# Logicality and Model Classes111The first author would like to thank the
Academy of Finland, grant no: 322488. The second author would like to thank
the Academy of Finland, grant no: 322795. This project has received funding
from the European Research Council (ERC) under the European Union’s Horizon
2020 research and innovation programme (grant agreement No 101020762).The
authors are grateful to Denis Bonnay, Dag Westerståhl and two anonymous
referees for comments.

Juliette Kennedy
Department of Mathematics and Statistics
University of Helsinki, Finland  Jouko Väänänen
Department of Mathematics and Statistics
University of Helsinki, Finland
ILLC, University of Amsterdam
Amsterdam, Netherlands

###### Abstract

We ask, when is a property of a model a logical property? According to the so-
called Tarski-Sher criterion this is the case when the property is preserved
by isomorphisms. We relate this to model-theoretic characteristics of abstract
logics in which the model class is definable. This results in a graded concept
of logicality in the terminology of Sagi [46]. We investigate which
characteristics of logics, such as variants of the Löwenheim-Skolem Theorem,
Completeness Theorem, and absoluteness, are relevant from the logicality point
of view, continuing earlier work by Bonnay, Feferman, and Sagi. We suggest
that a logic is the more logical the closer it is to first order logic. We
also offer a refinement of the result of McGee that logical properties of
models can be expressed in $L_{\infty\infty}$ if the expression is allowed to
depend on the cardinality of the model, based on replacing $L_{\infty\infty}$
by a “tamer” logic.

##  1 Introduction

To say that the syllogism

All $A^{\prime}s$ are $B^{\prime}s$.

Some $C^{\prime}s$ are $A^{\prime}s$.

Some $C^{\prime}s$ are $B^{\prime}s$.

is a valid argument form is to say that the conclusion follows from the
premises no matter which predicates are substituted for the nonlogical terms
$A$, $B$, $C$, whether ‘men’, ‘mortal’, ‘Greek’ or ‘tulips’, ‘bridges’ and
‘pious’, or whatnot, as long as the substitution is done uniformly. That the
conclusion follows from the premises is a matter of logic—“is obviously true
in a purely logical way,” to quote Carnap.222[16], Engl. translation in [15].

But what is it that makes a concept distinctively logical, as opposed to, say,
mathematical? How to circumscribe the logical? The question is an urgent one
for the logician, as the model-theoretic notion of consequence is parasitic on
the distinction between logical and nonlogical expressions. For on this
account of logical consequence, a sentence $\phi$ is held to be a semantic
consequence of a sentence $\psi$, if for every uniform substitution of the
nonlogical expressions in $\phi$ and $\psi$, if $\psi$ is true, then so is
$\phi$.

In 1968 in a (posthumously published) lecture called “What are logical
notions?” [52] Tarski proposed a definition of “logical notion,” or
alternatively of “logical constant,” modelled on the Erlanger Program due to
Felix Klein. The core observation is the following: for a given subject area,
the number of concepts classified as invariant are inversely related to the
number of transformations—the more transformations there are, the fewer
invariant notions there are. If one thinks of logic as the most general of all
the mathematical sciences, why not then declare “logical” notions to be the
limiting cases? Thus a notion is to be thought of as logical if it is
invariant under all permutations of the relevant domain.

As expected, the standard logical constants, namely conjunction, disjunction,
negation and quantification, together with the equality relation, are all
judged to be logical operations under this criterion, being all isomorphism
invariant. This has the consequence that first order definability is
classified as logical, tout court, being generated by these constants.333We
argue for this below. V. McGee [41] extends the observation to
$L_{\infty\infty}$:  Since the primitive connectives of $L_{\infty\infty}$ are
all intuitively clearly logical connectives, and since, intuitively, anything
definable from logical connectives is again a logical connective, this will
show that every operation invariant under permutations is describable by a
logical connective, so every operation invariant under permutations is a
logical operation.

As was clear to Tarski, the criterion overgenerates, in the sense that
concepts one does not immediately assimilate to logic turn out to be included,
such as the cardinality of the underlying domain, which is clearly isomorphism
invariant. A second problem has to do with domain relativity: Tarski’s
criterion identifies the logical operations on a fixed domain, but that a
domain consists of this or that type of object should not have anything to do
with logicality—shouldn’t a logical concept be domain independent? G. Sher
[48] repaired this problem by extending Tarski’s criterion to cover notions
invariant across isomorphic structures, and accordingly the criterion is now
known as the Tarski-Sher invariance criterion.

The Tarski-Sher criterion has generated a substantial literature, both for and
against. S. Feferman’s [22], based on his earlier [21], proposes a notion of
homomorphism invariance. His critique of the Tarski-Sher thesis is the
following:

> I critiqued the Tarski-Sher thesis in [21] on three grounds, the first of
> which is that it assimilates logic to mathematics, the second that the
> notions involved are not set-theoretically robust, i.e. not absolute, and
> the third that no natural explanation is given by the thesis of what
> constitutes the same logical operation over arbitrary basic domains.444[22],
> p. 1.

Feferman’s suggestion that Tarski assimilates logic to mathematics, in
particular to set theory, sets aside Tarski’s lifelong program to carry out
exactly what Feferman criticizes Tarski of here, namely expressing
metamathematical concepts in mathematical and set-theoretical terms.555The
first author has argued this point in [31]. Feferman’s second objection has to
do with overgeneration, in particular he resists the idea that a canonically
mathematical but non-absolute notion such as cardinality is rendered logical
under the criterion.666The fact that cardinality is construed as logical under
his criterion seemed to pose no problem for Tarski:  That a class consists of
three elements, or four elements …that it is finite, or infinite—these are
logical notions, and are essentially the only logical notions on this level.
[52], p. 151.

One way of thinking about topic neutrality is in terms of absoluteness, by
which we mean general independence from the background set theory. As is well-
known, a set may have one cardinality in one model of set theory but another
cardinality in another model.777Cardinality also emerges as an artifact in the
context of ramsification and Newman’s objection to epistemic structural
realism. See Ainsworth [1]. This shows that cardinality is not an absolute
notion. On the other hand, whether a given set is empty or not, whether it is
included in another given set, and whether it is the cartesian product of two
given sets are all independent of the (transitive) model in which such
questions are evaluated. Thus those notions are considered absolute. A
possible reason to consider absoluteness a necessary quality of logical
notions is that whatever depends on set theory could be considered a
mathematical rather than logical notion.

Quine has extensively argued against taking set-theoretic notions as
logical,888See for example [45]. and in his seminal paper [52] Tarski raises
the question whether mathematical notions are logical. According to Tarski, if
we construe mathematical notions in the framework of higher order logic, they
are logical, while if we use the set-theoretical framework, based on a single
binary relation $x\in y$, mathematical notions are not logical. Tarski
concludes that the question of logicality of mathematical notions is
unresolved and depends on how we construe mathematics. In this paper we take
the set-theoretical approach to mathematics. However, we operate with model
classes, namely classes of structures of the same similarity type, but
possibly of different cardinality, that are closed under isomorphism,999If we
work in set theory, classes are objects of the form $\\{a:\phi(a)\\}$.
Ordinary set theory does not have objects of this kind, so classes are simply
identified with their defining formulas, in this case $\phi(x)$. The defining
formula is allowed to have set parameters. Thus when we refer to the class of
all ordinals, the class of all structures, etc, we mean to refer to the
defining formula, which defines when a set is an ordinal, the formula which
defines when a set is structure, etc. If we worked in class theory, such as
the Mostowski-Kelley-Morse class theory, we would not need to assume that
classes are definable. which are part of the higher order logic framework. For
example, according to Tarski the notion of a binary relation $R$ being a well-
order is non-logical, as it is defined by reference to the $\in$-relation of
set theory, but the notion of a structure $(M,<)$ being a well-order is
logical, as it is second order definable. This example shows how delicate the
question of logicality of mathematical notions is.

Feferman proposes restricting the Tarski-Sher invariance criterion to
operations that are absolute with respect to set theories making no
assumptions about the size of the given domain. Feferman cites
[2],101010Feferman also cites an unpublished result of K. Manders. in which it
is shown that operations on relational structures that are definable in an
absolute way relative to KPU-Inf, i.e. Kripke-Platek set theory with
urelements and without the Axiom of Infinity, are exactly those expressible in
ordinary first-order predicate calculus with equality—another endorsement of
the idea that first order logic captures the notion of “logical concept.”

Feferman’s solution solves the overgeneration problem, while giving a nice
characterisation of first order logic along the way. But this comes at the
expense of a strong absoluteness assumption, and for restricted classes of
structures.

Setting overgeneration aside for the moment, and considering Feferman’s third
objection, namely the problem of domain relativity, it would seem that
following Feferman’s line we should consider model classes in general. How
then to formulate the concept of logicality for a model class? Originally the
question of logicality arises in connection with operations such as
connectives, quantifiers and their generalisations. As V. McGee explains in
[41], the question can be restated as a question about model classes, as the
question whether the property of a given model being in a given model class is
a logical property of the model.111111This property is essentially the same as
deeming the associated generalised quantifier, together with everything first
definable from it, as logical. See below section 2. In other words, any
relevant operation $O$ can be (re)formulated as a property $P$ of models (see
Section 2.1) and then the logicality of $O$ translates into the question
whether the property $P$ is a logical property of models. Typical examples of
logical properties of a unary model $(M,A)$, where $A\subseteq M$, would be
the non-emptyness of $A$, and the property of $A$ of being equal to $M$. On
the side of operators these example would correspond to the existential and
the universal quantifers.

Let’s back up and ask when would we call a property of an individual structure
${\mathcal{M}}$ logical. With Sher, certainly we should consider the elements
of ${\mathcal{M}}$ irrelevant as long as the arrangement of the elements
remains the same, as was mentioned. It would then be natural to call a
property of ${\mathcal{M}}$ logical only if the property is closed under
permutations of the elements of ${\mathcal{M}}$ in the sense that if $f$ is a
permutation of $M$ then the image of ${\mathcal{M}}$ as a structure under $f$
has the property. But then it is a short step to require closure under
isomorphisms.

A theorem due to McGee [41] characterises logicality for an arbitrary model
class, provided the cardinality of the models in the class is fixed in
advance. It is a weakness of this characterisation that it depends on the
cardinality of the models in the class. One might then ask, whether there is a
sentence in a logic $L^{*}$, perhaps other than $L_{\infty\infty}$, which
characterises the logical property, not just in one cardinality, but in many
cardinalities. This leads to an analysis of spectra, as we call them,121212See
also [46]. and Löwenheim-Skolem style properties of logics as expressed in
their Löwenheim and Hanf numbers, which we now define:

######  Definition 1.

Suppose $C$ and $D$ are classes of cardinal numbers. A logic $L^{*}$ satisfies
the _Löwenheim-Skolem Property_ $\mbox{\rm LS}(C,D)$ if every sentence in
$L^{*}$ which has a model of some cardinality in $C$ has a model of some
cardinality in $D$. The smallest $\kappa$ such that $L^{*}$ satisfies
$\mbox{\rm LS}([1,\infty),[1,\kappa])$ is called the Löwenheim number of
$L^{*}$ and denoted $\ell(L^{*})$. The smallest $\kappa$ such that $L^{*}$
satisfies $\mbox{\rm LS}([\kappa,\infty),[\lambda,\infty))$ for all $\lambda$
is called the Hanf number of $L^{*}$ and denoted $h(L^{*})$.

