bonnay invariance definability

8/14/2019 bonnay invariance definability

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arXiv:1308

.1565v1

[math.LO

]7Aug2013

INVARIANCE AND DEFINABILITY, WITH AND WITHOUT EQUALITY

DENIS BONNAY AND FREDRIK ENGSTROM

Abstract. The dual character of invariance under transformations and definability

by some operations has been used in classical work by for example Galois and Klein.

Following Tarski, philosophers of logic have claimed that logical notions themselves could

be characterized in terms of invariance. In this paper, we generalize a correspondence due

to Krasner between invariance under groups of permutations and definability inLso as

to cover the cases (quantifiers, logics without equality) that are of interest in the logicality

debates, getting McGees theorem about quantifiers invariant under all permutations and

definability in pure L as a particular case. We also prove some optimality results

along the way, regarding the kind of relations which are needed so that every subgroup of

the full permutation group is characterizable as a group of automorphisms.

1. Introduction. Permutations and relations may be seen as two ways ofencoding information about a domain of ob jects. In the case of relations, it ispretty obvious that some piece of information is encoded: a difference is madebetween objects between which the relation holds, and objects between which itdoes not. This is also true, if slightly less obvious, of permutations. By takinginto account a permutation, it is recognized that some differences between objects

do not matter: given the purpose at hand, one may as well consider that suchand such objects have been interchanged.Moreover, permutations and relations may be seen as two dualways of encod-

ing information about objects. Permutations tell us which ob jects are similar(they can be mapped to each other), relations tell us which objects are different(think of a unary relation true of one object and false of another). This duality isat work whenever mathematicians or physicists consider groups of isomorphismsand classes of invariants.1 A group of automorphisms is a group of permuta-tions that respect a given structure: permutations are extracted from a domainequipped with relations. A class of invariants consists of a class of relations thatrespect a given group of permutations: extensions for symbols to be interpretedin the domain of objects under consideration are extracted from permutations.

In logic, this duality has been extensively investigated within model theory.

Typical questions include the characterization of structures that can be described

Both authors were partially supported by the EUROCORE LogICCC LINT program and

the Swedish Research Council.1Kleins Erlanger Program [11] and the systematic study of the duality between groups of

transformations on a space and geometries on that space is a famous example. It was also adirect inspiration to Tarski [21].

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2 DENIS BONNAY AND FREDRIK ENGSTROM

in terms of their automorphism groups, or more generally how much automor-phism groups say about the structures they come from (see [4] on model theoryand groups). Within the philosophy of logic, the duality between permutationsand extensions comes to the forefront in debates regarding the characterizationof logical constants. Invariance under all permutations has been taken as theformal output of a conceptual analysis of what it is to be a logical notion. Thus,it has been proposed as a mathematical counterpart to the generality of logic (byTarski himself, [21]) and to its purely formal nature (by Sher [19] and MacFarlane[16]).

The aim of the present paper is to bring these two traditions closer together byadapting some results in model theory to the cases that are discussed by philoso-phers of logic interested in invariance as a logicality criterion. As a case in point,we will show how the much quoted theorem by McGee in [17] about invarianceunder all permutations and definability in the pure infinitary logic L can be

seen as following from a suitable extension of much earlier work by Krasner ([12],[13]). Our motivation for such a generalization is that a conceptual assessmentof a given logicality criterion stated in terms of invariance properties should begrounded in a prior evaluation of the significance of the general duality betweengroups of permutations and sets of relations and in an independent assessment ofwhat invariance under groups of permutations (or other kind functions) can andcannot do in the contexts of interest. From a technical perspective, our startingpoint will be Krasners work and his so-called abstract Galois theory. Appli-cation to cases of interest in the logicality debates involve two generalizations.In order to cover the interpretation of quantifiers, we will to need to considerstructures equipped with second-order relations. In order to cover scenarii inwhich the logicality of identity is not presupposed, we will to need to considerfunctions that are not injective, as suggested by Feferman [8].

Our main result (Theorem 11 below) provides a unified perspective on theduality by establishing a correspondence between certain monoids of relationsand sets of operations closed under definability in an infinitary language withoutequality. In Section2, we recall Krasners correspondence between classes of re-lations of possibly infinite arity closed under definability in Land subgroupsof the permutation group. We show how Krasners correspondence relates tothe standard model-theoretic characterization of groups of permutations whichare the automorphism groups of a first-order structure (with relations of finitearity) as closed groups in the topology of point-wise convergence. Section3 isdevoted to extending Krasners correspondence to second-order relations. Weprove some optimality results along the way: what kind of relations are reallyneeded so that every subgroup is the automorphism group of a class of relations?In the first-order case, infinitary relations were necessary. Second-order binary

relations suffice to do the job, but second-order unary relations do not. The nextsection is devoted to relaxing the assumption of injectivity, extending the corre-spondence to logics without equality. Finally, we draw some lessons in Section5for the logicality debate.

