## 1. Classical Quantificational Logic



What is now a commonplace treatment of quantification began with Frege
(1879), where the German philosopher and mathematician, Gottlob Frege,
devised a formal language equipped with quantifier symbols, which
bound different styles of variables. He formulated axioms and rules of
inference, which allowed him to represent a remarkable range of
mathematical argumentation. Classical quantificational logic is
sometimes known as "first-order" or
"predicate" logic, which is generally taken to include
functional and constant symbols. The vocabulary of classical
quantificational logic is often supplemented with an identity
predicate to yield the classical theory of quantification with
identity.


### 1.1 Pure Quantificational Logic



At the core of classical quantificational logic lies what we may call
pure quantificational logic, which makes no provision for any singular
terms other than variables. In pure quantificational logic, one may
still make use of Russell's theory of definite description to
simulate singular terms for which we have a method of contextual
elimination.


#### 1.1.1 The Language of Pure Quantificational Logic


 The vocabulary of pure quantificational logic contains the usual
propositional connectives: \(\lnot\), \(\wedge\), \(\vee\),
\(\rightarrow\), and \(\leftrightarrow\). It contains predicate
letters of different adicities: one-place predicate letters with or
without subscripts, \(P^{1}, Q^{1}, R^{1}, P^{1}\_{1}\), ...,
two-place predicate letters, \(P^{2}, Q^{2}, R^{2}, P^{2}\_{1}\),
..., and more generally, \(n\)-place predicate letters. Sentence
letters are 0-place predicates. There is, in addition to this, an
infinite stock of variables with or without subscripts: \(x, y, z,
x\_1, y\_1, z\_1\), ... , and two quantifiers \(\forall\) and
\(\exists\). In a classical framework, we need not treat all these
symbols as primitive; we may, for example, treat \(\lnot\),
\(\rightarrow\), and \(\forall\) as primitive and provide standard
definitions for the other propositional connectives and quantifiers in
terms of them. For example, the existential quantifier, \(\exists x
A\), may be defined: \(\lnot \forall x\lnot A\).



The definition of a formula of the language of pure quantificational
logic proceeds recursively as follows. First, one defines an
*atomic formula* to consist of an \(n\)-place predicate
followed by \(n\) variables: \(P^{n}\_{i} x\_{1}\), ..., \(x\_{n}\).
One then defines \(A\) to be a formula of the language of pure
quantificational logic if, and only if, (i) \(A\) is an atomic
formula, or (ii) \(A\) is of the form \(\lnot B\), where \(B\) is a
formula, or (iii) \(A\) is of the form \((B \rightarrow C)\), where
\(B\) and \(C\) are each a formula, or (iv) \(A\) is of the form
\(\forall x B\), where \(x\_i\) is a variable and \(B\) is a
 formula.[2]



All occurrences of variables in an atomic formula are *free*.
The free occurrences of variables in the negation of a formula \(\lnot
B\) are the free occurrences of variables in \(B\). The free
occurrences of variables in a formula of the form \((B \rightarrow
C)\) are the free occurrences of variables in \(B\) and \(C\).
Finally, the free occurrences of variables in a quantified formula
\(\forall xB\) are the variables other than \(x\) that are free in
\(B\). We call \(B\) the scope of the initial occurrence of the
quantifier \(\forall x\) in \(\forall xB\). If an occurrence of a
variable is not free in a formula \(A\), then it is bound; all
occurrences of \(x\) in \(B\) are bound in the quantified formula
\(\forall xB\). In fact, we will often write that they lie within
the scope of the initial quantifier.



A formula \(A\) is *closed* if, and only if, all of its
variables are bound. Otherwise, \(A\) is *open*. A variable
\(y\) is *free for* \(x\) if, and only if, no free occurrences
of \(x\) lies within the scope of a quantification on \(y\), \(\forall
y\). We write \(A(y/x)\) for the formula that results from \(A\) when
every free occurrence of \(x\) in \(A\) is replaced by an occurrence
of the variable \(y\). A sentence is a closed formula of the
language.


#### 1.1.2 Axioms for Pure Quantificational Logic



Frege's *Begriffsschrift* develops a formal system, which
includes axioms for quantification over individual objects and
concepts. For Frege, objects are the appropriate values for singular
variables, and a concept is what is referred to by a predicate. Notice
that despite Frege's use of the term, concepts, for him, are
nothing like mental constructs, but rather objective constituents of
the world. In particular, concepts are functions, which map objects
into truth values. Since a predicate like "is extended" is
true of some objects and false of others, Frege takes the concept
*being extended* to assign truth to an object if, and only if,
it is extended.



Frege's system is an axiomatization of what has come to be known
as second-order logic, which is an important extension of classical
quantificational logic. Hilbert & Ackermann (1928) contains the
first axiomatic treatment of quantificational logic as a subject of
its own, which they called "the narrower functional
calculus". In what follows, we look at what is now a common
axiomatization of pure quantificational logic. This axiom system
adopts all tautologies of propositional logic as axioms and *modus
ponens* as a rule of inference, and it supplements them with two
more axiom schemata and a rule of universal generalization. The first
two axiom schemata are:


* (\(\forall 1\)) \(\forall
xA \rightarrow A(y/x)\),
  

*provided \(y\) is free for \(x\) in \(A\)*.
* (\(\forall 2\)) \(\forall
x(A \rightarrow B) \rightarrow (A \rightarrow
\forall xB)\),
  

*provided \(x\) does not occur free in \(A\)*.



The first axiom, (\(\forall 1\)), is commonly known as the axiom of
*universal instantiation*. To complete the axiomatization, we
need to add a rule of *universal generalization*:


* (\(\forall 3\)) from
\(A\), infer \(\forall xA\).



Since \(\exists xA\) abbreviates \(\lnot \forall x \lnot A\), we
have a principle of *existential generalization*:


* \(A(y/x) \rightarrow
\exists xA\),
  

*provided \(y\) is free for \(x\) in \(A\)*.



This principle is not without consequence. Since \(Px \rightarrow Px\)
is a tautology, the combination with *modus ponens* and
existential generalization yields \(\exists x(Px \rightarrow Px)\) as
a theorem. For another example, consider the theorem \(\forall xPx
\rightarrow \exists xPx\), which is similarly derivable from the
axioms by combining universal instantiation and existential
generalization. Some have found these consequences objectionable on
the grounds that the existence of an object should not be derivable
from logic alone.


#### 1.1.3 Interpretations for Pure Quantificational Logic



The interpretation of the language of pure quantificational logic
requires one to specify both a domain for the variables to range over
and an extension for each non-logical predicate of the language.
Alfred Tarski developed what is now known as a model theory for a wide
range of formal languages. We recall some definitions from the entry
on
 model theory.
 A model for the language of pure quantificational logic is an ordered
pair, \(\langle D, I\rangle\), where \(D\) is a non-empty set, and
\(I\) is an interpretation function, which maps each \(n\)-place
predicate of the language into a set of ordered \(n\)-tuples of
objects in the set \(D\). This set provides the domain of discourse
for the quantifier, and \(I\) provides each predicate of the language
with an extension: one-place predicates are mapped into subsets of
\(D\), two-place predicates are mapped into sets of ordered pairs of
members of \(D\), and more generally, \(n\)-place predicates are
mapped into sets of \(n\)-tuples of members of \(D\).



Truth in a model is then defined in terms of satisfaction. A variable
assignment \(s\) for the model \(\langle D, I\rangle\) is a function
from the set of variables of the language into the domain \(D\). A
formula \(A\) is satisfied in a model \(\langle D, I\rangle\) by an
assignment \(s\) of objects to variables. If \(s\) is a variable
assignment for \(\langle D, I\rangle\), we write \(s[x/d]\) to specify
a variable assignment, which is just like \(s\) except perhaps for
assigning \(d\) in \(D\) to the variable \(x\).



We define a formula to be *satisfied* in a model \(\langle D, I
\rangle\) by an assignment \(s\) recursively as follows. An atomic
formula \(P^{n}\_{i} x\_{1}, \ldots , x\_{n}\) is satisfied in \(\langle
D, I\rangle\) by \(s\) if, and only if, \(\langle s(x\_{1}), \ldots,
s(x\_{n})\rangle \in I(P^{n}\_{i})\), which means that the \(n\)-tuple
of objects \(s\) assigns to the variables that occur in the formula is
in the interpretation of the predicate \(P^{n}\_{i}\) under \(I\). A
negation \(\lnot B\) is satisfied in \(\langle D, I\rangle\) by \(s\)
if, and only if, \(B\) is not satisfied in \(\langle D, I\rangle\) by
\(s\). A conditional \((B \rightarrow C)\) is satisfied in \(\langle
D, I\rangle\) by \(s\) if, and only if, \(B\) is not satisfied or
\(C\) is satisfied in \(\langle D, I\rangle\) by \(s\). Finally,
\(\forall xB\) is satisfied in \(\langle D, I\rangle\) by \(s\) if,
and only if, \(B\) is satisfied in \(\langle D, I\rangle\) by all
assignments of the form \(s[x/d]\), where \(d\) is a member of the
domain \(D\).



A formula \(A\) is *true in a model* \(\langle D, I\rangle\)
if, and only if, \(A\) is satisfied in \(\langle D, I\rangle\) under
all assignments for the model. A formula \(A\) is *valid* if,
and only if, \(A\) is true in all models.


#### 1.1.4 Metatheory for Pure Quantificational Logic



The Tarskian model theory for pure quantificational logic gives us
exactly what we want. Since the axioms of pure quantificational logic
are valid and the rules of inference preserve validity, all theorems
of pure quantificational logic are valid.
 (Soundness).[3]
 One may in addition, prove that all valid formulas are, in fact,
derivable from the axioms of pure quantificational logic.
(Completeness). The entry on
 classical logic
 outlines a proof.



Call a set of formulas \(\Gamma\) *satisfiable* if, and only
if, there is a model \(\langle D, I\rangle\) and a variable assignment
\(s\), which satisfies each formula \(A\) in \(\Gamma\) in the model.
A set \(\Gamma\) of formulas is satisfiable if, and only if, every
*finite* subset of \(\Gamma\) is satisfiable. (Compactness).
This is a simple corollary of Completeness.



A simple version of the Lowenheim-Skolem theorem states that if a
set of formulas \(\Gamma\) has an infinite model, then it has a
denumerable model, where a *denumerable* model is one whose
domain is no larger than the set of natural numbers. There is a
generalized form as well. If we let the cardinality \(\kappa\) of the
non-logical vocabulary be greater than \(\aleph\_0\), which is the
cardinality of the set of natural numbers, then if \(\Gamma\) has an
infinite model, then it has a model of cardinality \(\kappa\).



Model-theoretic interpretations are sets: a model is an ordered pair
of a non-empty set and an interpretation function. But since modern
 set theory
 proves that there is no universal set, no model can ever interpret
the quantifiers by means of a universal domain of discourse. It may
seem, then, that there are interpretations of the language of pure
quantificational logic to which no model corresponds. And this raises
the question of whether truth in all models may fall short of truth
under all interpretations of the language.



Fortunately, there is an elegant argument due to Georg Kreisel (1967),
which tells us that we can safely restrict attention to models for the
specification of the set of validities of pure quantificational logic:
a formula \(A\) of the language of quantificational logic is true
under every interpretation of the language if, and only if, it is true
in every model. The argument is a simple application of Completeness.
If \(A\) is true under every interpretation, whether it is a model or
not, then \(A\) will be true in every model. The challenge, however,
is to argue for the converse. But if \(A\) is true in every model,
then, by Completeness, \(A\) is a theorem of pure quantificational
logic. And since every theorem is presumably true under every
interpretation, not just model-theoretic ones, it follows that \(A\)
is true under every interpretation. Thus Kreisel's argument
suggests we can make do with models for the purpose of characterizing
validity in pure quantificational
 logic.[4]


### 1.2 Classical Quantificational Logic with Identity



Classical quantificational logic allows for singular terms other than
variables. There are, first, individual constants, and, second,
singular terms, which result from the combination of a function symbol
with an appropriate number of singular terms. In addition to this,
classical quantificational logic with identity makes provision for a
special identity predicate.



The vocabulary of the classical theory of quantification and identity
extends the vocabulary of pure quantificational logic with a set of
individual constants with or without subscripts, \(a, b, c, a\_1\),
..., and a set of function symbols of various kinds: one-place
function symbols, \(f^{1}, g^{1}, f^{1}\_{1}\), ..., two-place
function symbols, \(f^{2}, g^{2}, g^{2}\_{1}\),
 etc.[5]
 In addition to this, we may add a special two-place predicate, \(=\),
for identity.



