Home

Papers

Blog

Pictures

Contact

GitHub

KISS Linux



= Dilyn Corner =

= A Limited View on Everything =

= SECTION 1 – AN INTRODUCTION =

When classical logic is first taught, propositional logic is a relatively simple
place to begin from. Statements like ‘a is a cube’ and ‘b is to the
left of c’ are common statements, from which further complex propositions
can be expressed through the introduction of classical sentential connectives.
Once we begin to have a mastery of these more complicated logical statements, it
becomes further generalized to such statements as P(a), where a is the object
which P is true of. It is from this point that first-order logic arises. We can
further generalize the statement P(a) using the rule of existential
instantiation: $\exists \mathit\left\{xP\right\}\left\left(x\right\right)$, which says
‘there exists an x such that P is true of x’. Similarly, we can express
universal statements: $\forall \mathit\left\{xP\right\}\left\left(x\right\right)$, ‘for
every x P is true of x’. Such statements involving quantifiers are taken to
be about particular sorts of domains, the domains which the quantifiers range
over. The question we might be left to wrangle with is what exactly this domain
of discourse is. As ordinary language users, we take no real issue with the
statement that “everybody is in the car”. If someone were to respond and
say, “but the President is not”, we might give them an incredulous look;
we were, after all, only concerning ourselves with the relevant people. These
‘relevant people’ are determined by the context of my claim. Perhaps my
family is getting ready to go on vacation, and now that everyone in my immediate
family is in the car we may leave. In any case, it certainly doesn’t seem to
be reasonable to suggest that I meant all seven billion people on Earth were in
my vehicle. But there are other cases in which we certainly would take ourselves
to be broadly discussing everything there is. The statement “there are no
abstract objects” certainly seems to be about every kind of object; of all
the things which exist, none of them are abstract. If this domain were somehow
restricted – say, to be about concrete objects – the proposition
certainly wouldn’t express what the speaker might have hoped it would. The
aforementioned speaker is charged with speaking absolutely generally. They
purport to be quantifying over literally everything, and as such their
quantifiers can be thought of as being absolutely unrestricted. However, there
are an assortment of problems which arise from attempting to speak in such
absolutely general terms. Considering this, the ensuing discussion will consider
objections presented to such a view – that absolute generality and
unrestricted quantification are allowable in logic. Those who accept some form
of absolute generality or unrestricted quantification are known as absolutists;
their opponents, limitavists. With this perspective in mind, the premier
questions of this thesis are (1) what support does the absolutist position have
considering these objections, (2) do the limitavists have proper reservations
and responses towards these arguments, and (3) what then are we left to make of
absolutely unrestricted quantification? In what follows I will attempt to answer
these three questions in three parts. First, I will explain what some of the
problems absolute generality presents to us are. The three problems I will be
focusing on are the existence of indefinitely extensible concepts, the issue of
semantic indeterminacy for the size of a given domain of quantification, and the
problem of using sortal concepts which fail to provide an adequate criterion of
identity. Second, I will examine the resolutions to some of these problems which
have been proposed. Finally, I will discuss which of these positions, if any,
can put our worries at ease and be used as proper justification for our
unrestricted discourse.

We begin our examination of the
three problems for absolutely general inquiry with indefinite extensibility. The
development of this criticism arises primarily from arguments due to Michael
Dummett.See Dummett (1991). For a closer examination of
Dummett’s argument, see Clark (1998).  In short, a concept is
indefinitely extensible just in case its extension is not definite. That is to
say, there are certain concepts under which the totality of its extension is not
fully specified. The problem which this criticism points out is that, given any
particular domain our quantifiers range over which purports to be absolute, we
can always find an object which sits outside of the scope of our quantifiers.
The development of set theory in the 20th century led to one of the
most paradigmatic instances of indefinite extensibility. While the object in
question need not be setsOur domain need not be particularly
specific on this matter, but there are cases to watch for. As a matter of fact,
(proper) classes can be a totality which allows you to quantify over the object
language of set theory just fine. Models are also another proper way of
discussing certain kinds of totalities. For more on these points, refer to the
introduction in Peters & Westerståhl (2006)., they are a paradigmatic
instance of this sort of problem. Generally speaking, a set is a collection of
objects satisfying some given property. So$\left\lbrace x\right\vert \phi \left\left(x\right\right)$} is the collection of objects $x$which satisfy
$\phi \left\left(x\right\right)$. Given the schema $\exists y\forall x\left\left(x\in y\leftrightarrow \phi \left\left(x\right\right)\right\right)$, we can then
formulate the set for which $\forall x\left\left(x\in r\leftrightarrow \phi \left\left(x\right\right)\right\right)$, and in defining the formula $\phi \left\left(x\right\right)$as $x\notin x$, we have that $\forall x\left\left(x\in r\leftrightarrow x\notin x\right\right)$From this we can infer that
$r\in r\leftrightarrow r\notin r$.This is the famous
Russell’s Paradox. See Frege (2013).  What this means is that a set is
a member of itself if and only if it is not a member of itself. Usually referred
to as the universal set, it is seen as a major contradiction in set theory. It
can also be shown that other concepts are indefinitely extensible as well. Take
for instance the notion of a proper ordering of collections. Suppose I give you
some sort of set, say $\left\lbrace \right\rbrace$, and I tell you
that there is a number we can associate to it, say zero. We can say then that
the size of this set is zero. From this, I give you a few more sets, like
$\left\lbrace 0\right\rbrace$, $\left\lbrace 0,1\right\rbrace$, $\left\lbrace 0,1,2\right\rbrace$, and tell you that we can
associate one, two, and three to each of these. We can continue this as far as
we like, counting the size of each as we go. From this point, we can list out
the natural numbers $\left\lbrace 0,1,2,3,\dots \right\rbrace$. Can
we not also assign an order to this as well? Indeed, we can; call the size of
this set the ordinal . Suppose that I simply offer up a reordering of this
list; perhaps we can call  + 1 the associated ordinal to $\left\lbrace 1,2,3,\dots ,0\right\rbrace$. Likewise, with the natural numbers, it
seems that we should be able to continue this process; so, suppose we add more
numbers to our list like , such as  + 1,  + 2, and so on.
Eventually, we could arrive at  + ,  +  + 1, and even further if
we wanted to. These seem to be like numbers in a similar fashion to the real
numbersSome hesitation is required on this point; I have merely
brushed aside that we can add numbers to ordinals without properly defining
addition or anything like this. Perhaps just take my word for it; some such
definition can be made. , and we have no qualms with collecting all the
naturals or, indeed, the real numbers, into a single collection. So, suppose we
do, and call this collection of ordinals . But certainly, we can associate
to this collection of ordinals another ordinal, not originally in our
collection! It seems that our totality can be extended, and we were wrong to
suppose that we had in fact collected all of the ordinals in the first
place.For further explanation on ordinals and the Burali-Forti
Paradox, see Moore (2015). We are left with two possible avenues in light
of these contradictions: we can either reject the formula we chose to arrive at
the paradox, and thus rejecting the formulation of certain kinds of sets
(ordinals), or else we can admit that the given set (ordinal) lies outside of
the extension of the variable $x$(the collection ).Russell anticipated such a move in Russell (1906) for sets. Note
that, in rejecting the existence of the Russell set, the proponent of indefinite
extensibility need only point out that this is exactly what they are suggesting,
namely that the $x$in question ranges over a domain of sets for which
the Russell set is not a member – it does not exist in the collection.
The problem of indefinite extensibility in sum demonstrates that the idea
of an all-inclusive domain is false. After all, if we cannot have an
all-inclusive domain of sets, we certainly cannot have an all-inclusive domain
of objects which includes sets – and if it didn’t include sets, it would
seem to not be a domain we could call a totality! Alternatively, indefinite
extensibility suggests that we might simply always have to deal with restricted
quantifiers, and that they are restricted to some sort of domain which is not
all-inclusive.For a furthermore feature rich discussion of the
problems posed by ordinals and sets, along with a more generalized version of
the Dummettian and Russellian points, see Wright & Shapiro (2006).

