidity of names and insist on the de dicto reading of “ φ(a),” we may take “ φ(a)” as an abbreviation of “ axφ(x).” But, in this case we have to withhold (12) so as to allow the possibility that the modal sentence
“ αxφ→αx φ” may not hold, though “αx φ→ αxφ” holds.
One can see that our treatment not only explains how a name in a sentence gets its reference, namely via the posited constant quantification, but also shows at the syntactic level how the name in use preserves its reference without the appeal to purely semantic
stipulation that names are rigid designators. This meets a requirement for a satisfactory theory of reference that Evans set:
“Whatever explains how a word gets its reference also explains how it preserves it, if preserved it is” (1993: 216).28
(f) Apart from these, our quantificational treatment of names also provides alternative view of the functioning of names in propositional attitude sentences. Of course, a full analysis of the meaning of sentences involving propositional attitudes phrases and that of the functioning of names in contexts of this kind are rather complicated: a topic which, as it stands, is beyond the scope of this paper. However, some brief remarks are worth making. Consider an ordinary name, say, “Socrates,” occurs in a subordinate clause of a propositional attitude sentence, e.g.,
(13) John believes that Socrates is a philosopher.
If we insist that what John believes is something about Socrates, on our treatment, (13) can be reformulated as:
(14) Socratesx (John believes that x is a philosopher), in symbols,
(15) ax⌧Px,
where ⌧, taken as an operator, stands for the phrase “John believes that,” and a, for Socrates. One can see that (15) can basically be viewed as a special type of quantifying in de re sentence.
The failure of application of Leibniz’s law to contexts of this type is precisely the same as that in the contexts of quantifying in de re sentences.
(g) Finally, it is perhaps somewhat interesting to compare our
28 It is also of some interest to note that Kripke (1980: 94) claims that
“philosophical analyses of the concept of reference in completely different terms which make no mention of reference are very apt to fail.” I think our treatment at least offers an exception.
treatment of names with Quine’s elimination of names in favour of counterpart predicates. Following Russell’s quantificational analysis of descriptions and identification of ordinary proper names with descriptions, Quine suggests that for any name a in a sentence of the form “⋅⋅⋅a---,” we may replace a by some suitable definite description(s); and in case no suitable description(s) is available, we can always introduce to the language in use a new word, say “a-izes,” as the counterpart predicate of a, which can be satisfied solely by the unique object, to which a is supposed to refer, whatever it is. Here, the counterpart predicate “a-izes” of a can be read as “being a,” or “being identical with a.” Intuitively, a counterpart predicate will behave precisely as the original name intended to do, so Quine argues. In particular, it is perfectly sensible to rephrase an existential sentence of the form “a does not exist” in terms of “Nothing a-izes.” And an ordinary statement of the form “⋅⋅⋅a---” can be rephrases as “For some (or, for all) variable x, ‘x a-izes such that ⋅⋅⋅x---’,” in symbols,
(16) ∃x(a-izes(x) ∧ ⋅⋅⋅x---) or
(17) ∀x(a-izes(x)→⋅⋅⋅x---).
Quine claims that this procedure supplies a way out to a variety of persisting problems, such as an appropriate semantic treatment for truth-value gaps caused by the use of empty names, the failure of application of Leibniz Law to opaque contexts, and intelligibility of existential statements for non-beings.
Some have argued that since the new added counterpart predicate, say “Socratizes,” could only be used to stand for a concept, the referring role that the original name plays has to be deprived. For instance, Redmon (1978: 192-193) argues that if we rephrase the sentence
(18) Pegasus does not exist,
as “Nothing pegasizes” where pegasizing is the property, say being a fly horse, etc., which distinguishes Pegasus from other things, then how would we show that (18) is false? For Redmon, finding the remains of a flying horse, etc., does not falsify (18). Thus, the conditions which would show that “Nothing pegasizes” is false—finding a flying horse, etc.—do not make (18) false. The Quinean treatment of names with regard to the existence sentences must be mistaken, so Redmon concluded. Redmon apparently takes the name in question as a set of descriptions. However, it seems to me that what Quine intends to do is no more and no less than to confine the possible values of the variable attached with a counterpart predicate a-izes not only to the unique object that satisfies the given predicate, but also to the very object which solely and uniquely satisfies the predicate a-izes, if there is any. After all, it would be perfectly sensible to read a counterpart predicate a-izes(x) as “being a,” or “being identical with a.”
It strikes me that, from a semantic point of view, there is no significant difference between the Quinean treatment understood in this way and our quantificational treatment of names. Yet, there are several reasons for me not to follow Quine’s footsteps. For one thing, I prefer to adhere to Frege’s third principle: names should never be used to stand for concepts. Moreover, Quinean treatment looks as if there is a sense of name independent of the context in which the original name occurs. This would contradict Frege’s context principle. Be that as it may, I can see no sensible reason to suspend all ordinary proper names and then introduce some extra new counterpart predicates, especially when the quantificational account of names we proposed can be easily adapted for free semantics—semantics appropriate for free logic wherein the use of constant quantifiers (names) is free from the existential assumption, if we have to deal with empty names in ordinary discourse.
Compared with the semantic rule for constant quantifiers in classical predicate logic as I put forth (on page 208), we may stipulate the required interpretation of constant quantifiers
Compared with the semantic rule for constant quantifiers in classical predicate logic as I put forth (on page 208), we may stipulate the required interpretation of constant quantifiers