Before we modify the definition of *formula* for the expanded
language, we may give a recursive definition of a *term* of the
language of the classical theory of quantification and identity. An
expression \(t\) is a term if, and only if, (i) \(t\) is an individual
variable, or (ii) \(t\) is an individual constant, or (iii) \(t\) is
an expression of the form \(f^{n}\_{i}t\_{1}, \ldots, t\_{n}\), where
\(f^{n}\_{i}\) an \(n\)-place function symbol and \(t\_1\), ...,
\(t\_n\) are themselves terms. Given the definition of term, we can
reformulate the definition of *atomic formula* for the expanded
language to consist of all and only expressions \(A\) of the form
\(P^n\_i t\_1, \ldots, t\_n\), where \(P^n\_i\) is an \(n\)-place
predicate and \(t\_1\), ..., \(t\_n\) are each terms or \(t\_i =
t\_j\) in which \(t\_i\) and \(t\_j\) are terms. At this point, we may
adapt the usual recursive definition of *formula* for the
expanded language.



The axioms of classical quantificational logic with identity include
axioms for quantification and axioms for identity. The axioms for
quantification include a suitably modified variant of \((\forall 1)\)
in addition to \((\forall 2)\) and \((\forall 3)\). The minimal
adjustment in \((\forall 1)\) is meant to accommodate the presence of
function symbols and individual constants. We call a term \(t\)
*free for* a variable \(x\) if, and only if, no free occurrence
of \(x\) lies within the scope of a quantifier \(\forall y\) or
\(\exists y\), where \(y\) is a variable which occurs in \(t\). The
axiom of universal instantiation, \((\forall 1)\), now reads:


* \(\forall xA
\rightarrow A(t/x)\),
  

*provided \(t\) is free for \(x\) in \(A\)*.



The other set of axioms concerns identity. In particular, we will
supplement the axioms for quantification with an axiom, (I1), and an
axiom schema, (I2), designed to govern the identity predicate:


* (I1) \(t =
t\)
* (I2) \(t\_1 = t\_2
\rightarrow (A(t\_1/x) \rightarrow A(t\_2/x))\)
  

*provided \(t\_1\) and \(t\_2\) are free for \(x\) in
\(A\)*



The first axiom is sometimes known as Reflexivity, and the second is
known as the Indiscernibility of Identicals. When combined, they
enable one to derive the symmetry and transitivity of identity as
immediate
 consequences.[6]



We need only make minimal adjustments in the model theory in order to
accommodate the expanded vocabulary of the classical theory of
quantification and identity. A model \(\langle D, I \rangle\) will
still consist of a non-empty domain, \(D\), and an interpretation
function, \(I\). One difference, however, is that we now take \(I\),
in addition, to assign a member of the domain \(D\) to each individual
constant and to assign an \(n\)-place function from the set of ordered
\(n\)-tuples of members of \(D\), \(D^{n}\), into \(D\) to each
function symbol \(f^{n}\_{i}\).



We now define satisfaction in terms of *denotation*. Given a
model \(\langle D, I\rangle\) and an assignment \(s\) for the model,
we say that a term \(t\) *denotes* a member \(d\) of the domain
under assignment \(s\), in symbols \(Den\_{s}(t) = d\), if, and only
if, (i) \(t\) is a variable \(x\_i\) and \(s(x\_i) = d\), or (ii) \(t\)
is an individual constant \(c\) and \(I(c) = d\), or (iii) \(t\) is of
the form \(f^{n}\_{i}t\_1 , \ldots, t\_n\) and \(d =
I(f^{n}\_{i})(Den\_{s}(t\_1), \ldots, Den\_{s}(t\_n))\).



The definition of satisfaction in a model \(\langle D, I\rangle\) by
an assignment \(s\) proceeds much like before, except for the obvious
clauses for atomic formulas of the form \(P^n\_i t\_1, \ldots, t\_n\) and
\(t\_i = t\_j\): \(P^n\_i t\_1, \ldots, t\_n\) is satisfied by \(s\) in
\(\langle D, I\rangle\) if and only if \(\langle Den\_{s}(t\_1), \ldots,
Den\_{s}(t\_n) \rangle \in I(P^n\_i)\). And \(t\_i = t\_j\) is satisfied by
\(s\) in \(\langle D, I\rangle\) if and only if the denotation of
\(t\_i\) under \(s\) is identical to the denotation of \(t\_j\) under
\(s\).



The definitions of *truth in a model* and *validity*
carry over from pure quantificational logic.



We have Soundness and Completeness theorems for classical
quantificational logic with identity as well as Compactness and
Lowenheim-Skolem theorems. The entry on
 first-order model theory
 offers an in-depth examination of these and other meta-theoretic
results for classical quantificational logic with identity.


## 2. Departures from Classical Quantificational Logic



In what follows, we look at three rival accounts of quantification in
modern logic. They are departures from classical quantification logic
because they reject some of classical axioms of quantification or
because they question some aspect of the Tarskian model theory we have
used to interpret the language of classical quantification logic.


### 2.1 Inclusive and Free Quantificational Logic



In pure quantificational logic, one may, for example, derive the
conditional \(\forall xPx \rightarrow \exists xPx\) as an
immediate consequence of the axioms. But the derivability of this
sentence--and others like \(\exists x (Px \rightarrow
Px)\)--may give one pause. To the extent to which logic should
remain neutral in ontological matters, the axioms of pure
quantificational logic should not by themselves be able to prove that
there is something rather than nothing. From a model-theoretic
perspective, the exclusion of the empty domain as an eligible domain
for a model may likewise seem artificial. Since the validity of
formulas like \(\forall xPx \rightarrow \exists xPx\) or \(\exists
x(Px \rightarrow Px)\) is largely a byproduct of what may seem an
*ad hoc* stipulation, one may be motivated to expand the range
of models in order to allow for a model with an empty domain of
discourse. Quine (1954) used the label "inclusive" to
refer to alternatives to quantificational logic that make allowance
for such a model.



According to Quine (1954), an inclusive quantificational logic should
evaluate every universally quantified formula of the form \(\forall x
A\) to be vacuously true in a model \(\langle D, I\rangle\) in which
\(D\) is empty, and it should evaluate every existentially quantified
formula of the form \(\exists x A\) to be false in such a model.
\(\forall xPx \rightarrow \exists xPx\) and \(\exists x(Px
\rightarrow Px)\) are therefore not valid on the expanded model
theory. However, it is quite a delicate matter to provide a
compositional amendment of the Tarskian definition of truth in a model
\(\langle D, I \rangle\) under an assignment \(s\) that delivers
Quine's
 verdicts.[7]



An axiomatization of inclusive quantificational logic should weaken
the axiom of universal instantiation in order to prevent the
derivation of theorems such as \((\forall xPx \rightarrow \exists x
Px)\) or \(\exists x(Px \rightarrow Px)\). One option independently
explored by Kripke (1963) and Lambert (1963) is to replace (\(\forall
1\)) with a closed axiom schema:


* (\(\forall 1^{-}\)) \(\forall
y(\forall xA \rightarrow A(y/x))\),
  

*provided \(y\) is free for \(x\) in \(A\)*.



In the absence of identity, by itself, this change results in an
inadequate axiomatization of pure quantificational logic, one which
cannot yield every instance of a permutation principle discussed by
Fine (1983):


* \(\forall x\forall y
A \rightarrow \forall y\forall xA\).



The permutation principle, however, becomes redundant in the presence
of axioms for identity.



The axiomatization of classical quantification logic with identity
that emerges is discussed by Lambert (1963) with a different
motivation in mind. While classical quantificational logic assumes
singular terms denote members of the domain of quantification, Lambert
wanted to allow for systems that remain "free" from such
assumptions. Free quantificational logic makes room for empty singular
terms, which fail to denote any member of the domain. That means that
free logic need not prove every formula of the form:


* \(\exists x(x =
t)\).



These are all theorems of quantificational logic with identity. Given
Reflexivity (I1), the classical axiom of universal instantiation
(\(\forall 1\)) is all that is required for the proof of these
theorems. So, free logicians have another incentive for the retreat
from (\(\forall 1\)) to (\(\forall 1^{-}\)). It is not uncommon to use
an existence predicate in order to distinguish terms that denote
members of the domain from those that do not. In general, \(E!t\) is
read to be true if \(t\) denotes a member of the domain, and false if
it does not. This existence predicate is sometimes taken as primitive
and sometimes defined by the formula \(\exists x(x =t)\). That
suggests a different weakening of (\(\forall 1\)):


* \( \forall xA
\rightarrow (E!t \to A(t/x)) \),
  

*provided \(y\) is free for \(x\) in \(A\)*.



What is less obvious is how to modify the model theory for the
language in order to accommodate empty terms. One important choice
point is the evaluation of atomic formulas in a model of free
quantificational logic with identity. One approach is to declare as
false an atomic formula of the form \(P^n\_i t\_1, \dots, t\_n\) or \(t\_i
= t\_j\) if any of the terms involve fail to denote anything in the
domain. That would result in a *negative* model theory for the
language. On the other hand, we could make allowance for true atomic
formulas even if some of the relevant terms denote nothing in the
domain, e.g., we might declare \(t = t\) true even if nothing in the
domain of quantification is a denotation for the term in question.
That would result in a *positive* model theory for the
language. Finally, we could declare atomic formulas not of the form
\(E!t\) to be neither true nor false, which would result in a
*neutral* model theory for the language. The entry on
 free logic
 explores all these options in detail.


### 2.2 Intuitionistic Quantificational Logic



Intuitionistic propositional logic is weaker than classical
propositional logic. All intuitionistically valid formulas are
classically valid, but classically valid formulas like \((A \vee \lnot
A)\) and \((\lnot \lnot A \rightarrow A)\) are not intuitionistically
 valid.[8]
 The exclusion of these formulas from the range of intuitionistic
theorems can be motivated by the usual Brouwer-Heyting-Kolmogorov
interpretation of the connectives:


* A proof of \(\lnot A\) consists of a proof that there cannot be a
proof of \(A\).
* A proof of \((A \wedge B)\) consists of a proof of \(A\) and a
proof of \(B\).
* A proof of \((A \vee B)\) consists of a proof of \(A\) or a proof
of \(B\).
* A proof of \((A \rightarrow B)\) consists of a construction, which
turns a proof of \(A\) into a proof of \(B\).



Despite the differences in interpretation, there is an important
connection between intuitionistic and classical propositional logic.
For we know that a formula \(A\) is a theorem of classical
propositional logic if, and only if, \(\neg \neg A\) is a theorem of
intuitionistic propositional logic. Thus while \((A \vee \neg A)\) is
obviously not a theorem of intuitionistic propositional logic, \(\neg
\neg (A \vee \neg A)\) is in fact intuitionistically provable. This
fact is at the core of a familiar translation from classical into
intuitionistic propositional logic due to Kurt Godel and Gerhard
Gentzen. For a summary of this and related facts, the reader may
consult the entry on
 intuitionistic logic.



Now, intuitionistic quantificational logic may be motivated by the
Brouwer-Heyting-Kolmogorov interpretation of the quantifiers:


* A proof of \(\exists xA\) is given by a specification of some
\(d\) and a proof that \(A\) obtains for \(d\).
* A proof of \(\forall xA\) is given by a construction, which
given an object \(d\), produces a proof that \(A\) obtains for
\(d\).



The intuitionistic interpretation doesn't sanction the classical
equivalence between \(\exists x A\) and \(\neg \forall x \neg A\).
Indeed, the two quantifiers must remain part of the primitive
vocabulary of intuitionistic quantificational logic.


#### 2.2.1 Axioms for Intuitionistic Quantificational Logic



The intuitionistic axioms of quantification include counterparts of
the classical axioms of universal instantiation and existential
generalization:


* (\(\forall^{I}1\)) \(\forall
xA \rightarrow A(y/x)\)
  

*provided \(y\) is free for \(x\) in \(A\)*.
* (\(\exists^{I}1\)) \(A(y/x)
\rightarrow \exists x A\)
  

*provided \(y\) is free for \(x\) in \(A\)*.



In addition to this, we have two more principles:


* (\(\forall^{I}2\)) \(\forall
x(A \rightarrow B) \rightarrow (A \rightarrow
\forall xB)\),
  

*provided \(x\) does not occur freely in \(A\)*.
* (\(\exists^{I}2\)) \(\forall
x(A \rightarrow B) \rightarrow (\exists xA
\rightarrow B)\),
  

*provided \(x\) does not occur freely in \(B\)*.



We have a rule of universal generalization:


* (\(\forall^{I}3\))  from
\(A\), infer \(\forall xA\).