The second problem for the
absolutist is semantic indeterminacy. The argument for semantic indeterminacy is
that there is nothing in the thoughts or practices of language users which
determine the domain of our quantifiers to be absolutely general. The argument
is a consequence of the Löwenheim-Skolem theorem,
which Hilary Putnam used to demonstrate issues relating to moderate metaphysical
realism.See Putnam (1980). Quine (1968) also speaks to
this. The theorem states that a satisfiable first-order theory has a
countable model. This theorem gives rise to the Skolem paradox.For an amusing reformulation of the argument, See Lavine (2006),
105.  Take Zermelo-Frankel set theory as an example. We have a first-order
theory which includes two axioms: the axiom of infinity, and the power set
axiom. So, we have an infinite set. By the power set axiom, we have an
uncountable infinite set.As the power set of an infinite set
is uncountable; see Barker-Plummer (2011) for a plethora of details.  But
our model was supposed to be countable; thus, a paradox emerges. The lesson we
learn from this problem is that we cannot put our theory in a one-to-one
correspondence with the natural numbers ''within'' the model. What this means
for us is that the “intended interpretation… of a set… is not captured
by the formal systems”Putnam (1980), pp. 465.. Given
a stronger version of the theoremCalled the “downward
Löwenheim-Skolem
theorem”., we can demonstrate that a satisfiable first-order theory
given in some countable language has a countable model, which is a submodel of
any given model. From this, it becomes clear that the totality which our
quantifiers range over in some given model always is a proper subset of a larger
totality. This means that our formerly uncountable and all-inclusive domain has
a countable subdomain, and our usage of quantifiers for the former is also
compatible with the latter. The indeterminacy arises from the fact that we
cannot rule out the existence of such a model simply because we cannot determine
it. What this means is that we cannot ever be absolutely certain that the domain
which absolutists purport to be quantifying over is actually all-inclusive
because it is the case that they could simply quantify over some smaller domain
and still be using the same quantifiers.As Putnam (1980) and
Rayo & Uzquiano (2006) point out, this consequence requires some relatively
robust philosophical baggage. For starters, proving the downward Löwenheim-Skolem theorem
requires the Axiom of Choice. As a result, many objections are available. See
Lewis (1983).

The final problem I wish to
discuss is one which relates to sortal concepts. Suppose someone were to ask you
how many things were in your kitchen. Perhaps you’d begin by counting your
spoons and forks, bowls and plates, pots and pans, and other useful utensils.
But then you find a bag of flour; how do you suppose you should count this? Is
it a single item, should it be measured in cups, is it each individual grain? Do
we likewise splinter the microwave into its requisite parts, the bulb and
interior plate, the wires and circuit boards each counting individually? Perhaps
we can split these even further, dividing all the objects into their respective
atomic structures, and continuing, splitting atoms and protons until we’re
left with a rather large bundle of quarks, maybe energy states, and perhaps even
strings. We might return to our inquirer and inform them that they weren’t
clear enough in their question. The problem which we have uncovered is that of
an improper delineation of what counts as a ‘thing’. This issue has been
characterized as the lack of a substantival term by people like Dummett, and due
to this lack, we are faced with a failure to have a proper criterion of
identity.See Dummett (1981). Dummett takes the Fregean view on
identity; for what it means to be a criterion of identity, see Frege (1884).
Put in a more precise way, the question of how many ''objects'' there are
only makes sense if the term ''object'' has a way of properly carving up the
items in question. Contrast this with a term such as ‘car’. The question
of “how many cars are parked on my street” has a proper answer; I need
only step outside and count the parked cars I see. There is a proper criterion
by which I can tell if I have counted all of the cars. I will know that I have
done so by checking each car off, and I can tell all of the cars apart from each
other and verify that they are cars. This is far more difficult to do with
something like mud or snow. If someone were to ask me “how much mud is
there” I would be hard pressed to find a proper answer, unless we had of
course established some way of measuring mud, perhaps by volume or height. But I
cannot count mud like I can count cars, and that is the point.There’s an interesting point to be made over whether this is
merely a problem with our concepts or with the actual structure of the world.
For any absolutist who is also a realist, the metaphysical indeterminacy
presented in the latter case is quite worrisome; see Taylor (2015) for an
argument about how not all indeterminacies are semantic.  As object,
thing, and individual are not criteria for identity, they aren’t sortal
concepts and, as a result, don’t meaningfully articulate the scope or
content of the domain we’re meant to be quantifying over.