As G. Sagi notes in her [46], the Löwenheim number of a logic encodes the
degree to which the logic is indifferent to cardinality. This is due to the
fact that if a sentence satisfied by a model is also satisfied by a model of
size less than or equal to a given cardinality $\kappa$, then all the
validities131313Or for Sagi, the meaning, see [46]. which are captured in the
full range of models are captured already on an initial segment (of $V$). In
the case of first order logic the Löwenheim-Skolem Theorem tells us that
$\kappa$ can be taken to be $\aleph_{0}$, so in this sense first order logic
is indifferent to cardinalities above $\aleph_{0}$. First order logic is
already classified as logical because its logical constants are permutation
invariant; this is now witnessed by the Löwenheim-Skolem Theorem.

The strength of the Löwenheim-Skolem property of a given logic—or if you like,
the degree of its indifference to cardinality—is measured, then, by its
Löwenheim number. As an example, the quantifier $Q_{\alpha}$ associated with
the logic $\mathcal{L}(Q_{\alpha})$, having Löwenheim number
$\aleph_{\alpha}$, is in a sense indifferent to cardinalities greater than
$\aleph_{\alpha}$.141414This is a central example of [46]. First order logic
and $\mathcal{L}(Q_{0})$ are maximally logical under the criterion assigning
logicality according to Löwenheim number, and the degree of logicality
decreases as $\alpha$ increases. Thus if $\alpha\leq\beta$, then $Q_{\alpha}$
is more logical than $Q_{\beta}$.151515$\mathcal{L}(Q)$ denotes first order
logic with the generalised quantifier $Q$ appended. The expression
“$Q_{\alpha}x\phi(x)$” means that “there are at least $\aleph_{\alpha}$ many
$x$ such that $\phi(x)$.” By the Löwenheim-Skolem Theorem, the Löwenheim
number of first order logic is $\aleph_{0}$, as was noted; for each $\alpha$,
the Löwenheim number of the logic $\mathcal{L}(Q_{\alpha})$ is
$\aleph_{\alpha}$.

The criterion for logicality presented in Sagi’s paper is based on a
particular philosophical position, to wit: a metaphysical view of the
cumulative hierarchy of sets, as well as a view about how the meaning of the
terms of a logic are fixed.

> …the higher set-theoretic infinite is metaphysically loaded…The lower the
> cardinalities to which the meaning161616emphasis ours of a term may be
> sensitive, the more logical it is. …[We] view Löwenheim numbers as telling
> us how much structure a term [in this case the quantifiers $Q_{\alpha}$
> JK/JV] requires in order to be fixed in the context of a logic.”171717[46]

In short, the smaller the Löwenheim number of the logic, the less
metaphysically involved e.g. the quantifier $Q_{\alpha}$ is.

Sagi’s criterion partitions logical space in ways that differ from ours. In
this paper we will not take account of the possible metaphysical commitments
of set theory; and while her analysis of meaning is one we potentially
endorse, we will not take a stand on the meanings of the terms of a logic in
this paper. Our basic thesis is simply this: if first order logic is taken to
be the fundamental exemplar of logicality, then logics that resemble, to a
degree, first order logic in their model theoretic properties should be graded
as logical to that degree.

Why do we take first order logic to be the fundamental exemplar of logicality?
Considerations of space do not allow us to argue for this contested point
here.181818 Those who have contested the view that logic is first order logic
include Sher [48] and Barwise [8]. Many others have argued for the value of
nonclassical and higher order logics on grounds other than logicality such as
Boolos [13]. See also Sher’s 2016 [49].  Indeed those opposed to the thesis
that first order logic is maximally logical can perhaps find this paper useful
as a test of that very thesis, given the complexity of the landscape of higher
order logics laid out here, with respect to their (graded) degrees of
logicality. For example, we suggest in this paper that logics such as
$\mathcal{L}(Q_{1})$ have, arguably, a higher degree of logicality than
$\mathcal{L}(Q_{0})$, on the basis of the completeness of the former with
respect to the Keisler axioms (see (9) in Section 7).

As is well known, Quine was a forceful advocate for the thesis that logic is
first order logic on various grounds, including: First, the fact that first
order logic has a complete proof system. Second, first order logic is the
basis of the distinction between logic and set theory, as we mentioned above.
Finally, there is Quine’s view of semantics, in particular his view that first
order quantification is the optimal instrument for reading off the ontological
commitments of a given theory, being itself ontologically minimal.191919See
e.g. [45].

Apart from completeness, we do not take a stand on these stalwarts of Quinean
logical theory here,202020More recent advocates for the thesis that first
order logic is logic include Bonnay [10] and Feferman [22]. rather referring
to what seems to be a consensus among writers on the topic that the
connectives ($\wedge,\vee,\neg,\to$ , etc) as well as the quantifiers
($\exists,\forall$) are logical operations, being closed under bijections, and
also on the basis of what seems to be a strong intuition that they have the
same meaning across domains of even different sizes. In fact, in many cases
these operations are where the discussion on logicality starts. First order
logic consists of iterations of these operations. Why would logicality
disappear in this iteration? When we go beyond first order logic, e.g. to
infnitary connectives and generalized quantifiers, we recognise the emerging
problem whether we remain in the realm of logicality.

Thus while taking up Sagi’s suggestion that logicality may be calibrated by
Löwenheim numbers, we also give weight to the other model theoretic properties
of a logic, such as whether they have a Completeness Theorem or
not.212121Quine emphasises the Completeness Theorem as primary evidence of the
logicality of first order logic, asserting it as proof of the existence of “an
integrated domain [i.e. first order logic JK/JV] of logical theory with bold
and significant boundaries.” See p. 90-91 of [45]. As we will point out below,
logics that have, for example, a Completeness Theorem do not seem to be tied
to the $\aleph$-hierarchy in any obvious way.

Another drawback of McGee’s theorem which we address in this paper has to do
with the model-theoretic properties of the logic $L_{\infty\infty}$, namely
that it is highly non-absolute (see Definition 17), that it fails to have a
strong Löwenheim-Skolem theorem, and also that it is _unbounded 222222as it is
defined in [6]_ in the sense that it can define the concept of well-ordering
leading to very large Hanf numbers.232323See discussion following (8). This
leads us to ask whether $L_{\infty\infty}$ can be replaced by a “tamer” logic,
one closer to being first order in its model-theoretic properties. If such
were to exist, then even with the problem of dependence on the cardinality of
the models in the class unsolved, the relevant logicality claim would be
strengthened by virtue of its proximity to first order logic.

In sum: in this paper we develop further this aspect of logicality identified
by Sagi, namely that it is graded, albeit with a different philosophical
agenda than is laid out in [46]. We especially point out problems in the
Löwenheim-Skolem properties of $L_{\infty\infty}$, along with other ways in
which it diverges from first order logic. As was mentioned above, we take
first order definability as a particularly strong form of logicality.
Accordingly, our suggestion here is that definability in a logic which
resembles with its model theoretic properties first order logic should
represent an intermediate form of logicality. We calibrate these intermediate
forms according to the model theoretic properties of these logics, considering
mainly absoluteness, having a Löwenheim-Skolem theorem, and having a
completeness theorem.

While $L_{\infty\infty}$ is sufficient for McGee’s theorem, there is a weaker
logic, namely the so-called $\Delta$-extension of $L_{\infty\omega}$ which
does the job as well. We shall explain in which way this weaker logic is
better than $L_{\infty\infty}$ as a test of logicality, and also point out in
which respects it may be lacking. Looking ahead, our refinement of McGee’s
theorem replaces $L_{\infty\infty}$ by a logic which is absolute and which has
a good Löwenheim-Skolem theorem, along with a number of other desirable
properties. The improvement here is partial in the sense that it is still the
case that the definition is given relative to the size of the models in the
class. In section 6 below we give a theorem which does not rely on this
assumption, namely we present a logic in which any model class is definable
irrespective of the cardinality of the models in the class.

In spite of the seemingly clear intuition behind the syllogistic example with
which we opened this paper, logicality is an elusive concept. It is by no
means obvious what it should mean, even informally. But then, without such a
clear informal intuition of its meaning it is difficult to judge whether this
or that improvement of Tarski’s original definition hits the mark or not, or
even whether it is a step in the right direction. The only thing there seems
to be a consensus on in the wake of Tarski’s suggestion to identify logicality
with isomorphism invariance is that Tarski’s criterion, extended by Sher, is a
necessary but not sufficient one.242424As Bonnay and Westerståhl put it in
their [12]: “And topic-neutrality, in the precise form of invariance under
permutations of the universe, is almost universally agreed to be a necessary
condition for logicality. It guarantees that the logical core of a language is
general enough to carve out content in any conceivable situation of language
use, irrespective of what objects are being talked about.”

In this paper we create a base map of the landscape relevant for logicality.
Following Tarski’s, ours is a semantic approach. Beginning with McGee’s
Theorem [41] to the effect that logical operations in the sense of Tarski-Sher
can be described in each cardinality separately in $L_{\infty\infty}$, the
landscape that opens in front of us is the world of different logics and the
question, do the model-theoretic properties of these logics shed any light on
the logicality problem?

##  2 McGee’s Theorem

To describe McGee’s result and to put it into a more general framework we
define what we mean by a “logic”: A _logic_ (a.k.a. abstract logic) in the
sense of [33] is a pair $L^{*}=(\Sigma,T)$, where $\Sigma$ is an arbitrary set
(sometimes also a class) and $T$ is a binary relation between members of
$\Sigma$ on the one hand and structures on the other. Members of $\Sigma$ are
called $L^{*}$-sentences. Classes of the form

$\mbox{Mod}(\phi)=\\{{\mathcal{M}}:T(\phi,{\mathcal{M}})\\},$

where $\phi$ is an $L^{*}$-sentence, are called $L^{*}$-characterizable, or
$L^{*}$-definable, classes. Abstract logics are assumed to satisfy five axioms
expressed in terms of $L^{*}$-characterizable classes, corresponding to being
closed under isomorphism, conjunction, negation, permutation of symbols, and
“free” expansions.252525The free expansion to vocabulary $L$ of a model class
$K$ of a smaller vocabulary is the class of all expansions of elements of $K$
to the vocabulary $L$. A class $\mathcal{K}$ of models is said to be
_definable_ in a logic $L^{*}$ if there is a sentence $\phi$ in $L^{*}$ such
that

$\mathcal{K}=\mbox{Mod}(\phi).$

Every model class is definable in _some_ logic because we can take the model
class as a generalized quantifier in the sense of [32]: Suppose $\mathcal{K}$
is a model class with vocabulary $L$. For simplicity we assume $L=\\{R\\}$
where $R$ is a binary predicate symbol. We can associate with $\mathcal{K}$
the generalized quantifier $Q_{\mathcal{K}}$ with the semantics

${\mathcal{M}}\models Q_{\mathcal{K}}xy\phi(x,y,\vec{a})\iff(M,\\{(b,c)\in
M^{2}:{\mathcal{M}}\models\phi(b,c,\vec{a})\\})\in\mathcal{K}.$

Now $\mathcal{K}$ is trivially definable in the extension
$\mathcal{L}_{\omega\omega}(Q_{\mathcal{K}})$ of first order logic by the
quantifier $Q_{\mathcal{K}}$ by the sentence

$Q_{\mathcal{K}}xyR(x,y).$

Conversely, every class of models definable in
$\mathcal{L}_{\omega\omega}(Q_{\mathcal{K}})$, or indeed in any abstract
logic, is a model class i.e. is closed under isomorphisms. We obtain the
following simple and at the same time basic characterization:

######  Theorem 2 ([32]).

If $\mathcal{K}$ is a class of models of the same vocabulary, then the
following conditions are equivalent:

  1. 1.