It is not generally true that a formula \(A\) is a theorem of pure
quantificational logic if, and only if, \(\neg \neg A\) is a theorem
of intuitionistic quantificational logic, but there is a more
sophisticated double-negation interpretation of theorems of pure
quantificational logic. In particular, we have that a formula \(A\) is
a theorem of quantificational logic if, and only if, it is a theorem
of intuitionistic quantificational logic supplemented with a double
negation schema:


* (DNS) \(\forall x \neg \neg A \rightarrow \neg \neg \forall xA\)



Much like in the propositional case, one may exploit this fact to give
a more sophisticated translation from pure quantificational logic into
intuitionistic quantificational logic. Details are given in
Moschovakis (2010).



In intuitionistic quantificational logic, we cannot move from \(\neg
\forall x \neg A\) to \(\exists x A\), but we still have this:


* \(\exists x A
\rightarrow \neg \forall x \neg A\)



Moreover, since \(A\) intuitionistically entails \(\lnot \lnot A\), we
can infer:


* \(\forall x A
\rightarrow \lnot \lnot \forall x A\)



In intuitionistic quantificational logic, \(\forall x \lnot \lnot A\)
is strictly weaker than \(\lnot \lnot \forall x A\), since we
have:


* \(\lnot \lnot \forall
x A \rightarrow \forall x \lnot \lnot A\).


#### 2.2.2 Kripke Models for Intuitionistic Quantificational Logic



A variety of interpretations have been advanced for intuitionistic
quantificational logic, but the model theory developed by Kripke
(1965) is perhaps the most similar to the model theory of classical
quantificational logic. A Kripke model \(\mathcal{K}\) is based on a
frame \(\langle S, \preceq \rangle\), which consists of a set of
*stages* partially ordered by an accessibility relation
\(\preceq\). A Kripke model assigns an inhabited domain, \(D\_{s}\), to
each stage \(s \in S\), subject to the constraint that \(D\_s \subseteq
D\_{s'}\) when \(s \preceq s'\). We may assume that suitable names for
the members of each \(D\_s\) have been adjoined to the language.
Finally, for \(n>0\), a Kripke model assigns a set of \(n\)-tuples
of members of \(D\_s\) for each \(n\)-place predicate \(P^{n}\) at each
stage \(s\) in \(S\), subject to the constraint that if \(s \preceq
s'\) and \(P^n\) is true of \(\langle d\_1, \ldots, d\_n \rangle\) at
\(s\), then \(P^{n}\) is still true of \(\langle d\_1, \ldots, d\_n
\rangle\) at \(s'\).



The entry on
 intuitionistic logic
 provides a more formal and complete statement of Kripke's model
theory for intuitionistic quantificational logic and states Soundness
and Completeness Theorems for it. Very roughly, a negation \(\neg A\)
is true at a stage \(s\) if, and only if, \(A\) is not true at any
stage \(s'\) such that \(s \preceq s'\); an existentially quantified
formula \(\exists x A\) is true at a stage \(s\) if, and only if,
\(A\) is true of some \(d\) in \(D\_s\) at \(s\); and a formula
\(\forall xA\) is true at a stage \(s\) if, and only if, \(A\) is
true of every \(d\) in \(D\_{s'}\) for every \(s'\) such that \(s
\preceq s'\).



For a taste of what Kripke models can do, consider:


1. \(\neg \forall x \neg A \rightarrow \exists x A\)
2. \(\lnot \lnot \forall x A \rightarrow \forall x A\)
3. \(\forall x \lnot \lnot A \rightarrow \lnot \lnot \forall x A\)



Let \(\mathcal{K}\) be a Kripke model based on a frame in which we
have two stages \(s\_0 \preceq s\_1\). Assume \(\{d \}\) is the domain
associated to each stage by the model, and let a monadic predicate
\(P\) be true of \(d\) at \(s\_1\) but not at \(s\_0\). We may verify
that \(\neg \forall x \neg Px\) is true at \(s\_0\), even though
\(\exists x Px\) is not true at \(s\_0\). So, (i) is not
intuitionistically valid. One may likewise verify that \(\lnot \lnot
\forall x Px\) is true at \(s\_0\), even though \(\forall x Px\) is not
true at \(s\_0\). It follows that (ii) is not intuitionistically valid
either.



For (iii), consider, for example, a Kripke model based on a frame in
which we have an infinite number of stages \(s\_0 \preceq s\_1 \preceq
\ldots \preceq s\_n \preceq \ldots\). We may assume that \(D\_{s\_{n}} =
\{ d\_0 , \ldots, d\_n \}\) and that a monadic predicate \(P\) is true
at \(s\_n\) of every \(d\_m\) in the domain except for \(d\_n\). It turns
out that \(\forall x \lnot \lnot Px\) is true at \(s\_0\) even though
\(\lnot \lnot \forall x Px\) is not true at \(s\_0\). Therefore,
\(\forall x \lnot \lnot A \rightarrow \lnot \lnot \forall x A\) is not
intuitionistically valid either. These and similar examples are
discussed in Burgess 2009.


### 2.3 Substitutional Quantification



The model-theoretic interpretation of the language of quantificational
logic relied on the Tarskian definition of satisfaction in a model by
an assignment of values to the variables. But the assignment of an
object to a variable is never dependent on the availability of a term,
\(t\), for the object in the language for which one defines truth in a
model. Indeed, an existentially quantified formula \(\exists x A\)
may well be true in a model even if no atomic formula of the form
\(A(t/y)\) is true in the model.



Substitutional quantifiers \(\Pi \alpha \ A\) and \(\Sigma \alpha \
A\) are interpreted differently. An interpretation associates with
them not a domain of quantification, but rather a substitution class,
\(C\), of linguistic expressions of an appropriate syntactic category
in the initial language. The truth conditions for substitutionally
quantified sentences of the form \(\Sigma \alpha \ A\) and \(\Pi
\alpha \ A\) may be given in terms of the truth conditions of suitable
substitution instances of \(A(\epsilon / \alpha)\) in which each
occurrence of the substitutional variable \(\alpha\) has been replaced
with a linguistic expression, \(\epsilon\), of the appropriate
syntactic category in the substitution class for the quantifier:


* A sentence of the form \(\Sigma \alpha \ A\) is true relative to a
substitution class \(C\) if, and only if, some substitution of the
variable \(\alpha\) for an expression \(\epsilon\) in the substitution
class \(C\) for the quantifier, \(A(\epsilon / \alpha)\), is
true.
* A sentence of the form \(\Pi \alpha \ A\) is true relative to a
substitution class \(C\) if, and only if, every substitution of the
variable \(\alpha\) for an expression \(\epsilon\) in the substitution
class \(C\) for the quantifier, \(A(\epsilon / \alpha)\), is
true.



This characterization of substitutional quantification allows for
substitutional variables of different syntactic categories, whether
singular terms, predicates or sentences. Indeed, substitutional
quantification is often used to mimic quantification into predicate
and sentence position of the kinds discussed later in this entry.



Early work on substitutional quantification was developed in Marcus
(1961) but it soon became the subject of intense debate in the next
two decades as philosophers made use of substitutional quantification
in ontology and the philosophy of language and mathematics. Belnap and
Dunn (1968), Parsons (1971), Grover (1972), and Kripke (1976) are some
of the relevant papers in the debate. For a sense of some of the
purported applications of substitutional quantification, the reader
may consult the essays collected in Gottlieb 1981. For further
discussion, see Hand (2007).


#### 2.3.1 A simple example



Let us momentarily focus on singular terms. To keep matters simple,
consider an impoverished fragment of the language of arithmetic with
one constant, \(0\), read: "zero" and one functional
symbol, \(S\), read: "the successor of". The domain of the
intended interpretation consists of a set of natural numbers, which
are named by singular terms of the form \(0,\) \(SS0,\) \(SSS0,\)
..., etc. If we now associate the class of all such terms to the
substitutional quantifiers \(\Sigma \alpha\) and \(\Pi \alpha\), then
a sentence like \(\Sigma \alpha \ \alpha = \alpha\) is evaluated as
true in virtue of the truth of formulas like \(0 = 0\), whereas \(\Pi
\alpha \ \alpha < \alpha\) is evaluated as false in virtue of
falsity of formulas like \(0 < 0\). In general, a sentence of the
form \(\Sigma \alpha \ A\) exhibits the same truth conditions as an
infinitary disjunction of the form: 
\[A(0/\alpha) \vee A(S0/\alpha) \vee \ldots\vee A(SS
\overbrace{\ldots}^n S0/\alpha) \vee \ldots\]
 whereas a sentence of
the form \(\Pi \alpha \ A\) exhibits the same truth conditions as an
infinitary conjunction: 
\[A(0/\alpha) \wedge A(S0/\alpha) \wedge
\ldots\wedge A(SS \overbrace{\ldots}^n S0/\alpha) \wedge \ldots\]
 The case of arithmetic is optimal
because we have a name for each member of the intended domain. This
means that in general, a quantified sentence of the form \(\exists x
A\) will be true if, and only if its substitutional counterpart
\(\Sigma \alpha \ A\) is likewise true. But in this respect, however,
the language of arithmetic is the exception and not the rule. In real
analysis, for example, there are too many objects in the domain to
have a name in a countable language. In such a situation, we must
confront the risk that a sentence of the form \(\exists x A\) may be
true even if \(\Sigma \alpha \ A\) remains false in virtue of the lack
of a true substitution instance of the form \(A(t/ \alpha)\). This may
happen, for example, if the objects that satisfy the open formula
\(A\) are not denoted by any singular term in the language.



Call the initial language "the object language". And call
the language in which we explain the truth conditions of sentences of
the initial language "the metalanguage". In the
metalanguage, we have explained the truth condition for \(\Sigma
\alpha \ A\) in terms of what looks like objectual quantification over
linguistic expressions. To acknowledge this is of course not to claim
that the intended interpretation of substitutional quantification is
one on which it is merely objectual quantification over linguistic
expressions. But just what the intended interpretation of the
substitutional quantifier might be has been the subject of intense
controversy. Indeed, van Inwagen (1981) and Fine (1989), for example,
have each argued that there is no separate intended interpretation we
can understand independently from our grasp of objectual
quantification over linguistic expressions of the relevant sort.


#### 2.3.2 Kripke on Substitutional Quantification



Whatever the philosophical import of substitutional quantification,
Kripke (1976) makes plain that there is no technical obstacle to
introducing substitutional quantifiers into a given language. In what
is perhaps the canonical treatment of substitutional quantification,
he explained how to extend an interpreted language \(\mathcal{L}\)
into a substitutional language \(\mathcal{L}^{\Sigma}\) equipped with
substitutional quantification over a class of expressions in
\(\mathcal{L}\). One first expands the vocabulary of \(\mathcal{L}\)
with an infinite stock of variables \(\alpha\_1\), \(\alpha\_2\),
... and a substitutional quantifier \(\Sigma\). (Kripke defines
\(\Pi\) as the dual of \(\Sigma\), where \(\Pi \alpha \ A\)
abbreviates: \(\lnot \Sigma \alpha \lnot \ A\).)



An *atomic preformula* is an expression that results from a
sentence of \(\mathcal{L}\) when zero or more terms are replaced by a
substitutional variable. A *form* is an atomic preformula,
where the replacement of its substitutional variables with terms
yields back a sentence. We can now define a formula of
\(\mathcal{L}^{\Sigma}\) recursively. \(A\) is a formula of
\(\mathcal{L}^{\Sigma}\) if and only if either (i) \(A\) is a
*form* of \(\mathcal{L}^{\Sigma}\) or (ii) \(A\) is \(\lnot
B\), where \(B\) is a formula of \(\mathcal{L}^{\Sigma}\) or (iii)
\(A\) is \((B \rightarrow C)\), where \(B\) and \(C\) are each
formulas of \(\mathcal{L}^{\Sigma}\) or (iv) \(A\) is \(\Sigma \alpha
\ B\), where \(B\) is a formula of \(\mathcal{L}^{\Sigma}\). A
sentence of \(\mathcal{L}^{\Sigma}\) is a formula without free
substitutional variables.



Kripke defines truth for sentences of \(\mathcal{L}^{\Sigma}\)
recursively in terms of truth in \(\mathcal{L}\). If \(A\) is an
atomic sentence of \(\mathcal{L}^{\Sigma}\)--which may well be a
complex sentence of \(\mathcal{L}\), then \(A\) is true in
\(\mathcal{L}^{\Sigma}\) if, and only if, \(A\) is true in
\(\mathcal{L}\). If \(A\) is a sentence of the form \(\lnot B\), then
\(A\) is true in \(\mathcal{L}^{\Sigma}\) if, and only if, \(B\) is
not true in \(\mathcal{L}^{\Sigma}\). If \(A\) is of the form \((B
\rightarrow C)\), then \(A\) is true in \(\mathcal{L}^{\Sigma}\) if,
and only if, \(B\) is not true or \(C\) is true in
\(\mathcal{L}^{\Sigma}\). Finally, and more crucially, if \(A\) is of
the form \(\Sigma \alpha \ B\), then \(A\) is true in
\(\mathcal{L}^{\Sigma}\) if and only if \(B(\epsilon/\alpha)\) is true
in \(\mathcal{L}^{\Sigma}\) for some \(\epsilon\) in the substitution
class \(C\) associated with \(\Sigma\).



