Logicality and Model Classes∗

Juliette KennedyDepartment of Mathematics and Statistics

University of Helsinki, Finland

Jouko VaananenDepartment of Mathematics and Statistics

University of Helsinki, FinlandILLC, University of Amsterdam

Amsterdam, Netherlands

October 19, 2020

Abstract

We ask, when is a property of a model a logical property? Ac-cording to the so-called Tarski-Sher criterion this is the case whenthe property is preserved by isomorphisms. We relate this to model-theoretic characteristics of abstract logics in which the model class isdefinable. This results in a graded concept of logicality in the termi-nology of Sagi [42]. We investigate which characteristics of logics, suchas variants of the Lowenheim-Skolem Theorem, Completeness Theo-rem, and absoluteness, are relevant from the logicality point of view,continuing earlier work by Bonnay, Feferman, and Sagi. We suggestthat a logic is the more logical the closer it is to first order logic. Wealso offer a refinement of the result of McGee that logical propertiesof models can be expressed in L∞∞ if the expression is allowed todepend on the cardinality of the model, based on replacing L∞∞ bya “tamer” logic.

∗The first author would like to thank the Academy of Finland, grant no: 322488. Thesecond author would like to thank the Academy of Finland, grant no: 322795. The authorsare grateful to Denis Bonnay for comments.

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1 Introduction

To say that the syllogism

All A’s are B’s.x is an A.x is a B.

is a valid argument form is to say that the conclusion follows from thepremises no matter which predicates are substituted for the nonlogical termsA and B, whether ‘men’ and ‘mortal’ or ‘tulips’ and ‘pious’, or whatnot, aslong as the substitution is done uniformly. That the conclusion follows fromthe premises is a matter of logic—“is obviously true in a purely logical way,”to quote Carnap1.

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

In 1968 in a (posthumously published) lecture called “What are logicalnotions?” [47] Tarski proposed a definition of “logical notion,” or alterna-tively of “logical constant,” modelled on the Erlanger Program due to FelixKlein. The core observation is the following: for a given subject area, thenumber of concepts classified as invariant are inversely related to the numberof transformations—the more transformations there are, the fewer invariantnotions there are. If one thinks of logic as the most general of all the math-ematical sciences, why not then declare “logical” notions to be the limitingcases? Thus a notion is to be thought of as logical if it is invariant under allpermutations of the relevant domain.

As expected, the standard logical constants, namely conjunction, disjunc-tion, negation and quantification, together with the equality relation, are alljudged to be logical operations under this criterion, being all isomorphisminvariant. This has the consequence that first order definability is classifiedas logical, tout court, being generated by these constants.2

1[13], Engl. translation in [12].2V. McGee [37] extends the observation to L∞∞:

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As was clear to Tarski, the criterion overgenerates, in the sense thatconcepts one does not immediately assimilate to logic turn out to be in-cluded, such as the cardinality of the underlying domain, which is clearlyisomorphism invariant. A second problem has to do with domain relativity:Tarski’s criterion identifies the logical operations on a fixed domain, but thata domain consists of this or that type of object should not have anythingto do with logicality—shouldn’t a logical concept be domain independent?G. Sher [44] repaired this problem by extending Tarski’s criterion to covernotions invariant across isomorphic structures, and accordingly the criterionis now known as the Tarski-Sher invariance criterion.

The Tarski-Sher criterion has generated a substantial literature, both forand against. S. Feferman’s [18], based on his earlier [17], proposes a notionof homomorphism invariance. His critique of the Tarski-Sher thesis is thefollowing:

I critiqued the Tarski-Sher thesis in [17] on three grounds, the firstof which is that it assimilates logic to mathematics, the secondthat the notions involved are not set-theoretically robust, i.e. notabsolute, and the third that no natural explanation is given bythe thesis of what constitutes the same logical operation overarbitrary basic domains.3

Feferman’s objection that Tarski assimilates logic to mathematics, in par-ticular to set theory, overlooks Tarski’s lifelong program to carry out exactlywhat Feferman accuses Tarski of here, namely expressing metamathematicalconcepts in mathematical and set-theoretical terms.4 Feferman’s second ob-jection has to do with overgeneration, in particular he resists the idea thata canonically mathematical but non-absolute notion such as cardinality isrendered logical under the criterion.5

Since the primitive connectives of L∞∞ are all intuitively clearly logical con-nectives, and since, intuitively, anything definable from logical connectivesis again a logical connective, this will show that every operation invariantunder permutations is describable by a logical connective, so every operationinvariant under permutations is a logical operation.

3[18], p. 1.4The first author has argued this point in [28].5The fact that cardinality is construed as logical under his criterion seemed to pose no

problem for Tarski:

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One way of thinking about topic neutrality is in terms of absoluteness.Thus a logical notion, it is thought, should not be sensitive to the backgroundset theory, in particular a logical notion should not be entangled with car-dinality, which is famously non-absolute, because of forcing.6 AccordinglyFeferman proposes restricting the Tarski-Sher invariance criterion to opera-tions that are absolute with respect to set theories making no assumptionsabout the size of the given domain. Feferman cites [2],7 in which it is shownthat operations on relational structures that are definable in an absolute wayrelative to KPU-Inf, i.e. Kripke-Platek set theory with urelements and with-out the Axiom of Infinity, are exactly those expressible in ordinary first-orderpredicate calculus with equality—another endorsement of the idea that firstorder logic captures the notion of “logical concept.”

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

Setting overgeneration aside for the moment, and considering Feferman’sthird objection, namely the problem of domain relativity, it would seem thatfollowing Feferman’s line we should consider model classes in general, namelyclasses of structures of the same similarity type, but possibly of differentcardinality, that are closed under isomorphism8. How then to formulate theconcept of logicality for a model class? Originally the question of logicalityarises in connection with operations such as connectives, quantifiers and theirgeneralisations. As V. McGee explains in [37], the question can be restatedas a question about model classes, whether the property of a given model

That a class consists of three elements, or four elements . . . that it is finite, orinfinite—these are logical notions, and are essentially the only logical notionson this level. [47], p. 151.

6Cardinality also emerges as an artifact in the context of ramsification and Newman’sobjection to epistemic structural realism. See Ainsworth [1].

7Feferman also cites an unpublished result of K. Manders.8If we work in set theory, classes are objects of the form {a : φ(a)}. Ordinary set theory

does not have objects of this kind, so classes are simply identified with their definingformulas, in this case φ(x). The defining formula is allowed to have set parameters. Thuswhen we refer to the class of all ordinals, the class of all structures, etc, we mean to referto the defining formula, which defines when a set is an ordinal, the formula which defineswhen 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.

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being in a given model class is a logical property of the model.9

Let’s back up and ask when would we call a property of an individualstructure M logical. With Sher, certainly we should consider the elementsofM irrelevant as long as the arrangement of the elements remains the same,as was mentioned. This leads to the idea that we call a property ofM logicaliff the property is closed under permutations of the elements of M in thesense that if f is a permutation of M then the image of M as a structureunder f has the property. But then it is a short step to require closure underisomorphisms.

A theorem due to McGee [37] characterises logicality for an arbitrarymodel class, provided the cardinality of the models in the class is fixed inadvance. It is a weakness of this characterisation that it depends on thecardinality of the models in the class. This leads us to ask whether there isa sentence in a logic L∗, perhaps other than L∞∞, which characterises thelogical property, not just in one cardinality, but in many cardinalities. Thisleads in turn to an analysis of spectra, as we call them,10 and Lowenheim-Skolem style properties of logics as expressed in their Lowenheim and Hanfnumbers.11

As G. Sagi notes in her [42], the Lowenheim number of a logic encodesthe degree to which the logic is indifferent to cardinality.12 This is due tothe fact that if a sentence satisfied by a model is also satisfied by a modelof size less than or equal to a given cardinality κ, then all the validities13

which are captured in the full range of models are captured already on aninitial segment (of V ). In the case of first order logic the Lowenheim-SkolemTheorem tells us that κ can be taken to be ℵ0, so in this sense first order logicis indifferent to cardinalities above ℵ0. First order logic is already classifiedas logical because its logical constants are permutation invariant; this is nowwitnessed by the Lowenheim-Skolem Theorem.