$\mathcal{K}$ is closed under isomorphisms i.e. $\mathcal{K}$ is a model
class.

  2. 2.

$\mathcal{K}$ is definable in some extension of first order logic by a
generalized quantifier.

  3. 3.

$\mathcal{K}$ is definable in some logic.

###  2.1 Operations vs. model classes

In the literature, logical operations and model classes are considered to be
carriers of logicality. Our paper focuses on model classes and on the extent
to which they can be called logical, but here we take a moment to establish
the connection between model classes and logical operations. McGee [41]
defines an abstract concept of what he calls a _logical operation_. He goes on
to define what it means for a logical operation to be _described_ by a formula
of a logic. We review these definitions and establish a close connection
between model classes and connectives as well as between definability of a
model class and describability of a logical operation.

Suppose $M$ is a non-empty set. By the semantic value (on $M$) of a formula
(of any logic) we mean the set of assignments into $M$ that satisfy the
formula. Abstractly, these are just subsets $A$ of $M^{n}$ for some $n$. A
_(local) operation_ $f$ _on_ $M$ maps sequences $\langle
A_{\alpha}:\alpha<\beta\rangle$ of sets $A_{\alpha}\subseteq M^{n_{\alpha}}$
to sets $f(\langle A_{\alpha}:\alpha<\beta\rangle)\subseteq M^{n}$. Such a
local operation is a _logical operation_ , if it is closed under permutations
of $M$, i.e. if for all permutations $\pi$ of $M$

$f(\langle\pi^{\prime\prime}A_{\alpha}:\alpha<\beta\rangle)=\pi^{\prime\prime}f(\langle
A_{\alpha}:\alpha<\beta\rangle),$

where for any $s=(a_{1},\ldots,a_{n})\in A\subseteq M^{n}$ we define
$\pi(s)=(\pi(a_{1}),\ldots,\pi(a_{n}))$.

######  Example 3.

Suppose $A_{0},A_{1}\subseteq M^{n}$. _Conjunction_ is the logical operation
$f^{n}_{\wedge}(\langle A_{0},A_{1}\rangle)=A_{0}\cap A_{1}$. _Disjunction_ is
the logical operation $f^{n}_{\vee}(\langle A_{0},A_{1}\rangle)=A_{0}\cup
A_{1}$. Suppose $A\subseteq M^{n+1}$. The _existential quantifier_ with
respect to $n$ is the logical operation $f^{n}_{\exists}(\langle
A\rangle)=\\{s\upharpoonright\\{0,\ldots,n-1\\}:s\in A\\}$.

For the existential second (and higher) order quantifiers to be logical
operations would require an extension of the approach to include more general
assignments, which we however disregard in this paper. Still, every formula of
second (or higher) order logic gives rise (separately) to a logical operation
in the above sense. The situation is the same with the so-called team
semantics [56] where semantic values are sets of sets of assignments, rather
than sets of assignments as above.

More generally we have an operation $f_{M}$ for every non-empty set $M$. Here
we assume that $f_{M}$ is always an operation on $M$. We call the (class)
mapping $M\mapsto f_{M}$ a _global operation_ and denote it $\bar{f}$. We call
the global operation $\bar{f}$ a _logical operation_ if it is closed under
bijections, i.e. if for all bijections $\pi:M\to N$

$f_{N}(\langle\pi^{\prime\prime}A_{\alpha}:\alpha<\beta\rangle)=\pi^{\prime\prime}f_{M}(\langle
A_{\alpha}:\alpha<\beta\rangle),$

A formula $\phi(\vec{x})$ with an $n_{\alpha}$-ary predicate symbol
$P_{\alpha}$ for each $\alpha<\beta$ is said to _describe_ a local operation
$f$ on $M$ if in any model ${\mathcal{M}}$ with domain $M$ and
$P_{\alpha}^{\mathcal{M}}=A_{\alpha}\subseteq M^{n_{\alpha}}$ the semantic
value of $\phi(\vec{x})$ is $f(\langle A_{\alpha}:\alpha<\beta\rangle)$. The
concept of a formula describing a global operation is defined similarily.

######  Example 4.

The formula $P_{0}(\vec{x})\wedge P_{1}(\vec{x})$ describes the operation
$f^{n}_{\wedge}$. The formula $P_{0}(\vec{x})\vee P_{1}(\vec{x})$ describes
the operation $f^{n}_{\vee}$. The formula $\exists x_{n}P_{0}(\vec{x},x_{n})$
describes the operation $f^{n}_{\exists}$.

If $\phi(\vec{x})$ is a formula of a logic $L^{*}$ in the above sense, then
the operation described by $\phi(\vec{x})$ is a logical operation, since we
assume that truth in every logic $L^{*}$ that we consider is closed under
isomorphisms (cf. [33]).

An operation $\bar{f}$ can be represented alternatively as the model class in
the vocabulary $L$ which has in addition to the predicate symbols
$P_{\alpha}$, $\alpha<\beta$, a new predicate symbol $P$:

$\mathcal{K}_{\bar{f}}=\\{{\mathcal{M}}:\mbox{${\mathcal{M}}$ is an
$L$-structure and }P^{\mathcal{M}}=f_{M}(\langle
P^{\mathcal{M}}_{\alpha}:\alpha<\beta\rangle))\\}.$

It is easy to see that the class $\mathcal{K}_{\bar{f}}$ is closed under
isomorphisms if and only if the operation $\bar{f}$ is preserved by
bijections.

On the other hand, any model class $\mathcal{K}$ can be turned into a logical
operation $\bar{f^{\mathcal{K}}}$ as follows. Let for any non-empty set $M$
and relations $\vec{R}$ on $M$ of the right arity:

$f^{\mathcal{K}}_{M}(\vec{R})=\left\\{\begin{array}[]{ll}M&\mbox{if
$(M,\vec{R})\in\mathcal{K}$}\\\ \emptyset&\mbox{otherwise.}\end{array}\right.$

It is easy to see that the class $\mathcal{K}$ is closed under isomorphisms if
and only if the operation $\bar{f^{\mathcal{K}}}$ is preserved by bijections.

###  2.2 Cardinal dependent definability

The theorem of McGee improves “some logic” of Theorem 2 to a very specific
logic, namely $L_{\infty\infty}$, but at a price: we obtain a different
definition for each cardinality separately.262626We point out other weaknesses
in the below section 2.3 involving the model-theoretic properties of
$L_{\infty\infty}$, e.g. its non-absoluteness.

Note that Theorem 2 has some uniformity in the sense that the defining
sentence does not depend on the cardinality of the model in question, but it
is also lacking in uniformity in the sense that we do not have one single
logic but possibly a different logic for each model class.

We now isolate this property of cardinality dependence. For any cardinal
$\lambda$ let $\mathcal{K}_{\lambda}$ be the class of elements of
$\mathcal{K}$ with a domain of size $\lambda$.

######  Definition 5.

A model class $\mathcal{K}$ is _cardinal dependently definable_ , or _CD-
definable_ , in a logic $L^{*}$, or _cardinal dependently $L^{*}$-definable_,
if $\mathcal{K}_{\lambda}$ is $L^{*}$-definable for every $\lambda$.

Note that a model class can be CD-definable even in first order logic without
being definable even in $L_{\infty\infty}$: Consider the class
$\mathcal{K}_{\mbox{\tiny lim}}$ of models $(M,P)$, where either $|M|$ is a
limit cardinal and $P=\emptyset$ or else $|M|$ is a successor cardinal and
$P\neq\emptyset$. It is a consequence of the Löwenheim-Skolem Theorem of
$L_{\infty\infty}$ (see Proposition 14 and the proof of Theorem 15 below) that
this model class cannot be definable in it. The following theorem relates
isomorphism closure to CD-definability in the case of classes of models:

######  Theorem 6 ([41]).

If $\mathcal{K}$ is a class of models of the same vocabulary, then the
following conditions are equivalent:

  1. 1.

$\mathcal{K}$ is closed under isomorphisms.

  2. 2.

$\mathcal{K}$ is CD-definable in $L_{\infty\infty}$.

###### Proof.

Let us fix an infinite cardinal $\lambda$. Let ${\mathcal{M}}_{\alpha}$,
$\alpha<2^{\lambda}$, list all elements of $\mathcal{K}$ with domain
$\lambda$. Let $\theta_{\alpha}\in L_{\lambda^{+}\lambda^{+}}$ characterize up
to isomorphism the model ${\mathcal{M}}_{\alpha}$. The sentence
$\bigvee_{\alpha<2^{\lambda}}\theta_{\alpha}$ defines the model class
$\mathcal{K}_{\lambda}$. ∎

The fact that McGee’s theorem depends on the cardinality of the models in the
class means that the sentence of $\phi_{\lambda}$ of $L_{\infty\infty}$
defining the model class $\mathcal{K}_{\lambda}$ may very well depend on
$\lambda$, as in the above example $\mathcal{K}_{\mbox{\tiny lim}}$. That is,
the class size mapping $\lambda\mapsto\phi_{\lambda}$ may encode information
that has to be analysed as to its logicality. In the above example, when we
ask whether the property of a model of belonging to the model class
$\mathcal{K}_{\mbox{\tiny lim}}$ is logical or not, we essentially ask whether
the property of a model of having a limit cardinality is logical or not.
According to the Tarski-Sher criterion it undoubtedly is logical. On the other
hand, it is fair to say that it is a mathematical property rather than a
logical one. As we will see below, the logicality of membership in
$\mathcal{K}_{\mbox{\tiny lim}}$ manifests in our below analysis a rather low
degree of logicality.

As we pointed out earlier, McGee [41] considers describability of logical
operations rather than definability of model classes. Let us now draw a
further connection, using the $\Delta$-operation on logics, between
describability of logical operations and definability of model classes.

Suppose $\mathcal{K}$ is a class of models of vocabulary $L$ and
$L^{\prime}\subseteq L$. We use $\mathcal{K}\upharpoonright L^{\prime}$ to
denote the class of reducts ${\mathcal{M}}\upharpoonright L^{\prime}$ of
models ${\mathcal{M}}\in\mathcal{K}$. We call $\mathcal{K}\upharpoonright
L^{\prime}$ a projection of $\mathcal{K}$.

######  Definition 7.

A model class is $\Sigma(L^{*})$-definable, or in $\Sigma(L^{*})$, if it is a
projection of an $L^{*}$-definable model class. A model class is
$\Delta$-definable in $L^{*}$, or in $\Delta(L^{*})$, if both $\mathcal{K}$
and the complement of $\mathcal{K}$ are $\Sigma(L^{*})$.

It should be noted that $\Delta(L_{\omega\omega})=L_{\omega\omega}$ [19],
$\Delta(L_{\omega_{1}\omega})=L_{\omega_{1}\omega}$ [34] and
$\Delta(L_{\infty\omega})\subseteq L_{\infty\infty}$ [40].

######  Lemma 8.

Suppose $f$ is a logical operation on $M$ and $L^{*}$ is a logic. Then (1)
$\to$ (2) $\to$ (3), where:

  1. (1)

$f$ is describable in $L^{*}$.

  2. (2)

The model class $\mathcal{K}_{f}$ is definable in $L^{*}$.

  3. (3)

$f$ is describable in $\Delta(L^{*})$.