Kripke then defines a sentence \(A\) of the language of pure
substitutional quantificational logic to be valid if, and only if
\(A\) comes out true no matter what base language \(\mathcal{L}\) and
non-empty class of terms \(\mathcal{C}\) we take as input for the
substitutional expansion and what predicates of \(\mathcal{L}\) we
substitute for the predicates of \(A\). In particular, Kripke notes
that when the quantifiers of a valid sentence of pure quantificational
logic are suitably rewritten as substitutional quantifiers, we obtain
a valid sentence in the language of pure substitutional
quantificational logic.


## 3. Extensions of Classical Quantificational Logic



Each departure from classical quantificational logic we have
considered originated from an objection to either axioms of pure
quantificational logic or the Tarskian definition of satisfaction in a
model by an assignment of objects to the variables of the language. We
now take both for granted and look at proposed extensions of classical
quantificational logic with new styles of quantification. Each
extension will generally require us to add new axioms for the new
styles of quantification and to expand the Tarskian definition of
truth in a model in terms of satisfaction.


### 3.1 Generalized Quantifiers



Frege originally conceived of the quantifier \(\forall\) as a monadic
predicate that is true of a first-level concept \(F\) under which only
objects fall if, and only if, all objects fall under \(F\); likewise,
Frege assimilated \(\exists\) to a monadic predicate that is true of a
first-level concept if, and only if, at least one object falls under
\(F\). In modern terms, we may say that the truth conditions of
\(\forall x A\) are given in terms of a certain condition on the
extension of the formula \(A\). Given a model \(\langle D, I\rangle\)
and a formula \(A(x)\) with a single free variable \(x\), it will be
convenient to use \(A(x)[d]\) to abbreviate: \(A\) is satisfied in
\(\langle D, I\rangle\) by variable assignments on which \(d\) is the
value assigned to the variable \(x\). The extension of a formula
\(A(x)\) with a single free variable \(x\) in a model \(\langle D,
I\rangle\) is just \(\{ d \in D: A(x)[d] \}\), which we may abbreviate
\(I(A)\).


* \(\forall x A\) is
true in \(\langle D, I\rangle\) if, and only if, \(D \subseteq
I(A)\).



The contemporary study of generalized quantifiers begins with
Mostowski (1957) and Lindstrom (1966) (the entry on
 generalized quantifiers
 provides a survey of recent developments and applications in logic
and natural language). Lindstrom and Mowstoski began with a
generalization of Frege's conception of a quantifier in at least
two dimensions. First, they considered additional unary quantifiers
\(Q(A)\), which behave syntactically like \(\forall\) and \(\exists\)
but set a different condition on the extension of \(A\). Second, they
considered binary quantifiers \(Q(A, B)\), which set a given condition
based on a relation between the extensions of \(A\) and \(B\).


#### 3.1.1 Cardinality Quantifiers



Cardinality quantifiers provide a simple example of the first sort of
generalization. If we write \(card(S)\) to denote the cardinality of a
given set \(S\), we may introduce unary quantifiers of the form
\(Q(A)\), which set a cardinality condition on the extension of a
formula \(A\):


* \(\exists\_{\geq n} x
A\) is true in \(\langle D, I\rangle\) if, and only if, \(card(I(A))
\geq n\)
* \(\exists !\_{n} x A\)
is true in \(\langle D, I\rangle\) if, and only if, \(card(I(A)) =
n\)



\(\exists\_{\geq n}\) is read: "at least \(n\) objects" and
\(\exists !\_{n}\) is read: "exactly \(n\) objects." Both
quantifiers are definable in terms of a standard quantifier against
the background of classical quantificational logic.



Other cardinality quantifiers, however, do increase the expressive
power of classical quantifical logic. We may, for example, introduce
cardinality quantifiers of the form \(Q(A)\), which require the
extension of \(A\) to have an infinite cardinality:


* \(Q\_0 x \ A\) is true
in \(\langle D, I\rangle\) if, and only if, \(card(I(A)) \geq
\aleph\_0\)
* \(Q\_1 x \ A\) is true
in \(\langle D, I\rangle\) if, and only if, \(card(I(A)) \geq
\aleph\_1\)



\(\aleph\_0\) is the cardinality of the set of natural numbers, and
\(\aleph\_1\) is the next transfinite
 cardinal.[9]
 When we supplement classical quantificational logic with \(Q\_0\),
read: "at least \(\aleph\_0\) objects", we must relinquish
Compactness even if we can still retain the Lowenheim-Skolem
theorem. The opposite is the case, however, when we supplement
classical quantificational logic with \(Q\_1\), read: "at least
\(\aleph\_1\) objects" instead.



Two more cardinality quantifiers that have been studied in the
literature are the Chang quantifier and the Rescher quantifier:


* \(Cx \ A\) is true in
\(\langle D, I\rangle\) if, and only if, \(card(I(A)) =
card(D)\)
* \(R x \ A\) is true in
\(\langle D, I\rangle\) if, and only if, \(card(I(A)) > card(D -
I(A))\)



The Chang quantifier collapses into \(\forall\) only in the finite
case; if \(D\) is infinite, then the extension of a formula \(A\),
\(I(A)\), may match the cardinality of the domain, \(D\), even if the
two sets are different and \(I(A) \neq D\).


#### 3.1.2 Generalized Binary Quantifiers



The second dimension of generalization concerns the number of
arguments we let quantifiers take. We can, for example, introduce
binary quantifiers, \(Q(A, B)\), whose truth condition is given in
terms of a binary relation between the extension of the formula \(A\)
and the extension of the formula \(B\) in a model.


* \(All(A, B)\) is true in \(\langle D, I\rangle\) if, and only if,
\(I(A) \subseteq I(B)\)
* \(More(A, B)\) is true in \(\langle D, I\rangle\) if, and only if,
\(card(I(A)) > card(I(B))\)
* \(Most(A, B)\) is true in \(\langle D, I\rangle\) if, and only if,
\(card(I(A) \cap I(B)) > card(I(A)-I(B))\)



As defined above, \(Most(A, B)\), for example, is true if and only if
more objects are both \(A\) and \(B\) than there are objects that are
\(A\) and not \(B\).



Of special interest in the philosophy of language is the case of
Russellian definite descriptions, which can be subsumed under the
category of a binary quantifier:


* \(The(A, B)\) is true
in \(\langle D, I\rangle\) if, and only if, \(card(I(A)) = 1\) and
\(I(A) \subseteq I(B)\)



For a systematic discussion of these issues, the reader may consult
Peters & Westerstahl (2006).


### 3.2 Many-Sorted Quantification



We generally let quantifiers bind a single type of variable, but we
may find it helpful, on occasion, to let them bind additional types of
variables in order to quantify over different sorts of values. In
arithmetic, for example, we may want to be able to quantify over
natural numbers and sets thereof and we may find it convenient to
classify the individual variables of the language into at least two
categories. We may reserve lowercase variables \(x, y, z, x\_1\),
... to range over natural numbers, and we may use uppercase
variables \(X, Y, Z, X\_1\), ... to range over sets of natural
numbers. In a many-sorted language with functional symbols and
individual constants, each singular term would be assigned a single
sort. If we want identity, we need to include an identity predicate
for each sort of a variable and insist on each such predicate to be
flanked only by singular terms of the appropriate sort. Other
predicates may nevertheless take arguments of different sorts; for
example, in a two-sorted language in which lowercase variables range
over numbers and uppercase variables range over sets of numbers, \(x
\epsilon X\), for example, can take as arguments variables of
different sorts and may be read "\(x\) is in \(X\)".



To accommodate the new styles of variables in the model theory for
many-sorted logic, we may add a domain for each style of variable. A
model for a two-sorted language, for example, may consist of an
ordered triple \(\langle D\_1, D\_2, I\rangle\) in which \(D\_1\) and
\(D\_2\) specify the domain of quantification associated to each style
of variable, and \(I\) is an interpretation function, which assigns
appropriate values and extensions to each expression in the
non-logical vocabulary. (Since each singular term is assigned a sort,
\(I\) must be constrained to make sure singular terms of each sort
denote objects in the appropriate domain.) The metatheory for
many-sorted logic is closely related to the metatheory for classical
quantificational logic. One may, for example, prove many-sorted
versions of Compactness and the Lowenheim-Skolem Theorem. Details
may be found in Enderton 2001.



However, the move to a many-sorted language is largely a matter of
convenience. Many-sorted quantification may be analyzed in terms of
restricted one-sorted quantification. For each many-sorted language
with \(n\) styles of variable, we may introduce a one-sorted language,
which comes with a one-place predicate, \(S\_i\), for each sort in the
many-sorted language. We may set out to translate each formula of the
many-sorted language into a formula of the one-sorted language: we
translate each identity in terms of a single identity predicate; a
quantification on a variable of sort \(i\), \(\forall x^i A\), is
replaced with a formula of the form \(\forall x(S\_i x \rightarrow
A)\). We may, in addition, turn each model for the many-sorted
language \(\langle D\_1, \ldots, D\_n, I \rangle\) into a model for the
resulting one-sorted language \(\langle \bigcup\_{n} D\_n, I^\ast
\rangle\) in which we take the domain to consist in the union of the
domains associated to each original sort. The interpretation function,
\(I^\ast\) will coincide with \(I\) when it comes to predicates and
individual constants. However, \(I^\ast\) will generally have to
extend the interpretation of a functional symbol, \(f^n\_i\), by \(I\),
in one way or another. Finally, \(I^\ast\) will interpret the identity
predicate as usual.



Still, even if there is no real gain in expressive power, one reason
to flag the availability of many-sorted quantification is because some
of the extensions of classical quantificational logic to be examined
below are closely related to certain theories couched in the language
of two-sorted logic. In particular, second-order logic and the theory
of plural quantification will be each closely related to two
first-order two-sorted theories, which lack the expressive resources
often attributed to each extension of classical quantificational
logic.


### 3.3 Second-Order Quantifiers



Second-order logic is an extension of classical quantificational logic
in which we allow for quantification into predicate position.
Second-order logic originated with Frege (1879), which developed a
formal language equipped with predicate variables of the form
\(X^{n}\_{i}\). Predicate variables behave syntactically like
\(n\)-place predicates of the form \(P^{n}\_{i}\). In second-order
logic, a predicate variable \(X^{n}\_{i}\) followed by \(n\) variables
\(X^{n}x\_{1}, \ldots, x\_{n}\) counts as an atomic formula. We finally
let the quantifiers \(\forall\) and \(\exists\) bind predicate
variables as in \(\forall X^{n}\, A\) and \(\exists X^{n}\,A\).


#### 3.3.1 Axioms for second-order quantification



The axioms for second-order logic supplement the axioms of classical
quantificational logic with distinctive axioms for the second-order
quantifiers. Call a predicate variable \(Y^n\) *free for*
\(X^n\) in \(A\) if, and only if, no free occurrences of \(X^n\) lie
within the scope of a quantification on \(Y^n\), \(\forall Y^n\) or
\(\exists Y^n\):


* (\(\forall^{2}1\)) \(\forall
X^{n}\,A \rightarrow A(U^{n}\_{j}/X^{n})\),
  

*provided \(U^{n}\_{j}\) is free for \(X^{n}\) in
\(A\)*.
* (\(\forall^{2}2\)) \(\forall
X^{n}(A \rightarrow B) \rightarrow (A
\rightarrow \forall X^{n}\,B)\),
  

*provided \(X^{n}\) does not occur freely in
\(A\)*.



(\(\forall^2 1\)) is a second-order version of *universal
instantiation*. The second-order version of *universal
generalization* becomes:


* (\(\forall^{2}3\)) from
\(A\), infer \(\forall X^{n}\,A\).