The strength of the Lowenheim-Skolem property of a given logic—or ifyou like, the degree of its indifference to cardinality—is measured, then, byits Lowenheim number. As an example, the quantifier Qα associated with

9This property is essentially the same as deeming the associated generalised quantifier.together with everything first definable from it, as logical. See below section 2.

10See also [42].11See Definition 8 below.12For the definition of Lowenheim number see section 3.1. This observation is an im-

portant point of departure for this paper.13Or for Sagi, the meaning, see [42].

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the logic L(Qα), having Lowenheim number ℵα, is in a sense indifferent tocardinalities greater than ℵα.14 First order logic and L(Q0) are maximallylogical under the criterion assigning logicality according to Lowenheim num-ber, and the degree of logicality decreases as α increases. Thus if α ≤ β,then Qα is more logical than Qβ.15

The criterion for logicality presented in Sagi’s paper is based on partic-ular philosophical position, to wit: a metaphysical view of the cumulativehierarchy of sets, as well as a view about how the meaning of the terms of alogic are fixed.

. . . the higher set-theoretic infinite is metaphysically loaded. . . Thelower the cardinalities to which the meaning16 of a term may besensitive, the more logical it is. . . . [We] view Lowenheim numbersas telling us how much structure a term [in this case the quanti-fiers Qα JK/JV] requires in order to be fixed in the context of alogic.”17

In short, the smaller the Lowenheim number of the logic, the less meta-physically involved e.g. the quantifier Qα is.

Sagi’s criterion cuts across partitions of logical space in ways that differfrom ours. In this paper we will not take account of the possible metaphysicalcommitments of set theory; and while her analysis of meaning is one wepotentially endorse, we will not take a stand on the meanings of the termsof a logic in this paper. Our basic thesis is simply this: if first order logic istaken to be the fundamental exemplar of logicality, then logics that resemble,to a degree, first order logic in their model theoretic properties should begraded as logical to that degree.

Thus while taking up Sagi’s suggestion that logicality may be calibratedby Lowenheim numbers, we also give weight to the other model theoreticproperties of a logic. As we will point out below, logics that have, for example,a Completeness Theorem18 do not seem to be tied to the ℵ-hierarchy in any

14This is a central example of [42].15L(Q) denotes first order logic with the generalised quantifier Q appended. The ex-

pression “Qαxφ(x)” means that “there are at least ℵα many x such that φ(x).” By theLowenheim-Skolem Theorem, the Lowenheim number of first order logic is ℵ0, as wasnoted; for each α, the Lowenheim number of the logic L(Qα) is ℵα.

16emphasis ours17[42]18As Bonnay [8] points out, Quine [41] emphasises the Completeness Theorem as a

carrier of logicality in first order logic.

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obvious way—some of these logics are axiomatisable and some are not.Another drawback of McGee’s theorem which we address in this paper

has to do with the model-theoretic properties of the logic L∞∞, namely thatit is highly non-absolute (see Definition 14), that it fails to have a strongLowenheim-Skolem theorem, and also that it is unbounded19 in the sensethat it can define the concept of well-ordering leading to very large Hanfnumbers.20 This leads us to ask whether L∞∞ can be replaced by a “tamer”logic, one closer to being first order in its model-theoretic properties. If suchwere to exist, then even with the problem of dependence on the cardinalityof the models in the class unsolved, the relevant logicality claim would bestrengthened by virtue of its proximity to first order logic.

In sum: in this paper we develop further this aspect of logicality identi-fied by Sagi, namely that it is graded, albeit with a different philosophicalagenda than is laid out in [42]. We especially point out problems in theLowenheim-Skolem properties of L∞∞, along with other ways in which itdiverges from first order logic. As was mentioned above, we take first orderdefinability as a particularly strong form of logicality.21 Accordingly, oursuggestion here is that definability in a logic which resembles with its modeltheoretic properties first order logic should represent an intermediate formof logicality. We calibrate these intermediate forms according to the modeltheoretic properties of these logics, considering mainly absoluteness, havinga Lowenheim-Skolem theorem, and having a completeness theorem.

While L∞∞ is sufficient for McGee’s theorem, there is a weaker logic,namely the so-called ∆-extension of L∞ω which does the job as well. Weshall explain in which way this weaker logic is better than L∞∞ as a test oflogicality, and also point out in which respects it may be lacking. Lookingahead, our refinement of McGee’s theorem replaces L∞∞ by a logic whichis absolute and which has a good Lowenheim-Skolem theorem, along with anumber of other desirable properties. The improvement here is partial in thesense that it is still the case that the definition is given relative to the sizeof the models in the class. In section 7 below we give a theorem which doesnot rely on this assumption, namely a we present a logic in which any modelclass is definable irrespective of the cardinality of the models in the class.

In spite of the seemingly clear intuition behind the syllogistic example

19as it is defined in [7]20See discussion following (9)21A view endorsed by Bonnay [8] and Feferman [18].

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with which we opened this paper, logicality is actually 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 judgewhether this or that improvement of Tarski’s original definition hits the markor not, or even whether it is a step in the right direction. The only thingthere seems to be a consensus on in the wake of Tarski’s suggestion to identifylogicality with isomorphism invariance is that Tarski’s criterion, extended bySher, is a necessary but not sufficient one.22

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’sTheorem [37] to the effect that logical operations in the sense of Tarski-Shercan be defined in each cardinality separately in L∞∞, the landscape thatopens in front of us is the world of different logics and the question, do themodel-theoretic properties of these logics shed any light on the logicalityproblem?

2 McGee’s Theorem

To describe McGee’s result and to put it into a more general framework wedefine what we mean by a “logic”: A logic (a.k.a. abstract logic) in the senseof [30] is a pair L∗ = (Σ, T ), where Σ is an arbitrary set (sometimes also aclass) and T is a binary relation between members of Σ on the one hand andstructures on the other. Members of Σ are called L∗-sentences. Classes ofthe form

Mod(φ) = {M : T (φ,M)},

where φ is an L∗-sentence, are called L∗-characterizable, or L∗-definable,classes. Abstract logics are assumed to satisfy five axioms expressed in termsof L∗-characterizable classes, corresponding to being closed under isomor-phism, conjunction, negation, permutation of symbols, and “free” expan-sions.23 A class K of models is said to be definable in a logic L∗ if there is a

22As Bonnay and Westerstahl put it in their [10]: “And topic-neutrality, in the preciseform of invariance under permutations of the universe, is almost universally agreed to be anecessary condition for logicality. It guarantees that the logical core of a language is generalenough to carve out content in any conceivable situation of language use, irrespective ofwhat objects are being talked about.”

23The free expansion to vocabulary L of a model class K of a smaller vocabulary is theclass of all expansions of elements of K to the vocabulary L.

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sentence φ in L∗ such thatK = Mod(φ).

Every model class is definable in some logic because we can take themodel class as a generalized quantifier in the sense of [29]: Suppose K is amodel class with vocabulary L. For simplicity we assume L = {R} whereR is a binary predicate symbol. We can associate with K the generalizedquantifier QK with the semantics

M |= QKxyφ(x, y,~a) ⇐⇒ (M, {(b, c) ∈M2 :M |= φ(b, c,~a)}) ∈ K.

Now K is trivially definable in the extension Lωω(QK) of first order logic bythe quantifier QK by the sentence

QKxyR(x, y).

Conversely, every class of models definable in Lωω(QK), or indeed in anyabstract logic, is a model class i.e. is closed under isomorphisms. We obtainthe following simple and at the same time basic characterization:

Theorem 1. If K is a class of models of the same vocabulary, then thefollowing conditions are equivalent:

1. K is closed under isomorphisms i.e. K is a model class.

2. K is definable in some extension of first order logic by a generalizedquantifier.

3. K is definable in some logic.

Corollary 2. An operation is a logical operation if and only if it can beexpressed in some logic.