###### Proof.

Suppose $f$ is described on $M$ by the formula $\psi(\vec{x})\in L^{*}$.
$\mathcal{K}_{f}$ is defined by the sentence
$\forall\vec{x}(P(\vec{x})\leftrightarrow\psi(\vec{x},\vec{R})).$ This proves
the first implication. Suppose then $\mathcal{K}_{f}$ is defined by the
sentence $\phi\in L^{*}$ with vocabulary
$\\{P_{\alpha}:\alpha<\beta\\}\cup\\{P\\}$. Consider the formula
$\psi(\vec{x})\equiv\exists P(\phi\wedge P(\vec{x}))$ with the second order
quantifier $``\exists P"$. This clearly describes $f$ but is not, a priori, in
$L^{*}$. However $\psi(\vec{x})\equiv\forall P(\phi\to P(\vec{x}))$ and
therefore $\psi(\vec{x})$ is in $\Delta(L^{*})$. ∎

######  Lemma 9.

Suppose $\mathcal{K}$ is a model class. Then $\mathcal{K}_{\lambda}$ is
definable in $L^{*}$ if and only if the logical operation
$f^{\mathcal{K}}_{\lambda}$ is describable in $L^{*}$.

###### Proof.

Suppose $\mathcal{K}_{\lambda}$ is defined in cardinality $\lambda$ by the
sentence $\phi\in L^{*}$. $f^{\mathcal{K}}_{\lambda}$ is defined by the
formula $\phi$ (without free variables). Suppose then
$f^{\mathcal{K}}_{\lambda}$ is defined by the formula $\psi({x})\in L^{*}$.
Then $\mathcal{K}_{\lambda}$ is defined by the sentence $\exists{x}\psi({x})$.
∎

The above two lemmas demonstrate the close relationship between logicality of
operations on semantic values and logicality of properties of models. The next
lemma, based on the method of [33, Theorem 5], shows that the relationship is
not perfect.

######  Lemma 10.

There is a logical operation $\bar{f}$ such that $\mathcal{K}^{\bar{f}}$ is
definable in $L(Q_{0})$ but $\bar{f}$ is not describable in $L(Q_{0})$.

Bonnay [10] suggests that the concept of potential isomorphism is a better
criterion for logicality than isomorphism itself. For potential isomorphism
closure there is the following version of McGee’s Theorem:

######  Theorem 11 ([5]).

If $\mathcal{K}$ is a class of models of the same vocabulary, then the
following conditions are equivalent:

  1. 1.

$\mathcal{K}$ is closed under potential isomorphisms.

  2. 2.

$\mathcal{K}$ is CD-definable in $L_{\infty\omega}$.

A problem with closure under potential isomorphism is that well-ordered
structures $(\alpha,\in,P_{1},\ldots,P_{n})$ and
$(\beta,\in,P^{\prime}_{1},\ldots,P^{\prime}_{n})$ are never potentially
isomorphic if $\alpha\neq\beta$. Just as an isomorphism closed model class can
have models with domains of different cardinalities and the isomorphism
closure gives no information across the domains, a potential isomorphism
closed model class can have models $(\alpha,\in,P_{1},\ldots,P_{n})$ with
different $\alpha$ and the potential isomorphism closure also gives no
information across such domains.

###  2.3 The model-theoretic properties of $L_{\infty\infty}$

We now note the different respects in which $L_{\infty\infty}$ deviates from
first order logic. First of all, $L_{\infty\infty}$ is badly
nonabsolute.272727For the technical definition of absoluteness see Section 5.
For example, the sentence

$\begin{array}[]{l}\forall x\forall y(\forall z(E(z,x)\leftrightarrow
E(z,y))\to x=y)\wedge\\\ \forall x_{0}\forall x_{1}\ldots\exists y\forall
z(E(z,y)\leftrightarrow\bigvee_{n<\omega}z=x_{n})\end{array}$ (1)

of $L_{\infty\infty}$ which has models exactly in cardinalities $\mu$ such
that $\mu^{\omega}=\mu$, is highly non-absolute.282828By König’s Theorem ([26,
Theorem 5.10]) such a $\mu$ cannot have cofinality $\omega$ and cofinality is
not absolute in set theory. Other failures of absoluteness can be generated by
replacing $\omega$ here by any regular cardinal.

A second important model-theoretic property of first order logic is its
downward Löwenheim-Skolem theorem. Consider the above (1). Such a $\mu$ cannot
have cofinality $\omega$ and thus $L_{\infty\infty}$ does not have a
Löwenheim-Skolem theorem in the same strong sense as first order logic, or
even in the sense of, e.g., $L_{\infty\omega}$ (see Theorem 13). That is,
$L_{\infty\infty}$ can “omit” cardinals of cofinality $\omega$ in this special
sense given by the Löwenheim-Skolem theorem. In contrast, a typical
consequence of a Löwenheim-Skolem theorem of $L_{\kappa^{+}\omega}$ is that if
a definable model class has a model of cardinality $\lambda>\kappa$ then it
has models of all cardinalities $\mu$ such that $\kappa\leq\mu\leq\lambda$
(Theorem 13), whatever their cofinality.

The logic behind Theorem 6, $L_{\kappa^{+}\kappa^{+}}$, fails in a strong
sense to have a Completeness Theorem. By a result of Dana Scott (published in
[29]) the set of valid sentences of $L_{\kappa^{+}\kappa^{+}}$ is not
$L_{\kappa^{+}\kappa^{+}}$-definable over $H(\kappa^{+})$, the set of sets of
hereditary cardinality $\leq\kappa$. In contrast, the set of valid sentences
of $L_{\kappa^{+}\omega}$ is $\Sigma_{1}$-definable over $H(\kappa^{+})$ [29],
reminiscent of the Completeness Theorem of first order logic which implies
that the set of (Gödel numbers of) valid sentences of first order logic is
recursively enumerable i.e. $\Sigma_{1}$ over $HF$, the set of hereditarily
finite sets. In fact, C. Karp gives a complete axiomatization for
$L_{\kappa^{+}\omega}$ in [29].

In section 5 we replace $L_{\infty\infty}$ by a logic which is (almost)
absolute and which has a strong Löwenheim-Skolem theorem, together with other
desirable properties, though it is still the case that the definition is given
relative to the size of the models in the class.

##  3 Löwenheim-Skolem theorems viewed through the spectra of model classes

The fact that the class of all models of cardinality $\kappa$, for example,
satisfies the Tarski-Sher criterion, irrespective of what $\kappa$ is, raises
the question, is cardinality a logical property? We saw that in the more
subtle case of $\mathcal{K}_{\mbox{\tiny lim}}$ the property of a model of
being of a limit cardinality is, according to this criterion, logical; by
modifying $\mathcal{K}_{\mbox{\tiny lim}}$ one can generate many other cases.
For example, if $A$ is a property of natural numbers, we can consider the
class $\mathcal{K}$ of models of cardinality $\aleph_{n}$, such that $n$ has
property $A$. The property of a model of belonging to $\mathcal{K}$ is a
logical property of the model by the Tarski-Sher criterion, even though to
judge whether a particular model is in $\mathcal{K}$ one has to first
determine whether the size of the universe is some $\aleph_{n}$, and after
that one has to determine whether $n$ has property $A$. This stretches the
concept of logicality and entangles it with the concept of what is
mathematical. Earlier writers have observed this, so we are not saying
essentially anything new here.

Of Feferman’s three objections to the Tarski-Sher criterion of logicality, the
third is that it leaves unexplained how logicality is to be understood across
domains of different sizes. The existential quantifier, for example, is, in a
sense, the same on any domain of any cardinality, but this is not true of many
properties which satisfy the Tarski-Sher criterion, for example
$\mathcal{K}_{\mbox{\tiny lim}}$. We suggest that in order that a property of
models manifests a greater degree of logicality than is provided by the
Tarski-Sher criterion, it should have a definition by a sentence in
$L_{\infty\infty}$ or in some other logic in such a way that the same sentence
defines the model class in as many cardinalities as possible.292929Note that
$\mathcal{K}_{\mbox{\tiny lim}}$ has the same first order definition on the
(closed unbounded) class of _all_ limit cardinals. We explore this suggestion
below via the concept of a spectrum. Our second suggestion is that a higher
grade of logicality would be assigned to the degree to which the logic in
question had “first-order-like” model-theoretic properties: absoluteness, a
Löwenheim-Skolem theorem, and completeness. $L_{\infty\infty}$ would thus be
given a low grade on the logicality scale, being non-absolute, having a
limited Löwenheim-Skolem theorem and failing to have a Completeness Theorem.
We explore this suggestion in section 7303030For a different analysis of
domain relativity, see [58]..

###  3.1 Spectra

In analysing the structure of a model class, in order to determine whether it
is in some sense logical or not or to find out whether it is definable in some
interesting logic, it is useful to investigate the cardinalities of models in
the class. One might think that mere cardinalities are too rough a measure of
any form of logicality but this is, in fact, surprisingly informative. This
leads naturally to the concept of a spectrum:313131See also [46].

######  Definition 12.

If $\mathcal{K}$ is a model class, the _spectrum_ of $\mathcal{K}$ is the
class $\mbox{\rm sp}(\mathcal{K})$ of cardinalities of models in $\mathcal{K}$
i.e.

$\mbox{\rm sp}(\mathcal{K})=\\{|M|:{\mathcal{M}}\in\mathcal{K}\\}.$

Depending on $\mathcal{K}$, the spectrum can be a singleton, an interval of
cardinals, an initial (or final) segment of the class of all cardinals, or
something more complicated, such as the class of all limit cardinals, or all
limit cardinals of cofinality $\omega$ (see Figure 1). Even the patterns of
finite numbers in spectra of first order sentences is highly interesting [3,
27]. However, we are here concerned with infinite cardinals in a spectrum.

Figure 1: Spectra of some model classes.

The property of a logic which reflects regularity patterns in its spectra is
captured by the Löwenheim-Skolem Theorem. The spectrum gives indirect
information of the possibility that the model class is definable in some
logic. Roughly speaking, if the logic has a strong Löwenheim-Skolem property,
then the spectra of definable model classes reflect this. If every sentence in
the logic which has an infinite model has also a countably infinite model, the
most famous case of a Löwenheim-Skolem property, then every spectrum with an
infinite cardinal in it has also $\aleph_{0}$ in it.

Skolem proved that first order logic satisfies $\mbox{\rm
LS}([\aleph_{0},\infty),\\{\aleph_{0}\\})$ (see Definition 1) giving rise to
the so-called Skolem Paradox: countable first order theories such as set
theory have countable models if they have models at all. Now, one hundred
years after Skolem’s discovery, the indifference of first order logic to the
infinite cardinality of its models is often considered a positive rather than
negative aspect. Indifference to cardinality is of course closely related to
set-theoretical absoluteness and thereby to a desirable quality of logicality.
In consequence, the stronger form of $\mbox{\rm LS}(C,D)$ a logic satisfies
the more appropriate the logic is for expressing logical properties.