Observe that while \(\forall X(Xx) \rightarrow Px\) is an instance
of (\(\forall^2 1\)), the formula \(\forall X(Xx) \rightarrow (Px
\wedge Qx)\) is not an instance of the axiom. But whatever one takes
the value of a second-order variable to be, you may think that there
should be one under which an object falls if and only if it satisfies
the condition stated by the formula \((Px \wedge Qx)\). To make
sure, we may rely on an axiom of *second-order
comprehension*:


* \(\exists X^{n}
\forall x\_1, \ldots x\_n(X^{n}x\_1, \ldots, x\_n \leftrightarrow A(x\_1,
\ldots, x\_n))\)
  

*provided \(X^{n}\) does not occur freely in \(A(x\_1, \ldots ,
x\_n)\)*.



A formula like \(\forall X(Xx) \rightarrow (Px \wedge Qx)\) now
follows from the combination of second-order universal instantiation
and an instance of second-order comprehension: \(\exists X\forall
x(Xx \leftrightarrow (Px \wedge
Qx))\).[10]


#### 3.3.2 Standard Models for Second-Order Logic



To develop a model theory for a second-order language, we may still
take a model \(\langle D, I\rangle\) to consist of a non-empty domain
and an interpretation function. We now let an assignment \(s\) for
\(\langle D, I\rangle\) to be a function that assigns a member of
\(D\) to each individual variable and a set of \(n\)-tuples of members
of \(D\) to each predicate variable \(X^n\). We then modify the
definition of satisfaction to let a formula of the form \(\forall
X^n\,B \) be satisfied in \(\langle D, I \rangle\) by an assignment
\(s\) if, and only if, \(B\) is satisfied by all assignments of the
form \(s[X^n/R^n]\), where \(R^n\) is a set of \(n\)-tuples of members
of \(D\_1\). The definitions of truth in a model and validity proceed
exactly as in the first-order case.



The entry on
 second-order and higher-order logic
 provides more detail and gives some indication of the complexity of
the set of valid second-order formulas in the standard model theory
for second-order logic. No Completeness, Compactness, or
Lowenheim-Skolem theorem is available for second-order logic with
standard models. The entry on
 second-order and higher-order logic
 provides concrete illustrations of these facts.


#### 3.3.3 Henkin Models for Second-Order Logic



A Henkin model lets a second-order variable \(X^n\) range over some
subset of the set of \(n\)-tuples of members of the domain \(D\). Thus
a Henkin model consists of a domain \(D\) of individuals, a domain
\(D^n\_2\) of sets of \(n\)-tuples of objects in \(D\) for each \(n
\leq 1\), and an interpretation function \(I\). To make sure that
every instance of second-order comprehension is validated in the
model, we require each \(D^n\_2\) to contain every set of \(n\)-tuples
of members of \(D\) that is definable by a formula \(A\) of the
language.



This is what the entry on
 second-order and higher-order logic
 calls general semantics. The metatheory of second-order logic with
Henkin models is very much like the metatheory of classical
quantificational logic in that it is complete, compact, and subject to
a Lowenheim-Skolem theorem. One reason for this is that
second-order logic under the Henkin model theory is only a notational
variant of a two-sorted first-order theory in which the axioms of
quantificational logic are supplemented by suitable axioms of
comprehension.



Which model theory is more appropriate for second-order logic? There
is a trade-off between the attractive features of the metatheory of
second-order logic with Henkin models, on the one hand, and the
expressive power of second-order logic with standard models, on the
other. Frege's own interest in second-order quantification had
much to do with its ability to express the concept of the
*ancestral* of a dyadic relation \(R\), which allowed him in
turn to define the concept *natural number* in terms of \(0\)
and *successor*. For details, the reader may consult the entry
on
 Frege's Theorem and Foundations of Arithmetic.
 There are, in addition, *categorical* axiomatizations of
arithmetic, where a set of sentences is *categorical* if, and
only if, all of its models are isomorphic, which means that there is
only one model up to isomorphism. Dedekind (1893) famously proved, for
example, that the second-order formulation of the Peano Axioms
characterized the structure of the natural number system. There is a
similar result for real analysis. For second-order set theory, Zermelo
(1930) observed that given two models of second-order Zermelo-Fraenkel
set theory, one is isomorphic to a strongly inaccessible initial
segment of the other. The entry on
 set theory
 describes the axioms of set theory. Other examples of the expressive
resources afforded by second-order quantification with standard models
are discussed in Shapiro (1991). The adoption of a model theory based
on Henkin models requires one to surrender all these applications.


#### 3.3.4 Models and Interpretations



Much like in the case of pure quantificational logic, no
model--standard or otherwise--for the language of
second-order logic interprets the quantifiers to range over a
universal domain of discourse. But if there are interpretations of the
language of second-order logic to which no model corresponds, the
question arises again of whether truth in all models is tantamount to
truth under all interpretations of the language. Unfortunately, in the
absence of a Completeness theorem, Kreisel's argument is not
available for second-order logic with standard models.



One may nevertheless argue that a formula is true in all models only
if it is true under all interpretations of the language by appeal to a
second-order principle of reflection. This reflection principle makes
sure that a formula \(A\) of the language of second-order set theory
is true when the quantifiers are taken to range unrestrictedly over
the universal domain if, and only if, \(A\) is true when the
quantifiers are restricted to some set-sized initial segment of the
universe of a certain
 sort.[11]
 Shapiro (1987) discusses both the set-theoretic principle and its
role in the justification of the claim that if a sentence is true in
every standard model, then it is true under every interpretation of
the language. Unfortunately, second-order reflection takes us beyond
the scope of second-order Zermelo-Fraenkel set theory and the axiom of
choice (ZFC).



To make sense of interpretations over a universal domain, one may
conceive of an interpretation as a value of a second-order predicate
variable, \(I\), which applies to ordered pairs of a certain sort. An
interpretation, \(I\), applies to ordered pairs of the form \(\langle
\forall, o \rangle\), when \(o\) is in the domain of the
interpretation, and \(I\) applies to ordered pairs of the form
\(\langle P^{n}\_{i}, \langle a\_1, \ldots, a\_n\rangle \rangle\), when
\(P^{n}\_{i}\) is taken to apply to the objects in \(\langle a\_1,
\ldots, a\_n\rangle\). The usual definition of an assignment is then
replaced by one on which an assignment is given by the value of a
second-order variable \(S\), which applies to an ordered pair of the
form \(\langle x, o\rangle\), where \(o\) is an object, for each
first-order variable in the language, and to ordered pairs of the form
\(\langle X, o\rangle\), where \(o\) is an object, for each
second-order variable in the language. An assignment \(S\) is required
to pair each first-order variable with exactly one object, but it is
allowed to pair a second-order variable with any objects whatever. In
the presence of these definitions, we can generalize Tarski's
definition satisfaction in an interpretation by an assignment, truth,
and
 validity.[12]


#### 3.3.5 How to Understand Second-Order Quantification



The status of second-order quantification remains controversial. The
standard model theory for second-order quantification states the truth
conditions of a formula of the form \(\forall X\,A\) in terms of
ordinary quantification over subsets of the domain. This may have
inclined some to agree with Quine (1986) that second-order
quantification is only intelligible as covert first-order
quantification over sets. Second-order logic would, on this view, be
best conceived as a species of two-sorted first-order set theory. But
the model-theoretic interpretation is not the *only*, much less
*the* intended interpretation of second-order quantification.
Indeed, Boolos (1984) proposed an alternative interpretation of
*monadic* second-order logic in terms of an ingenious
translation into a fragment of natural language on which second-order
quantifiers became plural quantifiers on which more below. There is an
even more radical alternative, which is to deny that the
interpretation of second-order quantification should be determined by
either the model theory for second-order logic or by means of any
explicit or tacit translation into some fragment of natural language.
Instead, the thought continues, one may have to rely a more direct
method and learn what second-order quantifiers mean through immersion
and triangulation with natural language. For we understand what is
meant by quantification into the position of a singular term in a
predication in natural language. That constrains the interpretation of
second-order quantification, which stands to predicates as first-order
quantification stands to singular terms. This is similar to the
approach one finds in Prior 1971 (Ch. 3) and Williamson 2003.


### 3.4 Plural Quantifiers



Much of the recent interest in plural quantification in logic and
philosophy began with Boolos(1984), who provided a plural
interpretation of monadic second-order quantification. Boolos wanted
to defend second-order axiomatizations of set theory against the
charge that second-order quantification over the domain of all sets is
incoherent, since there is no set of all sets. In response, he
explained how to interpret the second-order quantifier, \(\forall X\),
over the domain of all sets in terms of the plural quantifier
"some sets", and how to interpret atomic formulas of the
form \(Xx\) in terms of the plural locution "is one of".
The fact that plural locutions are systematically used and perfectly
understood in natural language, Boolos thought, helped address the
charge of incoherence. The Boolos translation is described in detail
in the entry on
 plural quantification.



Whatever the merits of the Boolos translation, plural quantification
is nowadays generally distinguished from second-order quantification
on the grounds that they bind expressions of different syntactic and
semantic categories--while predicates are *unsaturated* in
Frege's terminology and contain argument places to be completed
by expressions of an appropriate syntactic category, plural terms are
often thought to be akin to *saturated* singular terms.
Different variations on this theme may be found, for example, in
Higginbotham 1998; Oliver & Smiley 2001; Rayo & Yablo 2001;
Williamson 2003, 2013. This is a book, which includes an extensive
discussion of important differences in the behavior of plural and
second-order quantification in modal contexts. Plural quantification
has consequently become the subject of its own study.



The language of plural quantification is carefully described in the
entry on
 plural quantification.
 Syntactically, it is closely related to a two-sorted first-order
language in which we have an infinite number of variables \(xx\),
\(yy\), \(zz\), \(xx\_{1}\), .... , and a two-place predicate
\(\prec\), read "one of," which will be flanked by
singular and plural terms, respectively. Identity predicate can only
flanked by singular terms.



The axioms for plural quantification extend the axioms of
quantificational logic with plural counterparts of (\(\forall 1\)),
(\(\forall 2\)), and (\(\forall 3\)). Like in the second-order case,
the axioms of quantification are supplemented with an axiom schema of
plural comprehension:


* \(\exists x A
\rightarrow \exists xx \forall y(y \prec xx \leftrightarrow A)\)
  

*provided \(xx\) does not occur freely in \(A\)*.



One important difference with respect to monadic second-order logic is
that instances of plural comprehension are conditionals that state
that if something satisfies a certain condition, then some objects are
all and only objects satisfying the condition. The antecedent of each
conditional is motivated by an interpretation of the plural quantifier
\(\exists xx\) to mean "one or more objects". Indeed,
Linnebo (2013) explicitly supplements the theory of plural
quantification with an axiom that states that no matter what some
objects may be, there is at least one of them:


* \(\forall xx \ \exists
y (y \prec xx)\)



Like in the case of second-order logic, we may provide at least two
sorts of models for the theory of plural quantification. In both
cases, we interpret plural quantification in terms of quantification
over non-empty sets of individuals in the domain \(D\). The difference
is that in standard models we let plural variables range over the full
power set of \(D\), whereas plural variables range over a set \(S\) of
non-empty subsets of \(D\), provided only that \(S\) contains all sets
definable by some formula.



Both approaches take place in a singular metalanguage. Since the
models are based on set theory, they require the domain of individuals
to form a set. Furthermore, the definition of satisfaction explains
what is for a formula \(\forall xx A\) to be satisfied in a model by
an assignment in terms of *singular* quantification over
non-empty subsets of the domain of individuals. This may perhaps
encourage the impression that plural quantification is only
intelligible as singular quantification over non-empty sets of
individuals. Note, however, that neither the use of set-based models
nor the use of singular quantification in the definition of
satisfaction for a formula of the form \(\forall xx\) is compulsory.
In a plural metatheory in which we have a theory of ordered pairs, one
may adapt the generalized conception of an interpretation developed in
Rayo & Uzquiano (1999) to provide a plural model theory in which
one uses neither. This model theory is developed, for example, in
Burgess 2004.



The status of plural quantification remains controversial. All parties
agree that plural locutions are systematically used in natural
languages, but there is no consensus as to whether or not the truth
conditions for plurally quantified sentences should ultimately involve
covert reference to complex objects of some kind or another. The entry
on
 plural quantification
 discusses some reasons generally offered in favor of the
irreducibility of plural quantification to singular quantification
over sets. But an influential argument against the identification of
plural quantification and singular quantification over sets traces
back to Boolos 1984, which claims that the following two sentences
differ in truth value when we let the domain of quantification include
all sets there are:


1. There is a set of all, and only, non-self-membered sets.
2. Some sets are all, and only, non-self-membered sets.



We know from Russell's paradox that there is no set of all, and
only, non-self-membered sets, which makes the first sentence
 false.[13]
 But this is not to deny of course that some sets are all and only
non-self-membered sets, even if they fail to compose a set.
Unfortunately, the argument depends on a certain interpretation of
Russell's paradox that is itself controversial and will fail to
convince philosophers who think that the moral of Russell's
paradox is that no singular quantifier can range unrestrictedly over
all the sets there are.



