Abstraction Principles and the Classification of Second-Order Equivalence Relations
aa r X i v : . [ m a t h . L O ] M a r Abstraction Principles and the Classification ofSecond-Order Equivalence Relations
Sean C. Ebels-DugganVersion submitted July 23, 2016.Please see forthcoming article in
Notre Dame Journal of Formal Logic for final version
Abstract
This paper improves two existing theorems of interest to neo-logicistphilosophers of mathematics. The first is a classification theorem due toFine for equivalence relations between concepts definable in a well-behavedsecond-order logic. The improved theorem states that if an equivalencerelation E is defined without non-logical vocabulary, then the bicardinalslice of any equivalence class—those equinumerous elements of the equiv-alence class with equinumerous complements—can have one of only threeprofiles. The improvements to Fine’s theorem allow for an analysis ofthe well-behaved models had by an abstraction principle, and this in turnleads to an improvement of Walsh and Ebels-Duggan’s relative categoric-ity theorem. Neo-logicist philosophers of mathematics are impressed by the fact that some abstraction principles can interpret interesting fragments of mathematics. Theseprinciples are sentences of an enriched second-order language of the followingform: for E an equivalence relation between second-order objects, and ∂ a func-tor taking second-order objects to first order objects, the abstraction principle A E [ ∂ ] is the sentence ( ∀ X, Y )( ∂X = ∂Y ↔ E ( X, Y )) (1)In virtue of their form, say neo-logicists, abstraction principles are eligible to be“analytic truths”—meaning that they are epistemically near enough to logicaltruths. And if abstraction principles count, in some sense, as near enough tological, then so should any mathematics they interpret. See most notably [26] and [10].
1o goes the argument, and not without objections and replies. Much turnson what would count as a “(near enough to) logical” abstraction principle. Buta plausible minimal requirement is this: an abstraction principle’s equivalencerelation must itself be “logical”, and so cannot require identification of particularobjects, concepts, or relations for its definition. The thinking is that logicis indifferent to particulars, whether objects, concepts (monadic second-orderobjects), or relations (polyadic second-order objects). So for an abstractionprinciple to be logical, it is necessary that its equivalence relation be accordinglyindifferent. A natural way to flesh out this notion of “indifference” is with thenotion of permutation invariance : the status of the relation doesn’t vary, nomatter how one exchanges the objects of its concern. Further complicating the neo-logicist’s hopes is the fact that not all ab-straction principles—even those based on permutation invariant equivalencerelations— should count as logical: so not only does said neo-logicist need anaccount of how an abstraction principle could be logical, but that account mustalso sort the good principles from the bad. So more needs to be said for whatkinds of logical equivalence relations there are on concepts, and which are aptto yield suitably logical abstraction principles. It is thus of interest to classifylogical equivalence relations on concepts.This paper proves a classification theorem for such equivalence relations.Though discovered independently, it happens that the classification theoremhere presented is a stronger version one already on the books. Kit Fine’s Theo-rem 4 of [7, p. 142] classifies infinite concepts in standard models. The theoremis then used to determine the finest abstraction principle satisfiable on all infinitestandard models [7, Theorem 6, p. 144].But Fine’s theorem in its given form hides its true power; hence our newpresentation. In our version the result is put in a deductive setting in whichcardinalities are well-behaved. This is relatively minor, but it allows us to applythe theorem to problems cast in just such deductive settings (about which morein a moment). More importantly, our version of the theorem sorts equivalencerelations, and their abstraction principles, more usefully. In other words: thereare abstraction principles well-discussed in the literature on neologicism, butas Fine states the theorem in [7], one needs to squint to see how their equiva-lence relations are classified. The version here given allows for more clear-eyedrecognition of this sorting, and allows for a generalization of the classificationto (Dedekind) finite concepts as well.It is something of a journey from this restatement and expansion of Fine’stheorem to its payoff, but the midpoint is a worthy stop. The improved theoremenables an analysis of when a given abstraction principle has a well-behaved We put aside concerns, like those of Quine in [18], that second-order languages are them-selves not “logical” in virtue of their quantifying over higher-order objects. This view is associated with Tarski’s identification of logical notions as those that arepermutation invariant in this sense (see [22]), for a more recent defense see the work ofGila Sher (see [21]). Other relevant discussions can be found in [2] and [16]. The so-calledTarski-Sher thesis identifying logicality with permutation invariance is controversial; thatpermutation invariance is a necessary feature of logical notions is not (see [15]). bicardinally equivalent ones. This at leastgives the neo-logicist a tool for analyzing the space of abstraction principles.The terminal payoff comes in the form of a more direct result: the restatedand expanded theorem allows an improvement of the relative categoricity theo-rem given by Walsh and Ebels-Duggan in [24]. Walsh and Ebels-Duggan showedthat abstraction principles are naturally relatively categorical when and onlywhen their equivalence relation is coarser than that of the neo-logicist’s favoredabstraction principle, HP . Using the improved version of Fine’s theorem, we showthat this result obtains even when we remove the qualifier “naturally”. This im-provement allows for a plausible case to be made that HP and its ilk pass aminimal threshold for logicality, since (unqualified) relative categoricity is ar-guably the correct notion of permutation invariance for abstraction principles.The remainder of this paper is organized as follows. Sections 2–4 are in-troductory, motivating the project and setting in place the formal machineryfor the theorems. Section 2 explains why Fine’s theorem could use improve-ment, and gives an informal characterization of the improved theorem, whichwe call the “Main Theorem”. Section 3 describes the second-order language inwhich we will work, and its structures; as well as listing our use of standardabbreviations and giving formal clarifications of the terms we have loosely de-fined above and in section 2. Here also we describe the cardinality assumptionswe adopt with slight modification from [24], explaining the sense in which ourbackground logic admits only “well-behaved cardinalities”. The next section 4explains why restricting our attention to bicardinally equivalent concepts is aptto our purposes.With preliminaries done, section 5 provides a rigorous statement and proofof the Main Theorem. The remaining sections discuss the relevance of theMain Theorem to the neo-logicist project. Our aim in these sections will not beto argue for or against a particular version of neo-logicism, but instead to letthe Main Theorem shed light on some technical questions of interest. Section 6will address the sorting of so-called “bad companions” by the Main Theorem;while section 7 proves and discusses the extension of the relative categoricitytheorem of [24]. For ease of exposition, we put off some of the more tedious orrepetitive proofs to appendices, we conclude with these. The restatement and expansion of Fine’s theorem is more than just a curiousexercise, for we believe there is a better statement of the theorem than theoriginal. We begin by giving Fine’s version.To state Fine’s theorem efficiently, we need a bit of notation which we willuse in the rest of the paper. Letting “ ∪ ”, “ ∩ ”, and “ − ” denote the usual booleanfunctions on sets of union, intersection, and difference, we will use “ X △ Y ” tomean the symmetric difference between X and Y , namely the set ( X − Y ) ∪ ( Y − ) . And we’ll denote the complement of X in a universe M by M − X . Finally,we’ll say that X and Y are bicardinally equivalent to mean that | X | = | Y | and | M − X | = | M − Y | ; that is, that X and Y are equinumerous (in a givenstructure M ) and that their complements are also equinumerous (also in thegiven structure). We’ll abbreviate bicardinal equivalence by writing “ X ⊟ Y ”.Note that ⊟ is (provably) an equivalence relation in second-order logic.Now, to Fine’s theorem: Let M be a standard model of second-order logic, X and Y be infinite concepts such that X ⊟ Y , and E be a permutation invariantequivalence relation. The concepts X and Y are representative just if M | = ( ∀ Z, W )( Z, W ⊟ X ∧ E ( X, Y ) → E ( Z, W )) (2) Fine’s Classification Theorem.
Given a standard model M , infinite concepts X and Y of M are representative if and only if they meet exactly one of thefollowing conditions.1. | X △ Y | = | X | < | M | . In Fine’s words, X and Y are “small but verydifferent”.2. ω ≤ | M − X | , | M − Y | < | X | = | M | and | ( M − X ) △ ( M − Y ) | = | ( M − X ) ∪ ( M − Y ) | . In Fine’s words this is that X and Y “are almost universalbut with infinite very different complements.”3. | X | = | M − X | = | M | = | M − Y | = | Y | , and | X △ Y | = | X △ ( M − Y ) | = | X | or vice versa. In Fine’s words, X and Y “are bifurcatory with one verydifferent from the other and from its complement.”This statement of Fine’s theorem is somewhat dizzying; Fine’s statement ismore succinct since he uses definitions we have spelled out. But the complexityof the theorem is not by itself a count against it. The reason it can be improved,however, is that its classes do not obviously organize the variations of equiva-lence relations at play in the investigation of abstraction principles. The abovestatement of Fine’s theorem also obscures a generalization of the theorem: infact there is a general version of Fine’s theorem that applies to finite, as wellas infinite, concepts. It is hard to see how this could be given the classificationFine offers.Restating Fine’s theorem will make it easier to use, and highlights the con-nections between the resultant classification and abstraction principles of par-ticular interest to neo-logicists. (Unfortunately, restating and expanding thetheorem to the finite case requires proving it anew. One would hope to relyon Fine’s proof, but in fact once the work is done for the extension, the initialtheorem is all but proved.)To improve on Fine’s classification theorem, we start by thinking about threetypes of abstraction principles and their similarities.We noted in the Introduction 1 that Frege’s logicist project was to show thatarithmetic is in some sense “really” logic by showing that arithmetic laws are infact logical laws. Frege executes his program in two steps: first by proving in4econd-order logic (without our cardinal assumptions) that BLV , ( ∀ X, Y )( εX = εY ↔ X = Y ) (3)implies HP : ( ∀ X, Y )( X = Y ↔ X ≈ Y ) (4)Here “ = ” indicates co-extensiveness of concepts, and “ ≈ ” indicates equinumer-ousity, the existence of a bijection between concepts. Note that in the notationof (1), BLV is A = and HP is A ≈ .Frege then shows that in second-order logic (with full comprehension), HP interprets (what is now called) second-order Peano Arithmetic. In fact, relativesof HP will do the same thing, including the bicardinality principle , BP , which is A ⊟ . So if BLV is logical, then so are HP and BP , and thus so is arithmetic.But BLV is no principle of logic, since it is quite famously inconsistent insecond-order logic, deriving the Russell paradox. Neo-logicism proceeds againstFrege’s objections in [8, §63ff], asserting arithmetic is logical from the secondpart of the program alone—that is, that as HP is plausibly logical, so then isarithmetic. At a minimum, abstraction principles must be consistent to count as logical.But that this minimum is not enough is the “bad company” problem: there areconsistent abstraction principles that are otherwise unacceptable, at least to theneo-logicist. The prototypical bad companion is NP : ( ∀ X, Y )( ηX = ηY ↔ | X △ Y | < ω ) (5)The principle NP is unwelcome to the neo-logicist because in contexts of well-behaved cardinalities, it implies that the universe is Dedekind finite. Another bad companion, discussed in [24, § 5.5], is the ComplementationPrinciple, CP , ( ∀ X, Y )( c (cid:13) X = c (cid:13) Y ↔ [( | X | = | Y | = | M − X | ∧ ( X = Y ∨ X = M − Y )) ∨ ( | X | 6 = | M − X | ∧ | Y | 6 = | M = Y | )]) This principle sorts only concepts the same size as their complements. Suchconcepts are grouped together with only their complements, and all other con-cepts are grouped in a “junk” equivalence class. As we will show below, thisprinciple counts as “bad company” as well (see section 6.3). Interpreting Peano Arithmetic does not require the full power of HP , but only that anabstraction principle match HP on all finite concepts. See [12], and [14] for further discussion. Wright’s [26] marks the staring point of neo-logicism, though the observation that Frege’sprogram proceeds in two parts was made in [17, p. 183, p. 194]. The first bad companions appeared in [3]; one of these was simplified into NP by Wrightin [27]. And this is unwelcome to neo-logicists because they claim an abstraction principleshouldn’t imply anything about sortal concepts unrelated to the abstracts; see [9, p. 415],[27, pp. 295–7] and [28, pp. 314–5], as well as [6]. HP and BP ,and three bad ones ( BLV , NP , and CP ). Can we sort these into useful classes?We can if, like Fine, we look at their behavior just on bicardinally equivalentconcepts. Clearly, when restricted to just bicardinally equivalent concepts, theequivalence relations of HP (equinumerousity) and BP are trivial : they includeall such concepts.Second, BLV and NP are, in a sense, of the same type: the equivalence rela-tions for both are concerned with the size of the symmetric difference betweenthe two related concepts. The principle BLV discriminates concepts if their sym-metric difference has non-zero size. Likewise NP discriminates if the symmetricdifference is Dedekind infinite. Say such equivalence relations that sort conceptsaccording to the size of their symmetric difference, are separations : two con-cepts with “few” objects falling under one but not both of the two concepts areequivalent; two concepts with “many” such objects are not equivalent.Lastly, we have CP . Such an equivalence relation is neither trivial, nor aseparation, but it has many of the drawbacks of separations. And such anequivalence relation can be generalized, as NP is a generalization of BLV : twobicardinally equivalent concepts can be grouped together if their symmetric dif-ference is “small”, or the complement of their symmetric difference is “small”. Solet us say that on bicardinally equivalent concepts, such relations are comple-mentations : they sort concepts by the size of their symmetric difference, or thesize of the complements of their symmetric difference.Finally, say that an equivalence relation E is a refinement of an equivalencerelation E ′ if whenever two concepts are E -equivalent, they are E ′ equivalent. We can now state the informal (though somewhat inexact) version of ourmain theorem: (Informal) Main Theorem.