The theorem of McGee improves “some logic” to a very specific logic,namely L∞∞, but at a price: we obtain a different definition for each cardi-nality separately.24

Note that Theorem 1 is more uniform in the sense that the defining sen-tence does not depend on the cardinality of the model in question, but alsoless uniform in the sense that we do not have one single logic but possibly adifferent logic for each model class.

We now isolate this property of cardinality dependence. For any cardinalλ let Kλ be the class of elements of K with a domain of size λ.

24We point out other weaknesses in the below section 2.1 involving the model-theoreticproperties of L∞∞, e.g., its non-absoluteness.

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Definition 3. A model class K is cardinal dependently definable, or CD-definable, in a logic L∗, or cardinal dependently L∗-definable, if Kλ is L∗-definable for every λ.

Note that a model class can be CD-definable even in first order logicwithout being definable even in L∞∞: Consider the class Klim of models(M,P ), where either |M | is a limit cardinal and P = ∅ or else |M | is asuccessor cardinal and P 6= ∅. It is a consequence of the Lowenheim-SkolemTheorem of L∞∞ (see Proposition 11 and the proof of Theorem 12 below)that this model class cannot be definable in it. The following theorem relatesisomorphism closure to CD-definability:

Theorem 4 ([37]). If K is a class of models of the same vocabulary, thenthe following conditions are equivalent:

1. K is closed under isomorphisms.

2. K is CD-definable in L∞∞.

Setting logicality considerations aside for the moment, the theorem isinteresting because the logic L∞∞ appears out of ‘thin air’. What the theoremseems to show is that arbitrary model classes have an implicit logic; that it ispossible to, in a sense, “read” a syntax and/or a logic off an arbitrary modelclass.

Returning to McGee’s theorem, the fact that it depends on the cardinalityof the models in the class means that the sentence of φλ of L∞∞ defining themodel class Kλ may very well depend on λ, as in the above example Klim.That is, the class size mapping λ 7→ φλ may encode information that has to beanalysed as to its logicality. In the above example, when we ask whether theproperty of a model of belonging to the model class Klim is logical or not, weessentially ask whether the property of a model of having a limit cardinalityis logical or not. According to the Tarski-Sher criterion it undoubtedly islogical. On the other hand, it is fair to say that it is a mathematical propertyrather than a logical one. As we will see below, the logicality of membershipin Klim manifests in our below analysis a rather low degree of logicality.

2.1 The model-theoretic properties of L∞∞

We now note the different respects in which L∞∞ deviates from first orderlogic. First of all, L∞∞ is badly nonabsolute (for the definition of absoluteness

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see Section 6). For example, the sentence

∀x∀y(∀z(E(z, x)↔ E(z, y))→ x = y)∀x0∀x1 . . . ∃y∀z(E(z, y)↔

∨n<ω z = xn)

(1)

of L∞∞ which has models exactly in cardinalities µ such that µω = µ, ishighly non-absolute.25 Other failures of absoluteness can be generated byreplacing ω here by any regular cardinal.

A second important model-theoretic property of first order logic is itsdownward Lowenheim-Skolem theorem. Consider the above (1). Such a µcannot have cofinality ω and thus L∞∞ does not have a Lowenheim-Skolemtheorem in the same strong sense as first order logic, or even in the sense of,e.g., L∞ω (see Theorem 7). That is, L∞∞ can “omit” cardinals of cofinalityω in this special sense given by the Lowenheim-Skolem theorem. In contrast,a typical consequence of a Lowenheim-Skolem theorem of Lκ+ω is that if adefinable model class has a model of cardinality λ > κ then it has modelsof all cardinalities µ such that κ ≤ µ ≤ λ (Theorem 7), whatever theircofinality.

The logic behind Theorem 4, Lκ+κ+ , fails in a strong sense to have aCompleteness Theorem. By a result of Dana Scott (published in [25]) theset of valid sentences of Lκ+κ+ is not Lκ+κ+-definable over H(κ+), the setof sets of hereditary cardinality ≤ κ. In contrast, the set of valid sentencesof Lκ+ω is Σ1-definable over H(κ+) [25], reminiscent of the CompletenessTheorem of first order logic which implies that the set of (Godel numbersof) valid sentences of first order logic is recursively enumerable i.e. Σ1 overHF , the set of hereditarily finite sets. In fact, C. Karp gives a completeaxiomatization for Lκ+ω in [25].

In section 6 we replace L∞∞ by a logic which is absolute and which has astrong Lowenheim-Skolem theorem, together with other desirable properties,though it is still the case that the definition is given relative to the size ofthe models in the class.

25By Konig’s Theorem ([22, Theorem 5.10]) such a µ cannot have cofinality ω andcofinality is not absolute in set theory.

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3 Lowenheim-Skolem theorems viewed through

the spectra of model classes

The fact that the class of all models of cardinality κ, for example, satisfies theTarski-Sher criterion for all κ, irrespective of what κ is, raises the question,is cardinality a logical property? We saw that in the more subtle case of Klim

the property of a model of being of a limit cardinality is, according to thiscriterion, logical; by modifying Klim one can generate many other cases. Forexample, if A is a property of natural numbers, we can consider the class Kof models of cardinality ℵn, such that n has property A. The property of amodel of belonging to K is a logical property of the model by the Tarski-Shercriterion, even though to judge whether a particular model is in K one has tofirst determine whether the size of the universe is some ℵn, and after that onehas to determine whether n has property A. This stretches the concept oflogicality and entangles it with the concept of what is mathematical. Earlierwriters have observed this, so we are not saying essentially anything newhere.

Of Feferman’s three objections to the Tarski-Sher criterion of logicality,the third is that it leaves unexplained how logicality is to be understoodacross 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 trueof many properties which satisfy the Tarski-Sher criterion, for example Klim.We suggest that in order that a property of models manifests a greater degreeof logicality than is provided by the Tarski-Sher criterion, it should have adefinition by a sentence in L∞∞ or in some other logic in such a way that thesame sentence defines the model class in as many cardinalities as possible.26

We explore this suggestion below via the concept of a spectrum. Our sec-ond suggestion is that a higher grade of logicality would be assigned to thedegree to which the logic in question had “first-order-like” model-theoreticproperties: absoluteness, a Lowenheim-Skolem theorem, and completeness.L∞∞ would thus be given a low grade on the logicality scale, being non-absolute, having a limited Lowenheim-Skolem theorem and failing to have aCompleteness Theorem. We explore this suggestion in section 8.

26Note that Klim has the same first order definition on the (closed unbounded) class ofall limit cardinals.

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Figure 1: Spectra of some model classes.

3.1 Spectra

In analysing the structure of a model class, in order to determine whetherit is in some sense logical or not or to find out whether it is definable insome interesting logic, it is useful to investigate the cardinalities of models inthe class. One might think that mere cardinalities are too rough a measureof any form of logicality but this is, in fact, surprisingly informative. Thisapproach leads us to the concept of a spectrum:27

Definition 5. If K is a model class, the spectrum of K is the class sp(K) ofcardinalities of models in K i.e.

sp(K) = {|M | :M∈ K}.

Depending on K, the spectrum can be a singleton, an interval of cardinals,an initial (or final) segment of the class of all cardinals, or something morecomplicated, such as the class of all limit cardinals, or all limit cardinals ofcofinality ω (see Figure 1). Even the patterns of finite numbers in spectraof first order sentences is highly interesting [3, 23]. However, we are hereconcerned with infinite cardinals in a spectrum.

The property of a logic which reflects regularity patterns in its spectra iscaptured by the Lowenheim-Skolem Theorem. The spectrum gives indirect

27See also [42].

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information of the possibility that the model class is definable in some logic.Roughly speaking, if the logic has a strong Lowenheim-Skolem property, thenthe spectra of definable model classes reflect this. If every sentence in thelogic which has an infinite model has also a countably infinite model, themost famous case of a Lowenheim-Skolem property, then every spectrumwith an infinite cardinal in it has also ℵ0 in it.