If a logic satisfies $\mbox{\rm LS}(C,D)$, there are consequences for the
spectra of definable model classes. Suppose $\mathcal{K}$ is definable in a
logic with $\mbox{\rm LS}(C,D)$. Then we can make the following conclusions:
If there is ${\mathcal{M}}\in\mathcal{K}$ with $|M|\in C$, then there is
${\mathcal{N}}\in\mathcal{K}$ with $|N|\in D$. On the other hand, if
$\mathcal{K}$ contains a model of cardinality $\kappa$ but no models of
cardinality $\lambda$, then $\mathcal{K}$ cannot be definable in a logic with
$\mbox{\rm LS}(C,D)$ such that $\kappa\in C$ and $\lambda\notin D$. The point
is that by looking at the spectrum of $\mathcal{K}$ we can make inferences
about its definability in different logics. Thus, if we are given a model
class $\mathcal{K}$ but no logic in which it would be definable (apart from
the trivial $L(Q_{\mathcal{K}})$), and we can discern regular patterns in
$\mbox{\rm sp}(\mathcal{K})$, we may take it as an indirect indication (albeit
not a proof) that $\mathcal{K}$ is definable in a logic with $\mbox{\rm
LS}(C,D)$ for some $C$ and $D$ explaining the found patterns. On the other
hand, if $\mbox{\rm sp}(\mathcal{K})$ does not seem to have regular patterns,
we may take it as an indication that no such logic can be found. Thus
$\mbox{\rm sp}(\mathcal{K})$ gives implicit information about the possibility
of finding a syntax for $\mathcal{K}$.

Let us look at the Löwenheim-Skolem Properties of the logics
$L_{\infty\omega}$ and $L_{\infty\infty}$.323232Recall that
$L_{\infty\omega}=\bigcup_{\kappa}L_{\kappa^{+}\omega}$ and
$L_{\infty\infty}=\bigcup_{\kappa}L_{\kappa^{+}\kappa^{+}}$. We first recall a
basic construction in infinitary logic, based on a class of important formulas
due to D. Scott [47]. For any ordinal $\alpha$ let the formula
$\eta_{\alpha}(x)$ with one free variable $x$ and a binary relation symbol $<$
be defined, by transfinite recursion, as follows:

$\eta_{\alpha}(x)\leftrightarrow\forall
y(y<x\to\bigvee_{\beta<\alpha}\eta_{\beta}(y))\wedge\bigwedge_{\beta<\alpha}\exists
y(y<x\wedge\eta_{\beta}(y)).$ (2)

Then for a linear order $(A,<)$ and $a\in A$ we have

$(A,<)\models\eta_{\alpha}(a)\iff(\\{b\in A:b<a\\},<)\cong(\alpha,<).$ (3)

Let

$\eta^{\prime}_{\alpha}\leftrightarrow\forall
y\bigvee_{\beta<\alpha}\eta_{\beta}(y)\wedge\bigwedge_{\beta<\alpha}\exists y\
\eta_{\beta}(y).$ (4)

For a linear order $(A,<)$ we have

$(A,<)\models\eta^{\prime}_{\alpha}\iff(A,<)\cong(\alpha,<).$ (5)

Now $\eta^{\prime}_{\alpha}$ is in $L_{\kappa\omega}$ whenever
$\alpha<\kappa$. It follows that by combining the sentences
$\eta^{\prime}_{\alpha}$, the logic $L_{\kappa\omega}$ can manifest totally
arbitrary patterns of spectra in cardinalities below $\kappa$, roughly for the
same reason that any finite set of finite numbers can be the spectrum of a
first order sentence. More exactly, if $X$ is an arbitrary set of cardinal
numbers below $\kappa$, then

$\bigvee_{\lambda\in X}\eta^{\prime}_{\lambda}\in L_{\kappa^{+}\omega}\mbox{
and }X=\mbox{\rm sp}(\bigvee_{\lambda\in X}\eta^{\prime}_{\lambda}).$

This shows that when dealing with extensions of $L_{\kappa\omega}$ it makes
sense to focus on cardinals $\geq\kappa$.

The basic facts about the $\mbox{\rm LS}(C,D)$ type properties of infinitary
languages are the following:

######  Theorem 13 ([20]).

Suppose $\kappa\leq\lambda$ are cardinals and $\kappa$ is regular.

  1. 1.

$L_{\kappa^{+}\omega}$ satisfies $\mbox{\rm
LS}([\lambda,\infty)\\},\\{\mu\\})$ for all $\mu$ such that
$\kappa\leq\mu\leq\lambda$.

  2. 2.

$L_{\kappa^{+}\kappa^{+}}$ satisfies $\mbox{\rm
LS}([\lambda,\infty),\\{\mu\\})$ for all $\mu$ such that
$\kappa\leq\mu\leq\lambda$ and $\mu^{\kappa}=\mu$.

  3. 3.

$L_{\kappa^{+}\omega}$ satisfies $\mbox{\rm
LS}([\beth_{(2^{\kappa})^{+}},\infty),[\mu,\infty))$ for all $\mu$.

  4. 4.

$L_{\omega_{1}\omega_{1}}$ does not satisfy “For all $\mu$ $\mbox{\rm
LS}([\lambda,\infty),[\mu,\infty))$” for any $\lambda$ below the first
inaccessible cardinal (assuming there are inaccessible cardinals).

For a class size logic such as $L_{\infty\omega}$ or $L_{\infty\infty}$ the
$\mbox{\rm LS}(C,D)$ properties hold via their connection to
$L_{\kappa\omega}$ or $L_{\kappa\lambda}$: If $\phi\in L_{\infty\infty}$, then
$\phi\in L_{\kappa\lambda}$ for some (least) $\kappa$ and $\lambda$ and then
Theorem 13 applies.

The above theorem shows that the spectra of $L_{\infty\infty}$ are much more
complicated than the spectra of $L_{\infty\omega}$. For example, let $\theta$
be the sentence (1) of $L_{\omega_{1}\omega_{1}}$ which has a model of
cardinality $\mu$ if and only if $\mu^{\omega}=\mu$. Thus $\mbox{\rm
sp}(\theta)$ (i.e. $\mbox{\rm sp}(\mbox{Mod}(\theta))$) is full of “holes” as
it misses all cardinals, such as e.g. each $\aleph_{\alpha+\omega}$, that are
$\omega$-cofinal. Whereas the spectra of sentences of $L_{\kappa\omega}$
cannot have such holes above $\kappa$. (See Figure 2.)

What does it reveal about logicality if the spectrum is full of ‘holes’? It
suggests that we are very far from the situation in which we can claim we have
the same logical operation independently of the domain. In the case of
$L_{\kappa^{+}\omega}$ the spectrum is (above $\kappa$) a homogeneous segment
of cardinals, either bounded by $\beth_{(2^{\kappa})^{+}}$ or unbounded, and
we are closer to having the same logical operation independently of the
domain. Thus we judge a model class definable in $L_{\infty\omega}$ as having
a greater degree of logicality than a model class definable in
$L_{\infty\infty}$.

Figure 2: A difference between $L_{\infty\omega}$ and $L_{\infty\infty}$.

In Definition 1 above we defined the Löwenheim the Hanf number of an arbitrary
logic. These numebrs always exist if the class of formulas of the logic is a
set. Thus they exist for $L_{\kappa\lambda}$ for any $\kappa$ and $\lambda$
but not for $L_{\infty\omega}$ and $L_{\infty\infty}$.

The Hanf-number and Löwenheim number of $L^{*}$ can be easily defined in terms
of spectra. Below a spectrum is called _bounded_ if it is a set (rather than a
proper class).

$\begin{array}[]{lcl}\ell(L^{*})&=&\sup\\{\min(\mbox{\rm sp}(\phi)):\phi\in
L^{*},\mbox{\rm sp}(\phi)\neq\emptyset\\}\\\ h(L^{*})&=&\sup\\{\sup(\mbox{\rm
sp}(\phi)):\phi\in L^{*}\mbox{ and $\mbox{\rm sp}(\phi)\neq\emptyset$ is
bounded}\\}.\end{array}$ (6)

We saw that Sagi employs Löwenheim numbers as a measure of logicality: the
smaller the Löwenheim number the greater the degree of logicality. Thus model
classes definable in $L(Q_{\alpha})$ have a stronger degree of logicality than
those definable in $L(Q_{\beta})$, for $\beta>\alpha$.

The logic $L(Q_{\alpha})$ satisfies for trivial reasons the Löwenheim-Skolem
property $\mbox{\rm LS}([\omega,\aleph_{\alpha}),\\{\mu\\})$ for all
$\mu<\aleph_{\alpha}$ and the rather strong Löwenheim-Skolem property
$\mbox{\rm LS}([\aleph_{\alpha},\infty)),\\{\mu\\})$ for all
$\mu\geq\aleph_{\alpha}$. Thus the spectra have no holes and
$\ell(L(Q_{\alpha}))=\aleph_{\alpha}$. It should be noted that the class size
logic $L(Q_{\alpha})_{\alpha\in{\rm On}}$ is a sublogic of $L_{\infty\infty}$.
Its spectra have a greater degree of regularity than those of
$L_{\infty\infty}$, and thus in that respect it resembles more
$L_{\infty\omega}$. In sum, these two logics, namely $L_{\infty\omega}$ and
$L(Q_{\alpha})_{\alpha\in{\rm On}}$, both manifest a greater degree of
logicality than $L_{\infty\infty}$, if regularity in spectrum patterns is used
as a criterion.

##  4 Is every model class definable in $L_{\infty\infty}$ irrespective of the
cardinality?

Why is it that we cannot replace condition (2) of Theorem 6 with the
apparently better condition

$\mathcal{K}$ is definable in $L_{\infty\infty}$? (7)

This would solve the problems of cardinality and domain-relativity—the logical
concept would have the same definition in terms of a formal language
independently of the cardinality of the domain. However, there is a very
simple reason why condition (2) of Theorem 6 cannot be improved to (7): Not
every model class is definable in $L_{\infty\infty}$.

Recall the definition of $\mathcal{K}_{\mbox{\tiny lim}}$ above in Section 2.
It is, strictly speaking, a generalized quantifier in the sense of [43] but a
rather curious one. It is the existential quantifier in models of successor
cardinality and the quantifier “for no $x$” in models of limit cardinality.

######  Proposition 14.

The model class $\mathcal{K}_{\mbox{\tiny lim}}$ is not definable in
$L_{\infty\infty}$.

###### Proof.

Similar to the proof of Proposition 15 below. ∎

A different kind of generalized quantifier is the class

$W_{2}=\\{(A,<):(A,<)\cong(\alpha+\alpha,\in)\mbox{ for an ordinal
$\alpha$}\\}.$

This is potential isomorphism closed and not definable even in
$L_{\infty\infty}$ [40]. Note that this model class is clearly second order
definable. The model class $W_{2}$ has a particularly nice spectrum, its
infinite part being simply the class of all infinite cardinals. This model
class is also set-theoretically absolute (see below section 5). In the light
of the spectrum criterion, a model being an element of $W_{2}$ is a logical
property of the model of a very high degree of logicality. Looking at a model,
can we say that its being a well-order, let alone of a well-order of type
$\alpha+\alpha$ for some $\alpha$, is a logical property? Undoubtedly some
would say it is a mathematical rather than a logical property. But it is
logical in the Tarski-Sher sense and it has a second order definition which is
the same definition in each cardinality. However much it feels like a
genuinely mathematical concept, it fulfils the Tarski-Sher criterion of
logicality, escapes two of Feferman’s criticisms, and fulfils Sagi’s as well
as Bonnay’s criteria. So we may conclude that it is logical of a very high
degree.

A third example of a model class undefinable in $L_{\infty\infty}$ is
particularly interesting, namely the Härtig quantifier.

###  4.1 The Härtig Quantifier

The Härtig quantifier is defined as follows:

$\begin{array}[]{lcl}Ixy\phi(x)\psi(y)&\iff&\mbox{ there are as many $x$
satisfying $\phi(x)$}\\\ &&\mbox{as there are $y$ satisfying
$\psi(y)$.}\end{array}$

######  Proposition 15 ([25]).

The class of models $(M,A,B)$, where $A,B\subseteq M$ and $|A|=|B|$, is not
definable in $L_{\infty\infty}$. Equivalently, the Härtig quantifier is not
definable in $L_{\infty\infty}$.