Let E be a purely logical equivalence relation,and let our background logic be a strong but natural version of second-orderlogic. If we look only at E on bicardinally equivalent concepts, then E is eitherthe trivial equivalence relation, or it is a refinement of a separation, or it is arefinement of a complementation.This version of Fine’s result makes more obvious the relationship betweenits classification scheme and the kinds of abstraction principles that have beenseen in the neo-logicist’s laboratory. In restating the theorem we observe astriking alignment: the problematic abstraction principles arise from non-trivial equivalence relations. This is no accident; as we will see in section 6, abstractionprinciples involving non-trivial equivalence relations limit the size of their modelsin just the way that is typical of bad company. See [7] and [1] for discussion of “finer” and “coarser” equivalence relations and their rele-vance to neo-logicism. Cook’s [5] standardizes the terminology and indicates relations betweendifferent kinds of invariance. It was Cook’s paper that made me see the connection to Fine’swork. The language L , comprehension, and cardinal-ity assumptions Our background logic is a basic, though robust, version of second-order logic,adopted with slight weakening from §2 of [24]. In short, the language L useslower-case letters to vary over first-order objects, and upper-case letters to varyover second-order objects of all finite arities (arity will be clear from context inour presentation)—such objects of singular arity are called “concepts” or “sets”,others are called “relations”. All terms of L are variables; we exclude constantsof either type. Formulae of L and their interpretation are as usual.Thus models of this language are of the form M = ( M, S [ M ] , S [ M ] , . . . ) (6)where M is non-empty and the members of S i [ M ] are subsets of M i for i ≥ .We do not require S i [ M ] to be the full power-set of M , for compatibility ofour results with those of [24]; that is, we work in the non-standard semantics.Likewise we require our models to satisfy comprehension axioms for all formulaein their signature; these axioms are of the form ( ∃ X )( ∀ y )( Xy ↔ Φ( y, ¯ p, ¯ R )) (7)where ¯ p is a sequence of parameters from M , and ¯ R is a sequence of relationparameters from S i ∈ N S i [ M ] . This ensures that all finite concepts, and manymore besides, are included in S [ M ] . The remaining axioms of our logic, whichwe will call “second-order logic” are those of D2 found in [19, pp. 65–7], butexcluding the axiom of choice. We will soon be augmenting this logic withchoice principles to ensure well-behaved cardinalities. .Most symbols used are standard; we rehearse a few: The symbol ⊔ , whichappears in Proposition 8, means the disjoint union; we will at times abusenotation using it to mean the union of two disjoint sets. As usual we use “ (cid:22) ”to assert the existence of an injection, and “ ≈ ” the existence of a bijection. Wewill also use expressions like “ | X | ≤ | Y | ” and “ | X | = | Y | ”, and we will use theconvention of writing, e.g., X f (cid:22) Y to mean that f is an injection from X into Y ; likewise with the other expressions. As above the expression “ ⊟ ” indicatesthe relation of bicardinality ; the expression “ X f ⊟ Y ” means that f is a bijectionfrom M to itself and f ( X ) = Y and f ( M − X ) = M − Y . Both ≈ and ⊟ areprovably equivalence relations in second-order logic; we will say in the obviouscircumstances that concepts are cardinally, or bicardinally, equivalent.We identify concepts as infinite in M if they are Dedekind infinite, that is,that there is in M an injection from the concept into a proper subconcept ofitself, and we write this with the expression | X | ≥ ω . We say that X is finite in M if it is not Dedekind infinite, writing it | X | < ω . These expressions are usedexclusively within the model M . When we wish to say that a set X is finite orinfinite in the metatheory, we will either say so explicitly, or use “ | X | = n forsome n ∈ N ” or “ | X | < | N | ”. 7e will abuse notation and use “ M ” for both the first order domain of M and the universal concept { x | x = x } ; and we’ll use functional notation forrelations that are functional in M . In general we will write f ∈ M to meanthat for some i ∈ N , f is an i -ary function and f ∈ S i +1 [ M ] ; similarly withconcepts and relations. We write f ( X ) to mean the image of X under f ; sucha relation is always in M by comprehension.Though L is expressively rich, it is not too rich, for (crucially for our results),any function π : M → M that extends to permute objects of all types in M isan automorphism of M . This can be seen by the fact that, for a given M and π : M → M ′ , the “push model of M under π ” is always isomorphic to M , with π the witnessing isomorphism. If π is also such as to make M the same structureas its “push model”, then such a π is an automorphism. By comprehnsion, if π : M → M is a bijection in M , then π meets the needed criteria. Thus, wemay state the relevant result in the following form for our use: Permutation Invariance PI . If π : M → M ∈ M is bijection and Φ( x, X ) isan L -formula, then M | = ( ∀ x, X )(Φ( x, X ) ↔ Φ( π ( x ) , π ( X ))) (8)Under the same hypotheses about π , if E ( X, Y ) is an L -definable equiva-lence relation on concepts of M , then M | = E ( X, Y ) ↔ E ( π ( X ) , π ( Y )) (9)Here, of course, E being L -definable means definable without parameters:the formula defining E has only the second-order variables X and Y free. De-finability via an L -formula is the technical correlate of the informal “purelylogical” used in Section 1. Our main results depend on PI , in the sense thatthey obtain for any equivalence relation that is permutation invariant, not justthose L -definable.The second way our background logic is robust is that we require that therelation of equinumerousity behave as it does in more familiar contexts, e.g., ZFC . In particular, we require the following to obtain in all models M underconsideration: Cardinal Comparability CC . M | = ( ∀ X, Y )( X ≤ Y ∨ Y ≤ X ) (10) Infinite Sums are Maxima
ISM . M | = ( ∀ X, Y )( | X | , | Y | ≥ ω → | X ⊔ Y | = max( | X | , | Y | ) (11) Infinite Products are Maxima
IPM . M | = ( ∀ X, Y )( | X | , | Y | ≥ ω → | X × Y | = max( | X | , | Y | ) (12) See [4, pp. 225–31] for more on this method. ardinalities are Well-Founded CWF . For any L -formula Φ( X ) , M | = ( ∃ X )(Φ( X )) → ( ∃ X )( ∀ Y )(Φ( Y ) → X (cid:22) Y ) (13)We take over these principles nearly directly from [24]; both for their hand-iness and as we will apply the Main Theorem to offer a solution to some openquestions from that paper (see section 7). Thus, in what follows and unlessotherwise noted, all structures satisfy full comprehension and these cardinalityassumptions.The principles CC and ISM are ubiquitous in the proof of our first version ofthe Main Theorem, Theorem 6. The schematic principle
CWF is used in movingfrom Theorem 6 to the second formal version of the Main Theorem, Theorem 7.The principle
IPM is deployed mainly in the form of a pairing function in sec-tion 7.We will need specific notation suited to dealing with the divergence, in non-standard semantics, between Dedekind finitude and finitude in the metathe-ory. As this notation pertains only to the proof of the formal version of theMain Theorem, we introduce it in section 5.
Our Main Theorem says that an L -definable equivalence relation, when re-stricted to bicardinally equivalent concepts, can have one of only three profiles.But why is it interesting to look at this particular restriction of such equivalencerelations? This portion of our paper will motivate this restriction.Our interest in L -definable equivalence relations is that, provided thatsecond-order languages are a part of logic, these relations meet at least onecriterion of logicality: they are permutation invariant. But one interesting factabout second-order logic is that there are L -definable properties that can dis-tinguish second-order objects. To see what this means, consider the formula Singleton ( X ) : ( ∃ x )( ∀ y )( Xy ↔ y = x ) (14)The formula Singleton ( X ) can distinguish between second-order objects in suf-ficiently rich structures for second-order languages: if a structure includes dis-tinct first-order objects a and b , then in that structure Singleton ( { a } ) while ¬ Singleton ( { a, b } ) . These second-order objects are thus distinguished by theformula “ Singleton ( X ) ”. Thus, while first-order objects are indistinguishableusing only L -definable notions, second-order objects are not. It makes sense,then, that we would want to pay special attention to collections of second-order In [24] the principles CC , ISM , and
IPM were deployed as consequences of the principle GC ,though GC ’s well-ordering was in the main results only to show that the restrictions to smallconcepts and to abstracts were required (see [24, equations 4.22–4.23 and following]). In thispaper we will use only the cardinal consequences of GC listed above. Our logic is also weakerthan that deployed in [24] in that we make no use of the principle AC . not distinguishable in this way. And we need not look far forconcepts that are so indistinguishable. As can be see from PI , concepts arebicardinally equivalent if and only if they cannot be distinguished using an L -formula. It is thus of interest to see how purely defined equivalence relationsbehave on these concepts in particular.Relatedly, bicardinal equivalence is indicative of permutations. Let E bean equivalence relation on concepts. We will say that E is an indicator ofpermutations just if, for any second-order structure M , and any function f : M → M such that if X ∈ M then f ( X ) ∈ M , if ( ∀ X, Y )( E ( f ( X ) , f ( Y )) → E ( X, Y )) (15)in that structure, then f is a permutation (it is a bijection from the first-orderdomain M to itself). Theorem 1.
Bicardinality is an indicator of permutations.
Proof.
Observe that if f is not injective, then (15) fails: let x, y be such that f ( x ) = f ( y ) , and note that ¬ ( { x, y } ⊟ { f ( x ) } ) f ( { x, y } ) = { f ( x ) } ⊟ { f f ( x ) } .So it remains only to show that if f is injective and (15) holds in M , then f issurjective.Further, if | M | is finite in M , then every injection on M is a permutation.So we may further assume that M is not finite in M .Working in M , we show first thatif | X | = | M | then X − f ( X ) ⊟ f ( X − f ( X )) (16)That the concepts are equinumerous follows from the fact the f is injective. Tosee that their complements are equinumerous, we have that: | M | ≥ | M − ( X − f ( X )) | (17) = | ( M − X ) ∪ f ( X ) | ≥ | f ( X ) | = | M | (18)where the last equality follows again by the injectivity of f . By the Schröder-Bernstein theorem, | M − ( X − f ( X )) | = | M | . An identical argument showsthat | M − f ( X − f ( X )) | = | M | .Thus by (16) we have M − f ( M ) ⊟ f ( M − f ( M )) (19)Since ⊟ is preserved under complementation, f ( M ) ⊟ M − f ( M − f ( M )) (20)which implies that | M | = | f ( M ) | = | M − f ( M − f ( M )) | (21) Not just those with well-behaved cardinalities. Note this is provable in unaugmented second-order logic, see [19, Theorem 5.2, p. 102ff]).
10o by (16) together with (20) and (21) we obtain that f ( M ) ⊟ M − f ( M − f ( M )) ⊟ f ( M − f ( M − f ( M ))) (22)By assumption we have (15), and so M ⊟ M − f ( M − f ( M )) from (22). Comple-mentation then gives that ∅ ⊟ f ( M − f ( M )) , which means that M − f ( M ) = ∅ ,and so f ( M ) = M . Thus f is surjective, and so a permutation.Bicardinally equivalent concepts are thus interesting in their own right. Soit makes sense to observe the behavior of purely defined equivalence relationson just concepts that are ⊟ -equivalent.For this reason we introduce the following notation. As is customary, givenan equivalence relation E and a concept X we denote the E -equivalence classcontaining X by [ X ] E . Looking at how E behaves on the sets ⊟ -equivalentto X , we say that the bicardinal slice of E is the set of equivalence classes of ⊟ -equivalent sets. More formally, Definition 2.
Let M be a model and E an equivalence relation on sets of M .Given a set X in M , we let E ( ⊟ )( X, Y ) ⇔ X ⊟ Y ∧ E ( X, Y ) (23)so that [ X ] E ( ⊟ ) = [ X ] ⊟ ∩ [ X ] E (24)Thinking of E as a collection of concepts, we can then write E ( ⊟ ) = { [ X ] E ( ⊟ ) | X ∈ M} (25)We can then think of E as consisting in bicardinal slices of the form E ( ⊟ ) X = { [ Y ] E ( ⊟ ) | X ⊟ Y } (26)since E ( ⊟ ) = [ X ∈M E ( ⊟ ) X (27)These notions will be used in stating Theorem 7, the formal version of theMain Theorem. We are now nearly in a position to state our main results formally. Our theoremwill classify equivalence relations at bicardinal slices. We first must formallystate the classes into which any relation can be sorted.Theorem 7 is the formal version of the Main Theorem, which we prove intwo stages. The intermediate stop is Theorem 6, the remaining step is deployingthe cardinality assumption
CWF to obtain the final result.11roving these theorems is onerous because non-standard models allow thatconcepts finite according to M might not be finite in the metatheory. Thisdistinction is important because the proofs of our main results depend on Lem-mata 22 and 24, which deploy, in the metatheory, finitely many permutationsthat “move” one concept onto another while preserving equivalence classes.Thus, we will at times wish to indicate for given sets X and Y that finitelymany copies of Y are enough to cover X . Thus we write n × | Y | ≥ | X | to meanthat for some set Z with | Z | = n ∈ N , M | = | Z × Y | ≥ | X | . Similar expressionswill be used in an associated way so that their meaning is obvious.However, since finiteness in M is not the same as finiteness in the metathe-ory, we cannot count on expected relations of cardinality obtaining. In partic-ular, in M we have the expected Archimedean property in that ( ∀ X, Y )( | X | , | Y | < ω → ( ∃ Z )( | Z | < ω ∧ | Z × Y | ≥ | X | )) (28)is a theorem of second-order logic. But in a non-standard model it is not ingeneral true that for any X, Y ∈ S [ M ] with M | = | X | , | Y | < ω , there is an n ∈ N such that M | = n × | Y | ≥ | X | . For this reason we introduce the followingdefinition: Definition 3.
Given
X, Y ∈ S i [ M ] , we write | X | E | Y | to mean there is an n ∈ N such that M | = n × | Y | ≥ | X | (29)Further, we write | X | ⊳ | Y | to mean | X | E | Y | and ¬ ( | Y | E | X | ) . Note that E and ⊳ are relations in the metatheory ; they are not, in general, expressiblein L .In what follows, all concepts are understood to be concepts in given struc-tures M . In accordance with the two steps towards our main result, we givetwo definitions for the purposes of classification. Definition 4.
Let E be an L -definable equivalence relation over concepts ofa given structure M . For any concept X , we say that E ( ⊟ ) X
1. is trivial if [ X ] E ( ⊟ ) = [ X ] ⊟ .2. is separative , or refines a separation , if for all Y, Z ∈ [ X ] ⊟ , E ( Y, Z ) ⇒ | Y △ Z | ⊳ | X | (30)3. is complementative , or refines a complementation , if for all Y, Z ∈ [ X ] ⊟ , E ( Y, Z ) ⇒ ( | Y △ Z | ⊳ | X | or | M − ( Y △ Z ) | ⊳ | X | ) (31)Obviously if E ( ⊟ ) X is separative then it is complementative; therefore we saythat E ( ⊟ ) X is properly separative or refines a separation if it is separative andnon-trivial, and properly complementative or refines a complementation if it iscomplementative, non-separative, and non-trivial.12 emark 5. If X is properly complementative and Y ∈ [ X ] E ( ⊟ ) , then M − Y ∈ [ X ] E ( ⊟ ) .We now state the intermediate and final results as: Theorem 6.