Definition 6. Suppose C and D are classes of cardinal numbers. A logic L∗

satisfies the Lowenheim-Skolem Property LS(C,D) if every sentence in L∗

which has a model of some cardinality in C has a model of some cardinalityin D.

Skolem proved that first order logic satisfies LS([ℵ0,∞), {ℵ0}) giving riseto the so-called Skolem Paradox: countable first order theories such as settheory have countable models if they have models at all. Now, one hundredyears after Skolem’s discovery, the indifference of first order logic to theinfinite cardinality of its models is often considered a positive rather thannegative aspect. Indifference to cardinality is of course closely related toset-theoretical absoluteness and thereby to a desirable quality of logicality.In consequence, the stronger form of LS(C,D) a logic satisfies the moreappropriate the logic is for expressing logical properties.

If a logic satisfies LS(C,D), there are consequences for the spectra ofdefinable model classes. Suppose K is definable in a logic with LS(C,D).Then we can make the following conclusions: If there is M ∈ K with |M | ∈C, then there is N ∈ K with |N | ∈ D. On the other hand, if K containsa model of cardinality κ but no models of cardinality λ, then K cannot bedefinable in a logic with LS(C,D) such that κ ∈ C and λ /∈ D. The pointis that by looking at the spectrum of K we can make inferences about itsdefinability in different logics. Thus, if we are given a model class K but nologic in which it would be definable (apart from the trivial L(QK)), and wecan discern regular patterns in sp(K), we may take it as an indirect indication(albeit not a proof) that K is definable in a logic with LS(C,D) for someC and D explaining the found patterns. On the other hand, if sp(K) doesnot seem to have regular patterns, we may take it as an indication that nosuch logic can be found. Thus sp(K) gives implicit information about thepossibility of finding a syntax for K.

Let us look at the Lowenheim-Skolem Properties of the logics L∞ω andL∞∞.28 We first recall a basic construction in infinitary logic, based on a

28Recall that L∞ω =⋃κ Lκ+ω and L∞∞ =

⋃κ Lκ+κ+ .

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class of important formulas due to D. Scott [43]. For any ordinal α let theformula ηα(x) with one free variable x and a binary relation symbol < bedefined, by transfinite recursion, as follows:

ηα(x)↔ ∀y(y < x→∨β<α

ηβ(y)) ∧∧β<α

∃y(y < x ∧ ηβ(y)). (2)

Then for a linear order (A,<) and a ∈ A we have

(A,<) |= ηα(a) ⇐⇒ ({b ∈ A : b < a}, <) ∼= (α,<). (3)

Letη′α ↔ ∀y

∨β<α

ηβ(y) ∧∧β<α

∃y ηβ(y). (4)

For a linear order (A,<) we have

(A,<) |= η′α ⇐⇒ (A,<) ∼= (α,<). (5)

Now η′α is in Lκω whenever α < κ. It follows that by combining the sen-tences η′α, the logic Lκω can manifest totally arbitrary patterns of spectra incardinalities below κ, roughly for the same reason that any finite set of finitenumbers can be the spectrum of a first order sentence. More exactly, if X isan arbitrary set of cardinal numbers below κ, then∨

λ∈X

η′λ ∈ Lκ+ω and X = sp(∨λ∈X

η′λ).

This shows that when dealing with extensions of Lκω it makes sense to focuson cardinals ≥ κ.

The basic facts about the LS(C,D) type properties of infinitary languagesare the following:

Theorem 7 ([16]). Suppose κ ≤ λ are cardinals and κ is regular.

1. Lκ+ω satisfies LS([λ,∞)}, {µ}) for all µ such that κ ≤ µ ≤ λ.

2. Lκ+κ+ satisfies LS([λ,∞), {µ}) for all µ such that κ ≤ µ ≤ λ andµκ = µ.

3. Lκ+ω satisfies LS([i(2κ)+ ,∞), [µ,∞)) for all µ.

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Figure 2: A difference between L∞ω and L∞∞.

4. Lω1ω1 does not satisfy “For all µ LS([λ,∞), [µ,∞))” for any λ below thefirst inaccessible cardinal (assuming there are inaccessible cardinals).

For a class size logic such as L∞ω or L∞∞ the LS(C,D) properties holdvia their connection to Lκω or Lκλ: If φ ∈ L∞∞, then φ ∈ Lκλ for some(least) κ and λ and then Theorem 7 applies.

The above theorem shows that the spectra of L∞∞ are much more com-plicated than the spectra of L∞ω. For example, let θ be the sentence (1) ofLω1ω1 which has a model of cardinality µ if and only if µω = µ. Thus sp(θ)(i.e. sp(Mod(θ))) is full of “holes” as it misses all cardinals, such as e.g. eachℵα+ω, that are ω-cofinal. Whereas the spectra of sentences of Lκω cannothave such holes above κ. (See Figure 2.)

What does it reveal about logicality if the spectrum is full of ‘holes’? Itsuggests that we are very far from the situation in which we can claim wehave the same logical operation independently of the domain. In the case ofLκ+ω the spectrum is (above κ) a homogeneous segment of cardinals, eitherbounded by i(2κ)+ or unbounded, and we are closer to having the same logicaloperation independently of the domain. This leads us to classifying a modelclass definable in L∞ω as having a greater degree of logicality than a modelclass definable in L∞∞.

We now define two concepts that are crucial for this paper, one of whichwe have seen already:

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Definition 8. The smallest κ such that L∗ satisfies LS([1,∞), [1, κ]) is calledthe Lowenheim number of L∗ and denoted `(L∗). The smallest κ such thatL∗ satisfies LS([κ,∞), [λ,∞)) for all λ is called the Hanf number of L∗ anddenoted h(L∗).

The Lowenheim number and the Hanf number of a logic always exist ifthe class of formulas of the logic is a set. Thus they exist for Lκλ for any κand λ but not for L∞ω and L∞∞.

The Hanf-number and Lowenheim number of L∗ can be easily defined interms of spectra. Below a spectrum is called bounded if it is a set (ratherthan a proper class).

`(L∗) = sup{min(sp(φ)) : φ ∈ L∗}h(L∗) = sup{sup(sp(φ)) : φ ∈ L∗ and sp(φ) is bounded}. (6)

We saw that Sagi employs Lowenheim numbers as a measure of logicality:the smaller the Lowenheim number the greater the degree of logicality. Thusmodel classes definable in L(Qα) have a stronger degree of logicality thanthose definable in L(Qβ), for β > α.

The logic L(Qα) satisfies for trivial reasons the Lowenheim-Skolem prop-erty LS([ω,ℵα), {µ}) for all µ < ℵα and the rather strong Lowenheim-Skolemproperty LS([ℵα,∞)), {µ}) for all µ ≥ ℵα. Thus the spectra have no holesand `(L(Qα)) = ℵα. It should be noted that the class size logic L(Qα)α∈On

is a sublogic of L∞∞. Its spectra have a greater degree of regularity thanthose of L∞∞, and thus in that respect it resembles more L∞ω. In sum, thesetwo logics, namely L∞ω and L(Qα)α∈On, both manifest a greater degree oflogicality than L∞∞, if regularity in spectrum patterns is used as a criterion.

4 Potential isomorphism as an alternative to

isomorphism

Bonnay [4] suggests that the concept of potential isomorphism is a bettercriterion for logicality than isomorphism itself. The point is that potentialisomorphism does not distinguish among infinite cardinalities (in the emptyvocabulary), as isomorphism does. Another asset of potential isomorphismin comparison to isomorphism is that it is absolute. In fact, as Bonnaypoints out, potential isomorphism is the strongest absolute subrelation ofisomorphism:

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Theorem 9 ([4]). Suppose R(M,N ) is a set-theoretical relation betweenstructures which is absolute with respect to ZF . Then

ZF ` ∀M∀N (M∼= N → R(M,N ))

impliesZF ` ∀M∀N (M'p N → R(M,N )).

In particular, if a class of models is absolute with respect to ZF andprovably in ZF closed under isomorphism, then it is closed under potentialisomorphism.