Even if you take plural quantification at face value as opposed to
covert singular quantification over sets, you may still think that the
specification of some objects in the plural is all it takes to
characterize a set whose members are exactly those objects. This type
of consideration has recently led Florio and Linnebo (2021) to develop
what they call "critical plural logic" as an alternative
to classical plural logic characterized by a restriction of the axiom
of plural comprehension above.


### 3.5 Propositional Quantifiers



It is not uncommon to let the vocabulary of classical quantificational
logic include 0-place predicates, which are treated as propositional
variables. This approach makes propositional logic a proper fragment
of pure quantificational logic, and it construes quantification into
the position of a sentence as a species of second-order
quantification, i.e., quantification into the position of a 0-place
predicate.



Or we may supplement the language of propositional logic with a
propositional quantifier, \(\forall\), which is allowed to bind
propositional letters such as \(p\), \(q\), \(r\), .... This is,
for example, the course of action taken by Prior (1971), who outlines
a system of propositional quantification. Prior insisted that
propositional quantification should not be confused with either
objectual quantification over propositions or substitutional
quantification over sentences. Rather, propositional quantification is
a species of non-nominal quantification, since it is quantification
into sentence position.


#### 3.5.1 Quantified Propositional Logic



The vocabulary of quantified propositional logic contains
propositional letters \(p\), \(q\), \(r\), \(p\_1\), ... and
propositional connectives \(\neg, \rightarrow, \wedge, \vee,
\leftrightarrow\). We may as usual treat \(\neg\) and \(\rightarrow\)
as primitive and take the others as defined. The usual recursive
definition of a formula for propositional logic is appropriately
supplemented with a clause that states that \(\forall p A\) is a
formula whenever \(A\) is a formula.


#### 3.5.2 Axioms for Propositional Quantification



The axioms for propositional quantification are direct counterparts of
\((\forall 1)-(\forall 3)\). Call a sentence \(B\) *free for* a
propositional variable \(p\) in \(A\) if, and only if, no free
occurrences of \(p\) lies within the scope of a quantification on
\(q\), \(\forall q\) or \(\exists q\), where \(q\) is a propositional
variable which occurs in \(B\). Now:


* (\(\forall^{P}1\)) \(\forall
p A \rightarrow A(B/p)\)
  

*provided \(B\) is free for \(p\) in \(A\)*.
* (\(\forall^{P}2\)) \(\forall
p(A \rightarrow B) \rightarrow (A \rightarrow
\forall pB)\)
  

*provided \(p\) is not free in \(A\)*.
* (\(\forall^{P}3\)) from
\(A\), to infer \(\forall p A\)



For a taste of what they can do, notice that they yield a succinct
justification of a rule of Uniform Substitution, whereby a formula of
the form \(A(B/p)\) is a theorem whenever \(A\) is a theorem, provided
that \(B\) is free for \(p\) in \(A\). So, for example, if
\(p\rightarrow (q \rightarrow p)\) is a theorem, then \(\neg r
\rightarrow (q \rightarrow \neg r)\) is likewise a theorem, since this
last formula is just \(p\rightarrow (q \rightarrow p)(\neg
 r/p)\).[14]



The axioms immediately deliver every instance of a propositional
comprehension schema as a theorem:


* \(\exists p(p
\leftrightarrow A)\),
  

*provided \(p\) is not free in \(A\)*.



This is one important difference with respect to second-order
quantification generally, since we are not automatically able derive
every instance of second-order comprehension from the other
quantificational axioms. On the other hand, a simple instance of
existential generalization allows one to move from suitable formulas
of the form \(A \leftrightarrow A\) to an instance of propositional
comprehension of the form \(\exists p(p \leftrightarrow A)\). 


#### 3.5.3 Propositional Quantification and Truth



Propositional quantification has been used in order to present a
deflationary theory of truth. Grover, Camp, & Belnap (1975) and
Grover (1992) have, for example, proposed to make use of propositional
quantification in order to provide a variation on Ramsey's
redundancy theory of truth. The main claim they make is that we can
conceive of phrases like "that is true" or "it is
true" as *protosentences*, which are supposed to stand to
sentences as pronouns stand to nouns; unlike pronouns like
"it", protosentences occupy sentence position, but just
like pronouns may be used for purposes of cross-reference to earlier
nouns, protosentences may be used for purposes of cross-reference to
sentences uttered earlier in a conversation. Thus phrases like
"that is true" or "it is true" are akin to
propositional variables, which may be bound by propositional
quantifiers.


#### 3.5.4 Intensional Paradoxes



Prior (1961) makes use of propositional quantification in order to
provide an intensional version of the Liar Paradox. In particular, if
\(T\) is a sentential operator of the appropriate sort, then the
following is a consequence of the axioms for propositional
quantification:


* (P) \(T \forall p(Tp
\rightarrow \neg p) \rightarrow (\exists p(Tp \wedge p) \wedge
\exists p(Tp \wedge \neg p))\)



If we take \(T\) as Prior did, to stand for the sentential operator
"It is said by a Cretan that", then we may read the
theorem as the claim that if it is said by a Cretan that whatever is
said by a Cretan is not the case, then there are at least two things
said by a Cretan. So, quite unexpectedly, we learn that it is
inconsistent to assume that the only thing said by a Cretan is that
nothing a Cretan says is the case.



Prior's observation makes no special appeal to special
principles governing the behavior of the sentential operator he chose,
and the result carries over to a range of propositional attitude
operators such as "S believes that", "S hopes
that", "S writes on the board at midnight that",
etc. It is, for example, inconsistent to suppose that the only thing S
writes on the board at midnight is that everything S writes on the
board at midnight is false. 



Kaplan (1995) makes use of a related observation to raise a problem
for the standard model theory for quantified propositional modal
logic. He observed, in particular, that if \(Q\) is a sentential
operator of the appropriate sort, then the sentence (K) below is not
satisfiable in anny possible worlds model \(\langle W, \mathcal{I}
\rangle\) consisting of a set of possible worlds \(W\) and an
interpretation function, \(\mathcal{I}\) assigning a set of sets of
possible worlds to the sentential operator:


* (K) \(\forall p
\Diamond \forall q (Qq \leftrightarrow p=q)\)



When we let the propositional quantifier range over sets of worlds in
\(W\), we find that Kaplan's sentence can only be satisfied in
such a model if there is a map from the set of sets of worlds in \(W\)
into \(W\), which is not possible on account of Cantor's
theorem. Kaplan thought this is a problem because the model theory for
quantified propositional model should not unduly constrain the space
of intuitive possibilities. If we interpret \(Qp\) to mean that \(p\)
is Queried, i.e., it is asked whether it is the case that \(p\), then
(K) states that for every proposition, \(p\), it is possible that
\(p\) and only \(p\) is Queried.



The two observations are related. Prior's theorem provides a
witness to the claim that some propositions cannot be Queried
uniquely. If the proposition expressed by the condition \(\forall
p(Qp\rightarrow \neg p)\) is the only proposition that is Queried,
then it is true if and only if it is false. So, it can only be Queried
if it is false. But it is false only if something true is
 Queried.[15]



#### 3.5.5 Quantified Propositional Modal Logic



Propositional quantification made an early appearance in Lewis,
Langford, & Lamprecht 1932, which is commonly acknowledged to have
pioneered the study of modal logic. They outlined a formal system of
modal logic with propositional quantification, which they used to
formulate an *Existence Postulate* designed to guarantee the
existence of at least two propositions that are consistent and
independent from each other. (This is postulate B9 in the entry on the
 modern origins of modal logic.)
 They used the postulate to make sure strict implication did not
collapse into material
 implication.[16]



Kripke (1959), Fine (1970), and Kaplan (1970) went on to consider
systems of propositional modal logic supplemented with propositional
quantifiers governed by appropriate axioms and rules of inference.
Moreover, they explained how to generalize the Kripkean model theory
for propositional modal logic in order to accommodate the presence of
propositional quantification. A Kripke model for the language of
propositional modal logic is an ordered triple, \(\langle W, R,
\mathcal{I} \rangle\), in which \(W\) is a non-empty set of worlds,
\(R\) is a binary relation on \(W\), and \(\mathcal{I}\) is an
interpretation function, which can be taken to assign a set of
possible worlds to each sentence letter in the language. This suggests
we interpret propositional quantification in terms of quantification
over sets of worlds in \(W\). An assignment of values to the
propositional variables \(s\) will map each propositional variable
into a set of worlds, and we define satisfaction at a world by an
assignment \(s\) as usual, except for an appropriate clause for the
quantifier: \(\forall p A\) is satisfied at a world \(w\) in \(W\)
in \(\langle W, R, \mathcal{I} \rangle\) by assignment \(s\) if, and
only if, \(A\) is satisfied at \(w\) by every assignment \(s[p/S]\) in
which \(S\) is a subset of \(W\). The definitions of truth at a world
and truth in a Kripke model then proceed as usual.



The axioms for propositional quantification are sound with respect to
the class of extended Kripke models for quantified propositional modal
logic. Unfortunately, they are not complete with respect to the same
class, since they do not allow us to derive the following as a
theorem:


* \(\exists p(p \wedge
\forall q(q \rightarrow \Box (p \rightarrow q)))\)



However, this formula is valid in every extended Kripke model for the
simple reason that its truth at a world in a model only requires the
existence of a maximally specific proposition, one which is true at
exactly one world.


#### 3.5.6 How to Understand Propositional Quantification



The question remains of how to understand quantification into sentence
position. One may in fact be tempted to assimilate propositional
quantification to either singular quantification over propositions or
to substitutional quantification over a range of sentences. While
Prior (1971) and Grover (1972), for example, take propositional
quantification to be importantly different from singular and
substitutional quantification, Richard (1996) suggests that
propositional quantification should at the end of the day be treated
as a species of singular quantification, e.g., singular quantification
over propositions. Indeed, one may be tempted by this option by
reflecting on what we did above. We expanded the language of
propositional modal logic with a new style of quantification, but the
model theory explained the truth conditions associated with the
propositional quantifier, \(\forall p A\), in terms of singular
quantification over sets of worlds in the model.



One question is how to develop a fruitful model theory for the
language of quantified propositional logic and another question is
whether there is an intelligible interpretation of the formalism on
which propositional quantification is construed as neither singular
quantification over propositions nor as substitutional quantification
over a range of sentences. One option at this point is to deny again
that the interpretation of propositional quantification should be
determined by either the model theory for the language or by
translation into some fragment of natural language. Indeed, one may
adopt the line we mentioned above in connection to second-order
quantification. The best method to learn what is meant by a
propositional quantifier is through immersion and triangulation with
natural language by reflection on the fact that propositional
quantifiers stand to sentences as first-order quantifiers stand to
singular terms. 


## 4. Quantification and Ontology



Much of contemporary ontology builds on the assumption that existence
is to be understood in terms of quantification: in a slogan, to exist
is to be something. Ontology is largely concerned with the domain of
the *existential* quantifier. This assumption can be traced
back to the work of Frege and Russell, both of whom analyzed
quantification in terms of predication, and plays a crucial role in
Quine's admonition to transform ontology into the study of the
ontological commitments of our global theory of the world regimented
in the language of quantificational logic and identity.


### 4.1 Quantification, Predication, and Existence: Frege and Russell



By the link between quantification and existence, we mean the claim
that to exist is to be identical to something. This view can be traced
back to Frege and Russell, who offered roughly the same broad analysis
of quantification in terms of predication. The entry on
 existence
 provides a helpful overview and discussion of their account of
existence against the background of the historical context in which
they produced it. In what follows, we highlight a distinction between
the Frege-Russell analysis of quantification in terms of predication,
on the one hand, and the link between quantification and existence, on
the other.


#### 4.1.1 Quantification and Predication



Frege (1980a, 1980b) explicitly analyzed quantification in terms of
predication. For Frege, first-level predicates express concepts under
which objects fall. A quantifier expresses a second-level concept
under which first-level concepts fall. In particular, he proposed to
take a sentence like (3) below to predicate of a first-level concept
that it has instances:


* (3) There is at least
one square root of 4.
* (4) The concept
*square root of 4* is instantiated.



The existential quantifier, \(\exists\), is, for Frege, a second-level
predicate, which expresses a second-level concept under which a
first-level concept such as *square root of 4* falls if and
only if it has some instances. Likewise, the universal quantifier,
\(\forall\), is a second-level predicate, which expresses a
second-level concept under which a first-level concept such as
*self-identical* falls if and only if it has all objects as
instances.