Let E be an L -definable equivalence relation over a given struc-ture M . Then for any concept X , E ( ⊟ ) X is either trivial, separative, or com-plementative.If | M | > then exactly one of these options holds.If X is finite in M and E ( ⊟ ) X is nontrivial, then either M | = E ( Y, Z ) ⇔ | Y △ Z | ⊳ | X | (32)or M | = E ( Y, Z ) ⇔ [ | Y △ Z | ⊳ | X | or | M − ( Y △ Z ) | ⊳ | X | ] (33) Theorem 7 (Formal Main Theorem) . Let E be an L -definable equivalencerelation over a given structure M . Then for any concept X , E ( ⊟ ) X is eithertrivial, separative, or complementative.If | M | > then exactly one of these options holds.Further, if M | = | X | < ω then [ X ] E ( ⊟ ) is either [ X ] ⊟ , or { X } , or { X, M − X } .As we have said above, this is a strong version of Fine’s theorem—as onecan see with the aid of a few pages of Venn Diagrams. The proof of Theorems 6 and 7 will require, for a given X , the use of finitely(in the metatheory) many permutations, each of which fixes some member of [ X ] E ( ⊟ ) . These permutations will “shuttle” portions of a given set Z onto itsimage under a bijection f : Z → X .We will deploy the relations E and ⊳ in the next section to prove Lemma 22,which is crucial to proving Theorem 7 in its full generality. For this we will needto show that certain expected “arithmetic” relations hold. Though tedious tostate, we provide them as follows. 13 roposition 8. The following hold for all
X, Y, Z, W ∈ M : | X | E | Y | or | Y | E | X | (34) | X | E | Y | and | Z | ≤ | W | ⇒ | X ⊔ Z | E | Y ⊔ W | (35) ( | Z | = | W | and | X ⊔ Z | ⊳ | Y ⊔ W | ⇒ | X | ⊳ | Y | ) (36) | X | E | Y | E | Z | ⇒ | X | E | Z | (37) | X | E | Y ⊔ Z | ⇒ ( | X | E | Y | or | X | E | Z | ) (38) | X | = | Y ⊔ Z | ⇒ | X | E | Y | or | X | E | Z | (39) | X | ⊳ | Y ⊔ Z | ⇒ ( | X | ⊳ | Y | or | X | ⊳ | Z | ) (40) | X | ⊳ | X ⊔ Y | ⇒ | X | ⊳ | Y | (41) | Y ⊔ Z | E | X | ⇒ | Y | E | X | and | Z | E | X | (42) | Y ⊔ Z | ⊳ | X | ⇒ | Y | ⊳ | X | and | Z | ⊳ | X | (43) | X | ⊳ | Y | ⇒ | X | ⊳ | Y ⊔ Z | (44) | X ⊔ Y | E | Z | and | W | = | Y | ⇒ | X ⊔ W | E | Z | (45) | X | E | Y ⊔ Z | and | W | = | Y | ⇒ | X | E | W ⊔ Z | (46) | X ⊔ Y | ⊳ | Z | and | W | = | Y | ⇒ | X ⊔ W | ⊳ | Z | (47) | X | ⊳ | Y ⊔ Z | and | W | = | Y | ⇒ | X | ⊳ | W ⊔ Z | (48)Their proofs are even more tedious, so we relegate them to Appendix 9.The following two propositions relate infinity, in both its meta- and intra-theoretic senses, to the relation ⊳ , and are easy to prove using (34) and IPM . Proposition 9.
For all
X, Y ∈ M , | X | ⊳ | Y | ⇔ for all n ∈ ω, M | = n × | X | < | Y | (49) Proposition 10. If M | = | X | ≥ ω , then | Y | ⊳ | X | ⇔ M | = | Y | < | X | Our route to proving Theorem 7 is via Lemma 22, which is proved in the nextsection. This lemma specifies sufficient conditions for when E ( ⊟ ) X is trivial.The main way that we prove Lemma 22 is by exploiting the following conse-quence of PI : Proposition 11.
Let π ∈ M be a permutation of M such that π fixes X . Thenfor any Y ∈ [ X ] E , π ( Y ) ∈ [ X ] E .For by PI , Y ∈ [ X ] E ⇔ π ( Y ) ∈ [ π ( X )] E = [ X ] E , since π fixes X . Intuitively,the idea is that we can show sets X and Y to be E -equivalent if X is equivalentto some Z , and there is a permutation fixing X but sending Z to Y .We introduce the following notation to shorten our exposition.14 efinition 12. Let f : X → Y be a bijection with X and Y disjoint. The wesay that the permutation induced by f , ι ( f ) : M → M , is the function definedby ι ( f )( x ) = f ( x ) x ∈ Xf − ( x ) x ∈ Yx otherwiseNotice that if f ∈ M then ι ( f ) ∈ M is a well-defined bijection.Our goal in this subsection is to prove Lemmata 13, 14, 15, and 16, theproofs of which exploit Proposition 11. Lemmata 13 and 14 are those with themost involved proofs. In the remainder of this section, where context makessubscripts unnecessary we will use “ [ X ] ” to indicate [ X ] E . Lemma 13.
Suppose there is a Y = X with Y ∈ [ X ] and let Z ⊂ Y − X with M | = Z f (cid:22) X − Y . Suppose further that at least one of the following obtains: | Z | E | X ∩ Y | (50) | Z | E | M − ( X ∪ Y ) | (51)Then there is a permutation π ∈ M such that [ πX ] = [ X ] and Z ⊂ π ( X − Y ) ⊂ π ( X ) , and π ( x ) = x ⇔ x ∈ Z ∪ f ( Z ) (52) Proof.
The proof is essentially the same whether (50) or (51) obtains. We set W to be X ∩ Y if the former obtains, and M − ( X ∪ Y ) if not. We reason in M .As in the proposition, let Z f (cid:22) X − Y , and let Z g (cid:22) n × W . The function g induces for each j < n a partial function h j : W → Z defined by h j ( w ) = g − ( j, w ) . Note that each h j is injective from its domain to its range, which aredisjoint; so ι ( h j ) is well-defined for each j < n .We define for each j three permutations, as follows: p j = ι ( h j ) (53) q j = ι ( f ↾ rng( h j )) (54) r j ( x ) = q j ( p j ( x )) x ∈ dom( h j ) p j ( q j ( x )) x ∈ f (rng( h j )) x otherwise (55)So p j switches rng( h j ) and dom( h j )) , q j switches rng( h j ) and f (rng( h j )) , and r j switches f (rng( h j )) and dom( h j ) .Note that p j fixes Y , q j fixes p j ( X ) , and r j fixes q j ( p j ( Y )) ; thus threeapplications of Proposition 11 yield that for s j = p j ◦ q j ◦ r j , s j ( X ) , s j ( Y ) ∈ [ s j ( X )] = [ X ] (56)Note further that for w ∈ dom( h j ) , s j ( w ) = r j ( q j ( p j ( w ))) = q j ( p j ( q j ( p j ( w )))) = p j ( p j ( w )) = w (57)15here the third equality is due to the fact that q j ↾ dom( h j ) is the identityfunction. It follows from this and the construction of p j , q j , and r j that if s j ( x ) = x , then x ∈ rng( h j ) ∪ f (rng( h j )) . Conversely, if x ∈ rng( h j ) ⊂ Y − X ,then s j ( x ) ∈ X − Y , and if x ∈ f (rng( h j )) ⊂ X − Y , then s j ( x ) ∈ Y − X . Thusif x ∈ rng( h j ) ∪ f (rng( h j )) then s j ( x ) = x . So we have established s j ( x ) = x ⇔ x ∈ rng( h j ) ∪ f (rng( h j )) (58)Set π = s ◦ . . . ◦ s n − , and note that Z = S j Suppose there is a Y = X with Y ∈ [ X ] , and let Z ⊂ M − ( X ∪ Y ) with Z f (cid:22) X ∩ Y . Suppose further that one of the following obtains: | Z | E | X − Y | (70) | Z | E | Y − X | (71)Then there is a permutation π ∈ M such that [ πX ] = [ X ] , Z ⊂ π ( X ∩ Y ) ,and satisfying (52). 16s the idea for the proof of Lemma 14 is essentially the same as that ofLemma 13, we omit it.Now for two easy lemmata: Lemma 15. Suppose Z ⊂ M − ( X ∪ Y ) , Z f (cid:22) X − Y with Y ∈ [ X ] and Y = X .Then there is a permutation π ∈ M such that [ πX ] = [ X ] , Z ⊂ π ( X − Y ) ,satisfying (52). Lemma 16. Suppose Z ⊂ X − Y , Z f (cid:22) X ∩ Y with Y ∈ [ X ] and Y = X .Then there is a permutation π ∈ M such that [ πX ] = [ X ] , Z ⊂ π ( X ∩ Y ) , andsatisfying (52). Proof of Lemmata 15 and 16. For Proposition 15, note that π = ι ( f ↾ Z ) fixes Y , so [ X ] = [ π ( X )] by Proposition 11. The rest is obvious and routine.For Lemma 16, proceed similarly but set π = ι ( f ↾ Z ) . The results of the previous section showed what functions are needed in or-der to, speaking loosely, “transform one set into another”, while staying in thesame equivalence class. In this section we define properties that determine suf-ficient conditions for the application of those results. In particular, we establishLemma 22, which says that as long as these properties obtain for a set of a givencardinality, then all sets of that cardinality are E -equivalent.We begin, then, by defining these properties. Definition 17. We say that sets X, Y are almost complementary in M if | X | = | Y | and none of the following is satisfied in M : | Y − X | E | X ∩ Y | (72) | X − Y | E | X ∩ Y | (73) | X − Y | E | M − ( X ∪ Y ) | (74) | Y − X | E | M − ( X ∪ Y ) | (75)Similarly, we say that X, Y are symmetric in M if | X | = | Y | and for some Z ∈ M distinct from X and Y , | Z | = | X | and neither of the following is satisfiedfor M : | Z − ( X ∪ Y ) | E | X − Y | (76) | Z − ( X ∪ Y ) | E | Y − X | (77)In many cases, we will establish that either (76) or (77) holds in M byestablishing that one of the following holds for M : | M − ( X ∪ Y ) | E | X − Y | (78) | M − ( X ∪ Y ) | E | Y − X | (79)This is licensed by (37).The following two propositions follow by (47) and (48) and CC .17 roposition 18. Suppose π : M → M is a permutation. Then:1. X, Y are almost complementary if and only if π ( X ) , π ( Y ) are almost com-plementary.2. X, Y are symmetric if and only if π ( X ) , π ( Y ) are symmetric. Proposition 19. Distinct sets X and Y cannot be both almost complementaryand symmetric. Definition 20. In what follows, distinct, E -equivalent sets that satisfy oneof (72–75) and one of (76–77) will play a special rôle. Thus we say that sets X, Y are opportune if they are distinct, Y ∈ [ X ] , and they are neither almostcomplementary nor symmetric. We moreover will say that a single set X isopportune if there is a Y such that the pair X, Y is opportune. Definition 21. Two sets X, Y are relatively finite in M just if | X − Y | = | Y − X | in M . (Note that if X, Y are relatively finite, then X ⊟ Y .) We set RF ( X ) tobe the set of sets Y that are relatively finite in M to X . Lemma 22. If X, Y are opportune then RF ( X ) ⊆ [ X ] E ( ⊟ ) . That is, every setrelatively finite to X in M is E -equivalent to X . Proof. Let Z ∈ RF ( X ) with Z f ≈ X in M ; we may assume that f fixes Z ∩ X pointwise. We show that there is a permutation π of M such that π ( X ) = Z and [ π ( X )] = [ X ] .First we partition Z : Z = f − ( X − Y ) ∩ Y − X Z = f − ( X − Y ) − YZ = f − ( X ∩ Y ) ∩ Y − X Z = f − ( X ∩ Y ) − YZ = f − ( X ) ∩ X = X ∩ Z where the equality in Z is due to the choice of f . It is easy to see that thesesets are pairwise disjoint and that their union is Z .Note that Z f ↾ Z (cid:22) X − Y . Since X and Y are not almost complementary, atleast one these holds: | Z | E | X ∩ Y | | Z | E | M − ( X ∪ Y ) | (80)Either way Lemma 13 provides a permutation π ∈ M such that [ π X ] = [ X ] and Z ⊂ π ( X − Y ) ⊂ π ( X ) and satisfying (52). Note that by (52), Z ⊂ π ( X ) , so Z ∪ Z ⊂ π ( X ) (81)Set X = π ( X ) , Y = π ( Y ) .Then since Z ⊂ M − ( X ∪ Y ) (by (52)), and since (by (52) again) Z f ↾ Z (cid:22) X − Y , by Proposition 15 there is a permutation π ∈ M with Z ⊂ π ( X − Y ) ⊂ π ( X ) ∈ [ X ] and satisfying (52). By (81) and (52), we have Z ∪ Z ∪ Z ⊂ π ( X ) ∈ [ X ] (82)18et X = π ( X ) , Y = π ( Y ) . Since Z ⊂ Y − X and Z f ↾ Z (cid:22) X ∩ Y (by(52) twice), there is by Lemma 16 a permutation π ∈ M satisfying (52) andwith Z ⊂ π ( X − Y ) ⊂ π ( X ) ∈ [ X ] . By (82) and (52) we have that Z ∪ Z ∪ Z ∪ Z ⊂ π ( X ) (83)Finally set X = π ( X ) , Y = π ( Y ) . Note that Z ⊂ M − ( X ∪ Y ) andthat Z f ↾ Z (cid:22) X ∩ Y (by (52) twice). Notice now that since π , π , and π ∈ M are permutations and since X, Y are not symmetric, X , Y are not symmetricby Proposition 18. Thus we have at least one of | Z | E | X − Y | | Z | E | Y − X | (84)obtains. So by Lemma 14, there is a permutation π ∈ M such that Z ⊂ π ( X ∩ Y ) ⊂ π ( X ) ∈ [ X ] (85)By (83) and again (52) we have that Z = Z ∪ Z ∪ Z ∪ Z ∪ Z = π ( X ) = π ( π ( π ( π ( X )))) = π ( X ) (86)for π = π ◦ π ◦ π ◦ π ; which was to be established.Dealing with opportune sets via Lemma 22 will enable us to deal with casesin which X is finite, both within and outside of M . We use a special class ofopportune sets to deal with X infinite in M . Definition 23. We will say that X, Y are ideally opportune if they are oppor-tune and either | X | E | M − X | or | Y | E | M − Y | . We will also say that the singleconcept X is ideally opportune if there is a Y such that X, Y are opportune and | X | E | M − X | .We now turn to proving another lemma that will address the more generalcase, in which X is not finite in the metatheory (though it may be finite in M ).We begin with another preparatory lemma. Lemma 24. Suppose X is ideally opportune, and X ⊟ Z . Then Z ∈ [ X ] . Proof. Note that if | X | < ω then X and Z are relatively finite, so Z ∈ [ X ] E ( ⊟ ) by Lemma 22. So we assume | X | ≥ ω .Thus by Proposition 10, and since X is ideally opportune, we may assumethat | Z | = | X | ≤ | M − X | = | M − Z | .Now if | X | = | M − X | , then | M − Z | ≥ | Z | = | X | = | M − X | = | M | If | Z − X | < | X − Z | , then as M − X = ( M − ( X ∪ Z )) ∪ ( Z − X ) , by ISM , weobtain | X | ≤ | M − ( X ∪ Z ) | (87)19imilarly for | X − Z | < | Z − X | .On the other hand if | X | < | M − X | , then as | X | = | Z | , | X ∪ Z | < | M − X | by ISM . Then by ISM again, we have that | M − ( X ∪ Z ) | = | M | ≥ | M − X | > | X | which again yields (87).Given (87) with f a witnessing injection, observe that f ( X ) and X aredisjoint and so relatively finite. By Lemma 22, since X is one of a pair ofopportune sets, f ( X ) ∈ [ X ] E ( ⊟ ) . But then f ( X ) , X are opportune as well.Moreover, as Z, f ( X ) are disjoint and equinumerous, they also are