For potential isomorphism closure we have the following version of McGee’sTheorem:

Theorem 10 ([4]). If K is a class of models of the same vocabulary, thenthe following conditions are equivalent:

1. K is closed under potential isomorphisms.

2. K is CD-definable in L∞ω.

Of course, logicality across domains remains a problem even if we getrid of cardinalities; that is, potential isomorphism does not settle Feferman’sthird objection to the Tarski-Sher criterion, that it leaves unexplained howlogicality is to be understood across domains of different sizes.

Another problem with closure under potential isomorphism is that well-ordered structures (α,∈, P1, . . . , Pn) and (β,∈, P ′1, . . . , P ′n) are never poten-tially isomorphic if α 6= β. Just as an isomorphism closed model class canhave models with domains of different cardinalities and the isomorphism clo-sure gives no information across the domains, a potential isomorphism closedmodel class can have models (α,∈, P1, . . . , Pn) with different α and the po-tential isomorphism closure also gives no information across such domains.So potential isomorphism closed model classes can have similar pathologiesas isomorphism closed ones. For example, the class of models isomorphicto (A,<, P ) where either < well-orders A and P = ∅, or else < does notwell-order A and P 6= ∅, is potential isomorphism closed and not definablein L∞ω [31]. For another example consider the class

W2 = {(A,<) : (A,<) ∼= (α + α,∈) for an ordinal α}. (7)

This is potential isomorphism closed and not definable even in L∞∞ [36].

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Potential isomorphism solves the problem of the cardinal dependence ofisomorphism closure but it retains a phenomenon of a similar character,namely dependence on well-order type. Isomorphism closure as a criterionfor logicality is criticised for rendering quantifiers such as “for ℵ5016 manyx” logical, raising the question, how can being of that particular cardinalitybe a logical property? Potential isomorphism closure criterion renders “theorder-type is ω5016” logical, as Bonnay acknowledges [8]. So we still have theovergeneration problem even if we resort to potential isomorphism closure.

5 Is every model class definable in L∞∞ irre-

spective of the cardinality?

Why is it that we cannot replace condition (2) of Theorem 4 with the appar-ently better condition

K is definable in L∞∞? (8)

This would solve the problems of cardinality and domain-relativity—the log-ical concept would have the same definition in terms of a formal languageindependently of the cardinality of the domain. However, there is a verysimple reason why condition (2) of Theorem 4 cannot be improved to (8):Not every model class is definable in L∞∞.

Recall the definition of Klim above in Section 2. It is, strictly speaking,a generalized quantifier in the sense of [38] but a rather curious one. It isthe existential quantifier in models of successor cardinality and the quantifier“for no x” in models of limit cardinality.

Proposition 11. The model class Klim is not definable in L∞∞.

Proof. Similar to the proof of Proposition 12 below.

A different kind of generalized quantifier is the class W2 of (7). Notethat this model class is clearly second order definable. The model classW2 has a particularly nice spectrum, its infinite part being simply the classof all infinite cardinals. This model class is also set-theoretically absolute(see below section 6). In the light of the spectrum criterion, a model beingan element of W2 is a logical property of the model of a very high degree oflogicality. Looking at a model, can we say that its being a well-order, let alone

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of a well-order of type α+ α for some α, is a logical property? Undoubtedlysome would say it is a mathematical rather than a logical property. But it islogical in the Tarski-Sher sense and it has a second order definition which isthe same definition in each cardinality. However much it feels like a genuinelymathematical concept, it fulfils the Tarski-Sher criterion of logicality, escapestwo of Feferman’s criticisms, and fulfils Sagi’s as well as Bonnay’s criterions.So we may conclude that it is logical of a very high degree.

A third example of a model class undefinable in L∞∞ is particularlyinteresting, namely the Hartig quantifier.

5.1 The Hartig Quantifier

The Hartig quantifier is defined as follows:

Ixyφ(x)ψ(y) ⇐⇒ there are as many x satisfying φ(x)as there are y satisfying ψ(y).

Proposition 12 ([21]). The class of models (M,A,B), where A,B ⊆M and|A| = |B|, is not definable in L∞∞. Equivalently, the Hartig quantifier is notdefinable in L∞∞.

Proof. Two sentences of the logic Lωω(I) are useful here: the sentence φlim

which has the class of limit cardinals as its spectrum, and the sentence φsucwhich has the class of infinite successor cardinals as its spectrum. The sen-tence φlim says that the universe is totally ordered by a linear order in whichevery element determines an initial segment which has smaller cardinalitythan the initial segment determined by some bigger element. The sentenceφsuc says that the universe is totally ordered by a linear order in which someelement determines an initial segment which has the same cardinality asthe initial segment determined by any bigger element, but smaller cardinal-ity than the entire universe. Suppose the Hartig quantifier was definable inLκ+κ+ , where κ is regular. Let λ = 2κ. If λ is a limit cardinal, we can use φlim

to derive a contradiction as follows: LetM be a model of φsuc of cardinalityλ+. By Theorem 7 there is a model N of cardinality λ such that N |= φsuc,a contradiction. The case that λ is a successor cardinal is similar, but usingφlim instead of φsuc.

While the logic L(I) with the Hartig quantifier is not a sublogic of L∞∞ ithas some of the properties of the latter that we associate with being very far

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from first order logic (see Section 8). For example, it is highly non-absolute.It is also unbounded i.e. capable of defining well-ordering with additionalpredicates [29].

Note that the equicardinality concept, built around the existence of abijection, lies at the very heart of the definition of logicality in the senseof Tarski! It is almost paradoxical then, that equicardinality itself, or moreprecisely the logic built on the equicardinality quantifier, represents a par-ticularly weak degree of logicality.

The moral here seems to be that logicality is a difficult concept, andentangled not just with cardinality but with other concepts.

5.2 Vopenka’s Principle

While it is not true that every model class is definable in L∞∞, there is aweaker result which depends on large cardinals. Tarski proved that a modelclass K is definable by a universal first order sentence if and only if K isclosed under substructures and any model all of whose finite substructuresare in K, is itself in K. The following result removes the assumption onfinite substructures but lifts the result to an infinitary level. Recall thatVopenka’s principle is the axiom schema stating that if {Aα : α ∈ On}is a (definable) proper class of structures of the same vocabulary, there areα 6= β such that Aα can be embedded into Aβ. Despite its formulation, whichsuggests no connection to large cardinals, this principle is actually a largecardinal axiom schema. It implies the existence of extendible (and hencesupercompact) cardinals [32], and its consistency follows from the existenceof an almost huge cardinal (see e.g. [24, p. 338]). Magidor has proved thefollowing characterisation of Vopenka’s Principle29:

Theorem 13 ([33]). The following are equivalent:

1. Every model class which is closed under substructures is L∞∞-definableby a universal sentence.

2. Vopenka’s Principle.

Proof. (2) implies (1): Suppose K is a model class which is closed undersubstructures. By Vopenka’s Principle there is a K-supercompact cardinal κ(see [46]). For any M of cardinality ≤ κ, let θM ∈ Lκ+κ+ be an existential

29We present the proof with Professor Magidor’s kind permission.

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sentence such that θM is true exactly in models which have an isomorphiccopy of M as a substructure. We show that for all N , N /∈ K if and only ifN |= ψ, where ψ is

∨{θM : M ⊆ κ,M /∈ K}. Suppose first N /∈ K. By the

K-supercompactness of κ there is an Lωω(QK)-elementary substructure Mof N of cardinality ≤ κ. Now N /∈ K implies M /∈ K. Trivially, N |= θM.Hence N |= ψ. Conversely, suppose N |= θM for some M /∈ K such thatM ⊆ κ. W.l.o.g. M⊆ N . By closure under substructures, N /∈ K.