Russell (1905) offered a similar account of quantification. On
Russell's analysis, the proposition expressed by (5) predicates
of a certain propositional function that it maps every object whatever
into a true proposition. (The entry on
 propositional functions
 provides more detail on the role of propositional functions in the
early development of modern logic.)


* (5)Everything is
self-identical.
* (6)The propositional
function \(x\) *is self-identical* is always true.



The propositional function \(x\) *is self-identical* maps the
Moon to the true proposition that the Moon is self-identical, it maps
the Sun to the true proposition that the Sun is self-identical, etc.
The thrust of (5) is the proposition that no matter what object we
supply as argument, the propositional function \(x\) *is
self-identical* will map it into a true proposition. The only
salient difference between Frege and Russell's account of
quantification lies in the choice of concepts and propositional
functions, respectively, as the objects of predication; while Frege
takes each quantifier to correspond broadly to a second-level concept,
Russell analyzes them in terms of certain properties of propositional
functions.


#### 4.1.2 Quantification and Existence



Both Frege and Russell combine their account of quantification in
terms of predication with the substantive thesis that existence is to
be understood in terms of quantification. For Frege, existence is to
be identified with the second-level concept expressed by the
second-level predicate expressed by \(\exists\). For Russell,
existence is identified with a certain property of propositional
functions, e.g., being sometimes true. This may seem overkill; if
true, it might seem that only first-level concepts exist, not the
objects they instantiate. For only first-level concepts are instances
of the second-level concept expressed by
 \(\exists\).[17]
 Instead, they are mostly concerned with the thesis that statements of
existence such as (7) are *not* cases in which a primitive
property of existence is predicated of an object, but rather should be
understood as an implicitly quantified statement such as (8), which is
to be analyzed in terms of predication in (9) and (10) for Frege and
Russell, respectively:


* (7) 2
exists.
* (8) Something is
identical to 2.
* (9) The concept
*identical to 2* is instantiated.
* (10) The propositional
function \(x\) *is identical to 2* is sometimes
true.



This point is perfectly compatible with the existence of a first-level
concept under which all and only those objects that exist fall. Take,
for example, the concept *being self-identical*. Likewise for
Russell. The propositional function \(x\) *is self-identical*
will map all objects that exist into a true proposition.



Note, however, that one could in principle retain their analysis and
deny, for example, that first-level concepts and propositional
functions, respectively, can only be saturated by objects that exist.
One could take the view that some great philosophers who once existed,
no longer exist. Socrates, for example, was a great philosopher who no
longer exists. He can nevertheless instantiate the first-level concept
*admired by many philosophers*. Or consider the bookcase I
would have built, had I finally assembled all the materials I
purchased according to the assembly instructions that came with them.
I have had concrete plans to build the bookcase for ages now; never
mind the fact that I may never find the skill, time or energy to
assemble it. My planned bookcase does not exist yet, and knowing
myself, it might well never exist. But one might take the view that we
can refer to it and that it instantiates many first-level concepts
such as *being a planned bookcase*. On a view like this, the
assertion that there are planned bookcases that do not exist would
remain true even if none of its instances exist.



A view like this would be classified as a form of Meinongianism by
Nelson (2012), and it would be subject to some of the difficulties
other forms of Meinongianism face. Suffice it to say, for present
purposes, that it would, however, require a radical departure from a
very influential approach to ontology advocated by Quine and endorsed
by philosophers like Peter van Inwagen and David Lewis and their
followers.


### 4.2 Quine's Criterion of Ontological Commitment



Quine (1948) explicitly characterizes ontology as an attempt to answer
the question "what is there?" As he observed, the question
may seem deceptively simple, since it may be answered in a word:
"everything". The problem with this answer is that it is
largely uninformative. All parties agree that everything is something,
but there is still plenty of room of disagreement as to what kinds of
objects are there. Are there mereologically composite objects? Are
there concrete possible worlds? Are there mathematical objects? If
there are such objects, then they will of course lie in the domain of
the quantifier "everything". However, philosophers are
still intensely divided as to whether they do.



Quine's strategy for the regimentation and resolution of
ontological disputes has informed much of contemporary ontology. He
advised philosophers to look at the ontological commitments incurred
by our best global theory of the word--best by ordinary
scientific standards or principled extensions thereof--when
appropriately regimented in the language of pure quantificational
logic with
 identity.[18]
 One may perhaps think of the ontological commitments of our best
global theory of the world as the demands that the truth of our total
theory imposes on the world. For example, we are committed to the
existence of objects of kind \(K\) if a proper regimentation of our
best global theory includes--or entails--a sentence of the
form \(\exists x Fx\), where \(F\) is a predicate under which only
objects of kind \(K\) fall. Quine's advice comes accompanied
with a criterion of ontological commitment whereby an appropriately
regimented body of doctrine is ontologically committed to objects of
kind \(K\) just in case objects of kind \(K\) *must* lie in the
range of our bound variables for the sentences of our theory to be
true. More details are given in section 5 of the entry on
 Quine.



As Cartwright (1954) observed, it is far from clear how one is
supposed to interpret the modality in Quine's official
formulation of the criterion. A purely extensional version of the
criterion is plainly inadequate: \(T\) is ontologically committed to
objects of kind \(K\) if, and only if, \(T\) is true *only if*
the variables range over objects of kind \(K\). But the material
conditional is much too weak for Quine's purposes as it makes
any false theory to be ontologically committed to all kinds of
 objects.[19]
 Quine's general hostility to non-extensional notions makes the
problem all the more delicate for him.



Under Quine's approach to ontology, an argument for the
existence of objects of kind \(K\)--whether mereologically
complex objects, or concrete possible worlds, or mathematical
objects--takes the form of an argument for the thesis that
quantification over such objects is indispensable to our best global
theory of the world. One of the most familiar instances of such
arguments is the Quine-Putnam argument for the existence of
mathematical objects. Since mathematical objects are indispensable for
scientific purposes, we should expect the quantifiers of our best
global theory of the world to range over them. But once we settle the
question of whether the quantifiers of our best global theory of the
world range over mathematical objects, we have settled the ontological
question of whether such objects exist. The entry on
 indispensability arguments in mathematics
 includes extensive discussion of this and related indispensability
arguments in mathematics. Similar arguments have been deployed outside
mathematics to argue for the existence of possible worlds,
mereologically complex objects, and the like.



This brings us to a family of applications for different styles of
quantification explored above. It is not uncommon to respond to
indispensability arguments for the existence of objects of a certain
kind by means of a paraphrase strategy, which if successful, would
show how to make do without the ontological commitment to objects of
the offending kind. Oftentimes, the paraphrase will involve a
different style of quantification, whether an alternative to classical
quantification or an extension thereof. For example, Gottlieb (1980)
attempted to respond to the Quine-Putnam indispensability argument for
the existence of numbers by dispensing with objectual quantification
in favor of substitutional quantification in arithmetic. Burgess &
Rosen (1997) provides an extensive catalogue of different expressive
resources philosophers have used in order to bypass apparent
commitment to mathematical objects. Or take the question, for example,
whether there are composite objects of various kinds. Dorr & Rosen
(2002), Hossack (2000); and van Inwagen (1995) have each argued that
plural quantification can be used to paraphrase away apparent
ontological commitment to composite objects--other than organisms
in van Inwagen's case: we need only replace apparent singular
quantification over composite objects with plural quantification over
its alleged parts, and replace apparent singular predications of a
composite object with plural predications of their
 parts.[20]


### 4.3 Unrestricted Quantification



When Quine answers "everything" to the fundamental
ontological question of what there is, he presumably intends the
quantifier to range unrestrictedly over all objects at once. This
observation applies to ontological inquiry more generally. When a
nominalist asserts that there are no mathematical objects, she does
not intend the thesis to be qualified by a restriction to a domain of
concrete objects; otherwise, the thesis would be devoid of interest.
But unrestricted quantification is not a very common phenomenon
outside highly theoretical contexts such as logic and ontology. Take a
typical use of a quantifier expression in English as exemplified in a
typical utterance of the sentence:


* (11) Everything is on
sale.



In a typical context, the use of the quantifier
"everything" is tacitly restricted to a domain of
contextually salient objects. In particular, it would be inappropriate
for another participant in the conversation to point out that the Moon
is not on sale. The Moon is not an exception to the statement made by
my utterance of (11) because the Moon does not lie in the domain of
quantification associated to my use of the quantifier.



But the fact that unrestricted quantification is relatively uncommon
is no reason to doubt it is attainable in certain contexts.
Unfortunately, many philosophers have recently doubted that genuinely
unrestricted quantification is even coherent, much less
 attainable.[21]
 Note that these philosophers face at least two preliminary
challenges, both of which are forcefully pressed by Williamson (2003).
First, they face the question of what to make of the prospects of
ontological inquiry without unrestricted generality. How should we
formulate substantive ontological positions such as nominalism, if we
cannot hope to quantify over all objects at once? The second challenge
for the skeptics is to state their own position. To the extent to
which the thesis that we cannot quantify over everything appears to
entail that there is something over which cannot quantify, skeptics
seem to find themselves in a bind by inadvertently quantifying over
what, by their own lights, lies beyond a legitimate domain of
 quantification.[22]
 Typical responses to each problem involve a surrogate for
unrestricted generality in the form of either schematic generality or
the use of certain modal operators to simulate quantification over a
series of ever more comprehensive domains of
 quantification.[23]



At the core of the problem lies the assumption that the set-theoretic
paradoxes cast doubt upon the existence of a comprehensive domain of
all objects. What they reveal, according to Dummett (1991, 1993), is
the existence of indefinitely extensible concepts like *set*,
*ordinal*, and *object*. For Dummett, the indefinite
extensibility of *set* is incompatible with the existence of a
comprehensive domain of all sets, since no matter what putative domain
of all sets we isolate, we find that we can employ Russell's
reasoning to characterize further sets that lie beyond the putative
domain of all sets with which we began. The set of all
non-self-membered sets in the initial domain cannot, on pain of
contradiction, be in that domain, which means that it must lie in a
more comprehensive domain of all sets. If there is no domain of all
sets, there is, the thought continues, no hope for a domain of all
objects. One may respond to this line of argument by contesting
Dummett's diagnosis of the set-theoretic antinomies. Boolos
(1993) suggests that we take Russell's paradox, for example, to
establish that not every condition determines a set. The moral of
Russell's paradox is that there is no set of all
non-self-membered sets, not that it lies beyond the initial domain of
quantification.



For another line of attack against the coherence of unrestricted
quantification, one may ask what exactly is a domain of quantification
supposed to be. Skeptics often take a page from modern model theory
and think of a domain of quantification as a set--or at least as
a set-like object, which contains as members all objects over which
the putatively unrestricted quantifier is supposed to range. They take
for granted that for a speaker to be able to quantify over some
objects, they must all be members of some set-like object, which
constitutes a domain of quantification. Cartwright (1994) called this
thesis the All-in-One Principle. Now, if there is unrestricted
quantification, then the domain of quantification associated to it
cannot be a set-like object. For we could then deploy a suitable
variation on Russell's paradox in order to obtain a
 contradiction.[24]
 Many have concluded from this that there is no genuinely unrestricted
quantification over all objects at once.



The All-in-One principle is not beyond doubt. One application of
plural quantification, which is explored by Cartwright (1994), is to
abandon the All-in-One principle and to understand talk of a domain of
quantification not as singular talk of collections but rather as
plural talk of its members. To speak of a domain of certain objects is
just to speak of the objects themselves--or to speak of a
first-level concept under which they all fall; and to claim of a given
object that it lies in the domain is to claim of the object that it is
one of them. Alternatively, one may opt for the expressive resources
of second-order quantification with the second-order quantifiers taken
to range over Fregean concepts. Williamson (2003) outlines a
conception of a domain of quantification as a Fregean concept under
which certain objects may fall. To speak of a domain of all objects on
this view is to speak of a Fregean concept under which all
objects--without restriction--fall, and to claim of a given
object that it lies in the domain is merely to claim that the object
falls under the relevant Fregean concept.


### 4.4 Quantification, Tense and Modality



The combination of quantification and modality traces back to Barcan
(1946, 1947) and Carnap (1946). Historically, however, this
combination came under sustained attack by Quine, who, like Carnap,
assumed that \(\Box\) could only interpreted in terms of logical
necessity or analyticity. As Carnap himself recognized, the linguistic
conception of necessity raises the question of how to assign
intelligible truth conditions to *de re* modal generalizations
of the form \(\exists x \Box
 Px\).[25]
 For the truth conditions for such a sentence appear to require one to
make sense of the truth of an open formula of the form \(\Box Px\)
with respect to an assignment of a value to the variable \(x\). If we
read \(\Box\) to mean: "it is analytic that", then one may
wonder what sense could be made of the claim that an open formula
\(Px\) is analytic with respect to an assignment of an object to
\(x\). Quine had anticipated the issue in Quine (1943), and he pressed
the criticism again in Quine (1953, 1960). Quine's objections
are commonly taken not to apply to other interpretations of \(\Box\)
in terms of metaphysical necessity as discussed, for example, in
Kripke (1980). When we move past Quine's criticisms and combine
pure quantificational logic with propositional modal logic, we obtain
quantified modal logic.