relativelyfinite. So by Lemma 22, Z ∈ [ f ( X )] E ( ⊟ ) = [ X ] E ( ⊟ ) . We are now nearly ready to prove Theorem 6. We first deal with degeneratecases, namely those in E ( ⊟ ) X that may be trivial and also separative. Thefollowing propositions follow from IPM and (40) and (44) of Proposition 8. Proposition 25. If M | = | M | ≤ ∨ | X | = 0 ∨ | X | = | M | < ω (88)then E ( ⊟ ) X is both a separation and trivial.Further, if | M | = 2 and | X | = 1 , then E ( ⊟ ) X is either both trivial and acomplementation, or it is a separation. Proposition 26. If Y, Z ∈ [ X ] ⊟ are symmetric, then | Y △ Z | ⊳ | X | . Proposition 27. If Y, Z ∈ [ X ] ⊟ are almost complementary, then | Y ∩ Z | , | M − ( Y ∪ Z ) | ⊳ | X | . Proposition 28. If M | = ω ≤ | X | ≤ | M − X | and E ( ⊟ ) X is either separativeor complementative, then it is not trivial.The crucial move in proving Theorem 6 is in the following Lemma: Lemma 29. Suppose M | = | M | > ∧ < | X | ≤ | M − X | and E is an L definable equivalence relation over M . Then E ( ⊟ ) X is exactly one of: trivial,properly separative, or properly complementative. Proof. We first establish the lemma for the case where | X | < ω in M . Obviouslyno more than one can obtain; we show at least one must. Since all equinumeroussets finite in M are relatively finite in M , by Lemma 22, if there are opportunesets in [ X ] E ( ⊟ ) , then [ X ] ⊟ = [ X ] ⊟ ∩ RF ( X ) ⊆ [ X ] E ( ⊟ ) and so E ( ⊟ ) X is trivial. So we may assume there are no opportune sets Y, Z ∈ [ X ] E ( ⊟ ) , and thus that for any Y, Z ∈ [ X ] E ( ⊟ ) , either Y, Z are symmetric, or Y, Z are almost complementary by Lemma 22.20uppose now that all pairs of set in [ X ] E ( ⊟ ) are symmetric. By Proposi-tion 26, all such pairs satisfy (30). Now by Proposition 18 this holds for everyconcept bicardinally equivalent to X ; thus E ( ⊟ ) X is properly separative.On the other hand, if there are Y, Z ∈ [ X ] E ( ⊟ ) that are not symmetric,then these are almost complementary. By Proposition 27, the non-symmetricconcepts in [ X ] E ( ⊟ ) satisfy (31). Clearly then (30) fails of them, so E ( ⊟ ) X isnot separative. By Proposition 26 all the symmetric concepts satisfy (30), andso (31). Proposition 18 ensures then that E ( ⊟ ) X is complementative, and since(as just shown) not separative, it is properly complementative. So the resultholds for all | X | finite in M .The case in which X is infinite in M is similar. By Proposition 28, at mostone of the options can hold; we show at least one must. By Lemma 24 if E ( ⊟ ) X is non-trivial then there are no ideally opportune concepts in [ X ] E ( ⊟ ) . Since byassumption | X | ≤ | M − X | it follows that there are no opportune concepts in [ X ] E ( ⊟ ) . The above argument establishes again that E ( ⊟ ) X is either properlyseparative or properly complementative. Corollary 30. Suppose M | = | M | > ∧ < | M − X | < | X | and E is an L definable equivalence relation over M . Then E ( ⊟ ) X is exactly one of: trivial,properly separative, or properly complementative. Proof. It is easy to see that if E is an equivalence relation, then so is the L definable E c ( X, Y ) ⇔ E ( M − X, M − Y ) Thus, by Lemma 29, E c ( ⊟ ) M − X is exactly one of trivial, properly separative,or properly complementative. But by the definition of E c , E c ( ⊟ ) M − X has thesame profile as E ( ⊟ ) X .Lemma 29 and Corollary 30 suffice to prove the first part of Theorem 6. Theremainder follows from the following lemma. Lemma 31. Suppose M | = | X | < ω and there is a W = X with W ∈ [ X ] E ( ⊟ ) ,and E ( ⊟ ) X non-trivial. Then for any Y, Z ∈ [ X ] ⊟ , either (32) or (33) holds. Proof. By the first part of Theorem 6, E ( ⊟ ) X is either properly separativeor properly complementative. By Lemma 22, we know that though X, W arerelatively finite (since they are finite), they are not opportune, so X, W areeither symmetric or almost complementary.We will first show that if X, W are symmetric, then for any Z ∈ [ X ] ⊟ , | X △ Z | ⊳ | X | ⇒ M | = E ( X, Z ) (89)Assume X, W symmetric, so for some H with | X | = | W | = | H | , | H − ( X ∪ W ) | ⊲ | X − W | , | W − X | (90)Assuming | X △ Z | ⊳ | X | (91)21e have that | X | = | H | ⊲ | H − ( X ∪ W ) | ; it then follows from (41) of Proposi-tion 8 that | X ∩ W | ⊲ | X − W | , | W − X | (92)So by (38) of Proposition 8, we have | Z − X | ≤ | X △ Z | ⊳ | X ∩ W | (93)Thus by Lemma 13, Z − X ⊆ π ( X ) with π ( X ∩ Z ) = X ∩ Z and E ( π ( X ) , X ) .Thus Z ∈ [ X ] E , and this establishes (32).Now we will show that if X, W are almost complementary, then for any Z ∈ [ X ] ⊟ , [ | X △ Z | ⊳ | X | or | X ∩ Z | , | M − ( X ∪ Z ) | ⊳ | X | ] ⇒ M | = E ( X, Z ) (94)So assume X, W are almost complementary.If (91) obtains, then we have | ( Z − X ) ∩ W | ≤ | Z − X | ≤ | Z △ X | ⊳ | X ∩ W | (95)by (41) of Proposition 8, and so Z ∈ [ X ] E by Lemma 13.On the other hand suppose | X ∩ Z | ⊳ | X | (96) | M − ( X ∪ Z ) | ⊳ | X | (97)By (96) and (41), | X ∩ Z | ⊳ | X − Z | . But then by (97) and (41) again, | ( W − Z ) ∩ X | ≤ | X ∩ Z | ⊳ | W ∩ Z | and (98) | W − ( X ∪ Z ) | ≤ | M − ( X ∪ Z ) | ⊳ | W ∩ Z | , so (99) | W − Z | ⊳ | W ∩ Z | (100)by (40). As similar argument shows that | Z − W | ⊳ | W ∩ Z | (101)and thus that | W △ Z | ⊳ | W | by (38). From here we can use the argument fromearlier in this proof establishing (32), and the fact that [ W ] E = [ X ] E , and wehave that Z ∈ [ X ] E .From Theorem 6 we obtain the following corollary, which we will use inproving Theorem 7. Corollary 32. If X is finite in the metatheory, then [ X ] E ( ⊟ ) is either [ X ] ⊟ , or { X } , or { X, M − X } . Proof. If X is finite in the metatheory then | X △ Y | ⊳ | X | if and only if | X △ Y | =0 . Likewise for | X ∩ Y | , | M − ( X ∪ Y ) | ⊳ | X | . The reader may wish to aid her reasoning in these following arguments using Venn Dia-grams; label a set A “S” for relatively small , and B “L” for relatively large , if A⊳B . Proposition 8basically asserts that the expected reasoning will obtain (e.g., if A is relatively large but A ∩ B is relatively small, it follows that A − B is relatively large). .5 The Classification Theorem 7 To prove Theorem 7 we need to use CWF to address some complications lingeringdue to our use of non-standard semantics. We show that under CWF all properseparations and proper complementations on finite concepts are the finest pos-sible, even if “finite” only means “finite in M ”. Theorem 7 follows from thetheorem proved in this section. Definition 33. Suppose that E is properly separative on [ X ] ⊟ . Then J ∈ M is called a measure of E on [ X ] ⊟ just if for all Y, Z ∈ [ X ] ⊟ , E ( Y, Z ) ⇔ | Y △ Z | ⊳ | J | Suppose that E is properly complementative on [ X ] ⊟ . Then J ∈ M is calleda measure of E on [ X ] ⊟ just if for all Y, Z ∈ [ X ] ⊟ , E ( Y, Z ) ⇔ | Y △ Z | ⊳ | J | or | M − ( Y △ Z ) | ⊳ | J | Theorem 34. Let M | = | X | < ω . If E is nontrivial at [ X ] ⊟ then its measureis ∅ . Proof. If E is nontrivial on [ X ] ⊟ then by Theorem 6 it is either a proper sepa-ration or a proper complementation. Suppose J is a non-zero measure of E on [ X ] ⊟ .Let L ( ξ ) abbreviate the formula ( ∀ U, V ∈ [ X ] ⊟ )( E ( U, V ) → ( | U △ V | < | ξ | ∨ | M − ( U △ V ) | < | ξ | ) Clearly L ( J ) , so by CWF , there is a smallest J ′ such that L ( J ′ ) . If J ′ = ∅ ,then observe that by Corollary 32 that | J ′ | 6 = n for any natural number n —that is, J is not finite in the metatheory. Thus, letting j ∈ J ′ we have that ¬ L ( J ′ − { j } ) , and so there are U, V ∈ [ X ] E ( ⊟ ) such that E ( U, V ) but | U △ V | 6 < | J ′ − { j }| (102)(If the witnessing U, V to ¬ L ( J ′ ) satisfy the second disjunct of the consequent,just choose U, M − V so that they satisfy (102)). As L ( J ) we have that | U △ V | = | J ′ − { j }| .By assumption E ( ⊟ ) X is non-trivial, so regardless of whether it is a com-plementation or a separation, there is a Z with U = V | U △ V | ⊳ | J | | Z − ( U ∪ V ) | E | J | (103)As | X | ≥ | J | , | U | = | V | = | X | , and | U △ V | ⊳ | J | , we have that | U ∩ V | D | X | by Proposition 8. Thus, with (103) we have that | U ∩ V | , | Z − ( U ∪ V ) | > (104)Thus, let a ∈ U ∩ V and b, c ∈ Z − ( U ∪ V ) , and set U ′ = ( U − { a } ) ∪ { b } V ′ = ( V − { a } ) ∪ { c } (105)23ote now that as | U △ U ′ | = | V △ V ′ | = 2 ⊳ | J | (106)we have E ( U, U ′ ) , E ( V, V ′ ) , and so E ( U ′ , V ′ ) . Thus, since L ( J ) , | U ′ △ V ′ | < | J | ,but it is easy to see that the construction of U ′ and V ′ ensure that | U ′ △ V ′ | = | J ∪ { a }| > | J | Theorem 34 proves Theorem 7 for X finite in M . For the rest one needs onlyconsider X such that M | = | X | ≥ ω . By Proposition 10, if E ( ⊟ ) X is separative,then M | = ( ∃ X ′ ⊆ X )( ∀ Y, Z ∈ [ X ] ⊟ )( E ( Y, Z ) → | Y △ Z | < | X ′ | ) (107)and likewise if E ( ⊟ ) X is complementative. This completes the proof of Theo-rem 7. The “bad company problem” faced by neo-logicists is the two part challenge offinding a plausuble criterion for the “logicality” of abstraction principles, andshowing that the good abstraction principles like HP are logical in this sense,while the bad ones like BLV , NP , and CP are not. It is thus relevant to note howTheorem 7 and its apparatus bear on the joint consistency of abstraction prin-ciples, since at the very least any abstraction principles qualifying as “logical”should be jointly consistent.Consistency results in the absence of well-behaved cardinalities are difficultto obtain. As such our discussion will be directed towards the neo-logicist whothinks “logical” abstraction principles must be jointly consistent in the presenceof well-behaved cardinalities . We’ll thus use “jointly consistent” as if it implicitlyhas the qualification on the behavior of cardinalities.Such a neo-logicist is committed to thinking that joint consistency with HP is a necessary condition for the logicality of abstraction principles. In theremainder of this section we apply Theorem 7 and its apparatus to discern whichabstraction principles are consistent with a Dedekind infinite universe. Theseobviously determine which abstraction principles are consistent with HP , andthus which must be ruled out from being “logical”.One reason why the bad company problem can appear vexing is that any“logical” equivalence relation can give rise to an abstraction principle. Facingan untamed menagerie of such equivalence relations, the neo-logicist would facean equally wild zoo of abstraction principles. The import of Theorem 7 is intaming the menagerie—or better, classifying the species. At each bicardinalslice of M , there are only three “classes” of equivalence relations on that slice.This enables us to see clearly why certain abstraction principles have the typesof models they do: at certain (finite or infinite!) bicardinal classes, non-trivial equivalence relations distinguish more equivalence classes than there are objects.24e have said that BLV , NP , and CP all count as bad companions. We thushave so far two bad companions deploying refinements of separations, and onedeploying a refinement of a complementation. Of the former, BLV involves thefiner (in fact finest) separation; as we will see in contexts with IPM it is in a sensethe paradigm case of a separative abstraction principle. We will also discuss theparadigm case of a complementative abstraction principle; it is not CP but aclose relative, the Liberated Complementation Principle ( LCP ): ( ∀ X, Y )( ℓX = ℓY ↔ X = Y ∨ X = M − Y ) Remark 35. The equivalence relation for BLV is the finest equivalence relationthat refines a separation for every X ∈ M (it is also the finest equivalencerelation full stop, see [1]).The equivalence relation for NP refines a separation at all bicardinal slices of M for Dedekind infinite sets.Finally, the equivalence relation for LCP refines a complementation at X ∈ M if | X | = | M − X | , and in fact it is the finest equivalence relation refining acomplementation at bicardinal slices for such X . In the remainder of this section we will need the following definitions: Definition 36. For M | = | X | , ω ≤ | M | = | M − X | , we will write Y ≤ [ X ] ⊟ asshorthand for | Y | < | X | ∨ Y ∈ [ X ] ⊟ .Let E be an equivalence relation. Let ϕ ( Y, Z ) be a formula. We say that ϕ is functional below X in M if M | = | X | ≤ | M − X | ∧ ( ∀ Y )( Y ≤ [ X ] ⊟ → ( ∃ ! Z ) ϕ ( Y, Z )) (108)We say that ϕ is functional at X if M | = ( ∀ Y )( Y ∈ [ X ] ⊟ → ( ∃ ! Z ) ϕ ( Y, Z ) (109)Often in the presence of such a ϕ we will use functional notation, using F ϕ ( Y ) for the unique object Z such that ϕ ( Y, Z ) , dropping the subscript whereit is unneeded for understanding.If A E [ ∂ ] is an abstraction principle, and ϕ a formula functional below X in M , then we say that θ translates A E ′ ( ≤ X ) via ϕ in M to mean that M | = ( ∀ Y, Z )( Y, Z ≤ [ X ] ⊟ → ( ∂ ( F ϕ ( Y )) = ( ∂ ( F