(1) implies (2): Suppose Vopenka’s Principles fails as witnessed by theclass C = {Mα : α ∈ On}. Let K be the class of structures N of the samevocabulary as eachMα such that N is isomorphic to an substructure ofMα

for arbitrarily large α. By (1) we may choose κ such that K is definable inLκ+κ+ . By our choice of C, every Mα satisfies Mα /∈ K. By [16, Theorem3.4.1] there is for each α an Lκ+κ+-elementary substructure Nα of Mα ofcardinality ≤ 2κ such that Nα /∈ K. There are only a set of non-isomorphicstructures of cardinality ≤ 2κ. Hence Nα ∼= Nβ for a proper class of α andβ. Let α be one of those. Then Nα ∈ K, a contradiction.

The moral of the story here is that we can obtain the McGee style char-acterization of logicality independently of the cardinality of the model if werestrict to model classes closed under substructures, but we have to makea strong set theoretical assumption, namely Vopenka’s Principle. This isnot to suggest that we propose closure under substructures as a criterion forlogicality.

6 Defining an arbitrary model class in an ab-

solute logic, but still cardinal dependently

We shall now offer a refinement of McGee’s result (Theorem 4), by replacingL∞∞ by a substantially weaker infinitary logic30. Let us first note that wecannot replace L∞∞ by L∞ω because the quantifier Q1 is not CD-definable

30McGee alludes to such a possibility on Page 574 of [37].

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in the latter.31 For another example, the well-ordering quantifier

M |= Wxyφ(x, y,~a) ⇐⇒ {(c, d) :M |= φ(c, d,~a)} well-orders M

is definable in Lω1ω1 (see (9)) but not in L∞ω [31].An important quality of L∞ω is its absoluteness. Let us recall:

Definition 14 ([6]). A logic (Σ, T ) is called an absolute logic if the predicateΣ(φ) is Σ1 in φ, and the predicate T (φ,M) is ∆1 in φ and M. A logic isabsolute with respect to a set theory S if the predicates Σ and T are ΣS

1 and∆S

1 , respectively.

For example, L∞ω and L(W ) are absolute logics but L∞∞ is not.We will now define the ∆-extension of a logic. Suppose K is a class of

models of vocabulary L and L′ ⊆ L. We use K � L′ to denote the class ofreducts M � L′ of models M∈ K. We call K � L′ a projection of K.

Definition 15. A model class is Σ(L∗)-definable, or in Σ(L∗), if it is aprojection of an L∗-definable model class. A model class is ∆-definable inL∗, or in ∆(L∗), if both K and the complement of K are Σ(L∗).

As argued for in [35], ∆(L∗) can be considered a logic in itself. There isa many-sorted version and a single-sorted version of the ∆-extension. In themany-sorted version the model class K may have a many-sorted vocabulary.If L∗ is the logic L∞ω, or Lκ+ω, for regular κ, there is no difference betweenthe two versions i.e. they coincide [40]. Therefore we continue with thesingle-sorted version.

Some of the nice properties of the ∆-operation are:

• Exactly the same classes of cardinals are spectra of ∆(L∗) as are of L∗.

• ∆(L∗) satisfies LS(C,D) iff L∗ does.

• ∆(L∗) satisfies the Compactness Theorem iff L∗ does.

• ∆(L∗) satisfies the (abstract) Completeness Theorem32 iff L∗ does.

31SupposeM is a model of cardinality ℵ1 with one unary predicate P and P is countable.Suppose N is another model of cardinality ℵ1 for the same vocabulary but now both P andits complement are uncountable. ClearlyM and N are partially (potentially) isomorphicand hence satisfy the same L∞ω-sentences. But Q1xP (x) is true in one but not in theother.

32I.e. the set of Godel numbers of valid sentences is r.e. This is only meaningful for L∗

where formulas are finite objects so that Godel numbering makes sense.

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• If L∗ satisfies the Craig Interpolation Theorem, then ∆(L∗) and L∗ areequivalent logics.

• `(∆(L∗)) = `(L∗).

• h(∆(L∗)) = h(L∗)33.

If L∗ is an absolute logic in the sense of [6], then ∆(L∗) is still absolute inthe weaker sense that every definable model class is ∆1-definable in set theory(with the defining sentence as a parameter). This is a direct consequence ofthe definition of ∆(L∗). However, ∆(L∗) has a complication which preventsus from concluding that the ∆-operation preserves absoluteness. Namely,there may be an absolute logic L∗ and model classes K0 and K1 such thatboth K0 and K1 are in Σ(L∗), K0 ∩K1 = ∅ but the proposition that everymodel (of the right type) is in K0 or K1 is not absolute34. Thus we may nothave a Σ1-definition for the set of sentences of ∆(L∗) even if we have for L∗.Despite this shortcoming of the ∆-operation, we do have a Σ1-definition forthe set of sentences of Σ(L∗) when L∗ is absolute because with Σ(L∗) theabove problem (K0 and K1) does not arise.

Theorem 16 ([28]). If K is a class of models of the same vocabulary, thenthe following conditions are equivalent:

1. K is closed under isomorphisms.

2. K is CD-definable in ∆(L∞ω).

Proof. The proof is similar to the proof of Theorem 4 in [37]. Recall theformulas η′α and η′α defined in (2) and (4), with the properties (3) and (5).Suppose A is a model of the vocabulary L and |A| = λ. Let fA : λ → A bea bijection and a <A b ⇐⇒ f−1A (a) < f−1A (b). For α, β < λ let

ρα,β(x, y) =

{R(x, y) if A |= R(fA(α), fB(β))¬R(x, y) if A 6|= R(fA(α), fB(β))

33This may fail in the many-sorted version [50].34For example, we may take K0 to be the class of trees of height and size ω1 with an

uncountable branch, and K1 to be the class of trees of height and size ω1 with a strict orderpreserving mapping into the rational numbers. It is a consequence of Martin’s Axiom thatevery such tree is in K0 ∪K1. However, if V = L, then there are Souslin trees which arenot in K0 ∪K1.

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LetΦA ↔ ∀x∀y

∧α,β<λ

((ηα(x) ∧ ηβ(y))→ ρα,β(x, y)).

Now (A, <A) |= η′λ and (A, <A) |= ηα(a) ⇐⇒ a = fA(α). Hence

(A, <) |= η′λ ∧ ΦA.

On the other hand,

(A′, <′) |= η′λ ∧ ΦA ⇒ A′ ∼= A,

for if (A′, <′) |= η′λ ∧ ΦA and g : (A′, <′) ∼= (A,<A), then g : A′ ∼= A. Let

ΘK ↔∨{ΦA : A ∈ K, A = λ}.

Now by the above,

A ∈ Kλ ⇐⇒ (A, <) |= η′λ ∧ΘK for some <⇐⇒ (A, <) |= η′λ → ΘK for all <.

Since η′λ,ΘK ∈ L(2λ)+ω, we are done.

Why is this an improvement? ∆(L∞ω) is a sublogic of L∞∞, a conse-quence of the result of Malitz [36] to the effect that for regular κ a validimplication in Lκω can be interpolated in L2(<κ)

+κ . It is a proper sublogicbecause well-ordering is definable in the latter but not in the former. Sec-ondly, the logic L∞ω is an absolute logic [6] and ∆(L∞ω) inherits much ofthe absoluteness of L∞ω (see above) while L∞∞ is badly non-absolute.

The improvement that Theorem 16 represents over Theorem 4 is alsobased on the fact that L∞ω, and thereby also ∆(L∞ω), is a much “tamer”logic than L∞∞. In particular:

• ∆(L∞ω), even Σ(L∞ω), has a strong Lowenheim-Skolem theorem. Thisis in contrast to L∞∞, which does not have a Lowenheim-Skolem The-orem in the same strong form (see Theorem 7).