###### Proof.

Two sentences of the logic $L_{\omega\omega}(I)$ are useful here: the sentence
$\phi_{{\rm lim}}$ which has the class of limit cardinals as its spectrum, and
the sentence $\phi_{suc}$ which has the class of infinite successor cardinals
as its spectrum. The sentence $\phi_{{\rm lim}}$ says that the universe is
totally ordered by a linear order in which every element determines an initial
segment which has smaller cardinality than the initial segment determined by
some bigger element. The sentence $\phi_{suc}$ says that the universe is
totally ordered by a linear order in which some element determines an initial
segment which has the same cardinality as the initial segment determined by
any bigger element, but smaller cardinality than the entire universe. Suppose
the Härtig quantifier was definable in $L_{\kappa^{+}\kappa^{+}}$, where
$\kappa$ is regular. Let $\lambda=2^{\kappa}$. If $\lambda$ is a limit
cardinal, we can use $\phi_{{\rm lim}}$ to derive a contradiction as follows:
Let ${\mathcal{M}}$ be a model of $\phi_{{\rm suc}}$ of cardinality
$\lambda^{+}$. By Theorem 13 there is a model ${\mathcal{N}}$ of cardinality
$\lambda$ such that ${\mathcal{N}}\models\phi_{{\rm suc}}$, a contradiction.
The case that $\lambda$ is a successor cardinal is similar, but using
$\phi_{{\rm lim}}$ instead of $\phi_{{\rm suc}}$. ∎

While the logic $L(I)$ with the Härtig quantifier is not a sublogic of
$L_{\infty\infty}$ it has some of the properties of the latter that we
associate with being very far from first order logic (see Section 7). For
example, it is highly non-absolute. It is also unbounded i.e. capable of
defining well-ordering with additional predicates [32].

Note that the equicardinality concept, built around the existence of a
bijection, lies at the very heart of the definition of logicality in the sense
of Tarski! It is almost paradoxical then, that equicardinality itself, or more
precisely the logic built on the equicardinality quantifier, represents a
particularly weak degree of logicality.

###  4.2 Vopěnka’s Principle

While it is not true that every model class is definable in
$L_{\infty\infty}$, there is a weaker result which depends on large cardinals.
Tarski proved that a model class $\mathcal{K}$ is definable by a universal
first order sentence if and only if $\mathcal{K}$ is closed under
substructures and any model all of whose finite substructures are in
$\mathcal{K}$, is itself in $\mathcal{K}$. The following result removes the
assumption on finite substructures but lifts the result to an infinitary
level. Recall that _Vopěnka’s principle_ is the axiom schema stating that if
$\\{A_{\alpha}:\alpha\in{\rm On}\\}$ is a (definable) proper class of
structures of the same vocabulary, there are $\alpha\neq\beta$ such that
$A_{\alpha}$ can be embedded into $A_{\beta}$. Despite its formulation, which
suggests no connection to large cardinals, this principle is actually a large
cardinal axiom schema. It implies the existence of extendible (and hence
supercompact) cardinals [36], and its consistency follows from the existence
of an almost huge cardinal (see e.g. [28, p. 338]). Magidor has proved the
following characterisation of Vopěnka’s Principle:333333We present the proof
with Professor Magidor’s kind permission.

######  Theorem 16 ([37]).

The following are equivalent:

  1. 1.

Every model class which is closed under substructures is
$L_{\infty\infty}$-definable by a universal sentence.

  2. 2.

Vopěnka’s Principle.

###### Proof.

(2) implies (1): Suppose $\mathcal{K}$ is a model class which is closed under
substructures. By Vopěnka’s Principle there is a $\mathcal{K}$-supercompact
cardinal $\kappa$ (see [51]). For any ${\mathcal{M}}$ of cardinality
$\leq\kappa$, let $\theta_{\mathcal{M}}\in L_{\kappa^{+}\kappa^{+}}$ be an
existential sentence such that $\theta_{\mathcal{M}}$ is true exactly in
models which have an isomorphic copy of ${\mathcal{M}}$ as a substructure. We
show that for all ${\mathcal{N}}$, ${\mathcal{N}}\notin\mathcal{K}$ if and
only if ${\mathcal{N}}\models\psi$, where $\psi$ is
$\bigvee\\{\theta_{\mathcal{M}}:M\subseteq\kappa,{\mathcal{M}}\notin\mathcal{K}\\}$.
Suppose first ${\mathcal{N}}\notin\mathcal{K}$. By the
$\mathcal{K}$-supercompactness of $\kappa$ there is an
$L_{\omega\omega}(Q_{\mathcal{K}})$-elementary substructure ${\mathcal{M}}$ of
${\mathcal{N}}$ of cardinality $\leq\kappa$. Now
${\mathcal{N}}\notin\mathcal{K}$ implies ${\mathcal{M}}\notin\mathcal{K}$.
Trivially, ${\mathcal{N}}\models\theta_{\mathcal{M}}$. Hence
${\mathcal{N}}\models\psi$. Conversely, suppose
${\mathcal{N}}\models\theta_{\mathcal{M}}$ for some
${\mathcal{M}}\notin\mathcal{K}$ such that $M\subseteq\kappa$. W.l.o.g.
${\mathcal{M}}\subseteq{\mathcal{N}}$. By closure under substructures,
${\mathcal{N}}\notin\mathcal{K}$.

(1) implies (2): Suppose Vopěnka’s Principles fails as witnessed by the class
$\mathcal{C}=\\{{\mathcal{M}}_{\alpha}:\alpha\in{\rm On}\\}$. Let
$\mathcal{K}$ be the class of structures ${\mathcal{N}}$ of the same
vocabulary as each ${\mathcal{M}}_{\alpha}$ such that ${\mathcal{N}}$ is
isomorphic to an substructure of ${\mathcal{M}}_{\alpha}$ for arbitrarily
large $\alpha$. By (1) we may choose $\kappa$ such that $\mathcal{K}$ is
definable in $L_{\kappa^{+}\kappa^{+}}$. By our choice of $\mathcal{C}$, every
${\mathcal{M}}_{\alpha}$ satisfies ${\mathcal{M}}_{\alpha}\notin\mathcal{K}$.
By Theorem 13 there is for each $\alpha$ an
$L_{\kappa^{+}\kappa^{+}}$-elementary substructure ${\mathcal{N}}_{\alpha}$ of
${\mathcal{M}}_{\alpha}$ of cardinality $\leq 2^{\kappa}$ such that
${\mathcal{N}}_{\alpha}\notin\mathcal{K}$. There are only a set of non-
isomorphic structures of cardinality $\leq 2^{\kappa}$. Hence
${\mathcal{N}}_{\alpha}\cong{\mathcal{N}}_{\beta}$ for a proper class of
$\alpha$ and $\beta$. Let $\alpha$ be one of those. Then
${\mathcal{N}}_{\alpha}\in\mathcal{K}$, a contradiction. ∎

The moral of the story here is that we can obtain the McGee style
characterization of logicality independently of the cardinality of the model
if we restrict to model classes closed under substructures, but we have to
make a strong set theoretical assumption, namely Vopěnka’s Principle. This is
not to suggest that we propose closure under substructures as a criterion for
logicality.

##  5 Defining an arbitrary model class in an absolute logic, but still
cardinal dependently

We shall now offer a refinement of McGee’s result (Theorem 6), by replacing
$L_{\infty\infty}$ by a substantially weaker infinitary logic343434McGee
alludes to such a possibility on p. 574 of [41].. Let us first note that we
cannot replace $L_{\infty\infty}$ by $L_{\infty\omega}$ because the quantifier
$Q_{1}$ is not CD-definable in the latter.353535Suppose ${\mathcal{M}}$ is a
model of cardinality $\aleph_{1}$ with one unary predicate $P$ and $P$ is
countable. Suppose ${\mathcal{N}}$ is another model of cardinality
$\aleph_{1}$ for the same vocabulary but now both $P$ and its complement are
uncountable. Clearly ${\mathcal{M}}$ and ${\mathcal{N}}$ are partially
(potentially) isomorphic and hence satisfy the same
$L_{\infty\omega}$-sentences. But $Q_{1}xP(x)$ is true in one but not in the
other. For another example, the well-ordering quantifier

${\mathcal{M}}\models
Wxy\phi(x,y,\vec{a})\iff\\{(c,d):{\mathcal{M}}\models\phi(c,d,\vec{a})\\}\mbox{
well-orders $M$}$

is definable in $L_{\omega_{1}\omega_{1}}$ (see (8)) but not in
$L_{\infty\omega}$ [35].

An important quality of $L_{\infty\omega}$ is its absoluteness. Let us recall:

######  Definition 17 ([4]).

A logic $(\Sigma,T)$ is called an _absolute logic_ if the predicate
$\Sigma(\phi)$ is $\Sigma_{1}$ in $\phi$, and the predicate
$T(\phi,{\mathcal{M}})$ is $\Delta_{1}$ in $\phi$ and ${\mathcal{M}}$. A logic
is absolute with respect to a set theory $S$ if the predicates $\Sigma$ and
$T$ are $\Sigma_{1}^{S}$ and $\Delta_{1}^{S}$, respectively.

For example, $L_{\infty\omega}$ and $L(W)$ are absolute logics but
$L_{\infty\infty}$ is not.

We defined the $\Delta$-operation above in Definition 7. As argued for in
[39], $\Delta(L^{*})$ can be considered a logic in itself. There is a many-
sorted version and a single-sorted version of the $\Delta$-extension. In the
many-sorted version the model class $\mathcal{K}$ may have a many-sorted
vocabulary. If $L^{*}$ is the logic $L_{\infty\omega}$, or
$L_{\kappa^{+}\omega}$, for regular $\kappa$, there is no difference between
the two versions i.e. they coincide [44]. Therefore we continue with the
single-sorted version.

Some of the nice properties of the $\Delta$-operation are:

  * •

Exactly the same classes of cardinals are spectra of $\Delta(L^{*})$ as are of
$L^{*}$.

  * •

$\Delta(L^{*})$ satisfies $LS(C,D)$ iff $L^{*}$ does.

  * •

$\Delta(L^{*})$ satisfies the Compactness Theorem iff $L^{*}$ does.

  * •

$\Delta(L^{*})$ satisfies the (abstract) Completeness Theorem363636I.e. the
set of Gödel numbers of valid sentences is r.e. This is only meaningful for
$L^{*}$ where formulas are finite objects so that Gödel numbering makes sense.
iff $L^{*}$ does.

  * •

If $L^{*}$ satisfies the Craig Interpolation Theorem, then $\Delta(L^{*})$ and
$L^{*}$ are equivalent logics.

  * •

$\ell(\Delta(L^{*}))=\ell(L^{*})$.

  * •

$h(\Delta(L^{*}))=h(L^{*}).$373737This may fail in the many-sorted version
[55].