ϕ ( Z )) ↔ E ′ ( Y, Z ))) (110)If ϕ is functional at X , then we say that A E [ ∂ ] translates A E ′ (= X ) via ϕ in M to mean that M | = ( ∀ Y, Z )( Y, Z ∈ [ X ] ⊟ → ( ∂ ( F ϕ ( Y )) = ∂ ( F ϕ ( Z )) ↔ E ′ ( Y, Z ))) (111)Finally we may say that A E [ ∂ ] translates A E ′ in M via ϕ just if M | = ( ∀ Y, Z )( ∂ ( F ϕ ( Y )) = ∂ ( F ϕ ( Z )) ↔ E ′ ( Y, Z )) (112)25hese definitions afford us the capacity to talk of the restricted abstractionprinciples A E [ ∂ ]( ≤ X ) and A E [ ∂ ](= X ) in the sense that M satisfies one ofthese just if “ ( ∀ x )( x = x ) ” translates that principle in M via “ Y = Z ”.In what follows of this section we show that if < | X | = | M − X | and E ( ⊟ ) X is non-trivial, then (in the presence of IPM ) A E is inconsistent. The engine ofthese results will be the following two propositions. Proposition 37. Let ̟ be an L -sentence. Suppose that for all M | = A E [ ∂ ] , A E [ ∂ ] ∧ ̟ translates A E ′ (= M ) in M via ϕ , and that A E ′ (= M ) and ̟ arejointly inconsistent (in our background logic). Then A E and ̟ are jointlyinconsistent (in our background logic) as well. Proof. If A E ∧ ̟ is consistent then by the completeness of second-order logicfor the non-standard semantics, there is a structure M = ( M, S [ M ] , . . . , ∂ ) | = A E ∧ ̟ . As ϕ ( Y, Z ) is functional at M , define ∂ ′ Y = ∂ ( F ϕ ( Y )) ; ∂ ′ is thena function in M from concepts to objects. We now augment M to M ′ =( M, S [ M ] , . . . , ∂, ∂ ′ ) , and note that since A E translates A E ′ (= M ) , M ′ | = A E ′ [ ∂ ′ ](= M ) . Since ̟ is an L -sentence (with parameters from M ), M ′ | = ̟ .A routine induction on the complexity of formulas shows that M ′ satisfies thecomprehension axioms in the expanded language L [ ∂ ′ ] . Thus if A E [ ∂ ] ∧ ̟ hasa model then so does A E ′ [ ∂ ′ ](= M ) ∧ ̟ .For the next proposition, observe that if M | = | M | ≥ ω , then IPM ensuresthat there is a bijection h· , ·i : M × M → M . We will use this notation in whatfollows, as well as writing h X, Y i to mean {h x, y i | x ∈ X, y ∈ Y } . We alsoobtain: Proposition 38. For | M | ≥ ω , let h· , ·i : M × M → M be a bijection, whoseexistence is assured by IPM . For X, Y , if either | Y | = 1 ≤ | X | ≤ | M − X | (113)or | Y | , ω ≤ | X | ≤ | M − X | (114)then h Y, X i ∈ [ X ] ⊟ . Lemma 39. Let M | = | X | ≥ ω . If M | = | X | ≤ | M − X | and E ( ⊟ ) X refines aseparation, then A E translates BLV ( ≤ X ) via ̺ ( Y, Z ) := Z = h Y, X i (115)Under the same conditions, if E ( ⊟ ) X refines a complementation (and doesnot refine a separation), then A E translates LCP via ϕ ( Y, U ) := ( ∃ V )((( ∂ ∅ ∈ Y ∧ V = h Y, X i ) ∨ ( ∂ ∅ ∈ M − Y ∧ V = h M − Y, X i )) ∧ (( V = h M, X i ∧ V = U ) ∨ ( V = h M, X i ∧ U = ∅ ))) (116) The arguments of this section elaborate on the one given in [6]. roof. For both assertions, note: By hypothesis, M | = | M | ≥ | X | ≥ ω , so by IPM the bijection h· , ·i exists. Notice that since E ( ⊟ ) X is non-trivial, ∂ ∅ 6 = ∂Y for any Y ∈ [ X ] ⊟ , since otherwise by PI E ( ⊟ ) X would be trivial, as E ( ∅ , X ) holds whenever E ( ∅ , f ( X )) holds for | X | = | f ( X ) | and | M − X | = | f ( M − X ) | .Further if this Y = ∅ then h Y, X i ∈ [ X ] ⊟ by Proposition 38 (under the conditionsof the second assertion this follows since | X | = | M − X | = | M | .)We prove the first assertion in brief: It is easy to verify that ̺ is functional(and so is functional below X ). Let | Y | , | Z | ≤ | X | .For distinct Y, Z = ∅ , |h Y, X i△h Z, X i| = | X | by IPM , so if Y = Z then since E ( ⊟ ) X is a separation, ¬ E ( h Y, X i , h Z, X i ) , and so by A E , ∂ h Y, X i 6 = ∂ h Z, X i .Conversely if Y = Z then h Y, X i = h Z, X i . Since E is an equivalence relation,it follows that E ( h Y, X i , h Z, X i ) , and so, by A E , that ∂ h Y, X i = ∂ h Z, X i .For the second assertion, it is easy to verify that ϕ is functional (and so isfunctional below X ). Towards verifying the consequent of (110) and working in M , we need only show that ∂U ϕ ( Y ) = ∂U ϕ ( Z ) ↔ Y = Z ∨ Y = M − Z (117)for all concepts Y, Z ∈ M .By the definition of U ϕ , U ϕ ( ∅ ) = U ϕ ( M ) = ∅ . So clearly if X and Y arechosen from ∅ , M , then (117) obtains.Now, if Y = ∅ then U ϕ ( Y ) = ∅ , and if Y = M then U ϕ ( Y ) = ∅ . On theother hand if Y = ∅ , M , then U ϕ ( Y ) ∈ [ X ] ⊟ . Thus by A E [ ∂ ] , ∂ ( U ϕ ( Y )) = ∂ ( U ϕ ( ∅ )) = ∂ ( U ϕ ( M )) . Thus: ∅ 6 = Z → ∂ ( U ϕ ∅ ) = ∂ ( U ϕ Z ) (118) M = Z → ∂ ( U ϕ M ) = ∂ ( U ϕ Z ) (119)Now if Y, Z = ∅ , M then we have Y = Z → | U ϕ ( Y ) △ U ϕ ( Z ) | = | X | (120) Y = M − Z → | M − ( U ϕ ( Y ) △ U ϕ ( Z )) | = | X | (121)So, since E ( ⊟ ) X refines a complementation we have that ∂ ( U ϕ ( Y )) = ∂ ( U ϕ ( Z )) → Y = Z ∨ Y = M − Z (122)by A E [ ∂ ] .Clearly by the construction of U ϕ : Y = Z → E ( U ϕ ( Y ) , U ϕ ( Z )) (123) Z = ∅ 6 = Y = M − Z → E ( U ϕ ( Y ) , U ϕ ( Z )) (124)So we have, again by A E [ ∂ ] and the E ( ⊟ ) X refining a complementation: Y = Z ∨ Y = M − Z → ∂ ( U ϕ ( Y )) = ∂ ( U ϕ ( Z )) (125)Which completes the proof of the translation of LCP .27e know now some of which abstraction principles translate restrictions of BLV and LCP . We now show when, in the presence of IPM , these restrictions areinconsistent for each such principle. Each of the following two subsections willconclude by applying Lemma 39; the final subsection applies Theorem 7 to unifythose results. BLV (= X ) The utility of Lemma 39 is in relating equivalence relations that are non-trivialon bicardinal slices to what might be called their prime examples. In thissection we treat BLV and its restrictions as the prime examples of separations,and show that in infinite structures, refinements of separations translate certainrestrictions of BLV . The next section proves analogous results for LCP and itsrestrictions.From the Lemma we obtain very quickly: Corollary 40. Suppose M | = | X | = | M − X | = | M | . Then M 6| = BLV (= X ) . Proof. Given X as in the hypothesis, if M | = BLV (= X ) then E ( ⊟ ) X is a separa-tion, and so refines a separation. Thus BLV (= X ) translates both BLV [ ε ]( ≤ X ) and BLV [ ε ]( ≤ M − X ) . With M f ≈ X and M g ≈ M − X , set εY = ( f ( ε Y ) | Y | ≤ Xg ( ε Y ) | M − Y | < | M − X | Clearly then BLV (= X ) and | X | = | M − X | = | M | translate BLV , which isinconsistent. Thus BLV and | X | = | M − X | = | M | are jointly inconsistent, andby Proposition 37, BLV (= X ) and | X | = | M − X | = | M | are jointly inconsistent,which was to be demonstrated.The sitation is no better for BLV (= X ) if M is finite (in M ): Lemma 41. Suppose M | = 0 < | X | < | M | < ω . Then M 6| = BLV (= X ) . Proof. Let X be given, and since | M | > , we establish the following abbrevia-tions: for x ∈ X and a X , X x = X − { x } (126) a + X x = X x ∪ { a } (127)Notice that for all x ∈ X and a X , | a + X x | ∈ [ X ] ⊟ but a + X x = X . Thus if M | = BLV (= X ) , then εX = ε ( a + X x ) . Moreover, for distinct x, y ∈ X , a + X x and a + X y are distinct (thus so are their abstracts), and for distinct a, b X , a + X x and b + X x are distinct (thus so are their abstracts).Now |{ a + X x | x ∈ X }| = | X | for each a X , and |{ a + X x | a X }| = | M − X | for each x ∈ X . Further for fixed x ∈ X, a X , { a + X x | x ∈ X } ∩ { a + X x | a X } = { a + X x } ε ∅ 6 = εX, ε ( a + X x ) for any a X, x ∈ X . Thus, and since M | = | M | < ω , and letting A = { ε ( a + X x ) | x ∈ X } B = { ε ( a + X x ) | a X } (128)we have | M | = | M − X | + | X | (129) = | A | + | ( B − { ε ( a + X x ) } ) ∪ { εX }| (130) = | A ∪ B ∪ { εX }| (131) < | A ∪ B ∪ { εX }| + |{ ε ∅}| (132) ≤ | rng( ε ) | ≤ | M | (133)which means that | M | < | M | , a contradiction.The preceding results on refinements of separations and on restrictions of BLV yield: Corollary 42. If E ( ⊟ ) X refines a separation and M | = | X | = | M − X | , then M 6| = A E . Proof. Corollary 40 and Lemma 41 imply that (in the presence of IPM ) M | = | X | = | M − X | → ¬ BLV (= X ) (134)If E ( ⊟ ) X is a separation and M | = | X | < ω then by Theorem 7 M | = BLV (= X ) if M | = A E . But this contradicts (134).On the other hand, if M | = | X | ≥ ω , then by Lemma 39, A E translates BLV (= X ) . So A E cannot hold in M as again this contradicts (134). LCP (= X ) The situation is quite similar for LCP . Here we handle the finite case first: Lemma 43. LCP (= X ) | = | X | = | M − X | < ω → | M | ≤ (135) LCP | = | M | < ω → | M | ≤ (136)Here, abusing notation, “ | = ” indicates the derivability relation in second-order logic with well-behaved cardinalities.We begin with the following: Proposition 44. The following is a theorem of second-order logic: ( ∀ X, Y )( | X | > , | Y | ≥ → | X ∪ Y | ≤ | X × Y | ) (137)with a strict inequality in the consequent if X ∪ Y is Dedekind finite.29 roof. We assume without loss of generality that X and Y are disjoint, let a, b, c ∈ X and d, e ∈ Y be pairwise distinct. Then define f : X ∪ Y → X × Y by f ( x ) = ( a, x ) x ∈ Y, x = d ( x, d ) x ∈ X ( b, e ) x = d Clearly f is injective; it is not surjective since ( c, e ) rng( f ) . Further, if g : X × Y → X ∪ Y is a bijection, then g ◦ f witnesses that X ∪ Y is Dedekindinfinite. Proof of Lemma 43. For the first assertion, suppose the antecedent. Reasoningdeductively, suppose now that | M | > , then | X | , | M − X | > . Set X = A and M − X = B , then for each ( a, b ) ∈ A × B , set A b = ( A − { a } ) ∪ { b } and B a = ( B − { b } ) ∪ { a } . Now for ( a, b ) , ( c, d ) ∈ A × B distinct (as a pair), we havec (cid:13) A b = c (cid:13) B a c (cid:13) A d = c (cid:13) B c c (cid:13) A b = c (cid:13) A c (138)So f ( x, y ) = c (cid:13) A y = c (cid:13) B x defines an injection from A × B into M = A ∪ B ,contradicting Proposition 44. Thus, | M | ≤ .For the second assertion, suppose | M | > . Then f ( x ) = ℓ { x } shows thatfor rng( ℓ ) = { ℓX | | X | = 1 } , | rng( ℓ ) | = | M | and so by finiteness rng( ℓ ) = M . However for any m ∈ M , ℓ { m } 6 = ℓ ∅ by LCP so ℓ ∅ 6∈ rng( ℓ ) = M , acontradiction.Lemma 43 would make it appear that LCP (= X ) must have mostly structuresof infinite size. But it has no structures of infinite size: Proposition 45. In the presence of IPM , LCP (= X ) implies the universe isDedekind finite. Proof. Given IPM , if | M | ≥ ω , there is a concept X such that | X | = | M − X | = | M | . We show that LCP (= X ) ∧ | M | ≥ ω translates BLV (= X ) , so then byCorollary 40 and Proposition 37, the former is inconsistent.Let f : M → X and g : M → M − X be bijections, and define ε by εY = ( f ( ℓY ) ℓ ∅ ∈ Yg ( ℓY ) ℓ ∅ 6∈ Y Clearly ε is functional at X . Now if Y, Z ∈ [ X ] ⊟ , ℓ ∅ ∈ Y, Z and Y = Z ,then by LCP (= X ) and the injectivity of f , εY = εZ . If ℓ ∅ ∈ Y − Z , then as f ( M ) ∩ g ( M ) = ∅ , εY = εZ . Lastly if ℓ ∅ 6∈ Y, Z , then by LCP (= X ) and theinjectivity of g , εY = εZ .Together with Lemmata 39 and 43, from Proposition 45 we obtain Corollary 46. If M | = | X | = | M − X | and E ( ⊟ ) X refines a complementation,then M | = A E → | M | ≤ . 