• The class of well-orderings is not definable in ∆(L∞ω), while it is de-finable in Lω1ω1 (see (9) below). This is a kind of threshold difference.In general, logics in which the concept of well-order is not definable aremuch better behaved (in many different respects) than those in which

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it is. If well-order is definable, the logic can talk about well-foundedmodels of set theory, and thereby about transitive models of set theory.In this way the logic can break through the object theory/metatheorybarrier and have access to the background set theory. For a concreteexample, consider a sentence φ which is the conjunction of a sufficientlylarge finite part T0 of the ZFC axioms, the first order set-theoreticalstatement φ0 that there are no inaccessible cardinals, and the Lω1ω1-sentence

∀x0∀x1 . . .∨n<ω

¬xn+1 ∈ xn. (9)

Suppose κ is the smallest inaccessible cardinal. Then φ has a model ofcardinality κ, namely Vκ, but none of bigger cardinality. For supposeM is a model of φ of cardinality > κ. By Mostowski’s CollapsingLemma [39] we may assume M is a transitive model of T0. Since thecardinality of M is bigger than κ, the ordinal κ is in M and is thereforeinaccessible in M by the downward persistency of inaccessibility. Butthis contradicts the fact that M is a model of φ0. This argument dueto Silver [45] demonstrates the power of (9) to penetrate the objecttheory/metatheory barrier, which sentences of ∆(L∞ω) are unable todo [31].

• The Hanf number of ∆(Lκ+ω) is only moderately large, namely <i(2κ)+ , while the Hanf number of Lω1ω1 is bigger than the first weaklyinaccessible cardinal35 (and consistently bigger than the first measur-able cardinal). The smallness of the Hanf number is another indicationof regularity of patterns in spectra (see (6)).

A weakness of ∆(L∞ω) as compared to L∞∞ is that the former does nothave as explicit a syntax as the latter, although we can overcome this byreplacing ∆(L∞ω) by Σ(L∞ω). A strength is that its spectra are as regularas those of L∞ω, it is equally completely axiomatizable as L∞ω, and it isabsolute in the sense that its definable model classes are absolute in settheory. Thus it avoids Feferman’s second and third criticisms.

35See argument in the previous bullet.

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7 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 cardinaldependently? We know already L∞∞ is not that kind of a logic. This isbecause the isomorphism type of every individual structure is definable inL∞∞, but by the results in Section 2.1 there are model classes that arenot definable in L∞∞. McGee points out that we could take a class sizedisjunction of L∞∞ sentences and obtain a single ‘sentence’ which works inall cardinalities. As he points out, if we accept class size formulas we shouldalso accept class size logical operations and then we are back in the startingpoint: to account for class size logical operations we need classes of classesand we start climbing up the type hierarchy beyond first order set theory. Ifwe stick to set size logical operations we can operate within first order settheory.

We could, of course, take every model class as a generalized quantifier36

but there is a more canonical logic for this task.In [51] a logic Ls called sort logic was introduced. It is a kind of many-

sorted version of second order logic L2, which allows quantification not onlyover subsets and relations on the domain of the model but also over subsetsand relations on new domains outside the current one. Such quantificationhappens when we ask of a given group, whether it is the multiplicative groupof a field? We have to “guess” the addition of the field but also the neutralelement 0 and those are both outside the multiplicative group.

More exactly, sort logic arises from L2 by repeated applications of oper-ation L∗ 7→ Σ(L∗) (see Definition 15) and negation. Let us define ∆0 = L2.If ∆n has been defined, let ∆n+1 consist of model classes K such that bothK and the complement of K are Σ(∆n(L2))-definable. We get an increasinghierarchy of logics

∆1 ≤ ∆2 ≤ ∆3 ≤ . . .

the union of which is sort logic. A fine point is that every level ∆n of sortlogic is definable in set theory but there is no single formula that defines theentire sort logic.

Theorem 17 ([51]). If K is a class of models of the same finite vocabulary,definable in set theory without parameters,37 then the following conditions

36In the sense of section 2 and of [29].37We may allow parameters if we extend sort logic so that it includes L∞ω.

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are equivalent:

1. K is closed under isomorphisms.

2. K is definable in sort logic.

The above characterization of isomorphism closure is not as elegant asMcGee’s Theorem 4 simply because the definition of sort logic is more com-plicated than that of L∞∞. However, this theorem has the remarkable ad-vantage over Theorem 4 that the definition of the model class in sort logicholds for models of all cardinalities. As McGee points out, if he wanted toobtain the same level of generality with his method, he would have to takea disjunction of a proper class of L∞∞-formulas. Sort logic may be compli-cated but at least its formulas are sets and not proper classes. In McGee’scase we may need a formula which is not even a set. In the case of sort logiceach formula is a set, but the whole logic is not definable, only each level ∆n

separately is.

8 Is having a Completeness Theorem a marker

of logicality?

We saw that first order logic and L(Q0) are maximally logical under Sagi’scriterion, grading the logicality of these logics by their Lowenheim numbers;we also saw that the degree of logicality of the logics L(Qα) decreases as αincreases: if α ≤ β, then Qβ is less logical than Qα.

As for other kinds of quantifiers, the quantifiers “more” or Rescher quan-tifier

Jxyφ(x)ψ(y) ⇐⇒ there are at least as many x satisfying φ(x)as there are y satisfying ψ(y)

and the equicardinality or Hartig quantifier (see Section 5.1) both have thesame (very high) Lowenheim number, and are thus less logical than Qα atleast for α below the first α such that α = ℵα.38 We suggested earlier

38If `I is the Lowenheim number of the Hartig quantifier, then `I is always bigger thanthe first fixed point of the ℵ-hierarchy [19]. If V = L, then `I is bigger than the firstinaccessible (if any exist) [49]. If V = Lµ, then `I is bigger than the first measurablecardinal [48]. If Con(“there is a super compact cardinal”), then Con(`I < the first weaklyinaccessible) [34]. If Con(ZFC), then Con(`I < 2ω) [49].

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that the Hartig quantifier is a singular case, in seeming to express the coreconcept of isomorphism invariance, which is defined in terms of the conceptof equicardinality. Under our criteria, the Hartig quantifier is classified as(only) weakly logical even so, not only due to its high Lowenheim numberbut also because it lacks very seriously a complete axiomatization [49].

We remarked earlier that Sagi’s demarcation of logicality for a fixed classof logical constants, which is based on Lowenheim numbers, partitions logicalspace differently than ours. As we observed, logics that have a completenesstheorem do not seem to be tied to the ℵ-hierarchy in any obvious way, inthat some of these logics are axiomatisable and some are not. For example,Keisler’s axioms

Axiom 0 Axiom schemes for LωωAxiom 1 ¬Qx(x = y ∨ x = z)Axiom 2 ∀x(φ→ ψ)→ (Qxφ→ Qxψ)Axiom 3 Qxφ(x)→ Qyφ(y)Axiom 4 Qy∃xφ→ (∃xQyφ ∨Qx∃yφ)

are complete for L(Q1), and if GCH is assumed, then they are completefor all ℵα+1 for which ℵα is regular (hence for ℵn for all n > 0). Whereaswhile L(Q0) satisfies the Keisler Axioms, these (or any other recursive set ofaxioms) are not a complete axiomatisation of L(Q0).

39

The suggestion here is that having a complete axiomatisation should bea marker of logicality on the simple ground that logics of this kind resemblefirst order logic in one of its essential, if not most essential property, namelycompleteness. Except for Q0, one would then classify “there are very (i.e.uncountably) many” as a logical constant, as dictated by Keisler’s axioms;in particular, the quantifier Q1 would be graded as having a higher degree oflogicality than Q0, as L(Q1) is complete with respect to the Keisler Axiomswhereas L(Q0) is not complete in this respect. In short, our criterion oflogicality, which turns on the degree to which a logic resembles first orderlogic in its model theoretic properties, comes apart from Sagi’s already atthe level of the logics L(Q0) and L(Q1).

A possible objection might come from the fact that Keisler’s axioms aresatisfied by many different quantifiers, in fact by every Qα, where ℵα is reg-ular. On the other hand, we may consider Q1 logical admitting that fromthe point of view of logicality it cannot be separated from some other similar

39See [38].