If $L^{*}$ is an absolute logic in the sense of [4], then $\Delta(L^{*})$ is
still absolute in the weaker sense that every definable model class is
$\Delta_{1}$-definable in set theory (with the defining sentence as a
parameter). This is a direct consequence of the definition of $\Delta(L^{*})$.
However, $\Delta(L^{*})$ has a complication which prevents us from concluding
that the $\Delta$-operation preserves absoluteness. Namely, there may be an
absolute logic $L^{*}$ and model classes $\mathcal{K}_{0}$ and
$\mathcal{K}_{1}$ such that both $\mathcal{K}_{0}$ and $\mathcal{K}_{1}$ are
in $\Sigma(L^{*})$, $\mathcal{K}_{0}\cap\mathcal{K}_{1}=\emptyset$ but the
proposition that every model (of the right type) is in $\mathcal{K}_{0}$ or
$\mathcal{K}_{1}$ is not absolute383838For example, we may take
$\mathcal{K}_{0}$ to be the class of trees of height and size $\omega_{1}$
with an uncountable branch, and $\mathcal{K}_{1}$ to be the class of trees of
height and size $\omega_{1}$ with a strict order preserving mapping into the
rational numbers. It is a consequence of Martin’s Axiom that every such tree
is in $\mathcal{K}_{0}\cup\mathcal{K}_{1}$. However, if $V=L$, then there are
Souslin trees which are not in $\mathcal{K}_{0}\cup\mathcal{K}_{1}$.. Thus we
may not have a $\Sigma_{1}$-definition for the set of sentences of
$\Delta(L^{*})$ even if we have for $L^{*}$. Despite this shortcoming of the
$\Delta$-operation, we do have a $\Sigma_{1}$-definition for the set of
sentences of $\Sigma(L^{*})$ when $L^{*}$ is absolute because with
$\Sigma(L^{*})$ the above problem ($\mathcal{K}_{0}$ and $\mathcal{K}_{1}$)
does not arise.

######  Theorem 18 ([31]).

If $\mathcal{K}$ is a class of models of the same vocabulary, then the
following conditions are equivalent:

  1. 1.

$\mathcal{K}$ is closed under isomorphisms.

  2. 2.

$\mathcal{K}$ is CD-definable in $\Delta(L_{\infty\omega})$.

###### Proof.

The proof is similar to the proof of Theorem 6 in [41]. Recall the formulas
$\eta^{\prime}_{\alpha}$ and $\eta^{\prime}_{\alpha}$ defined in (2) and (4),
with the properties (3) and (5). Suppose $\mathfrak{A}$ is a model of the
vocabulary $L$ and $|A|=\lambda$. Let $f_{A}:\lambda\to A$ be a bijection and
$a<_{A}b\iff f_{A}^{-1}(a)<f_{A}^{-1}(b)$. For $\alpha,\beta<\lambda$ let

$\rho_{\alpha,\beta}(x,y)=\left\\{\begin{array}[]{ll}R(x,y)&\mbox{ if
$\mathfrak{A}\models R(f_{A}(\alpha),f_{B}(\beta))$}\\\ \neg R(x,y)&\mbox{ if
$\mathfrak{A}\not\models R(f_{A}(\alpha),f_{B}(\beta))$}\end{array}\right.$

Let

$\Phi_{\mathfrak{A}}\leftrightarrow\forall x\forall
y\bigwedge_{\alpha,\beta<\lambda}((\eta_{\alpha}(x)\wedge\eta_{\beta}(y))\to\rho_{\alpha,\beta}(x,y)).$

Now $(\mathfrak{A},<_{A})\models\eta^{\prime}_{\lambda}$ and
$(\mathfrak{A},<_{A})\models\eta_{\alpha}(a)\iff a=f_{A}(\alpha)$. Hence
$(\mathfrak{A},<)\models\eta^{\prime}_{\lambda}\wedge\Phi_{\mathfrak{A}}.$ On
the other hand,
$(\mathfrak{A}^{\prime},<^{\prime})\models\eta^{\prime}_{\lambda}\wedge\Phi_{\mathfrak{A}}$
implies $\mathfrak{A}^{\prime}\cong\mathfrak{A},$ for if
$(\mathfrak{A}^{\prime},<^{\prime})\models\eta^{\prime}_{\lambda}\wedge\Phi_{\mathfrak{A}}$
and $g:(A^{\prime},<^{\prime})\cong(A,<_{A})$, then
$g:\mathfrak{A}^{\prime}\cong\mathfrak{A}$. Let

$\Theta_{\mathcal{K}}\leftrightarrow\bigvee\\{\Phi_{\mathfrak{A}}\ :\
\mathfrak{A}\in\mathcal{K},A=\lambda\\}.$

Now by the above,

$\begin{array}[]{lcl}\mathfrak{A}\in\mathcal{K}_{\lambda}&\iff&(\mathfrak{A},<)\models\eta^{\prime}_{\lambda}\wedge\Theta_{\mathcal{K}}\mbox{
for some $<$ }\\\
&\iff&(\mathfrak{A},<)\models\eta^{\prime}_{\lambda}\to\Theta_{\mathcal{K}}\mbox{
for all $<$}.\end{array}$

Since $\eta^{\prime}_{\lambda},\Theta_{\mathcal{K}}\in
L_{(2^{\lambda})^{+}\omega}$, we are done. ∎

Why is this an improvement? $\Delta(L_{\infty\omega})$ is a sublogic of
$L_{\infty\infty}$, a consequence of the result of Malitz [40] to the effect
that for regular $\kappa$ a valid implication in $L_{\kappa\omega}$ can be
interpolated in $L_{2^{(<\kappa)^{+}\kappa}}$. It is a proper sublogic because
well-ordering is definable in the latter but not in the former. Secondly, the
logic $L_{\infty\omega}$ is an absolute logic [4] and
$\Delta(L_{\infty\omega})$ inherits much of the absoluteness of
$L_{\infty\omega}$ (see above) while $L_{\infty\infty}$ is badly non-absolute.

The improvement that Theorem 18 represents over Theorem 6 is also based on the
fact that $L_{\infty\omega}$, and thereby also $\Delta(L_{\infty\omega})$, is
a much “tamer” logic than $L_{\infty\infty}$. In particular:

  * •

$\Delta(L_{\infty\omega})$, even $\Sigma(L_{\infty\omega})$, has a strong
Löwenheim-Skolem theorem. This is in contrast to $L_{\infty\infty}$, which
does not have a Löwenheim-Skolem Theorem in the same strong form (see Theorem
13).

  * •

The class of well-orderings is not definable in $\Delta(L_{\infty\omega})$,
while it is definable in $L_{\omega_{1}\omega_{1}}$ (see (8) below). This is a
kind of threshold difference. In general, logics in which the concept of well-
order is not definable are much better behaved (in many different respects)
than those in which it is. If well-order is definable, the logic can talk
about well-founded models of set theory, and thereby about transitive models
of set theory. In this way the logic can break through the object
theory/metatheory barrier and have access to the background set theory. For a
concrete example, consider a sentence $\phi$ which is the conjunction of a
sufficiently large finite part $T_{0}$ of the ZFC axioms, the first order set-
theoretical statement $\phi_{0}$ that there are no inaccessible cardinals, and
the $L_{\omega_{1}\omega_{1}}$-sentence

$\displaystyle{\forall x_{0}\forall x_{1}\ldots\bigvee_{n<\omega}}\neg
x_{n+1}\in x_{n}.$ (8)

Suppose $\kappa$ is the smallest inaccessible cardinal. Then $\phi$ has a
model of cardinality $\kappa$, namely $V_{\kappa}$, but none of bigger
cardinality. For suppose $M$ is a model of $\phi$ of cardinality $>\kappa$. By
Mostowski’s Collapsing Lemma [42] we may assume $M$ is a transitive model of
$T_{0}$. Since the cardinality of $M$ is bigger than $\kappa$, the ordinal
$\kappa$ is in $M$ and is therefore inaccessible in $M$ by the downward
persistency of inaccessibility. But this contradicts the fact that $M$ is a
model of $\phi_{0}$. This argument, due to Silver [50], demonstrates the power
of (8) to penetrate the object theory/metatheory barrier, which sentences of
$\Delta(L_{\infty\omega})$ are unable to do [35].

  * •

The Hanf number of $\Delta(L_{\kappa^{+}\omega})$ is only moderately large,
namely $<\beth_{(2^{\kappa})^{+}}$, while the Hanf number of
$L_{\omega_{1}\omega_{1}}$ is bigger than the first weakly inaccessible
cardinal393939See argument in the previous bullet. (and consistently bigger
than the first measurable cardinal). The smallness of the Hanf number is
another indication of regularity of patterns in spectra (see (6)).

A weakness of $\Delta(L_{\infty\omega})$ as compared to $L_{\infty\infty}$ is
that the former does not have as explicit a syntax as the latter, although we
can overcome this by replacing $\Delta(L_{\infty\omega})$ by
$\Sigma(L_{\infty\omega})$. A strength is that its spectra are as regular as
those of $L_{\infty\omega}$, it is equally completely axiomatizable as
$L_{\infty\omega}$, and it is absolute in the sense that its definable model
classes are absolute in set theory. Thus it avoids Feferman’s second and third
criticisms.

##  6 Sort Logic

We can ask, is there a logic in which every model class whatsoever is
definable, irrespective of the cardinality of the domain? That is, not just
cardinal dependently? We know already $L_{\infty\infty}$ is not that kind of a
logic (Section 4). McGee points out that we could take a class size
disjunction of $L_{\infty\infty}$ sentences and obtain a single ‘sentence’
which works in all cardinalities. As he points out, if we accept class size
formulas we should also accept class size logical operations and then we are
back in the starting point: to account for class size logical operations we
need classes of classes and we start climbing up the type hierarchy beyond
first order set theory. If we stick to set size logical operations we can
operate within first order set theory.

We could, of course, take every model class as a generalized
quantifier404040In the sense of section 2 and of [32]. but there is a more
canonical logic for this task.

In [57] a logic $L^{s}$ called _sort logic_ was introduced. It is a kind of
many-sorted version of second order logic $L^{2}$, which allows quantification
not only over subsets and relations on the domain of the model but also over
subsets and relations on new domains outside the current one. Such
quantification happens, for example, when we ask of a given group, whether it
is the multiplicative group of a field? We have to “guess” the addition of the
field but also the neutral element $0$ and those are both outside the
multiplicative group.

More exactly, sort logic arises from $L^{2}$ by repeated applications of
operation $L^{*}\mapsto\Sigma(L^{*})$ (see Definition 7) and negation. Let us
define $\Delta_{0}=L^{2}$. If $\Delta_{n}$ has been defined, let
$\Delta_{n+1}$ consist of model classes $\mathcal{K}$ such that both
$\mathcal{K}$ and the complement of $\mathcal{K}$ are
$\Sigma(\Delta_{n}(L^{2}))$-definable. We get an increasing hierarchy of
logics

$\Delta_{1}\leq\Delta_{2}\leq\Delta_{3}\leq\ldots$

the union of which is sort logic. A fine point is that every level
$\Delta_{n}$ of sort logic is definable in set theory but there is no single
formula that defines the entire sort logic.

######  Theorem 19 ([57]).

If $\mathcal{K}$ is a class of models of the same finite vocabulary, definable
in set theory without parameters,414141We may allow parameters if we extend
sort logic so that it includes $L_{\infty\omega}$. then the following
conditions are equivalent:

  1. 1.

$\mathcal{K}$ is closed under isomorphisms.

  2. 2.

$\mathcal{K}$ is definable in sort logic.