30 roof. Propositions 43 and 45 together give that M | = | X | = | M − X | ≥ → ¬ LCP (= X ) (139)If M | = | X | < ω and E ( ⊟ ) X refines a complementation, then by Theorem 7 M | = LCP (= X ) if M | = A E . But then (139) gives that | X | = | M − X | < .On the contrary if M | = | X | ≥ ω , then as E ( ⊟ ) X refines a complementation, Having dealt individually with each kind of non-trivial equivalence relation, wenow apply Theorem 7, for putting together Corollaries 42 and 46 with Theorem 7gives the following: Theorem 47. If M | = 2 < | X | = | M − X | and E ( ⊟ ) X is nontrivial, then M 6| = A E . Proof. By Theorem 7, if E ( ⊟ ) X is not trivial it either refines a separation or re-fines a complementation. In the former case A E is inconsistent by Corollary 42,and in the latter case A E is inconsistent by Corollary 46 and by our assumptionthat | M | > . Remark 48. The converse of Theorem 47 fails, for let A E be the abstractionprinciple NewV : ( ∀ X, Y )( ςX = ςY ↔ ( | X | = | Y | = | M | ∨ ( | X | , | Y | < | M | ∧ X = Y )) But by König’s theorem, NewV is has no standard models whose first-order do-main is a singular limit cardinal. To put Theorem 47 in the terms used earlier: If M is large enough (havinggreater than four elements), then M satisfies no abstraction principle whoseequivalence relation is non-trivial on concepts which evenly divide the universe.So Theorem 7 gives a mark by which to identify (at least) many bad companions:these insufferable principles limit the size of the universe by non-trivially carvingconcepts of maximally large bicardinality. There is in fact a small error on just this point at [24, p. 596] in stating the results of [20,p. 315]: “For instance, the claim that the abstraction principle New V is strongly stable isequivalent to the generalized continuum hypothesis.” The correct statement is this: New V isnot strongly stable, and whether it is stable depends on whether κ = κ + for unboundedlymany κ . Theorem 47 is in some ways a companion to Fine’s Characterization Theorem, mentionedearlier: that what he calls the basal abstraction principle is the finest satisfiable in all infinitedomains. As Theorem 7 is a deductive analogue of Fine’s Classification Theorem, a deductiveanalogue of his Characterization Theorem can also be obtained from Theorem 7, though weleave this for Appendix 10. Classification and relative categoricity In [24] Walsh and Ebels-Duggan introduce a criterion that distinguishes theabstraction principle HP . The so-called “Julius Caesar problem” arises fromthe following fact. There are abstraction principles, among them HP , with thefollowing unfortunate property: for a given base L -structure, it is possible tointerpret ∂ in two ways, both satisfying the abstraction principle, but such thatthe two expanded structures are neither isomorphic nor elementarily equivalent.That this can be done with the equivalence relation ≈ and HP was noted byFrege himself [8, § 56, §§66ff]. Walsh and Ebels-Duggan showed that thoughthese abstraction principles lack these properties, some of them have weaker butstill noteworty equivalence properties.Abusing notation somewhat, let A E [ ∂ ] be not only the abstraction principlecorrelated with E , but the theory containing as axioms that abstraction prin-ciple and the axioms of our background logic (including those for well-behavedcardinalities). With similar abuse, we’ll use A E [ ∂ , ∂ ] be a theory contain-ing axioms for our background logic and the abstraction principles A E [ ∂ ] and A E [ ∂ ] . In other words, A E [ ∂ , ∂ ] is a theory with two copies of the abstractionprinciple A E , one for each abstraction operator. To introduce Walsh and Ebels-Duggan’s weakened equivalence notions we first introduce the relevant notionof an isomorphism between induced models: Definition 49. Given a structure M = ( M, S [ M ] , S [ M ] , . . . , ∂ , ∂ ) satisfying A E , for i = 1 , , let M i = (rng( ∂ i ) , S [ M ] ∩ P (rng( ∂ i )) , S [ M ] ∩ P (rng( ∂ i ) × rng( ∂ i )) . . . , ∂ i ) (140)We say that M i is the model induced by ∂ i .As noted in [24], for Γ to be an isomorphism between M and M it issufficient for Γ to satisfy the following condition: For all X ∈ S [ M ] ∩ P (rng( ∂ )) , Γ ∂ X = ∂ (Γ X ) (141) Definition 50. The theory A E is naturally relatively categorical just if forany model M , and any ∂ , ∂ such that M | = A E [ ∂ , ∂ ] , the natural bijection Γ : rng( ∂ ) → rng( ∂ ) , defined by Γ ∂ X = ∂ X (142)is an isomorphism between the induced models M and M .The theory A E is relatively categorical just if for any model M , and any ∂ , ∂ such that M | = A E [ ∂ , ∂ ] , M ∼ = M .The theory A E is relatively elementarily equivalent just if for any model M ,and any ∂ , ∂ such that M | = A E [ ∂ , ∂ ] , M | = ϕ ∂ ⇔ M | = ϕ ∂ for every sentence (i.e., closed formula) ϕ in the language of the theory A E [ ∂ ] ,and ϕ ∂ i is the result of replacing every occurrence of ∂ in ϕ with ∂ i . See [8, § 55], and for example [13] and [11, Chapter 14]. natural relative categoricity of an abstraction principle; some ofwhich we state as: Natural Relative Categoricity Theorem NRCT ([24]) . Let A E be an ab-straction principle. The following are equivalent:1. A E is naturally relatively categorical.2. A E [ ∂ ] | = ( ∀ X, Y )( | Y | = | X | ≤ | rng( ∂ ) | → E ( X, Y )) .where again “ | = ” means the deductive consequence relation for our strong back-ground logic. Abstraction principles satisfying this second condition are said tobe cardinality coarsening on abstracts .As can be seen plainly, ≈ is the finest equivalence relation such that A E is cardinality coarsening on abstracts; thus HP is the finest naturally relativelycategorical abstraction principle. This gives the neo-logicist a criterion by whichto distinguish HP from other abstraction principles; notably, the abstractionprinciple NewV , which is equivalent to BLV ( < | M | ) , is not naturally relativelycategorical (see [24, § 5]).But there is more than can be asked. In [23, Proposition 14, p. 1687] Walshshows that HP is relatively categorical in the stronger, unqualified sense. More-over the foregoing establishes the following implication relations:Natural relative categoricity ⇔ Cardinality coarsening on abstracts ⇓ Relative categoricity ⇓ Relative elementary equivalence (143)Walsh and Ebels-Duggan thus raised two questions in [24]: Question 51. Are the conditions for relative categoricity the same as for natu-ral relative categoricity? Can we drop the specification of the natural bijectionin the NRCT ? More precisely is (2) equivalent to the relative categoricity of A E ? Question 52. What is the relation between relative categoricity, natural rela-tive categoricity, and relative elementary equivalence?It is a consequence of Theorem 7 that Question 51 can be answered in theaffirmative, and that, in answer to Question 52, all the implication arrows of(143) can be reversed.From here onward it will be helpful to recall the definitions of restrictedabstraction principles from page 26. 33 emma 53. Suppose A E is an abstraction principle, and that A E [ ∂ ] translates BLV ( ≤ via ϕ ( Y, Z ) . Then A E [ ∂ ] is not relatively elementarily equivalent, andso not relatively categorical. Proof. Let ∂ be given such that M | = A E [ ∂ ] . Note first that as A E translates BLV ( ≤ , | rng( ∂ ) | = | M | ≥ ω . We may thus assume that rng( ∂ ) = M . The ∂ we will construct will have the same range; this allow us to avoid worries thatthe translation ϕ of BLV ( ≤ in M , which may contain nested quantifiers, maychange its behavior on the induced models M and M .For ϕ translating as above, let x ∈ U ( X ) if and only if ϕ ( X, Y ) and x ∈ Y .Consider now ψ i = ( ∃ a )( a = ∂ i U ( { a } )) (144)On the one hand, if M | = ¬ ψ , then select a ∈ M , and note that thereare b, c ∈ M with b = ∂ U ( { a } ) and a = ∂ U ( { c } ) . Let f ( b ) = a , f ( a ) = b ,and f the identity map on M − { a, b } . As f ∈ M since it is definable, setting ∂ X = f ( ∂ X ) we have that M | = A E [ ∂ , ∂ ] . And yet M | = ψ .On the other hand, if M | = ψ , let A = { ∂ X | ( ∃ a )( a = ∂ X = ∂ U ( { a } )) } (145)Suppose first that | A | = | rng( ∂ ) | . As | A | ≥ ω , there is a B ⊂ A such that | B | = | A − B | = | A | by IPM ; let g : B → A − B be a bijection. Then let ∂ X = g ( ∂ X ) ∂ X ∈ Bg − ( ∂ X ) ∂ X ∈ g ( B ) ∂ X ∂ X A (146)Again we have that M | = A E [ ∂ , ∂ ] , and M | = ¬ ψ : for if ∂ U ( { a } ) = a ∈ B then ∂ U ( { a } ) ∈ A − B , and so ∂ U ( { a } ) = a . And likewise if ∂ U ( { a } ) = a ∈ A − B .Lastly suppose | A | < | rng( ∂ ) | , then as | rng( ∂ ) | ≥ ω , then by ISM there isan injection g : A → rng( ∂ ) − A . Setting B = A , the construction as in (146)again delivers the verdict that M | = ¬ ψ .Lastly, we prove a parallel to Lemma 39. Lemma 54. Suppose M | = 0 < | X | < ω ≤ | M | and E ( ⊟ ) X refines a separation.Then A E translates BLV ( ≤ . Proof. By Proposition 38, for any singleton Y , h Y, X i ∈ [ X ] ⊟ . Further, if Y = Z and both are singletons, then h Y, X i and h Z, X i are disjoint, so |h Y, X i△h Z, X i| 6 < | X | . Thus, setting εY = ∂ h Y, X i , we have for singletons Y and Z : εY = εZ ⇔ E ( h Y, X i , h Z, X i ) ⇔ h Y, X i = h Z, X i ⇔ Y = Z We now answer Questions 51 and 52:34 heorem 55. An abstraction principle A E is relatively categorical if and onlyif it is cardinality coarsening on abstracts.Further, an abstraction principle is relatively elementarily equivalent if andonly if it is relatively categorical. Proof. The right-to-left direction of the second assertion is trivial; for the right-to-left direction of the first, suppose A E is cardinality coarsening on abstracts,and let M | = A E [ ∂ , ∂ ] . Let Γ be the natural bijection defined by Γ ∂ X = ∂ X ;note that as | Γ X | = | X | , since A E is cardinality coarsening on abstracts, in A E [ ∂ , ∂ ] it follows that ∂ i X = ∂ i Γ X . Thus Γ ∂ X = ∂ X = ∂ Γ X (147)so Γ is an isomorphism. For the left-to-right direction of both assertions, observe that the followingexhaust the possibilities for the cardinal placement of any two equinumerousconcepts: M | = | M | = | Y | = | X | = | M − X | = | M − Y | (148) M | = | M | = | Y | = | X | = | M − X | > | M − Y | (149) M | = | M | = | Y | = | X | > | M − X | ≥ | M − Y | (150) M | = ω ≤ | X | = | Y | < | M | (151) M | = | X | = | Y | < ω ≤ | M | (152) M | = | X | = | Y | ≤ | M | < ω (153)To prove the second assertion, then it suffices to show that on each of thesepossibilities, if M | = A E [ ∂ , ∂ ] then either (i) M and M are isomorphic(and so elementarily equivalent), or (ii) there are ∂ , ∂ such that M and M are not elementarily equivalent (and so not isomorphic). From the right-to-leftdirection of the first assertion, we know that if E is cardinality coarsening onabstracts, then (i) holds for all possibilities. Thus to prove the second assertion,it suffices to prove that if E is not cardinality coarsening on abstracts, theneither M 6| = A E , or there are ∂ , ∂ with M | = A E [ ∂ , ∂ ] but M and M arenot elementarily equivalent. This will prove the first assertion as well.So suppose that A E is not cardinality coarsening on abstracts; let M | = A E witness this failure. So there are X, Y ∈ M such that M | = | Y | = | X | ≤ | rng( ∂ ) | ∧ ¬ E ( X, Y ) (154)Clearly then M | = 0 < | X | .The first case in which (148) holds can be ruled out since if E ( ⊟ ) X is non-trivial, then by Corollaries 42 and 46, M 6| = A E . Thus for the second case(149) we must assume that E ( ⊟ ) X is trivial. Without loss of generality assume M | = X ⊂ Y . Since by assumption (154), M | = | M | = | X | = | Y | ≤ | rng( ∂ ) | ,let f, g ∈ M be respective witnesses of | X | = | rng( ∂ ) | and | Y | = | rng( ∂ ) | , then This direction of the proof generalizes the proof in [23, Proposition 14]. ∂ = f ◦ ∂ and ∂ = g ◦ ∂ . Since ∂ , ∂ are both definable injections in M ,it follows that M | = A E [ ∂ , ∂ ] . Note, however, that for all Z ⊂ X , if | Z | = | X | then Z ⊟ X . Hence, since E ( ⊟ ) X is trivial, E ( X, Z ) . Thus, as M | = ( ∀ x )( Xx ↔ x = x ) it follows that M | = ( ∀ U, W )(( ∀ x )( U x ↔ x = x ) ∧ | W | = | U | → ∂ U = ∂ W ) (155)However, notice that M | = ( ∀ x )( Y x ↔ x = x ) and that f ◦ g ∈ S [ M ] and f ◦ g ⊂ Y × Y . Thus f ◦ g ∈ M , so M | = | X | = | Y | ∧ ∂ X = ∂ Y as by assumption X ⊂ Y and M | = A E [ ∂ ] ∧ ¬ E ( X, Y ) . Consequently, we seethat M = ( ∀ U, W )(( ∀ x )( U x ↔ x = x ) ∧ | W | = | U | → ∂ U = ∂ W ) So by (155) A E is not relatively elementarily equivalent, and so not relativelycategorical.For (150) observe that by IPM if | M | = | X | ≥ ω then there is Z ⊂ X suchthat | Z | = | M − Z | = | M | ; by (148) E ( ⊟ ) Z is trivial, and so by (149) both X and Y are in [ Z ] E .If (151) holds, then by Theorem 7, E ( ⊟ ) X is non-trivial, but then by Lemma 39 A E translates BLV ( ≤ X ) on M . Likewise if (152), then by Lemma 54 we seeagain that A E translates BLV ( ≤ on M . But then by Lemma 53 A E is notrelatively elementarily equivalent, and so not relatively categorical.Finally if (153) holds then by Theorem 7, as < | X | < | M | , by Theorem 47and M | = A E , we have that | M | ≤ and M | = LCP (= X ) . If | M | = 2 then if E ( ⊟ ) X refines a complementation then it is trivial (and so cardinality coarseningon abstracts), so | M | = 4 . But then by the argument of [24, § 5.5, p. 594], A E is not relatively elementarily equivalent, and so not relatively categorical.Walsh and Ebels-Duggan prove another theorem [24, Theorem 1.2] for an-other weakened equivalence property: that if all objects are abstracts, thenthe induced models for A E are isomorphic via the natural bijection if and onlyif bicardinally equivalent concepts are E -equivalent. They ask an analogousquestion as well: can one drop the assumption that the natural bijection is anisomorphism and still obtain the biconditional? The proof of Theorem 55 indi-cates an affirmative answer here also: to adapt the proof one must observe that(149) is irrelevant on the assumption that X and Y are bicardinally, but not E -, equivalent. The other cases remain the same.Thus we have: Theorem 56. Suppose that M | = A E [ ∂ , ∂ ] ∧ rng( ∂ ) = rng( ∂ ) = M . Then M ∼ = M if and only if A E is bicardinality coarsening on abstracts.36t may seem surprising to see in Theorem 55 the equivalence of relativeelementary equivalence with relative categoricity. But reflection suggests thatthis shouldn’t be so striking after all: for Theorem 55 depends on Theorem 7,and so on the equivalence relations in play being permutation invariant because L -definable. The equivalence is in a sense an artifact of this restriction. It isan open question as to whether other