= SECTION 2 – THE ABSOLUTIST POSITION EXAMINED =

Given the above three troubles for the absolutist position, the question then
becomes whether there are ways in which the absolutist can properly capture
unrestricted quantification and either resolve the problems and emergent
paradoxes, or avoid them altogether. In what follows, we will examine some
arguments offered by some of those who accept unrestricted quantification to see
how they tackle these three problems. Vann McGee is one such defender of the
absolutist position, and offers an illuminating take and break down two of the
three problems on offer. One tactic McGeeAnd others; see for
instance Williamson (2003); an explanation can be seen in Fine (2006).
utilizes is to turn the argument of indefinite extensibility against the
limitavists and argue that they themselves are unable to formulate their own
position against the absolutist, as it would require them to use unrestricted
quantification themselves. Indeed, if the limitavist thesis is that ‘no
quantification can be unrestricted’, this prospect is troubling. For
certainly they would mean to be speaking of all uses of quantification. In fact,
they are attempting to quantify over all instances of quantification. But this
is precisely what the limitavist says is illegitimate! Indeed, this issue points
out the lack of a definite boundary for the domain of discourse those opposed to
unrestricted quantification would have us use. Take our quantifiers to range
over a small domain of objects – restricted to particular kinds of objects
we have access to. Suppose we were then to find an object which is new to us. It
seems difficult, McGee says, to “specify the new objects when the logical
devices at our disposal are limited to quantifiers that range over the old
objects”McGee (2006), 184. . McGee uses this point
to argue that, in fact, the only real way to escape the problems associated to
vagueness or understanding hinges on the idea that unrestricted quantification
is more natural or foundational than that of restricted quantification.McGee (2006), 183. McGee argues this point in conjunction
with a point against indefinite extensibility. For the limitavist who thinks
extensibility is an issue for the absolutist position, this issue becomes even
more salient. The argument is essentially in the form of a dilemma. In order to
avoid issues regarding the universal set in a set-theoretic construction of
mathematics, we must abide by the vicious circle principle.The
vicious circle principle states that no function on objects can be defined until
the domain of the function is established. The vicious circle principle is
employed in Whitehead & Russell (1910).  However, in order to further our
construction of mathematics, we require the least upper bound property for
sets.The least upper bound property is that, given a
well-ordered set $S$, $x$is an upper bound of
$S$just in case $x\ge s$for all $s\in S$and
$x$is the least upper bound of $S$if $x\le y$for every upper bound $y$of $S$.  But our
only way of defining the least upper bound of a set is with reference to the
totality of upper bounds. Thus, in order to formulate this necessary property
for analysis in mathematics, we must violate the vicious circle principle. To
avoid this problem, we can follow what Russell and Whitehead did in response: we
can adopt the Axiom of Reducibility, which says that for any impredicative class
C, we can always find a predicative class C’ under it.What
it actually means is that in the hierarchy of types, for a predicate at any type
level there is a predicate at the first level that is equivalent to it. It was
formally introduced on primarily pragmatic grounds so as to make the type theory
construction of numbers easier to handle (to put it lightly).  What this
Axiom effectively does is it collapses the stratification of types, functions,
predicates, etc. In this way, one of the primary purposes of the vicious circle
principle is violated; no longer do we have to worry about our function sitting
over and above the domain on which it is specified, as the Axiom of Reducibility
guarantees we get what we want. So we can either reject the vicious circle
principle, in which case indefinite extensibility is no longer a cogent strike
against the absolutist, or we can reject the Axiom of Reducibility, thus making
the construction and use of mathematical objects almost unbearably complex.What’s interesting about this argument is that McGee does not
take it from Dummett, the major espouser of indefinite extensibility as a
problem for quantification. For another detailed discussion about Russell and
Dummett’s own philosophies of mathematics from an ontological perspective
(in which both of the issues I have highlighted here are thoroughly explored),
see Shapiro (2011). While this second choice certainly doesn’t act to
deny the problem itself, accepting this option is less than ideal under
axiom in the first place.

McGee puts forward another
argument, this time responding to the second problem of semantic indeterminacy.
In responding to the Löwenheim-Skolem theorem,
McGee rightly notes that the theorem applies to first-order languages.McGee (2006), 185 As English is a natural language and is not
a first-order language as such, it seems like an insufficient charge for the
limitavist to have for the absolutist. However, models can be extended to
second-order languagesKnown as Henkin models, which are
countable models in which the nonstandard interpretation of the second-order
quantifiers make all the right sentences true with the second-order variables
range over a countable collection of collections. For a basic introduction to
Henkin models, see Barker-Plummer (2011). and the theorem can be extended
in a similar fashion. It seems, McGee points out, that it’s difficult to
imagine that the theorem would not be able to apply to more logically complex
languages such as English in some form or another. McGee offers a possible
avenue of objection through the fact that the rules of logical inference must be
open-ended. In first-order languages we have a fixed vocabulary. New names are
introduced in formal languages because they exist in the domains of our
quantifiers, for instance. But in the case of more logically complex languages
like English, we do not have a fixed vocabulary. Indeed, our words change all of
the time. Our best scientific theories introduce terms that did not exist prior;
space-time points were not spoken of before Einstein. With each new name we
introduce, our language expands. It’s important to note that this expansion
does not change the prior names in the language. Mathematics is stable in that
we need not reevaluate all of our mathematical facts because a person names
their child or a new theory of the mind arises. This is a problem for the
objection because, for the theorem, the subdomain requires that the names exist
in the domain. If we introduce a new name into our language, this means that it
must not have been in the domain previously. What this means is that our
quantifiers did not range over it previously, and this new expansion of our
language will not preserve truth in the relevant way for the Löwenheim-Skolem theorem
to apply. To put it more concretely: suppose that $\forall x$quantifies over all the objects in our language $L$. From
this, we could infer a certain property that might apply to every object in our
language, say $\forall x\phi \left\left(x\right\right)$. Supposing that we
introduce a new name, c, into our domain for the language, it isn’t the case
that we can infer $\phi \left\left(c\right\right).$Because the expansion
doesn’t preserve truth, the Löwenheim-Skolem theorem
does not apply to our language $L$. But this problem only exists for
restricted quantification. Note that, without any sort of domain expansion, the
restricted and unrestricted quantifiers appear to behave the same in the
original language. But after the introduction of a new name not previously in
the language, we can recognize the problem. This does not happen to our
unrestricted quantifiers; they always quantified over the new name. To get
around issues relating to semantic indeterminacy, the position that McGee puts
forward is that “the semantic values of the quantifiers are fixed by the
rules of inference”McGee (2006), 191. This claim is not
necessarily extendable to natural languages like English. However, McGee does
suggest that this point is meant to assuage our skeptical worries.. Our
first point to flesh out is what exactly it is about indeterminacy that could
worry us. If something is indeterminate, there are multiple possible candidates
available to us. McGee points out that this is not the case for our
quantifiers.McGee in particular cites the work of J. H. Harris
demonstrating that two quantifiers, $\forall$1 and
$\forall$2, are interderivable and thus they are
logically equivalent.  Supposing that the quantifiers preserve truth in
the expansion of a domain, and that the quantifiers do not lapse into any sort
of incoherenceMcGee suggests that this might look like the
quantifiers not actually contributing to the truth conditions of a sentence.
, then the quantifiers themselves must pick out a unique and optimal
candidate for their semantic value. This semantic value is not determined by the
quantifiers themselves, but is instead acting to limit a range of choices. This
is not the same as restricting the domain, however. Domain restriction results
in imprecise object choices, and our quantifiers are not at all imprecise.McGee stresses that this all relies on the strength of Harris’
work.  In order to avoid any such ambiguities, we should allow our domain
to be unrestricted. The biggest reason for suggesting that this is the case is
due to simplicity. If our quantifiers were contextually limited themselves, then
we would be using some sort of precise variant of the quantifiers. Such a thesis
would be wildly complicated and speculative, and we are supposed to be pragmatic
about these things. A precisified domain is, as it were, untenable simply
because it is overly extravagant and speculative, and we have simpler solutions
to the problem. Namely, unrestricted quantification.Here McGee
is heavily relying on certain tenets of hypothesis acceptance and rejection that
he doesn’t not provide a positive argument for; as such, I do not intend on
arguing for or against this part of the analysis, if only because I find myself
unconvinced by “Occam’s Razor”.