The above characterization of isomorphism closure is not as elegant as McGee’s
Theorem 6 simply because the definition of sort logic is more complicated than
that of $L_{\infty\infty}$. However, this theorem has the remarkable advantage
over Theorem 6 that the definition of the model class in sort logic holds for
models of _all_ cardinalities. As McGee points out, if he wanted to obtain the
same level of generality with his method, he would have to take a disjunction
of a proper class of $L_{\infty\infty}$-formulas. Sort logic may be
complicated but at least its formulas are sets and not proper classes. In
McGee’s case we may need a formula which is not even a set. In the case of
sort logic each formula is a set, but the whole logic is not definable, only
each level $\Delta_{n}$ separately is.

##  7 Is having a Completeness Theorem a marker of logicality?

We saw that first order logic and $\mathcal{L}(Q_{0})$ are maximally logical
under Sagi’s criterion, grading the logicality of these logics by their
Löwenheim numbers. We also saw that the degree of logicality of the logics
$\mathcal{L}(Q_{\alpha})$ decreases as $\alpha$ increases: if
$\alpha\leq\beta$, then $Q_{\beta}$ is less logical than $Q_{\alpha}$.

As for other kinds of quantifiers, the quantifiers “more” or Rescher
quantifier

$\begin{array}[]{lcl}Jxy\phi(x)\psi(y)&\iff&\mbox{ there are at least as many
$x$ satisfying $\phi(x)$}\\\ &&\mbox{as there are $y$ satisfying
$\psi(y)$}\end{array}$

and the equicardinality or Härtig quantifier (see Section 4.1) both have the
same (very high) Löwenheim number, and are thus less logical than $Q_{\alpha}$
at least for $\alpha$ below the first $\alpha$ such that
$\alpha=\aleph_{\alpha}$.424242If $\ell_{I}$ is the Löwenheim number of the
Härtig quantifier, then $\ell_{I}$ is always bigger than the first fixed point
of the $\aleph$-hierarchy [23]. If $V=L$, then $\ell_{I}$ is bigger than the
first inaccessible (if any exist) [54]. If $V=L^{\mu}$, then $\ell_{I}$ is
bigger than the first measurable cardinal [53]. If Con(“there is a super
compact cardinal”), then Con($\ell_{I}<$ the first weakly inaccessible) [38].
If Con(ZFC), then Con($\ell_{I}<2^{\omega}$) [54]. We suggested earlier that
the Härtig quantifier is a singular case, in seeming to express the core
concept of isomorphism invariance, which is defined in terms of the concept of
equicardinality. Under our criteria, the Härtig quantifier is classified as
(only) weakly logical even so, not only due to its high Löwenheim number but
also because it lacks very seriously a complete axiomatization [54].

We remarked earlier that Sagi’s demarcation of logicality for a fixed class of
logical constants, which is based on Löwenheim numbers, partitions logical
space differently than ours. As we observed, logics that have a completeness
theorem do not seem to be tied to the $\aleph$-hierarchy in any obvious way.
For example, Keisler’s axioms

$\left\\{\begin{array}[]{ll}\mbox{Axiom $0$}&\mbox{Axiom schemes for
$\mathcal{L}_{\omega\omega}$}\\\ \mbox{Axiom $1$}&\neg Qx(x=y\vee x=z)\\\
\mbox{Axiom $2$}&\forall x(\phi\to\psi)\to(Qx\phi\to Qx\psi)\\\ \mbox{Axiom
$3$}&Qx\phi(x)\to Qy\phi(y)\\\ \mbox{Axiom $4$}&Qy\exists x\phi\to(\exists
xQy\phi\vee Qx\exists y\phi)\end{array}\right.$ (9)

are complete for $\mathcal{L}(Q_{1})$, and if $GCH$ is assumed, then they are
complete for all $\aleph_{\alpha+1}$ for which $\aleph_{\alpha}$ is regular
(hence for $\aleph_{n}$ for all $n>0$). Whereas while $\mathcal{L}(Q_{0})$
satisfies the Keisler Axioms, these (or any other recursive set of axioms) are
not a complete axiomatisation of $\mathcal{L}(Q_{0})$.434343See [43].

The suggestion here is that having a complete axiomatisation should be a
marker of logicality on the simple ground that logics of this kind resemble
first order logic in one of its essential, if not most essential property,
namely completeness. Except for $Q_{0}$, one would then classify “there are
very (i.e. uncountably) many” as logical, as dictated by Keisler’s axioms; in
particular, the quantifier $Q_{1}$ would be graded as having a higher degree
of logicality than $Q_{0}$, as $\mathcal{L}(Q_{1})$ is complete with respect
to the Keisler Axioms whereas $\mathcal{L}(Q_{0})$ is not complete in this
respect.444444However, $\mathcal{L}(Q_{0})$ is the only one among the logics
$\mathcal{L}(Q_{\alpha})$ which satisfies the same logical consequences as
$\mathcal{L}(Q_{0})$. See [59]. In short, our criterion of logicality, which
turns on the degree to which a logic resembles first order logic in its model
theoretic properties, comes apart from Sagi’s already at the level of the
logics $\mathcal{L}(Q_{0})$ and $\mathcal{L}(Q_{1})$.

A possible objection might come from the fact that Keisler’s axioms are
satisfied by many _different_ quantifiers, in fact by every $Q_{\alpha}$,
where $\aleph_{\alpha}$ is regular. On the other hand, we may consider $Q_{1}$
logical admitting that from the point of view of logicality it cannot be
separated from some other similar quantifiers. What is logical, according to
this view, is not so much the cardinality $\aleph_{1}$, as, simply,
uncountable cardinality—or “very many”—in general, while the failure of
$Q_{0}$ to permit a recursive axiomatization is an indication that there is
something mathematical, as against logical, in $Q_{0}$. The permutation
invariance characterization of logicality does not differentiate between
$Q_{1}$ and $Q_{0}$, as the Completeness Theorem criterion does. By the
invariance criterion every $Q_{\alpha}$ is logical, which may seem
unintuitive. By further applying the Completeness Theorem criterion we can
make finer distinctions and see a difference: $Q_{1}$ is “more logical” than
$Q_{0}$.

For singular strong limit $\aleph_{\alpha}$ Keisler [30] proved a Completeness
Theorem, but with different axioms (no simple set of axioms is currently
known). Thus there are (at least) two different “logical” concepts of “very
many,” one for $\aleph_{\alpha}$ the successor of regular and one for the
singular case. (Successor of singular is open.) The two concepts have
different logical content: Keisler’s Axioms for $Q_{\alpha}$ are valid for all
regular $\aleph_{\alpha}$, and in some cardinalities (successor of regular)
they have a Completeness theorem, modulo the $GCH$, as was noted above. In
some other cardinalities (singular) different axioms have to be employed in
order to get a completeness theorem. In the base case $\alpha=0$ no
effectively given additional axioms can be added to give a completeness
theorem.

According to Sagi, “We should distinguish between a logic $\mathcal{L}(Q)$
used to measure the logicality of $Q$ and the logic we ultimately use for
validity and logical consequence.”454545ibid, p. 22. But if metaphysical
involvement is inversely related to logicality, then having a completeness
theorem should possibly be considered as a marker of logicality also under the
Sagi criterion. This is because what the completeness theorem precisely does
is to enable the conversion of semantic content, which is metaphysically
involved (from the Sagi point of view, presumably), into syntactic content
which, presumably, is not. In this connection, the following caveat is
important: As Carnap observed [18] the rules do not fix the interpretation of
the logical constants, they have non-standard interpretations. In their [12]
Bonnay and Westerståhl present a way to sidestep the problem:

> Our take on Carnap’s Problem is that it is made artificially difficult by
> considering all possible interpretations, no matter how bizarre. As
> speakers, we know that our language is going to be compositional, that it
> will have some true and some false sentences, and that its logical
> constituents will be topic-neutral. Therefore attention may be restricted to
> interpretations which satisfy these principles. Following Church’s advice,
> this amounts to explicitly factoring out the role of semantic principles and
> the role of inference rules in fixing the interpretation of logical
> constants, rather than covertly using semantic notions to make sense of
> extended inference rules. This strategy proves successful both for
> propositional connectives and for quantifiers.

There is also the issue of expressive power, with respect to which the logics
built on the quantifiers “most”, “more” and the Härtig quantifier differ. In
certain models with an equivalence relation the Härtig quantifier can be seen
to be eliminable while the Rescher quantifier is not [24]. Thus the Rescher
quantifier is strictly stronger than the Härtig quantifier from the point of
view of expressive power. The point here is that expressive power should be
inversely related to logicality.

Bonnay also ties logicality to syntax via a completeness theorem, as is
spelled out in some detail in [11].464646See also the recent [9]. Here Bonnay
proposes a modification of the program of Carnap’s Aufbau [17], calling for
logical expressions to be defined syntactically. It is a feature of his
treatment that the absoluteness of a logic plays a crucial role in
guaranteeing the robustness of the syntactic definition. Thus $L(Q_{0})$ is an
absolute logic because it has a recursive syntax, just like first order logic,
and its semantics is absolute in transitive models of (even weak) set
theory.474747This is essentially because finiteness is absolute, i.e.
$\Delta_{1}$-definable. See Barwise “Admissible sets and Structures” [7], p.
38. On the other hand, $L(Q_{1})$ is not absolute, even though it has a
recursive syntax.484848This is due to the well-known fact that countability is
not absolute: a set can be uncountable in a model of set theory and countable
in a transitive extension. On the relevance of absoluteness in this context
see also Bonnay’s [10].

Burgess’s [14] forges a link between absoluteness and the idea of having a
proof procedure. He exhibits a quasi-constructive complete proof procedure
involving rules with $\aleph_{1}$ premisses for the hereditary countable part
of any absolute logic.

In sum, if one classifies the first order existential quantifier “there is at
least one,” as inherently logical, and if as such this quantifier is thought
of as having minimal or no semantic content, then is there a principled way to
determine when and how higher quantification acquires semantic content, e.g.
at what level in the cumulative hierarchy? Sagi’s answer is that logicality
diminishes the higher up we are in cumulative hierarchy; while we suggest that
logicality kicks in arbitrarily high up, e.g. for all $\aleph_{\alpha+1}$ for
which $\aleph_{\alpha}$ is regular, with the $\aleph_{0}$ case an anomaly,
again because of completeness. The intuition here is that logics which have a
completeness theorem are close to first order logic in this special sense.
This is because completeness enables the conversion of semantic consequence
into syntactic consequence via the two Keisler axiomatisations—albeit
conditioned on the continuum hypothesis in certain important cases.494949See
[31] for a similar treatment of the relation between logicality and
completeness.

##  8 Conclusions

McGee’s Theorem translates the Tarski-Sher criterion into definability in a
logic, obtaining cardinal dependent definability in $L_{\infty\infty}$. The
cardinal dependency aspect of the theorem leads to criticism of logicality
across domains (Feferman’s third critique), to criticism of entanglement with
mathematics (Feferman’s first critique) and of non-absoluteness (Feferman’s
second critique). Maintaining cardinal dependency we lowered
$L_{\infty\infty}$ to $\Delta(L_{\infty\omega})$ which is more palatable from
the point of view of model theoretic properties. A stronger degree of
logicality is obtained by abandoning cardinal dependency and investigating
logics in which a candidate for logicality is definable. Since every class of
models which is closed under isomorphisms is definable in some logic, this
seems reasonable. Following and expanding on Sagi’s suggestion to delineate
degrees of logicality according to their Löwenheim numbers, we delineate
logicality according to their Löwenheim numbers but also according to a wider
spectrum of model theoretic properties of logics, such as Completeness
Theorems and absoluteness properties, together with Löwenheim-Skolem
properties, all considered from a logicality point of view.

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