invariance conditions (see [7], [1]), [5])would yield the same results.However, reflection on this equivalence could be taken as philosophicallyrelevant, and indeed good news for the neo-logicist. For it is trivial, by PI , thatin an L -structure M , if π is a permutation in M , then the structure resultingfrom that permutation is both isomorphic and elementarily equivalent to M .This is trivial, of course, because the permuted structure just is the originalstructure. But it is noteworthy that this triviality hides the coinstantiation ofisomorphism and elementary equivalence.In light of this, one might ask: is this coinstantiation preserved under ab-straction? In more detail: given two abstraction operators, ∂ , ∂ for an ab-straction principle A E , the natural bijection Γ( ∂ X ) = ∂ will always be in anymodel of A E [ ∂ , ∂ ] . Therefore if ∂ and ∂ share the same range, then M canbe regarded as the structure resulting from the permutation Γ of the structure M . Is it the case that under any such permutation, the resulting models will beisomorphic if and only if elementarily equivalent? Indeed, answers Theorem 55,the answer is affirmative: this property of coinstantiation is, in the given sense,preserved under permuations of abstractions. One might argue on behalf of theneo-logicist (though we will not) that this shows abstraction is in some sense a“logical” operation. We began with the promise that strengthening Fine’s Classification Theoremwould advance our understanding of the demarcation of logical abstraction prin-ciples. The strengthened version of Fine’s theorem yields a strengthened relativecategoricity theorem; the relevant advancement thus comes in the option pro-vided by the latter result.Walsh and Ebels-Duggan note that relative categoricity as we have describedit cannot rule out bad companions like NP : since NP has only Dedekind finitemodels, it (the principle ) is cardinality coarsening on abstracts. They alsonote, however, that the equivalence relation underlying NP is not cardinalitycoarsening on small concepts —a notion that applies not to abstraction principlesbut to equivalence relations. (See [24, section 5.4].)The confluence of results presented in this paper offers a stronger statementof this point. As we can see from the results in sections 6 and 7, when nontrivialequivalence relations are manifest in a structure, they generate failures of rel-ative categoricity. It is curious that NP is relatively categorical as a principle.But it is more relevant that the equivalence relation deployed by NP is nontrivialon all infinite bicardinal slices—it has non-trivial manifestations. And wherever37ontriviality appears in a structure satisfying an associated abstraction princi-ple, it generates a failure of relative categoricity. Thus, even though NP is bythe letter relatively categorical, it deploys an equivalence relation that generatesfailures of relative categoricity. On this line of thinking, NP can thus be ruledout. The same goes for other bad companions.This is a sketchy argument for several reasons, but for our purposes it needn’tbe more. As we said at the outset, our goal is to provide and clarify options.All things considered, this sort of account may be rejected. But the results hereadduced, along with those in [24] and [23], indicate that the sort of answer justoffered is one of the things that should be considered.In light of this, it may be worth considering relative categoricity as kin topermutation invariance. Both are motivated by the idea that logic is indifferentto the particulars of objects. Permutations represent this indifference by ex-changing objects arbitrarily, while relative categoricity represents it by treatingof arbitrarily selected abstraction assignments. Here, more issues loom. Ourdiscussion has been organized around permutation invariance, but there arestronger conditions claiming to be necessary for logicality. Permutation invari-ance is intra-modular : permuations simply re-order or re-organize the elementsof a given structure. Recent work has focused on the trans-modular notion of isomorphism between structures . This is arguably a better mark of the logicalthan permutation invariance, and better matches the intuitive notion of indif-ference to objects. Though the language L is logical in this stronger sense,we have not addressed how a notion like relative categoricity would apply in a38rans-modular setting. One last point is worth making. The results of section 6 can be recast. Say Jack Woods, in [25], has addressed the question of indefinite abstraction principles in atrans-modular setting. Given a domain D and an equivalence relation E on P ( D ) , let F ( D ) E = { f : P ( D ) → D | ( ∀ X, Y ⊂ D )( fX = fY ⇔ E ( X, Y )) } In this setting of [25], the members of F ( D ) E are called abstraction functions for E , whilean abstraction operator for E is a function σ taking domains D as an argument, outputtinga set σ D ⊆ F ( D ) E . If ζ : D → D ′ is a bijection, then ζ can be extended to an isomorphism ζ + of all types over D and D ′ . As such, ζ + ( σ D ) will be { ζ + ◦ f ◦ ( ζ + ) − | f ∈ σ D } (see [25,pp. 281ff]). The abstraction operator σ is then isomorphism invariant just if ζ + ( σ D ) = σ D ′ for all D, D ′ , and bijections ζ : D → D ′ . An abstraction operator is said to be full just if forany D , σ D = F ( D ) E . Woods endorses the view that indifference to particular objects is themark of the logical, and thus that isomorphism invariance is the mark of the logical.Woods offers as evidence for this claim the proposition on [25, p. 298] that all and onlyabstraction operators logical in the sense of isomorphism invariance are both full and associ-ated with an isomorphism invariant (in a broader sense) equivalence relation. However, thisproposition, and the lemma used to prove it (presented on [25, p. 296]), are incorrect. Thelemma in question claims that if σ is (non-empty and) invariant then it is full; but the proofmakes two incorrect assumptions: the first is that if f, g ∈ F ( D ) E and rng( f ) ≈ rng( g ) then M − rng( f ) ≈ M − rng( g ) [25, p. 296]. The second is that if ζ ( f ( A )) = g ( A ) for f, g ∈ F ( D ) E and all A ⊆ D , then ζ + ( f ) = g . But this is not in general true, for the extension of ζ to ζ + has ζ + ( f ) = ζ + ◦ f ◦ ( ζ + ) − , and while this implies that ζ + ( f )( A ) = g (( ζ + ) − ( A )) , the last g (( ζ + ) − ( A )) = g ( A ) holds only, in general, when E is coarser than ≈ ; see [24, Theorems 1.1and 1.2] and Theorem 55.The lemma and resulting proposition can be shown false by the example of ≈ for E and σ D = { f ∈ F ( D ) E | rng( f ) = D } . Clearly this is not full, and it can be shown isomorphisminvariant using the techniques of the lemma on [25, p. 297]. This would suggest suitablerestrictions on the cardinality or identity of the range of the abstraction functions mightrepair the lemma and proposition. But more robust counterexamples can be generated by theresults of the present paper. For these we will first need the following observation.Suppose M [ ∂ , ∂ ] with first-order domain M is standard and witnesses the fact that A E is not relatively categorical. Then if ζ + : M → M ′ is an isomorphism, then M ′ [ ζ ( ∂ ) , ζ ( ∂ )] also witnesses that A E is not relatively categorical, for M i ∼ = M ′ i for i = 1 , .The main idea of the counterexample is to choose a pair ∂ , ∂ of surjective abstractionfunctions such that M = M [ ∂ ] = M = M [ ∂ ] . Then for M the domain of M , we let σ M be the set of all abstraction functions ∂ such that M [ ∂ ] is isomorphic to M [ ∂ ] . Clearly ∂ will not be among these. Then, for every domain D , we set σ D to be the set containing ζ + ( ∂ ) for each ∂ ∈ σ M and each bijection ζ : M → D . This ensures σ will be isomorphisminvariant, but also ensures, by the observation given above, that it will not be full.More formally: it is a consequence of Theorem 56 that there are abstraction principles withsurjective abstraction functions ∂ , ∂ on a domain M such that for no bijection ζ : M → D is there a bijection π : D → D such that π ( ζ + ( ∂ )( A )) = ζ + ( ∂ )( π ( A )) (That is, the bijection π does not commute with ζ + ( ∂ ) and ζ + ( ∂ ) .) This is true of NewV , asdemonstrated by Theorem 56 in conjunction with the results of [24, Section 5.2].Thus for each D ≈ M , set σ D = { ∂ ∈ F ( D ′ ) E | ∂ = ζ + ◦ ∂ ◦ ( ζ + ) − for some bijection ζ : M → D } The indefinite abstraction operator σ is then isomorphism invariant by construction, but ζ + ( ∂ ) σ ζ ( D ) for any bijection ζ , so σ is not full. Applied to the cited case of NewV developed in [24], we might choose ∂ such that the natural membership relation derived fromit is well-founded. Thus for any D and any f ∈ σ D , f will foster such well-founded relationsas well. But for each D there will always be ∂ ∈ F ( D ) E such that the natural membershiprelation is not well-founded, and thus such ∂ will be omitted from each σ D . A E proves the universe set-like —in symbols, A E | = ZFC ( M ) —just if CC , ISM , IPM , and every instance of CWF is provable in the theory A E . The results ofsection 6, put into this context, say that if A E proves the universe set-like, andif E is provably non-trivial at “large enough” bicardinalities, then A E is incon-sistent. For if A E | = ZFC ( M ) and E is non-trivial at a large enough cardinality,then since ZFC ( M ) , at that cardinal E is either separative or complementative.Then following the theorems of that section, IPM renders the principle inconsis-tent. This can be summed up by: Corollary 57. If a theory A E proves | M | > then A E is inconsistent (withoutthe help of our cardinality principles) if and only if A E proves that there is a well-ordering of M and [ X ] E ( ⊟ ) is non-trivial for some X with | X | = | M − X | = | M | .The non-trivial direction follows because our cardinality principles followfrom a well-ordering of the universe.Similarly we obtain necessary and sufficient conditions for satisfiability (pos-session of a standard model): Corollary 58. If a theory A E proves | M | > then A E is unsatisfiable if andonly if [ X ] E ( ⊟ ) is non-trivial for some X with | X | = | M − X | = | M | .The non-trivial direction again follows by our results, since the cardinalityprinciples are true in all standard infinite models.These are not, however, the best results available, since a weaker conditionon the right-hand side may also be sufficient for inconsistency (respectively, un-satisfiability). We can already obtain such improvements understanding “largeenough” to mean not “universe-sized”, but “exponentially large” in the sense ofFine (see Appendix 10). That is, replace “for some X with | X | = | M − X | = | M | ”on the right with “for some X such that T OP ( X ) ”, and the corollaries still hold.In any case, as an explanation of the consistency and satisfiability profilesof abstraction principles it is worth considering.Outside of the discussion on neo-logicism, we note that the classificationtheorems proved herein, being completely general, may have applications of in-terest in other research areas related to second-order logic. Having answeredsome questions, it is thus apt to raise one in conclusion. We showed in Theo-rem 6 that for concepts finite in M , the conditional arrows “ ⇒ ” reverse for bothseparative and complementative bicardinal slices. Is this true for more than justfinite concepts? In other words, are there L -definable equivalence relations inwhich the arrow of implication does not reverse? Though the lemma and proposition are false, the above counterexamples do not obviouslydetermine the correctness of Woods’s overall assertion: that isomorphism invariance (for ab-straction operators) is the mark of logic’s characteristic indifference to the particularities ofobjects. The second counterexample does depend on the choice of an abstraction function,but not on the choice of any first-order object. This accords with Woods’s prediction. Appendix: The proof of Proposition 8 Proof. Suppose | X | | Y | , so by Cardinal Comparability, for all n ∈ N , M | = n × | Y | < | X | . Thus × | X | ≥ | Y | , so | Y | E | X | , establishing (34).For X ∩ Z = Y ∩ W = ∅ and | Z | g ≤ | W | , suppose | X | E | Y | . Choose n , U ,and f such that | U | = n (in the metatheory), and | X | f ≤ | U × Y | . Then choose u ∈ U , and then let h ( v ) = ( f ( v ) v ∈ X ( u, g ( v )) v ∈ Z (156)The function h is an injection establishing (35).Toward establishing (36), assume | Z | ≤ | W | and | X ⊔ Z | E | Y ⊔ W | , but ¬ ( | Y ⊔ W | E | X ⊔ Z | ) . For a contradiction, assume ¬ ( | X | ⊳ | Y | ) . By (34), | Y | E | X | . But then since | W | ≤ | Z | , by (35) we have | Y ⊔ W | E | X ⊔ Z | ,contradicting our assumption.For (37), choose n, m such that | X | f ≤ n × | Y | and | Y | g ≤ m × | Z | . Thendefine h : X → nm ( n + 1) × Z by h ( x ) = ( nmf ( x ) + g ( f ( x )) , g ( f ( x )) (157)where f and g output the left value of f and g respectively, and f and g output the right value of f and g , respectively. Verifying that h is injective isroutine, h ∈ M by comprehension.To establish (38), suppose M | = X f ≤ n × ( Y ⊔ Z ) . By (34), either Y E Z or Z E Y ; assume the former, letting g witness the injection in M . Then h ( x ) = ( f ( x ) f ( x ) ∈ Z ( n + f ( x ) , g ( f ( x ))) f ( x ) ∈ Y (158)injects X into n × Z ; h ∈ M by comprehension. So X E Z ; a similar argumentshows that if Z E Y then X E Y . Note that (39) follows immediately.For (40), suppose M | = | Y | f ≤ n × | X | and M | = | Z | g ≤ m × | X | . Then h ( u ) = ( f ( u ) u ∈ Y ( n + i, x ) u ∈ Z, g ( u ) = ( i, x ) (159)injects | Y ⊔ Z | into ( n + m ) × | X | in M . So if | X | ⊳ | Y ⊔ Z | , | Y ⊔ Z | ⋪ | X | ,and so | Y | | X | or | Z | | X | . By (34) this implies that either | X | ⊳ | Y | or | X | ⊳ | Z | holds. Note that (41) follows immediately.To establish (42), assume | Y ⊔ Z | f E | X | , then f ↾ Y and f ↾ Z are therequired injections. If | X | E | Y | then by (35) and (37) | X | E | Y ⊔ Z | ; so (43)follows from (42).For (44), assume | X | ⊳ | Y | , so by the definition of ⊳ , we have that | Y | | X | .Thus by (42), | Y ⊔ Z | | X | . Then by (34), | X | ⊳ | Y ⊔ Z | .41or (45), assume | X ⊔ Y | f ⊳ | Z | and | W | g = | Y | . Then h ( u ) = ( f ( u ) u ∈ Xf ( g ( u )) u ∈ W (160)witnesses | X ⊔ W | ⊳ | Z | . A similar argument establishes (46). Then (47) followsfrom (45), (46) and (34), as does (48). 