The final concern which was
raised in the previous section is that of questions of identity in the context
of an absolutely unrestricted quantifier. The problem raised seems to be one of
an epistemic character; how is it that I would know if someone else were to use
identical in the same sense as I am, but it is likewise a constitutive problem:
in virtue of what do I mean ''identical'' by ‘identical’?Williamson (2006) offers such a characterization of the question of
identity.  In certain contexts, such an answer is easy to supply. For
instance, when I ask whether two books are identical we may go about verifying
that they have the same titles, authors, publication dates, pages, etc.We can also speak to numerical identity, which Williamson focuses
on. However, such a question has a less obvious answer in other cases.
When I ask whether everything is self-identical$\forall x\left\left(x=x\right\right)$., or how many things there are in the
universeThere is some question as to whether or not
quantification actually requires identity; I do not entertain such arguments
here, but some places to find the discussion would be in Geach (1967) and,
responding, Dummett (1981). , the question is in principle difficult to
answer. However, this does not mean that it itself is problematic. The notion of
sortal concepts helps to pacify the problem in the easier cases – two things
are identical just in case if one object Fs the other object Fs for some
property F – but with no such sortal available to us, identity is a
difficult topic to wrangle. Williamson suggests that any problems we may have
with absolute generality apply analogously to absolute identity. Given some
identity predicate $I$used by some speaker $S$of some
language $L$, the claims that ‘everything is self-identical’
and ‘if two things are identical then whatever applies to one also applies
to the other’ become the following expressions:

(1) $\forall x\left\left(\mathit\left\{xIx\right\}\right\right)$and

(2) $\forall x\forall y$

For a speaker $S$* of another language
$L$* with an identity predicate
$I$*, the claims are:

(1*)
$\forall x\left\left(\mathit\left\{xI\right\}\ast x\right\right)$and

(2*)
$\forall x\forall y$

If we merge the two languages spoken by each speaker into
$L+L$*, it can be demonstrated that these
identity statements can be interderived.See Williamson (2006),
378 for such a demonstration. See the entire discussion in section 13.1 for a
more thorough treatment of what exactly is happening in his construction – I
will provide only a discussion of his final points on the matter, but there are
many technical details that Williamson explores.  These identity
predicates are thus coextensive over their shared domains of objects. As a
result, identity can be uniquely characterized. Timothy Williamson points out
that in this context of absolute identity, we need not worry about whether we
“have a conception of all predicates in all possible extensions of the
language”Williamson (2006), pointing out possible semantic
paradoxes.. Indeed, we need only concern ourselves with subsets over the
domain if identity is taken to be ranging over a certain domain. Supposing that
these domains are sets then, under Zermelo-Frankel set theory, we should have no
real issue considering such subsets of our domain.Under the
usual axioms of Zermelo-Frankel set theory, our formalization is closed under
the power set function and Cartesian products. For the axioms of ZF, see
Barker-Plummer (2011).  Like identity, universal quantifiers are
interderivable and have an open-ended reading available to them.An open-ended commitment with respect to universal quantifiers
reflects a “general disposition to accept instances of universal
instantiation for the second-order quantifier in extensions of our current
language”. For this discussion and how it also relates to identity, see
Williamson (2006), 377-80. As a result, the problems posed to one are
analogously problems posed to the other. For instance, one can generalize the
standard inferential rules in first-order logic of $\forall$-Introduction and $\forall$-Elimination used in a language
$L$for the quantifier $\forall$to another language
$L$* with quantifier $\forall$* with parallel rules. It can be shown that
the two quantifiers are logically equivalent in some language
$L+L$*. Following Williamson in supposing
that the open-ended nature of the quantifiers saves such a position from
accusations of sophistry, it can be defended that the two quantifiers are
logically valid on an unrestricted reading. However, since our real question was
originally about identity, we should be worried that our initial arguments
against identity, open-ended or not, threaten such an unrestricted sense of the
quantifier. As it turns out, Williamson suggests that it is unrestricted
quantification which threatens identity, not the other way around. Indeed, one
of the largest charges against the absolutist is that it is inarticulate; to all
things which the absolutist assents or dissents, the limitavist interpretation
will likewise assent to truths and dissent from falsehoods. Supposing then that
the absolutist position can be rearticulated in such a way that the limativist
will agree with all the claims of the absolutist and maintain that the
quantifiers are restricted, the same paradoxes which the limativist might claim
the absolutist falls to also apply likewise to the limitavist. For suppose that
such contradictions are the case and the absolutist position can be
reinterpreted in terms of the limitavist one. Then any derivation of the
contradiction put forward by the limitavist on the absolutist’s view can be
restated in terms of statements which the relativist will accept. Thus, the
limativist view is itself inconsistent. If, however, the limitavist stops
identity is absolute, then the limitavist will necessarily have to maintain that
absolute generality is fine and give up their own restrictions on
quantification, as absolute identity supports absolute generality by
Williamson’s argument.