The interaction of quantification and modality turns out to raise
difficult philosophical problems. As we will see, some of these
difficulties have, in fact, inclined some philosophers to rethink the
scope of pure quantificational logic. The problem is that certain
theorems of quantified modal logic suggest that everything is
necessarily something. The subject matter of ontology is necessary:
nothing could have failed to be something.


#### 4.4.1 Quantification and Modality



Quantified modal logic is the system that combines pure
quantificational logic with propositional modal logic. Propositional
modal logic investigates the logic of the modal operators \(\Box\) and
\(\Diamond\) and is discussed in the entry on
 modal logic.
 The language of propositional modal logic supplements the language of
propositional logic with the sentential operator \(\Box\), and
\(\Diamond\) may then be explicitly defined as \(\neg \Box \neg\). A
weak propositional modal logic called K logic includes as axioms all
tautologies and all instances of schema K:


* (K)\(\Box(A \rightarrow
B) \rightarrow (\Box A \rightarrow \Box B)\)



The rules of inference of the system include *modus ponens* and
what is often known as a rule of necessitation:


* (RN)from \(A\), infer
\(\Box A\)



The combination of propositional modal logic and pure quantificational
logic is often called quantified modal logic and is discussed in
section 13 of the entry on
 modal logic
 and the entry on the
 possibilism-actualism debate.
 This combination immediately delivers what we may call the
*necessity of being*:


* (NB)\(\forall x \Box
\exists y(x=y)\)



The key to this observation is that the open formula \(\exists y (x =
y)\) is a theorem of pure quantificational logic. Successive
applications of the rule of necessitation and universal generalization
are all it takes to complete the proof.



On the alethic interpretation of \(\Box\) in terms of metaphysical
necessity, this theorem reads as the claim that everything is
necessarily something. Take the Griffith Observatory, for example. It
is a consequence of the theorem that it is necessarily something. The
Griffith Observatory would have been something even if no funds had
been available for construction and nothing had ever been built in the
site it occupies. And what is true of the Griffith Observatory is true
of us: each of us is necessarily something. When we combine this
result with the link between quantification and existence, we conclude
that everything exists necessarily, which flies in the face of the
assumption that we exist only contingently.



With the help of schema K, one may prove the Converse Barcan
Formula:


* (CBF)\(\Box \forall x A
\rightarrow \forall x \Box \ A\)



On the alethic interpretation of the modality, this states that if
necessarily, everything is \(A\), everything is necessarily \(A\). The
proof is given, for example, in one of the supplements to the entry on
the
 possibilism-actualism debate.
 But this provides another route to the necessity of being: since
necessarily, everything is something, everything is necessarily
something. In the presence of a further plausible modal principle, B,
we may even prove the Barcan Formula. The Brouwerian principle, B,
states that whatever is the case is necessarily, possibly the
case:


* (B)\(A \rightarrow \Box
\Diamond A\)



The Barcan Formula states that if everything is necessarily \(A\),
then necessarily, everything is \(A\):


* (BF)\(\forall x \Box
A \rightarrow \Box \forall x A\)



Or, equivalently:


* \(\Diamond \exists x
A \rightarrow \exists x \Diamond A\)



The entry on the
 possibilism-actualism debate
 includes a supplement with a proof of BF against the background of
(B). But this formula is deemed unacceptable by many philosophers.
Consider my plans to build a bookcase. Even if I never manage to build
one, it is certainly possible for me to build one. So, it is possible
for there to be some bookcase I have built. By the Barcan Formula, it
seems to follow that something is possibly a bookcase I build. The
Barcan Formula does not settle whether or not this object is in fact a
concrete object, much less a bookcase one may in fact use to store
one's books. In fact, to deny this is one way to come to terms
with the Barcan Formula.


#### 4.4.2 Quantification and Tense



What is true of the combination of pure quantificational logic with
propositional modal logic is *mutatis mutandis* true of the
combination of pure quantificational logic and propositional tense
logic. Certain theorems of quantified tense logic suggest that
everything has always been, is, and will always be something. The
domain of ontology is immutable: nothing ever fails to be
something.



Propositional tense logic investigates the logic of tense operators
like \(H\), read: "it has always been the case that",
\(P\), read: "it was the case that", \(G\), read:
"it will always be the case that", and \(F\), read:
"it will be the case that". In place of K, we have
distribution axioms:


* \(H(A \rightarrow B)
\rightarrow (HA \rightarrow HB)\)
* \(G(A \rightarrow B)
\rightarrow (GA \rightarrow GB)\)



And in place of the rule of necessitation, we have rules of temporal
generalization:


* from \(A\), infer
\(HA\)
* from \(A\), infer
\(GA\)



More details are provided in the entry on
 temporal logic.



The combination of propositional tense logic and pure quantificational
logic immediately delivers the eternality of being:


* (EB)\(\forall x
(G\exists y(x = y) \wedge \exists y(x = y) \ \wedge F\exists y(x =
y))\),



which states that everything has has always been, is, and will always
be, something. The Griffith Observatory was something even before it
was built in 1935--and will always be something--even
billions of years from now. And what is true of the Griffith
Observatory is true of us: each of us has always been, is, and will be
something--even before our birth and long after our death.



Quantified tense logic proves temporal versions of CBF:


* \(H \forall xA
\rightarrow \forall x H A\)
* \(G\forall x A
\rightarrow \forall x G A\)



And, finally, we have two temporal versions of BF:


* \(\forall x H A
\rightarrow H \forall x A\)
* \(\forall x G A
\rightarrow G \forall x A\)



In order to appreciate the difficulty, we may take the contrapositive
of the temporal version of BF for \(H\), where \(P\) is defined as
usual, as: \(\neg H \neg\):


* \(P \exists x A
\rightarrow \exists x P A\)



We infer, for example, that there were dinosaurs only if there are
past dinosaurs. If to exist is to be something, then past dinosaurs
exist. But whether this commits us to flesh-and-blood dinosaurs is of
course a different matter, since being a past dinosaur need not
require being a dinosaur.


#### 4.4.3 From Logic to Ontology



The interaction between pure quantificational logic and tense and
modality suggests the domain of ontology is immutable and necessary.
But we can only draw these conclusions by making use of some auxiliary
assumptions, some of which are far from obvious. The highly
counterintuitive character of the conclusions has consequently led
many philosophers to place some of the auxiliary assumptions under
close scrutiny. Other philosophers have followed argument where it
leads, and they have sought to reconcile the eternal and necessary
character of the domain of ontology with the apparent temporary and
contingent character of existence. In fact, Williamson 2013 set out to
provide a separate battery of arguments for the necessity and
immutability of ontology.



One auxiliary assumption one may question is the traditional link
between quantification and existence. Even if there is no change in
the domain of quantification, you may nevertheless think that
existence is only temporary. Socrates did not exist either before 490
BCE or after 399 BCE, and the mere fact that he is, has and will
always be something is not reason to attribute existence to him. One
variant of this move is a direct descendant of Meinong's view on
existence as a mere species of being. But to the extent to which
Meinongianism may be accompanied with the implausible thesis that no
matter what condition we substitute for \(F\) in "the \(F\) is
\(F\)" gives rise to a true sentence, it would seem to entail
that "the round square is a round square" is true, which
seems hopeless. But the Meinongian thesis is not compulsory for more
plausible developments of the view that not everything exists, even
though everything is
 something.[26]
 What is not negotiable, though, is the thesis that existence should
not be taken to be expressed by a quantificational expression, but
rather by a predicate which may or not apply to objects in the domain.
The moral of the temporal versions of BF and CBF is only that
unrestricted quantification ranges over an immutable and necessary
domain of objects, whether or not they enjoy temporary and contingent
existence. The difficult question remains of how to answer
Quine's question "what is there?" once one abandons
the thesis that no matter what condition \(F\) may be, there are
objects that satisfy the condition. For more on different variants of
Meinongianism, the reader may consult the entry on
 existence.



There is, in the second place, the assumption that the axioms of pure
quantificational logic remain true when we expand the language to
include other sentential operators, whether temporal, modal or
 otherwise.[27]
 What is perhaps the most common response to our predicament is to
recoil from pure quantificational logic into a suitably weakened
variant of it. This is, for example, the path taken in Kripke (1963),
where Kripke proposes to weaken the axiom of universal instantiation
in line with the first version of inclusive quantificational logic
discussed earlier. Since \(\exists y(x = y)\) is not a theorem of
inclusive quantificational logic, we are not in a position to
necessitate, which blocks the derivation of the necessity of
 being.[28]
 Kripke's quantified modal logic is discussed in the entry on
the
 possibilism-actualism debate.
 Others have fallen back into alternative forms of free logic, but
there appears to be no consensus as to which one is the best
alternative to pure quantificational logic. It may be helpful to note,
however, that some of the relevant free logics merely restrict the
axiom of universal instantiation by means of an existence predicate
and do nothing to block the derivation of other allegedly problematic
principles such as the necessity of identity, which is the thesis that
everything is necessarily self-identical. Since, presumably, an object
can only be necessarily self-identical if it exists necessarily,
opponents of CBF will also want to have some resources to block the
necessity of identity.



The derivability of the necessity of identity from the axioms of
identity and the rule of necessitation may perhaps invite one to
consider another option. Maybe the culprit is not pure
quantificational logic, but rather the indiscriminate use of the rule
of necessitation. The rationale for necessitation is that every
provable sentence should be necessarily true, but it is not clear how
this thought is supposed to generalize to open formulas, which,
strictly speaking, are not true or false. Open formulas are merely
true or false under an assignment of values to their free variables.
But the fact that \(x = x\), for example, is true under an assignment
\(s\) on which \(x\) is assigned a table is no reason to think that
\(x = x\) is necessarily true under such an assignment if you think
that the table could have not been
there.[29]
One problem for
this strategy is that it is of little help when we allow closed terms
such as constant symbols into the language of quantified modal
logic. In the presence of constant symbols, classical quantificational
logic allows us to derive the sentence \(\exists x (a = x)\) as a
theorem, whence, by necessitation for complete sentences, we infer
\(\Box \exists x(a = x)\), which seems objectionable if \(a\) is
suppose to regiment a name for an object that exists only
contingently. One could avoid the problems by barring closed terms
from the language, but such a radical exclusion seems *ad hoc*
and artificial. So, if one chooses to restrict necessitation, one must
provide a different rationale for it. Deutsch (1990) provides an
example of this strategy.



The fourth and final option to consider is to take the derivability of
CBF in tense and modal logic at face value, and embrace the conclusion
that existence is indeed immutable and necessary. These are the theses
that Williamson (2013) calls "permanentism" and
"necessitism". The task for each approach is to explain
our initial reluctance to embrace them in the first place. Take the
apparent resistance to accept the claim that Socrates will always be
something despite the fact that he died in 399 BCE. The permanentist
may respond that if we are initially disinclined to accept this claim,
it is only because we mistakenly think that because a person, for
example, is a concrete object, a past or a future person must be
concrete as well. Socrates, which is a past person, was not a person
either before 490 BCE or after 399 BCE; indeed, Socrates is now not a
person, nor will he be one in the future. Likewise, a merely possible
person need not be a person either and neither the Barcan nor the
Converse Barcan Formula threaten the temporary and contingent
character of concreteness. Since I could have found the time, skill
and energy to build a bookcase, some object is possibly a bookcase
built by me. But the necessitist will be at pains to add that to be a
possible bookcase built by me is not to be a bookcase built by me,
much less a bookcase. This outlook is defended by Linsky & Zalta
(1994) and Williamson (2013). But there are other points of contact.
For example, all versions of necessitism must reject unqualified
"essentialist" theses such as the view that a person is
necessarily a person; at most, one can perhaps maintain the weaker
thesis that a person is necessarily if concrete, a person. Likewise,
all versions of necessitism seem committed to the actual existence of
many more nonsets than can form a set. Sider (2009), for example,
argues that necessitism is committed to the actual existence of at
least \(\kappa\) nonsets for each set-theoretic cardinal \(\kappa\).
And Hawthorne & Uzquiano (2011) raises other cardinality problems
for a combination of necessitism with certain common mereological
assumptions.