2. Abstract Galois theory and permutation groups.


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INVARIANCE AND DEFINABILITY 3

2.1. Krasners abstract Galois theory. To begin with, we shall recall thefundamentals of Krasners abstractGalois theory, following in particular Poizat[18]. The starting point of Krasner isclassicalGalois theory, in which a particularinstance of the general duality we have outlined is at play. Classical Galois theoryis the study of the duality between the subgroups of the automorphism group ofcertain types of field extensions, called Galois extensions, and the intermediatefields in the field extension.

A field is an algebraic structure with addition, multiplication, inverses andunits. Ifk Kare both fields then we say that K :k is a field extension; it isGalois if it is algebraic, normal, and separable (definitions may be found in [14]).Given a field extension K :k letG be the Galois group ofK :k, i.e., the groupof automorphisms ofKfixing the elements ofk point-wise. Define the followingpair of mappings:

Fix(H) = { a K| ha = a for all h H}Gal(A) = { g G | ga = a for all a A }

whereHis any subset ofGandAany subset of the domain ofK. The fundamen-tal theorem of Galois theory says that, for finite Galois extensions, Fix(Gal(A))is the smallest subfield ofKincludingk Aand that Gal(Fix(H)) is the smallestsubgroup ofG includingH. There is thus a one-to-one correspondence betweenfieldsK such thatk K Kand subgroups ofG.

According to Krasner, the true origin of Galois theory [does not lie] in algebra,in the strict sense of the word, but in logic ([13], p. 163). This claim is to besubstantiated by exhibiting a general duality, between automorphism groups andrelations definable over a structure. Consider a domain , S the full symmetricgroup on and a set r of relations on ; these relations may include infinite

arity relations regarded as subsets of where is some ordinal number. Wenow define the following pair of mappings:

Inv(H) = { R |hR = R for all h H , || }

Aut(r) ={ g S| gR= R for all R r }

The logic L is the infinitary generalization of the predicate calculus whereformulas are built by means of arbitrarily long conjunctions and disjunctions andby means of arbitrarily long universal and existential quantifiers sequences (seee.g. [10]). The L-closure ofr is the set of relations definable in L fromrelations in r. The following Theorem is essentially2 shown by Krasner in [12]:

Theorem1. Let andS be as above, HS any set of permutations and

r any set of relations on of arities at most ||.

1. Inv(Aut(r)) is theL-closure ofr.2. Aut(Inv(H)) is the smallest subgroup ofS includingH.

2The qualification essentially is due to the fact that Krasner thinks directly in terms

of closure under logical operations on relations rather in terms of definability in a formallanguage.


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Thus, there is a one-to-one correspondence between the subgroups of the fullsymmetric group S of and the sets of relations closed under definability inL.

How does this relate with the original correspondence of classical Galois the-ory? The first half of the fundamental theorem of Galois Theory can be derivedfrom the first half of Theorem 1.3 Given a field extension K: k andA K, letr be the set of addition, multiplication, inverse and the relation Ra = { a } foreverya k A. By definition, we have that Gal(A) = Aut(r). Now Inv(Aut(r))and Fix(Gal(A)) are not exactly on a par, because Krasners theory takes setsofrelationsas values instead of, as in the classical Galois theory, sets ofpoints.However, the set of fixed points can be extracted from the set of fixed relationsby observing that an elementa is fixed by a permutation iff the singleton re-lation{ a }is fixed. Given that Inv(Aut(r)) is closed under definability in L,it is immediate that this set of fixed points is a subfield ofK. In order to get the

first half of the fundamental Galois theorem, one also needs to show that it isincluded in any subfield extendingk A. Krasner proves this in [13] by showingthat ifRa is definable in L over r, then a can be reached from elements ink A by means of the field operations.4

2.2. Permutation groups and topology. Krasners result shows that forany group of permutations of there is some set of relations r on such thatAut(r) is exactly that group. This is not true if we restrict the relations to beof finite arity. Ifr is a set of relations of finite arity then (, r) is a standardfirst-order relational structure and, obviously, the group Aut(r) coincides withthe automorphism group of the first-order structure (, r). However it is not truethat every subgroup of the full permutation group is the automorphism group ofa first-order structure (with only finite arity relations). We will give a detailedproof of the well known characterization of autormophism groups of first-order

structures in terms of topological properties. This is done partly to set up thescene for the next section (but see also [4]).