= SECTION 3 – THE LIMITAVIST’S RESPONSES =

What we have seen thus far is a defense of the absolutist view from the three
problems along with an additional criticism of the limitavist view, namely that
it is self-defeating or inarticulate. Considering these defenses and criticisms,
what then can the proponent of restricted quantification offer in response? In
what follows, I will explore the responses and arguments several limitavists
propose. I turn first to the semantic indeterminacy argument discussed earlier
and deflected by McGee.I am jumping ahead a bit here and
skipping indefinite extensibility for now; we shall return to it near the end of
this section.  Shaughan Lavine is one such proponent of the limitavist
view who attempts to argue against McGee’s open-ended quantifier solution to
the semantic indeterminacy argument. If the issue under debate is whether
unrestricted quantification is genuine, then finding a more primitive feature of
logic than quantification will defuse the tension posed. As it turns out, Lavine
argues that such is the case for full schemes. The point of a full scheme is
that we can take particular logical rules like $\phi ,\neg \phi \vdash \psi$to be schemes which we can declare valid in any instance.It is very important to note that ‘any’ is “sharply
distinguished” from ‘every’. This distinction means that we are not
relying on quantification; simply a presentation of a scheme is sufficient, and
we can accept that any such scheme presented to us is valid without being led to
say that all such schemes are valid.  $\phi$and $\psi$are schematic letters, and this scheme is open-ended just in case these
schematic letters are full schematic variables. They are full in the sense that
their acceptable instances of substitution “automatically expands as the
language in use expands”Lavine (2006), 118.. As
such, Lavine’s use of full schemes runs parallel to McGee’s insistence
that the logical quantifiers we employ in our language be open-ended in much the
same way: they act to quantify over names even when they are not in our original
domain. Of course, McGee says that our quantifiers are open-ended and as such
are able to fulfill this role because they quantify over an unrestricted domain.
Lavine, on the other hand, supposes that we need only be committed to sentences
that are closed instances of the scheme; we need not have totalities as a domain
that we quantify over. This distinction relies on the different commitments we
have for schemes versus quantifiers. Lavine uses the example of the successor
operation on the natural numbers.Let $0$be a
natural number. Then $S0$is a natural number and we call
$S0$the “successor of zero”, and we call $S$the
“successor function”. This is the general construction of the natural
numbers using the axioms of Peano Arithmetic. Most straightforwardly,
given that $n$is some schematic letter, from $S0\ne 0$one
can infer that $\mathit\left\{Sn\right\}\ne 0$. If $n$is, on the other
hand, a quantifiable variable, such a move is not allowed; unless, of course,
$n$did not appear free in any of the premises. What must be stressed
is that the full scheme user being committed to $\phi \left\left(n\right\right)\rightarrow \phi \left\left(\mathit\left\{Sn\right\}\right\right)$does not commit
them by itself to the claim that $\forall x\left\left(\phi \left\left(x\right\right)\rightarrow \phi \left\left(\mathit\left\{Sx\right\}\right\right)\right\right)$.For explicit instances of how this works in other cases (like for a
mathematical finitist), see Lavine (2006), 121.  What Lavine has
essentially argued for in saying that full schemes operate in this way is that
they are different from using quantifiers. Indeed, full schemes offer a complete
characterization of the logical operators we use in our formal languages.Based on the work done by Harris which McGee (2006) builds his own
argument from. Lavine also provides an argument for this fact in Lavine (2006),
132-4. Lavine utilizes the statement McGee references in his argument:
$\forall x\exists y\left\left(y=x\right\right)$. Given that $a$is in
the domain of our quantifiers, we can give it a name $c$and conclude
$\exists y\left\left(y=c\right\right)$. We could perform all of this from a
schematic perspective, arriving at the same statement with the full schematic
sentence $s$: $\exists y\left\left(y=s\right\right)$.Lavine refers to this as the “everything axiom”. I shall
follow suite.  From this, we can show that any two languages for which the
everything axiom is a sentence which has a respective domain have the same
extension.Lavine demonstrates the actual proof of this
throughout his paper; See Lavine (2006), 125.  Lavine goes on to argue
that, in fact, the usage by McGeeWilliamson also speaks to
this point in Williamson (2003).  of the everything axiom fails to show
what he wants. The scheme given by the everything axiom can be taken to have an
extension that coincides will all of its instances. McGee argues that, by the
open-endedness of our quantifiers, the everything axiom can capture any objects
that could be named in our language – that is to say, under expansions of
our language the axiom captures those objects – and, as a result, in some
instances of the everything axiom it takes on as a value any of the names in our
structures. Lavine presses the point, saying that this argument put forward by
McGee and others is simply begging the question: so long as we allow such an
extension of the everything principle as being open-ended and unrestricted, we
may as well allow unrestricted quantification. The method of quantifying and
assimilating all the instances of the scheme is already supposing that
unrestricted quantification is fine to begin with, and as a result this argument
holds no weight in establishing the desired conclusion. Suppose that the
everything axiom were to pick out the universe of discourse of unrestricted
quantification. Because schemes only commit us to particular instances of
general claims, we only get a set of instances of the axiom.Not a proper class, as McGee would demand.  We can then apply
the Löwenheim-Skolem theorem
to this set and, because it is not a proper class, it does not form a
totalityAn entire domain; everything., and thus it fails
to pick out a unique domain of everything.Lavine calls the set
we form from the theorem the “Hollywood set”, coming from the excellent
Hollywood Analogy he employs to discuss McGee’s objection in greater detail
in Lavine (2006), 105. Additionally, Williamson’s prior analogy
between unrestricted quantification and absolute identity seems to fail. Indeed,
without the hefty assumption that the everything axiom characterizes a universal
domain of discourse by considering ''every'' instance of the scheme,
open-endedness does not supply the absolutist with the proper extension of the
argument about absolute identity to unrestricted quantification.In fact, it can be (and has been) argued that the question of
identity is an ambiguous one; $\forall x\left\left(x=x\right\right)$is ambiguous
in a way over and above the usual ways for sets or properties. If our domain is
taken to be a kind of collection like a set, then the fact that we do not have a
proper conception of set (as argued earlier) only serves to undermine the notion
that we have a proper conception of identity when we expand our domain. For more
on this, see the discussion (and references) in Parsons (2006).

The question we might be pressed
to answer is how exactly we may determine what our domain ''is'', if not
everything. Lavine says that he takes the universe of discourse to be given by a
context of use.Lavine (2006), 139.  We can see that this
context of use might fall out of the predicate at hand in a particular instance
of the scheme we are considering. Given the power of full schemes in handling
arguments relating to the everything axiom and recognizing that full schemes are
not reducible to quantification, we would like to examine how they might be used
to better understand other charges levied by the absolutist against the
limitavist. Indeed, much in the same way semantic indeterminacy is turned
against the limitavist, so too is the argument relating to sortals. Suppose the
limitavist argues that unrestricted quantification is not genuine quantification
because we lack a proper principle of individuation; ‘object’ and
‘thing’ are not sortal concepts and, as such, do not allow us to carve
up the domain of quantification properly. However, if the limitavist is asked to
support even more mundane questions than ‘how many things are there’
like ‘there are no talking donkeys’, they seem to run into a problem
without allowing for unrestricted quantification. Indeed, to say that ‘there
are no talking donkeys’ is to say that $\forall x\left\left(D\left\left(x\right\right)\rightarrow \neg T\left\left(x\right\right)\right\right)$. Without
unrestricted quantification, there seem to be serious issues with this
formulation.See Williamson (2003) for an analysis of all the
ways in which this goes wrong.  The limitavist might put forward a
different statement of this, saying instead that for any object which the sortal
‘donkey’ applies to, none of them talk. A further generalization of this
statement can be made: ‘Every F, x, is $\phi$’ is true under
assignment A if and only if any ''compliant ''of ''F'' under A, d, is such that
$\phi$is true under A[x/d].This generalization is
provided by Hellman (2006), 89.  The problem with this generalization,
Geoffrey Hellman points out, is that ‘compliant’ is itself a sortal, but
does not have a proper principle of individuation! If it were, then it seems
that the charge against the absolutist likewise disappears; we have a proper
principle of individuation that individuates all objects in a domain. Hellman
offers a solution in the form of distinguishing between kinds of predicates. We
can consider some predicate F to be ‘unlimited’ just in case it is (1)
indefinitely extensible and (2) if E is also unlimited and ‘All E are F’
is taken as true, then F is unlimited.A predicate is
‘limited’ if it is concrete; he considers such predicates to be those
which appear in an earlier section of the paper relating to ontologies.
‘Limited’ predicates are, quite conveniently, ‘not unlimited’.
Note that this definition of unlimited is inductive.  Thus, if we have a
limited predicate, the notion of ‘compliant’ is not terribly
troublesome. As Hellman puts it, the compliant of ‘donkey’ is merely
‘the donkeys’. In this case, our reformulation is just to say that
‘no donkey talks’ is true under A if and only if among the donkeys, d,
‘x doesn’t talk’ is true under A[x/d]. This generalization falls
under a scheme for limited predicates. Under this consideration,
‘compliant’ is itself not a limited predicate, and so our formulation
does not apply to it; thus, we have no real problems with answering the
absolutist about whether donkeys talk. Indeed, the context which the question
provides to us is sufficient for determining exactly what it is our scheme
should be talking about: limited predicates.