10 Appendix: Fine’s Characterization Theorem We have noted throughout that our Main Theorem is a version of Fine’s Classi-fication Theorem; additionally Theorem 47 bears a striking similarity to Fine’sCharacterization Theorem [7, Theorem 6, p. 144]. In this last appendix we showhow to present Fine’s Characterization Theorem in the deductive setting of ourpaper.Following Fine [7, p. 143], say that a concept is exponentially large in M just its subconcepts outnumber the objects of M . For short-hand, now say thata concept X is “Top” just if X and its complement are both exponentially large.The basal equivalence relation E ( X, Y ) is the equivalence relation such that forany M and X, Y ∈ M , E ( X, Y ) holds (in the metatheory) just if either both X and Y are Top and M | = X ⊟ Y , or neither X nor Y is Top and M | = X = Y . Fine’s Characterization Theorem. The basal relation E is the finest equiv-alence relation satisfying Permutation Invariance such that for any infinite stan-dard model M , M | = A E .The most noticable differences between Fine’s Characterization Theoremand Theorem 47 are that the latter concerns only infinite standard models,while the former addresses all models with well-behaved cardinalities. Fine’sproof of the Characterization theorem uses his Classification theorem. As wehave stressed, this classification theorem is a version of what we have proved asour Main Theorem. But it is hard to see how, given that Fine’s terminologydoesn’t neatly capture the array of possibilities. But some reflection, with Fine’ssuggested aid of a Venn Diagram, shows that the Fine’s Classification theoremcan be restated as follows. Fine’s Classification Theorem (Restated) . For X infinite and M standard:(1) If | X | < | M | and E ( ⊟ ) X is does not refine a separation, then E ( ⊟ ) X istrivial.(2) If | X | = | M | > | M − ( X ∪ Y ) | and E ( ⊟ ) X refines neither a separationnor a complementation, then E ( ⊟ ) X is trivial.(3) If | X | = | M − X | and E ( ⊟ ) X refines neither a separation nor a comple-mentation, then E ( ⊟ ) X is trivial. Fine’s results are cast in terms of partitions of P ( M ) , so at this point we won’t talk about M satisfying E ( X, Y ) , since Fine’s presentation doesn’t ensure that E is expressible. Wewill show how to express something like E in L below, see Definition 60. E ( ⊟ ) X must satisfy at least one of being trivial, refining aseparation, or refining a complementation.Our Theorem 7 can thus be regarded as sharpening and expanding Fine’sachievement in his Classification Theorem, except for one easily ironed wrinkle: Remark 59. In the statement of our theorem: our main concern has been withidentifying L -definability as a necessary condition on an equivalence relation’sbeing logical, so our conditions on E have been that it be L -definable. However,inspection of the proofs of our results will show that L -definability is used onlyin invoking Permutation Invariance. Thus all of our results can be put as Fine’sare: about equivalence relations satisfying Permutation Invariance.As we will show below, Fine’s Characterization Theorem can also be sharp-ened and expanded, with the help of Theorem 7. We prove the CharacterizationTheorem in the deductive setting of Theorem 7.To do so, we first need to use L , rather than the metatheory, to describeconcepts being exponentially large, “Top”, and so to describe the basal equiv-alence relation E . Given a relation R ( x, y ) , we let R [ x ] = { y | R ( x, y ) } . If R ∈ M then so is R [ x ] by comprehension. Using this we can express, in L , theclaim that R “injects” from equivalence classes of subconcepts of a given concept W to objects with the following formula (see also [24, p. 588], [19, p. 105]): ( ∀ U ⊆ W )( ∃ x )( R [ x ] ⊆ U ∧ E ( R [ x ] , U )) ∧ ( ∀ x, y )( ¬ E ( R [ x ] , R [ y ]) → x = y ) (161)We will use the expression “ (cid:12)(cid:12)(cid:12) S [ M ] ↾ WE (cid:12)(cid:12)(cid:12) R ≤ | M | ” to abbreviate (161). We nowhave the material required for expressing the basal equivalence relation: It is worth noting that since we are allowing non-standard models, for some E there maybe “false negatives” (where W = M we omit the restriction “ ↾ M ”): There are models M andequivalence relations E such that M | = ¬ ( ∃ R ) (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) S [ M ] E (cid:12)(cid:12)(cid:12)(cid:12) R ≤ | M | (cid:19) even though there is, in the metatheory, an injection from the E -partition of S [ M ] to M . Asketch of the proof is as follows:The witnessing equivalence relation will be ≈ . Let L be a language expanding L bycountably many constants c i for first-order objects, and the T be the theory containing asaxioms all comprehension axioms in the signature, as well as all sentences of the form: c i = c j for each i = j ¬ ( ∃ R )( ϕ ( R ) ∧ (cid:12)(cid:12)(cid:12)(cid:12) S [ M ] ≈ (cid:12)(cid:12)(cid:12)(cid:12) R ≤ | M | ) for each formula ϕ ( R ) of L Every finite subset of T is consistent (and in fact satisfiable), so T is consistent, and has amodel whose first-order domain M is infinite in the metatheory . The resulting model witnessesthe truth of the theorem. efinition 60. Given E an L -definable equivalence relation, abbreviate asfollows: EXP L ( X ) := ¬ (cid:12)(cid:12)(cid:12)(cid:12) S [ M ] ↾ X = (cid:12)(cid:12)(cid:12)(cid:12) ≤ | M | (162) T OP ( X ) := EXP L ( X ) ∧ EXP L ( M − X ) (163) E ( X, Y ) :=( T OP ( X ) ∧ T OP ( Y ) ∧ X ⊟ Y ) ∨ (164) ( ¬ ( T OP ( X ) ∨ T OP ( Y )) ∧ X = Y ) The abstraction principle A E is the correlate of Fine’s basal abstractionprinciple. Theorem 61 (Generalization of Fine’s Characterization Theorem) . The ab-straction principle A E is consistent, and for any expressible equivalence rela-tion E satisfying Permutation Invariance on all models, if A E is consistent then E is finer than E .(Recall that E is finer than E means that | = ( ∀ X, Y )( E ( X, Y ) → E ( X, Y ) .)Note that Theorem 61 implies Fine’s Characterization Theorem. Proof. We give only the portion of the proof not available elsewhere. To seethat A E is consistent, see [7, p. 144]. Towards establishing the second assertion,note that extensional equality of concepts is the finest Permutation Invariantequivalence relation ([1, Theorem 2, p. 281] also uses this fact). So let M | = A E ;if E is strictly finer than E , there are W, S such that M | = T OP ( W ) ∧ T OP ( S ) , W ⊟ S and ¬ E ( W, S ) . Thus E ( ⊟ ) W is non-trivial, and by Theorem 7, it eitherrefines a separation or refines a complementation.If it refines a complementation then | W | = | M − W | . If | M | ≤ then M | = ( ∀ X, Y )( E ( X, Y ) ↔ X ≈ Y ) and so A E is HP . But HP has no models of finite size, and since E is finer than E , A E has no finite models either. Thus M is infinite. But then by Theorem 47, M 6| = A E .If E refines a separation, then by Lemma 39, A E and | W | ≤ | M − W | trans-late BLV ( ≤ W ) . Set R ( x, y ) to be ( ∃ Y )( x = εY ∧ Y y ) ; R exists by comprehensionand R [ ǫY ] = Y . It is easy to verify using BLV ( ≤ W ) implies (cid:12)(cid:12)(cid:12)(cid:12) S [ M ] ↾ W = (cid:12)(cid:12)(cid:12)(cid:12) R ≤ | M | and this contradicts the assumption that T OP ( W ) . Fine’s Characterization results can in fact be generalized neatly to more types of invari-ance, as Fine does himself; see [7, Corollary 7, p. 146], and [5]. cknowledgements The author wishes to thank Sean Walsh, Roy Cook, Jack Woods, and EileenNutting for helpful discussions, comments, and encouragement on this paper.All mistakes, however, are my own. The work on this paper was done much inanticipation of showing it to the author’s late friend and former teacher, AldoAntonelli. That Aldo could not see the conclusion of his inspiration and supportmarks the completion of this project with great sadness. References [1] G. Aldo Antonelli. Notions of Invariance for Abstraction Principles. Philosophia Mathematica , 18(3):276–292, 2010.[2] Denis Bonnay. Logicality and Invariance. Bulletin of Symbolic Logic ,14(1):29–68, 2008.[3] George Boolos. The Standard Equality of Numbers. In Meaning andMethod: Essays in Honor of Hilary Putnam , pages 261–277. CambridgeUniversity Press, Cambridge, 1990.[4] Tim Button. The Limits of Realism . Oxford University Press, Oxford,2013.[5] Roy T. Cook. Abstraction and four kinds of invariance (or: What’s sological about counting). Philosophia Mathematica , 2016.[6] Sean C. Ebels-Duggan. The Nuisance Principle in Infinite Settings. Thought: A Journal of Philosophy , 4(4):263–268, December 2015.[7] Kit Fine. The Limits of Abstraction . The Clarendon Press, Oxford, 2002.[8] Gottlob Frege. The Foundations of Arithmetic: A Logico-MathematicalEnquiry into the Concept of Number . Northwestern University Press,Evanston, second edition, 1980. Translated by John Langshaw Austin.[9] Bob Hale. Reals by Abstraction. Philosophia Mathematica , 8(3):100–123,2000. Reprinted in [10].[10] Bob Hale and Crispin Wright. The Reason’s Proper Study . Oxford Univer-sity Press, Oxford, 2001.[11] Bob Hale and Crispin Wright. To Bury Caesar. In The Reason’s ProperStudy , pages 335–396. Clarendon Press, Oxford, 2001.[12] Richard G. Heck, Jr. Finitude and Hume’s principle. Journal of Philosoph-ical Logic , 26(6):589–617, 1997. 4513] Richard G. Heck, Jr. The Julius Caeser Objection. In Language, Thought,and Logic: Essays in Honour of Michael Dummett , pages 273–308. OxfordUniversity Press, Oxford, 1997. Edited by Richard G. Heck Jr.[14] Fraser MacBride. On Finite Hume. Philosophia Mathematica , 8:150–159,2000.[15] John MacFarlane. Logical constants. In Edward N. Zalta, editor, TheStanford Encyclopedia of Philosophy . Fall 2015 edition, 2015.[16] Vann McGee. Logical operations. Journal of Philosophical Logic , 25:567–580, 1996.[17] Charles Parsons. Frege’s Theory of Number. In Max Black, editor, Philos-ophy in America , pages 180–203. Cornell University Press, Ithaca, 1965.[18] W.V. Quine. Philosophy of Logic . Harvard University Press, Cambridge,1970.[19] Stewart Shapiro. Foundations without Foundationalism: A Case forSecond-Order Logic , volume 17 of Oxford Logic Guides . The ClarendonPress, New York, 1991.[20] Stewart Shapiro and Alan Weir. New V, ZF and Abstraction. PhilosophiaMathematica , 7:293–321, 1999.[21] Gila Sher. The Bounds of Logic: A Generalized Viewpoint . MIT Press,1991.[22] Alfred Tarski. What are Logical Notions? History and Philosophy of Logic ,7(2):143–154, 1986.[23] Sean Walsh. Comparing Hume’s Principle, Basic Law V and Peano Arith-metic. Annals of Pure and Applied Logic , 163:1679–1709, 2012.[24] Sean Walsh and Sean Ebels-Duggan. Relative Categoricity and AbstractionPrinciples. The Review of Symbolic Logic , 8(3):572–606, September 2015.[25] Jack Woods. Logical indefinites. Logique et Analyse , 227:277–307, 2014.[26] Crispin Wright. Frege’s Conception of Numbers as Objects , volume 2 of Scots Philosophical Monographs . Aberdeen University Press, Aberdeen,1983.[27] Crispin Wright. On the Philosophical Significance of Frege’s Theorem.In Richard G. Heck Jr., editor, Language, Thought, and Logic: Essaysin Honour of Michael Dummett , pages 201–244. Oxford University Press,Oxford, 1997.[28] Crispin Wright. Is Hume’s Principle Analytic?