First we need to introduce the topology, the product topology, on the groupof permutations that we will be using. Let be any infinite set equipped withthe discrete topology, i.e., the topology in which every set is open. LetG bethe full symmetric groupS on , i.e., the group of all permutations of . ThestabilizerGa of a finite tuple a k is the set of all permutations in G keepingall elements in a fixed, i.e.,

Ga = { g G | ga= a } .

The symmetric group S on is a subset of ; let G inherit the product

topology from , i.e., a basis for this topology are the (right or left) cosets ofstabilizers of finite tuples, that is, sets of the following form:

Ga,b= { g G | ga=b } ,

3To our knowledge, the question regarding the other halves is still open.4This amounts to a property of eliminability of quantifiers. Krasner gives a general char-

acterization of structures having this property. See Poizat [18] regarding connections withcontemporary model theory.


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where a, b k.5

This is the topology of pointwise convergence: The (topological) closure of aset A G consists of all permutations g G such that for every k and everya k there isfA such that f(a) = g(a).6

Given a subgroup H of G let OHa = { ha| h H} be the orbit of a underH. Orbits provide a canonical way of translating a group of permutationson a domain into relations on that domain. (, OHa )ak is thus a first-orderrelational structure, called the canonical structure ofH.

The following well-known folklore result characterizes the subgroups of Gwhich are automorphism groups of a first-order structure (with only finite arityrelations):

Proposition2. LetH be a subgroup ofG. The following are equivalent:

1. There is a first-order structure M with domain whose automorphismgroup isH.

2. His the automorphism group of the canonical structure ofH,(, OHa )ak.3. H is closed.

Proof. (1) implies (3): Letg G be in the closure ofH, i.e., for any a k

there is h Aut(M) which is identical to g on a. Let R be a relation in M,a k and h Aut(M) such that ga = ha, then Ra iff Rha iff Rga and soR is invariant under g. Similarly we can show that any relation, constant andfunction inMis invariant underg; thusg Aut(M); and Aut(M) = His closed.

(2) implies (1): Obvious.(3) implies(2): LetMbe the canonical structure ofH. ClearlyHAut(M)

since Hrespects all its orbits, i.e., for all h H, a OHb

iffha OHb

. Assume

g Aut(M) and a k, then ga OHa and so there is an h H such thatga= ha. This shows that g is in the closure ofH; but sinceHis closed we have

Aut(M) H, proving thatHis the automorphism group ofM.

The proof actually shows that for any subgroup H of G the automorphismgroup of the canonical structure ofHis the closure ofH.

In order to see that not all subgroups ofG are automorphism groups of first-order structures, it is, using the proposition, sufficient to find a subgroup ofGwhich is not closed. Let the support of a permutationg of be the set of pointsmoved by g, i.e., the set { a | ga =a }. The set H of permutations with

5This topology makes G into a topological group which means that the operations of mul-

tiplication (in this case composition) and taking inverses are continuous. As with topologicalgroups in general, all open subgroups are also closed: let H be an open subgroup of G thenthe complement ofH is the union of all cosets gH of Hwhich are disjoint from H, and thus

the complement ofH is open.6

If ={ a0, a1, . . . } is countable then the topology is completely metrizable by the metricd(g, h) = 1/2k ifg =h and k is the least number such that gak =hak or g1ak =h

1ak, andd(g, g) = 0. Thus, a sequence of permutations g0, g1, . . . converges to g if and only if, for any

k, gnak = gak for all sufficiently large n. It should also be noted that G is not a closed subsetof which means that there are sequences gi of permutations of converging to a function

fwhich is not a permutation: Let = N, the sequence

(0, 1), (0, 1, 2), (0, 1, 2, 3), . . .

converges to the function f: n n + 1 which is not a permutation.


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finite support is a non-closed subgroup ofG: Take any permutationg G andany a k we can find h H such that ga = ha by extending the bijectiong|a : { a } { ga } to a permutation ofA = { a } { ga }. Let thenh be thispermutation on A and the identity function outside A. This shows that theclosure ofH is the full group G, and since not every permutation has a finitesupport, His not closed.