Given this definition of
unlimited predicates, what then of indefinite extensibility? If indefinite
extensibility is incoherent on a limitavist view, so much the worse for our
schematic approach to sortals, identity, and semantic indeterminacy. As McGee
has rightfully pointed out earlier, the limitavist construction of the
indefinite extensibility argument is quite unsatisfactory. For instance, the
Burali-Forti paradox requires reference to ‘a totality of the ordinals’.
While this assumption of such a totality is generally for the purpose of a
reductio argument, the premise is concerning in that to make sense of it, one
must have an idea of what exactly it articulates. If the limitavist is correct,
then it doesn’t actually articulate anything. What then, is the limitavist
actually saying in their argument? It seems that, without an absolutist to argue
against, the argument falls flat and does not shed light on the problem for the
generalist. Kite FineAlong with many others; see Shapiro &
Wright (2006) for another such take on it, appealing to Russell’s notion of
an injective mapping onto a subset of the ordinals.  recognizes the issue
and attempts to offer a more precise formulation of the argument in such a way
that it may be articulated without an absolutist position to argue against.Fine points this out quite explicitly after reformulating the
argument in terms of quantification over interpretations. I do not wish to
reiterate the entire argument, as it is quite similar to the one I provided
earlier in section 1 (albeit more technical). Suffice it to say that his
demonstration arrives at the following point: “The universalist seems
obliged to say something false in defense of his position… The limitavist, on
the other hand, can say nothing to distinguish his position from his
opponent’s – at least if his opponent does not speak… Both the
universalist and the limitavist would like to say something true but, where the
one ends up saying something indefensible, the other ends up saying nothing”
Fine (2006), 28.  Fine himself recognizes the strengths of the scheme
approach to resolving the difficulties in absolutely unrestricted
quantification. In the case of indefinite extensibility, the scheme proponent
like Lavine would commit themselves to the scheme $\exists$r(I)$y\forall$I$x\neg \left\left(x=y\right\right)$: given the Russell interpretation r(I), there is
something which is not under a given interpretation I. The problem, Fine
suggests, is that it is hard to make sense of what it might mean to be committed
to the truth of each instance of this scheme if one might know what it is for
such a scheme to be true, and also unwilling to commit themselves to the claim
that any instance of it is true.Fine gives this argument in
Fine (2006) 29. Lavine himself recognizes this point in Lavine (2006), and
agrees with the argument. However, he doesn’t see it as applying to his own
position.  In light of this trouble, Fine suggests that we consider
introducing modality into our position. The view that the limitavist would
maintain is that, given that any interpretation can be extended in principle, we
can say that some interpretation $J$extends (or properly extends) an
interpretation $I$and that the interpretation $I$is
extendible if it is possible that some interpretation extends it.In symbols, $\left\left(I\subset J\right\right)$and
$E\left\left(I\right\right)$, respectively.  In
modal terms, this would look like ◊$\exists J\left\left(I\subset J\right\right)$. Thus, in these terms, the
limitavist position becomes $\forall \mathit\left\{IE\right\}\left\left(I\right\right)$- for any interpretation, it is
extendible. A stronger version of this thesis is that □$\forall \mathit\left\{IE\right\}\left\left(I\right\right)$. Given our prior definitions,
this is identical to □$\forall I◊\exists J\left\left(I\subset J\right\right).$ It is
important to note that these modal symbols are not the usual sorts of modal
symbols. Fine calls them ‘postulational modalities’, and they are meant
to be distinct from both logical and metaphysical necessity and
possibility.There are good reasons for demanding this; namely that if we took
these modalities to be logical or metaphysical, we would be saying something too
strong or too weak. As such, we should take them to be something else. Fine
provides an argument in Fine (2006), 30. Indeed, the divide becomes clear when
we recognize that the modalities Fine is describing are possibilities for the
actual world and not references to other possible worlds; in this way, they are
not circumstantial modalities. Indeed, Fine (2006), 33 suggests that they are
not genuine modalities at all! It is comparable to something like epistemic
modality.
There are other issues posed to our understandings of the quantifiers in play
here. For instance, it is hard given this view to understand what it might mean
to reinterpret the quantifiers in the sense required. Again, much like for our
modal operators, the usual sense in which we might think about these things is
either too strong or too weak to actually capture the limitavist position.
Fine properly explains many of the worries, one of which being that if
our interpretation has the usual quantifiers $\forall x$,
$\exists x$and we expand that interpretation using our condition to
arrive at the quantifiers $\forall$+$x$,
$\exists$+$x$, how are we to understand these
new quantifiers? Presumably in terms of the original quantifiers; appealing to
the Russell set as a motivation, it seems that these reinterpretations give us
too much in the case of $\exists$+$x\forall y\left\left(y\in x\right\right)$, which should only add to our discourse the object
which is all of the objects in the range of $\forall y$- the
universal set. Instead it returns all the sets that have objects in the range
$\forall y$. Given the problems posed, Fine
urges us to loosen our notion of an interpretation of the quantifier. In order
to fix such a problem, we can approach the issue from another direction.
Supposing our only goal in doing this was to say that there is a (proper)
expansion of our interpretation of the quantifiers, we can think about how an
operator should extend the range of the quantifiers. We can postulate an object
such that $!x\forall y\left\left(y\in x\right\right)$. This postulated object
$x$has
members who are the objects $y$of the given domain. This is a
newly distinct account of interpreting our quantifiers. Both the absolutist and
the limitavist imposed restrictions on our quantifiers, the only difference
being whether these restrictions were on an absolute domain or not. On this
view, we are not restricting the quantifiers but instead offering a mechanism by
which their domain is to be expanded.Fine (2006)
surprisingly names this “the expansionist account”.  But now the cogency of our
newly postulated argument must be brought into question. We are not obviously
guaranteed that such an object exists, after all; it is only after a successful
reinterpretation of our quantifiers – seeing that the object postulated is
now in the range – that we see such an object exists. But what makes our
move valid in the first place? As it turns out, the indefinite extensibility of a
set is what can act to guarantee the existence of such an object!Fine refers to
this as the “Russell jump” we make on a given collection, referring to
the Russell set and the method by which we arrived at it.  Given this final sort of
understanding of what exactly it is the expansionist purports to be claiming, we
can further examine cases used by Williamson. For instance, we have seen how the
scheme view would deal with “no donkeys talk”. The expansionist can
utilize their newfound modal operators to properly strengthen their claim in
saying this: □$\forall x\left\left(D\left\left(x\right\right)\rightarrow \neg T\left\left(x\right\right)\right\right)$. Necessarily, no donkey talks
– under any expansion of the domain of our quantifiers, there will be no new
object in it such that it is a talking donkey. Likewise, for other universal
claims like everything is self-identical: instead of saying $\forall x\left\left(x=x\right\right)$,
one can instead say that necessarily, any postulated object will be
self-identical, or □$\forall x\left\left(x=x\right\right)$.