29

quantifiers. What is logical, according to this view, is not so much the cardi-nality ℵ1, as, simply, uncountable cardinality—or “very many”—in general.While the failure of Q0 to permit a recursive axiomatization is an indicationthat there is something mathematical, as against logical, in Q0. The permu-tation invariance characterization of logicality does not differentiate betweenQ1 and Q0, as the Completeness Theorem criterion does. By the invariancecriterion every Qα is logical, which may seem unintuitive. By further apply-ing the Completeness Theorem criterion we can make finer distinctions andsee a difference: Q1 is “more logical” than Q0.

For singular strong limit ℵα Keisler [27] proved a Completeness Theorem,but with different axioms (no simple set of axioms is currently known). Thusthere are (at least) two different “logical” concepts of “very many,” one for ℵαthe successor of regular and one for the singular case. (Successor of singularis open.) The two concepts have different logical content: Keisler’s Axiomsfor Qα are valid for all infinite α, and in some cardinalities (successor ofregular) they have a Completeness theorem, modulo the GCH, as was notedabove. In some other cardinalities (singular) new axioms have to be addedin order to get a completeness theorem. In the base case α = 0 no effectivelygiven additional axioms can be added to give a completeness theorem.

According to Sagi, “We should distinguish between a logic L(Q) used tomeasure the logicality of Q and the logic we ultimately use for validity andlogical consequence.”40 But if metaphysical involvement is inversely relatedto logicality, then having a completeness theorem should possibly be consid-ered as a marker of logicality also under the Sagi criterion. This is becausewhat the completeness theorem precisely does is to enable the conversionof semantic content, which is metaphysically involved (from the Sagi pointof view, presumably), into syntactic content which, presumably, is not. Inthis connection, the following caveat is important: As Carnap observed [15]the rules do not fix the interpretation of the logical constants, they havenon-standard interpretations. In their [10] Bonnay and Westerstahl presenta way to sidestep the problem:

Our take on Carnap’s Problem is that it is made artificially dif-ficult by considering all possible interpretations, no matter howbizarre. As speakers, we know that our language is going to becompositional, that it will have some true and some false sen-tences, and that its logical constituents will be topic-neutral.

40ibid, p. 22.

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Therefore attention may be restricted to interpretations whichsatisfy these principles. Following Church’s advice, this amountsto explicitly factoring out the role of semantic principles and therole of inference rules in fixing the interpretation of logical con-stants, rather than covertly using semantic notions to make senseof extended inference rules. This strategy proves successful bothfor propositional connectives and for quantifiers.

There is also the issue of expressive power, with respect to which the logicsbuilt on the quantifiers “most”, “more” and the Hartig quantifier differ. Incertain models with an equivalence relation the Hartig quantifier can be seento be eliminable while the Rescher quantifier is not [20]. Thus the Rescherquantifier is strictly stronger than the Hartig quantifier from the point ofview of expressive power. The point here is that expressive power should beinversely related to logicality.

Bonnay also ties logicality to syntax via a completeness theorem, as isspelled out in some detail in [9]. Here Bonnay proposes a modification of theprogram of Carnap’s Aufbau [14], calling for logical expressions to be definedsyntactically. It is a feature of his treatment that the absoluteness of a logicplays a crucial role in guaranteeing the robustness of the syntactic definition.Thus L(Q0) is an absolute logic because it has a recursive syntax, just likefirst order logic, and its semantics is absolute in transitive models of (evenweak) set theory.41 On the other hand, L(Q1) is not absolute, even thoughit has a recursive syntax.42

Burgess’s [11] forges a link between absoluteness and the idea of having aproof procedure. He exhibits a quasi-constructive complete proof procedureinvolving rules with ℵ1 premisses for the hereditary countable part of anyabsolute logic. The proof procedure is based on a judicious use of the so-calledVaught formulas (~x, ~y is short for x0, y0, . . . , xn, yn, ~k is short for k1, . . . , knand ~l is short for l1, . . . , ln)

∀x0∨

k0∈K0

∧l0∈L0

∃y0∀x1∨

k1∈K1

∧l1∈L1

∃y1 . . .∧n<ω

φ~l~k(~x, ~y)

41This is essentially because finiteness is absolute, i.e. ∆1-definable. See Barwise “Ad-missible sets and Structures” [5], p. 38.

42This is due to the well-known fact that countability is not absolute: a set can beuncountable in a model of set theory and countable in a transitive extension. On therelevance of absoluteness in this context see also Bonnay’s [8].

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and their approximations (see [52])

Φ~l0,~k

(~x, ~y) =∧m≤n

φl0,...,lmk0,...,km(x0, y0, . . . , xm, ym)

Φ~lα+1,~k

(~x, ~y) = ∀xn+1

∨kn+1∈Kn+1

∧ln+1∈Ln+1

∃yn+1Φ~l,ln+1

α,~k,kn+1(~x, ~y, xn+1, yn+1)

Φ~lν,~k

(~x, ~y) =∧α<ν

Φ~lα,~k

(~x, ~y),

where α is an ordinal, ν is a limit ordinal, the formulas φ~l~k(~x, ~y) are from L∞ω,

and Kn, Ln, n < ω, are sets. Note that the approximations are L∞ω-formulas.Burgess first finds a representation of sentences of a given absolute logic inthe form of Vaught sentences. Every Vaught sentence is equivalent to theproper class size “conjunction” of its approximations. If we restrict ourselvesto the hereditary countable sentences of the absolute logic, approximationsin Lω1ω will suffice. Then we can appeal to the Completeness Theorem forLω1ω given in [43].

In sum, if one classifies the first order existential quantifier “there is atleast one,” as inherently logical, and if as such this quantifier is thought ofas having minimal or no semantic content, then is there a principled wayto determine when and how higher quantification acquires semantic content,e.g. at what level in the cumulative hierarchy? Sagi’s answer is that logicalitydiminishes the higher up we are in cumulative hierarchy; while we suggestthat logicality kicks in arbitrarily high up, e.g. for all ℵα+1 for which ℵα isregular, with the ℵ0 case an anomaly, again because of completeness. Theintuition here is that logics which have a completeness theorem are close tofirst order logic in this special sense. This is because completeness enables theconversion of semantic consequence into syntactic consequence via the twoKeisler axiomatisations—albeit conditioned on the continuum hypothesis incertain important cases.43

9 Conclusions

It has been generally acknowledged that the Tarski-Sher criterion for logical-ity is necessary but not sufficient. McGee’s Theorem translates the criterioninto definability in a logic, obtaining cardinal dependent definability in L∞∞.

43See [28] for a similar treatment of the relation between logicality and completeness.

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Cardinal dependency leads to criticism of logicality across domains (Fefer-man’s third critique). The nature of L∞∞ leads to criticism of entanglementwith mathematics (Feferman’s first critique) and of non-absoluteness (Fefer-man’s second critique). Maintaining cardinal dependency we lowered L∞∞to ∆(L∞ω) which is more palatable from the point of view of model theoreticproperties. A stronger degree of logicality is obtained by abandoning cardi-nal dependency and investigating logics in which a candidate for logicalityis definable. Since every class of models which is closed under isomorphismsis definable in some logic, this seems reasonable. Following and expand-ing on Sagi’s suggestion to delineate degrees of logicality according to theirLowenheim numbers, we delineate logicality according to their Lowenheimnumbers but also according to a wider spectrum of model theoretic proper-ties of logics, such as Completeness Theorems and absoluteness properties,together with Lowenheim-Skolem properties, all considered from a logicalitypoint of view.

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Criteria of logicality: weak to strong Sources

Closed under isomorphisms Tarski [47], Sher [44]CD-definable in:L∞∞ McGee [37]∆(L∞ω) Vaananen, see [28]L∞ω Barwise/Karp [6, 26]

Definable in:Some logic Lindstrom [29]L(Q) for some Q Lindstrom [29]Sort logic Vaananen [51]L∞∞ for downward closed Magidor [33]First order logic Feferman [18]

Desirable properties:LS-theorem Sagi [42]Absoluteness Barwise [6]Axiomatizable Quine [41]Closed under potential isomorphisms Bonnay [8]Closed under homomorphisms Feferman [18]

Table 1: Criteria of logicality

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