From a logical point of view, the idea to treat names as quantifiers of some type is not a new one. Hodges (1977: 19) briefly proposes that logicians have been aware of the possibility of using names as a straightforward method of quantification, namely, instantiation.20 Still, a quantificational treatment of names sounds outrageous due to the lack of a method to illustrate that names behave in a way similar to what the universal/existential quantifier in a first-order theory does, both syntactically and semantically.21 Some might therefore wonder how this treatment could be adapted
20 I always feel uneasy about the adequacy of the terminology “instantiation”
here because it seems to me that the term is not appropriate for representing the constancy or rigidity of names. At first glance, the term suggests that the formula/sentence ϕ(a) is an instance of the formula ϕ(x) or the sentence
∀xϕ(x). But, a sentence of the form ϕ(a) is intended to express by and large something more than just an instance of ϕ(x) or ∀xϕ(x). It also displays explicitly that what we are talking about is precisely the very object a.
Moreover, when two sentences containing the same name, say ϕ1(a) and ϕ2(a), they are not merely to serve as two instances of ∀xϕ1(x) and ∀xϕ2(x), respectively. They are intended to show that the two sentences are about the same object. I hope that the term “constant quantifiers” could deprive us of such uneasiness. More personally, I prefer to call “instantiation quantifier” the ordinary existential quantifier ∃x. The reason is quite obvious. For a formula of the form ∃xϕ(x) is precisely intended to represent that there is at least one instance of ϕ(x). And more importantly, the new terminology may keep the notation ∃x semantically neutral to the two distinct interpretations of quantification, namely objectual and substitutional interpretation. Clearly, on the objectual interpretation, we say that ∃xϕ(x) is true if there is such and such an object a such that ϕ(a) holds;
while, on the substitutional interpretation, if there is such and such an instance ϕ(a), for some name letter (or individual constant) a.
21 It is noteworthy that Westerståhl (1998) also mentions the possibility of treating proper names as quantifier owing to the possibility of interpreting all noun phrases, i.e., Nps, in the syntactic structure of sentences in a unified way. But, he is rather pessimistic about how to give a formal treatment for names taken as quantifiers.
into a logical system. In what follows I shall show that syntactically, all that is required is to extend the usual formation rule for the universal/existential quantifiers in first-order language to cover names. Meanwhile, semantically, there should be no difficulty in adapting the use of constant quantifiers in place of individual constants to the standard semantics for predicate logic, either on objectual interpretation of quantification or on the substitutional semantics.
Let us recall that syntactically, an occurrence of a quantifier in a formula/sentence of a first-order language always affixes with a variable so that we can use an xi-binding quantifier (e.g., ∀xi or ∃xi) to turn an open formula φ(x1...xi...xn) with free variable xi amongst n distinct free variables into one with n-1 free variables. In particular, we can use an x-binding quantifier, either ∀x or ∃x, to turn an open formula φ(x) with the sole free variable x into a sentence (i.e., “∀xφ(x)” or “∃xφ(x)”). By the same token, a name
“a,” taken as a (constant) quantifier, can be used as an x-binding quantifier, for any arbitrary variable x. To imitate the formulation of usual quantifiers, we may use the notation “ax” as an x-binding constant quantifier a. We can then prefix such an x-binding quantifier notation to an open formula (e.g., φ(x)) to form a new formula/sentence of the form “axφ(x)).” By this procedure, names, taken as quantifiers, will behave exactly in the same way as usual universal/existential quantifiers do. The usual formation rule for
“∀” and “∃” can be thereupon applied to names.
More formally, let L be a usual first-order language for predicate logic, the alphabet of which contains denumerable sets of name letters {ak | k∈ℕ}, predicate letters {Pi | i∈ℕ}, variables {xn| n∈ℕ}, and usual logical operators and auxiliary signs {¬, →, ∀, ∃, (, ) , }. Now, on the given alphabet, we can construct a first-order language L* wherein the primitive symbols ak’s will be treated as constant quantifiers. Let us use meta-symbols v0, . . ., vn, . . . to stand for variables; α, α1, . . ., αk, . . ., for constant quantifiers, and φ, ψ, θ, σ,..., for formulae. Note that, as the language in use contains no function letters, only variables count as terms. The set
of atomic formulae of L* will be defined by the following formation rule:
(At) For any n-place predicate letter P, Pv0,...,vn -1 is an atomic formula.
The set of formulae F will be defined by the following formation rules:
(i) φ∈ F; for any atomic formula φ;
(ii) if φ∈F and ψ∈F, then ¬φ, (φ→ψ), ∀viφ, ∃viφ, αviφ, for any constant quantifier α, are members of F.