''SECTION 4 – WHAT TO MAKE OF IT ALL''

What I would like to
conclude with is a consideration of the responses the limitavist has for the
various charges levied against them by the absolutist, as well as whether we can
suspect that the limitavist has reasonably supported their position against the
proponent of unrestricted quantification or if the absolutist has managed to
convince us. McGee’s principle argument for an unrestricted domain of
quantification is that the universal quantifiers governed by universal
instantiation and universal generalization are interderivable and that, as this
is the case, they can pick out an optimal candidate in the domain for the
semantic value of the quantifier. Given this, we can conclude that unrestricted
quantification is the preferred form of quantifying over a domain because it
preserves truth in every instance where our language expands to include new
names that the restricted position on the quantifiers cannot support. What is
worth considering in McGee’s argument is that he explicitly presumes a
two-valued, classical logic. Given a logic which isn’t classical, is the
defense which McGee proposes satisfactory? As far as intuitionist logic is
concernedI consider intuitionist logic only here simply because I am more
sympathetic to the view myself and Dummett himself was an intuitionist logician;
I think it’s at least semi appropriate to look at McGee’s argument how
Dummett might have. , there do not seem to be too
many problems with the argument. When introducing quantifiers to intuitionist
logic, it behaves somewhat like a modal logic.In that given
our interpretation in the intuitionist logic with quantifiers, we consider the
truth of universals and existentials with respect to truth at possible worlds.
For a breakdown of this logic, see Priest (2008), 421-37. Indeed, in a one-world
interpretation for quantified intuitionist logic, it reduces to a classical logic
with quantifiers. In the case of a many-world interpretation, that the
quantifiers are interderivable is less obvious, but the claim is nonetheless
true.I wanted to provide my own proof of this, but I was beaten to the
punch; see Lavine (2006), 112.  As a result, the assumption of
classical truth is admissible, and the argument could be in principle extended
to nonclassical logics. However, where McGee’s position does falter is how
it attempts to defuse the semantic indeterminacy argument. Recall that
McGee’s primary point was that the Löwenheim-Skolem theorem
requires that names already exist in the domain; under any expansion of our
language through the introduction of new names – through a new scientific
theory, perhaps – the skeptic cannot support themselves and the argument is
subdued. However, we should recognize that there are a small number of names we
can add to any language in our scientific theories.The other
example McGee gives, that of naming a child or a new pet, make this point even
more obvious. So suppose that we have a language $L$. Expand
$L$by introducing all of the names that I could add in McGee’s
manner, and call this language $L$+. Is there anything
it? This does not seem to be the case – it is a perfectly acceptable
language in just the same way our unexpanded language was. Thus, given any
expansion of your language through the introduction of new names, I should
always be able to present the exact same problem back to you. However, I think
it is worth noting what McGee rightfully points out to us: the Löwenheim-Skolem theorem
is a theorem of first-order logic. While it can be extended to higher order
logics and McGee grants that such an extension could be performed in principle
to higher and higher order logics, we might be concerned with whether a
sufficiently complex logic would begin to capture the ways in which we use
natural language in our everyday discourse. That is to say, if it is determined
that natural language is somehow above and beyond what can be captured in a
formal language, it is not clear if we could extend the Löwenheim-Skolem theorem
to any kind of natural language.While I myself am skeptical of
the claim that natural languages are extensions of formal languages, I do not
think I am in any sort of position to offer an argument to this end. It
certainly wouldn’t fit here. Finally, McGee’s final point that the
open-endedness of our quantifiers give strong support to the conclusion that
absolutely unrestricted quantification is the preferred interpretation of our
quantifiers seems to not quite establish itself. Indeed, if Lavine is correct,
full schemes seem more than capable of satisfying the exact same requirements
McGee stresses in his thesis. However, that isn’t to say that the full
scheme position is saying anything which the absolutist would not likewise
accept. Lavine is careful to say that we are only committed to ''any'' instance
of our given scheme, not ''every'' instance. This distinction, while small, does
a large amount of work; for instance, we avoid quantifying over our schemes,
which would only serve to push the problem back a step.Fine
(2006) makes a similar sort of argument; Lavine rejects that it defeats his
position because it isn’t a proper characterization of his position.
However, as Fine points out earlier, it is unclear what it might mean to be
committed to the truth of each instantiation of a scheme but not be committed to
the commitment that they are true. As it were, it is unclear what I am to make
of any scheme’s truth if I cannot endorse that any instance is true.
However, if we grant that this does not make the scheme endorser’s argument
any less cogent, it does provide us with a certain strength to push back against
Williamson’s analogy between absolute identity and unrestricted
quantification. Indeed, given the sorts of commitments we have to a scheme like
the everything axiom, the question of identity might be taken to itself be
ambiguous. As a result, insofar as identity and unrestricted quantification are
closely connected, if identity is ambiguous so much the worse for
quantification. But if we do not endorse the scheme view because of its possible
problems, Fine’s position still seems attractive – so long as we are
willing to accept some modal baggage. The primary concern one might have with
Fine’s expansionism is how we are to properly understand these modal
operators as being different from the usual metaphysical and logical ones.
Indeed, we have excellent reasons for supposing that they aren’t; for
starters, they are inadequate or ill-equipped to provide an argument against the
absolutist with respect to indefinite extensibility without being charged with
committing themselves to unrestricted quantification, as Williamson and McGee
have previously argued such arguments open the limitavist up to. The operators
are not circumstantial, but interpretational. But if there is no way of properly
fleshing out how this works like David Lewis did with modality and possible
worldsKit Fine recognizes his lack of producing an adequate
kind of semantics. , it seems we are faced with having to take these
operators as a kind of primitive. However, this does not seem so bad. Indeed, it
seems that the scheme view also demands of us to take schemes as a primitive.
But if we take either of
these as primitives, for what purpose? Why choose one over the other? Both seem
relatively unloaded in terms of ontological or logical commitments;I have spent
very little time discussing ontological commitments. However, several people
referenced here such as Putnam (1980), Glanzberg (2004), (2006), and Lavine
(2006) have attempted to argue that the absolutist is committed to some form of
metaphysical realism, primarily relating to the semantic indeterminacy argument.
As it stands, I have attempted to focus more on the logical character of the
arguments and objections as opposed to getting bogged down in questions of
ontology. However, the two are intimately related, especially when we are
concerned with everything. The views presented by Lavine and Fine are both free
of the ontological commitments the absolutist might be committed to.
perhaps the
only reason to support one beyond another is on pragmatic grounds. Lavine sees
the full scheme view as being simpler and more readily understood than
postulational modalities; I myself see postulational modality as being a far
simpler version of the usual modal semantics we acquire in a Lewisian modal
logic. I agree with Fine in that it seems unclear what the commitments on the
part of the full scheme view amount to. Alternatively, the expansionist account
of the limitavist view seems to make it so that I am always ready, able, and
willing to inform the absolutist that they are not quite capturing everything.