Hence the use of relations of infinite arity in Krasners correspondence is indeednecessary: For non-closed H, ifH= Aut(M) then Mincludes relations of infinitearity.

There is a connection between the allowed arities of relations in the structuresand the topological closure property characterizing automorphism groups as thefollowing proposition shows. To see this, let us define the-topology, for aninfinite ordinal, on G in the following manner. The basic open sets are of theform

Ga,b { g G | ga=b } ,

where|a|= |b|< . The -topology coincides with the ordinary product topol-ogy described above. It should be noted that the -topology is a refinement ofthe-topology if , which means that an open set in the -topology is alsoopen in the -topology. If we relax the definition of a first-order structure toinclude relations of arity less than and define -orbits of a subgroupHto be

OHa ={ ha| h H} ,

where now a ; then the proof of Proposition 2 goes through with thecanonical structure ofHreplaced by (, OHa )aand closed replaced by closedin the -topology, giving the following generalization:

Proposition3. LetHbe a subgroup ofG and an infinite ordinal, then thefollowing are equivalent:

1. There is a first order structureMwith domain and relations of arity lessthan whose automorphism group isH.

2. H is the automorphism group of the relational structure(, OHa )a .3. H is closed in the-topology.

Krasners result that every subgroup ofG is Aut(M) for some relational struc-ture M when relations of infinite arity are allowed (first half of Theorem 1)follows from Proposition3. The (||+1)-topology is the discrete topology, i.e.every set is open: Ifg G and a is an enumeration of then

Ga,ga = { g }

is a basic open set. Thus every set of permutations is the union of basic opensets, and therefore open; and therefore also closed. Hence, if relations of arity|| are allowed, every group of permutations will be the automorphism group ofsome relational (infinitary) structure.

3. Quantifiers come into play.


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3.1. Second-order relations, L and Krasners correspondence.

We shall now extend Krasners correspondence to second-order operations, inorder to account for quantifier extensions, which have been the traditional focusof the debates regarding logicality and invariance. In this subsection, we statethe corresponding generalization of Theorem1 and prove its first part.

A finite second-order relationQ of type (i1,...,ik) on is a subset ofP(i1)

... P(ik) for finite k and finite i1,...,ik. A second-order structure q on adomain is a set of (finite-ary) first-order and second-order relations on .A permutation g on preserves a second-order relation Q of type (i1,...,ik)if (Ri1 ,...,Rik) Q iff (gRi1 ,...,gRik) Q where Rij , 1 j k, are first-order relations of aritiesij. The mappings Aut and Inv admit a straightforwardgeneralization to the present setting: Aut(q) is the group of permutations whichpreserve all first-order and second-order relations in q, and Inv(H), for HG,is the set of first-order and second-order relations which are preserved by all

permutations in H.Given a second-order structure q, L(q) is an interpreted language in thelogic L: it is the language whose signature matches the structure q andwhose predicate and quantifier symbols are interpreted by the relations in q.The syntax and semantics for symbols interpreted by second-order relations isfamiliar from generalized quantifier theory (see [15]). As a case in point, let usrecall what are the intended syntactic and semantic clauses for a second-orderrelation of type (2): LetQ in qbe of type (2), L(q) is then equipped witha matching quantifier symbol Q. The syntactic clause for Q has it that if is aformula ofL(q), so is Qx1x2 . The satisfaction clause for Q is given by

, q Qx1x2 []

iff||||x1,x2 Q

where is an assignment over and

||||x ={(a1, a2) 2 | , q [[x1 := a1, x2 := a2]]}.

We shall need to consider definability in L(q) for a given q. Without lossof generality, let Q be a second-order relation of type (2). We say that Q isdefinable in L(q) iff there is a sentence Q(R) in L(q) expanded with abinary predicate symbol R such that

, q , R Q(R)

iff

R Q

whereR is a binary first-order relation on interpreting R. The L-closureof a second-order structure q is the set of finite first-order and second-orderrelations on which are definable in L(q).

We can now state the generalization of Theorem1 to second-order structuresand automorphism groups thereof:


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Theorem4. Let be a domain and G = S the symmetric group on ,HG any set of permutations andqa second-order structure on.

1. Q Inv(Aut(q))iffQ is definable inL(q). The same holds for relationsR.

2. Aut(Inv(H)) is the smallest subgroup ofG includingH.

We shall postpone the discussion and the proof of the second part of thetheorem till the next subsection. Regarding the first part, the proof that anyrelation definable in L(q) is invariant under the automorphisms of q is astraightforward induction on the complexity of formulas ofL(Q). Whatremains to be proven is the following lemma:

Lemma 5. LetQ be a first-order or second-order relation on, ifQ Inv(Aut(q)),thenQ is definable inL(q).