Note that the role that constant quantifiers play can be further characterized by virtue of the following equivalences:
(3) ⊢ αvi¬φ ↔ ¬αviφ;
(4) ⊢ αvi(φ→ψ) ↔ (αviφ → αviψ);
(5) ⊢ αvi∀vjφ ↔ ∀vjαviφ;
(6) ⊢ αvi∃vjφ ↔ ∃vjαviφ.
In so far as the syntax is concerned, there is no substantial difference between a usual first-order language L and the language L* so constructed. As a matter of fact, corresponding to each atomic sentence of L, say P(a1. . . an) (for a sequence of n name letters a1, . . . , an), there is a sentence of the form a1v1...anvn P(v1,...,vn) in L* for a sequence of n constant quantifiers a1,...,an. Let us call a formula an atomic constant quantified sentence, or simply an atomic sentence, of L*, if it results by prefixing a suitable sequence of n constant quantifiers to an n-place predicate P(v1,...,vn) so that no free occurrence of variables remains. Moreover, let
“φ(α/vi)” stand for the formula resulting from αviφ(vi) by deleting the vi-binding constant quantifier αvi and then substituting the constant quantifier α for each occurrence of vi in φ(vi). Obviously, if we write “φ(α/vi),” instead of “αviφ,” for any atomic sentence φ, the language L* will be exactly the same as L
defined in the usual way.
We next turn our attention to appropriate semantics for L*. It can be shown that both the well-established objectual interpretation and the substitutional interpretation of quantification can be easily adapted to the required semantic treatment for the language L*. On the objectual interpretation of quantification, a quantifier is used as an operator on an open formula (or the associated predicate(s)) to indicate how many things of a certain type in the given domain would satisfy the given open formula (or associated predicate(s)). Thus, the existential quantification
(7) ∃x(x is a philosopher)
(also, the universal quantification, e.g., ∀x(x is a philosopher), respectively) indicates that the predicate “x is a philosopher” is satisfied in a given structure, provided that at least one object (or, every object, respectively) in the given domain is in the extension of the given predicate. Now if the predicate “x is a philosopher” is used to stands for a concept being a philosopher, as Frege so construed, the occurrence of the variable x here is used to indicate that the object, assigned as the value of x, whatever it is, is supposed to be an object falling under the concept being a philosopher. The quantifier ∃ (∀, respectively) is used to bind the variable x so to assert that at least one object (or every object, respectively) in the given domain can be assigned to x as its value such that the very object falls under the concept expressed by the predicate “x is a philosopher.” By the same token, when we wish to assert that a certain specified object, say Socrates, in the given domain falls under the concept being a philosopher, all that is required is to formulate that the variable x associated with the predicate “x is a philosopher” will take the very object as its value.
Now, we may set as an interpretation of constant quantifiers in what follows:
(SCQCL) To each constant quantifier a, an object a is associated so that whenever a occurs in a formula, or subformula of some formula, as an v-binding quantifier, any occurrence of the variable v in the specified scope will accept the very object a as its value.
Obviously, with (SCQCL), we do have a truism: for any constant quantifier a,
(CQ) ax∃yx = y
For example, assuming that to the constant quantifier “Socratesx,”
the person Socrates is associated, the sentence (8) Socratesx (x is a philosopher).
then asserts that the variable x in (8) will accept as its semantic value the person Socrates which falls under the concept being a philosopher. More generally, for any name a, taken as a constant quantifier, we may say that a constant quantification
(9) ax (φx)
is satisfied in a structure if associated to the (x-binding) constant quantifier a, there is a certain specified object in the given domain so that x will accept the very object as its value and the very object satisfies the open formula φ(x). Corresponding to the informal reading of ∀—“for all objects,” and that of ∃—“for some object(s),” the constant quantifier “ax” can be informally read: “for the very object a.”
Of course, if constant quantifiers are to be interpreted in this way, we need to define an interpretation (a structure) wherein, to each constant quantifier, a unique object in the give domain will be associated. The required semantic rule for constant quantifiers follows immediately:
(S) A formula of the form αviφ is satisfied in the given structure if φ is satisfied by the unique object associated to the constant quantifier .α
It is somewhat interesting to see that the substitutional interpretation of quantification can be easily adapted as well. All that is required is to show that a structure is merely an assignment of truth values to each (constant quantified) atomic sentence, and the required semantic rules for connectives and the universal/
existential quantifiers are the same as the standard ones: e.g., ∃viφ is true in a given structure if there is some constant quantifier α such that αviφ is true in the structure.22
The foregoing treatment suffices to show that it is formally adequate to treat names as constant quantifiers, both syntactically and semantically.