All things considered, I find
that the point on natural languages is less than appealing; generally speaking,
natural language users can disambiguate their own claims with ease and don’t
tend to worry about such issues as “does the domain of my quantification
include itself or not?” Instead, we usually tend to know what it is
we’re talking about – the claim that “there are no talking
donkeys” makes perfectly good sense to us not because we grant the use of
unrestricted quantification or don’t, but rather because we know that its
truth stands or falls with the claim that no animals talk, and that such animals
exist on Earth in certain regions and are members of a particular family of
species which evolved in a certain way. We know what we mean by donkeys, and it
certainly isn’t ‘everything’. Indeed, it seems that we only need to
quantify over a small region of space and consider all animals which exist
there. No animal that isn’t a donkey doesn’t share a perfect
evolutionary history with donkeys, and there seems to be something about donkeys
that commit us to their Earth-bound existence. If we found donkeys on Mars, it
isn’t obvious that we would consider them to be donkeys. The point on
Skolemization is likewise less satisfying. While McGee is probably correct in
arguing that higher and higher degrees of complexity in our logics could
similarly run into issues like a first-order logic does, we have the tools we
require in first-order logics to reduce the “Skolem paradox” to a fluke
in our understanding of the concept itself. Likewise, it does not seem to be an
issue that such a Skolemization of our natural languages wouldn’t likewise
allow us to utilize similar resolutions, albeit they would perhaps be more
complicated – but that is simply a consequence of the nature of our
language. Finally, on indefinite extensibility. As far as I don’t consider
absolute generality or unrestricted quantification to be a necessary or obvious
problem for a natural language, I think that indefinite extensibility is the
only real problem for an absolutist and only as it applies to more formal
languages. Indeed, it seems to be the only criticism which both satisfies and
convinces me of the limitavist position. Kit Fine’s argument on the topic
likewise convinces me in terms of how the limitavist can go about treating logic in the
usual way we would like, without accepting untenable positions like unrestricted
quantification ''considering indefinite
extensibility concerns''. What is most
interesting to point out here is that many of the defenses of a limitavist
perspective hinge on in some way utilizing what might be mistaken as a
universalist view. But these reformulations rely on a very particular claim:
their statements do not rely on quantification, but instead on some other aspect
of logic. For Fine, it is modal operators, which aren’t
‘traditional’ modal operators. For Lavine, it is schemes, which
aren’t reducible to quantifiers. What we are left to grapple with is whether
or not we are comfortable accepting these new tools or different ways of
understanding our commonplace discourse. Either way we go with the debate, we
are left with unsettling things to consider.

= REFERENCES =

Barker-Plummer, D., Barwise, J., Etchemendy, J. (2011) ''Language, Proof, and
Logic''. Stanford,

CA: CSLI Publications.

Clark, P. (1998) ‘Dummett’s Argument for the Indefinite Extensibility of
Set and Real Number’,

''Grazer Philosophische Studien ''55, 51-63.

Dummett, M. (1981) ''Frege: Philosophy of Language'', Harvard, Cambridge, MA,
second edition.

Dummett, M. (1991) ''Frege: Philosophy of Mathematics'', Duckworth, London.

Dummett, M. (1993) ‘Does Quantification Involve Identity?’, in Dummett,
''The Seas of ''

''Language''. Oxford: Clarendon Press.

Fine, K. (2006) ‘Relatively Unrestricted Quantification’. In A. Rayo &
G. Uzquiano (Authors),

''Absolute Generality'' (pp. 20-44). Oxford: Clarendon Press.

Frege, G. (1884) ''Die Grundlagen der Arithmetik''. English Translation by J. L.
Austin, ''The ''

''Foundations of Arithmetic'', Northwestern University Press, Evanston, IL,
1980.

Frege, G. (2013) ''Basic Laws of Arithmetic ''(P. A. Ebert & M. Rossberg,
Trans.). Oxford: Oxford

University Press.

Geach, P. T. (1967) ‘Identity’, ''Review of Metaphysics'' 21:3-12.

Glanzberg, M. (2004) ‘Quantification and Realism’, ''Philosophy and
Phenomenological Research''

69, 541-72.

Hellman, G. (2006) ‘Against ‘Absolutely Everything’!’. In A.
Rayo & G. Uzquiano (Authors),

''Absolute Generality'' (pp. 75-97). Oxford: Clarendon Press.

Lavine, S. (2006) ‘Something About Everything: Universal Quantification in
the Universal Sense

of Universal Quantification’. In A. Rayo & G. Uzquiano (Authors), ''Absolute
Generality''

(pp. 98-148). Oxford: Clarendon Press.

Lewis, D. (1983) ‘New Work for a Theory of Universals’, ''Australasian
Journal of Philosophy ''61,

343-77.

McGee, V. (2006) ‘There’s a Rule for Everything’. In A. Rayo & G.
Uzquiano (Authors), ''Absolute ''

''Generality'' (pp. 179-202). Oxford: Clarendon Press.

Moore, A. W. (2015) ''The Infinite''. London: Routledge.

Parsons, C. (2006) ‘The Problem of Absolute Universality’. In A. Rayo &
G. Uzquiano (Authors),

''Absolute Generality'' (pp. 203-19). Oxford: Clarendon Press.

Peters, S., & Westerståhl, D. (2006) ''Quantifiers in Language and Logic''.
Oxford: Clarendon Press.

Priest, G. (2008) ''An Introduction to Non-Classical Logic: From If to Is''.
Cambridge: Cambridge

University Press.

Putnam, H. (1980) ‘Models and Reality’, ''The Journal of Symbolic
Logic'' 45:3, 464-82.

Quine, W. V. (1968) ‘Ontological Relativity’, ''Journal of Philosophy''
65, 185-212.

Rayo, A., & Uzquiano, G. (2006) ''Absolute Generality''. Oxford: Clarendon
Press.

Russell, B. (1906) ‘On some Difficulties in the Theory of Transfinite
Numbers and Order Types’,

''Proceedings of the London Mathematical Society'' 4, 29-53.

Shapiro, S. (2011) ''Thinking About Mathematics: The Philosophy of
Mathematics''. Oxford: Oxford

University Press.

Shapiro, S., & Wright, C. (2006) ‘All Things Indefinitely Extensible’.
In A. Rayo & G. Uzquiano

(Authors), ''Absolute Generality'' (pp. 255-304). Oxford: Clarendon Press.

Taylor, D. E., & Burgess, A. (2015) ‘What in the World is Semantic
Indeterminacy?’ ''Analytic ''

''Philosophy, ''56:4, 298-317.

Whitehead, A. N., & Russell, B. (1910) ''Principia Mathematica''. Cambridge: Cambridge

University Press.

Williamson, T. (2003) ‘Everything’, ''Philosophical Perspectives'' 17,
415-65.

Williamson, T. (2006) ‘Absolute Identity and Absolute Generality’. In A.
Rayo & G. Uzquiano

(Authors), ''Absolute Generality'' (pp. 369-90). Oxford: Clarendon Press.

----

________________________________________________________________________________

Dilyn Corner (C) 2020-2022