Proof. For the sake of simplicity, we shall give the proof for Qa second-orderrelation of type (2). So let Q be invariant under the automorphims of q. Wefirst construct the sentence Q(R) which defines it. Let us fix an enumerationI of the elements in and, given a set of variables Xof cardinality||, anenumerationIXof the variables inX. Let be a partial assignment overXdefined by (xi) = ai fori I.

For any relationSon and for any symbol Tmatching the type ofS, let thedescription S(T) ofSby Tbe as follows. IfSis a first-order relation of arityk,S(T) is the conjunction of formulas of the form T xi1 ...xik orT xi1 ...xik whichare satisfied in , Sunder . Similarly, ifSis a second-order relation, say againof type (2), S(T) is the conjunction of formulas of the form

T y1, y2 (i,j)K

(y1 = xi y2 = xj)

or

T y1, y2

(i,j)K

(y1 = xi y2= xj)

for K II, which are satisfied in , S under . Moreover, let be thefollowing formula

i,jI,i=j

xi =xj yiI

y= xi

that is, says that is a bijection from Xonto .Q may now be written down as

X

Sq

S(S)

UQ

U(R)

whereXis short for the infinite sequence of universal quantifiers x1,..., xi,...quantifying over every variable in X.


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We now prove that , q , R Q(R) iffR Q. From left to right, ifQ(R) istrue in , q , R, we have in particular that

Sq

S(S) UQ

U(R)(1)

is satisfied under . But, by construction, the premise of (1) is satisfied under, hence there is a relationUQ such that U(R) is satisfied under when Ris interpreted by R. This forces U=R, hence R Q as required.

From right to left, let R be a first-order relation in Q and a partial assign-ment onX. Assume that

, q , R Sq

S(S) [].

We need to show that , R UQU(

R) [].

Since is satisfied under , is a bijection, hence there is a permutation f

on such that f =. Moreover, since

SqS(S) is satisfied under , fis an automorphism ofq, hence fpreservesQ.

By definition, , f R fR (R) []. Since = f, this implies , R fR (R) . Since fpreserves Q and R Q, we also have f R Q. ThereforefR (R) is one the disjuncts in

UQU(

R) and the formula is satisfied under

as required.

In [17], McGee proved that, on a given domain, an operation is invariantunder all permutations if and only if it is described by some formula ofL(p. 572). This result follows quite directly from what we have:7 Consider theempty structure. Aut() is the group G of all permutations. By the first partof Theorem4, Inv(G) is the L-closure of, which is exactly McGees result.It should be noted that the proof of Lemma 5itself is a rather straightforward

generalization of Krasners proof in [12]. The possibility to describe every subsetof k by means of formulas of L is sufficient to handle the extension tosecond-order relations.

3.2. A topology for second-order structures. We shall now turn to thesecond half of Theorem4and ask when a group of permutations is the automor-phism group of a structure, in a context where second-order relations are allowedbut all relations are finitary. As earlier with Proposition2,we shall base our an-swer on topological properties of groups of permutations. We will see that if weallow quantifiers of type (2) then all groups of permutations are automorphismgroups of some second-order structure. However, the case is quiet different whenwe only allow monadic quantifiers, i.e., quantifiers of type (1, 1, . . . , 1). Thereis a natural topology such that exactly the closed groups are the automorphismgroups of monadic second-order structures.

We say a second-order structure is monadic if all relations are relations on thepower set of . A monadic second-order structure on is nothing else than afirst-order structure on together with a relational first-order structure on the

7McGees operations are actually defined as functions from sequences of sets of assignments

to sets of assignments. This difference does not matter in the present context, since McGeesoperation may be translated into second-order relations and back in such a way that invarianceunder sets of permutations is preserved.


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power set P() of . LetSP()be the symmetric group on P(). The symmetricgroup G = S on embeds in a natural way in S

P(), by the mapping g g

wheregA= { ga | a A } .

LetSP() be equipped with the product topology.The second-order topology on G is defined by letting the basic open sets be

GA,B = { g G | gAi = Bi for all i }

where A= A0, . . . , An and B= B0, . . . , Bn are subsets of . As is seen directlyfrom the definition of the basic open sets the second-order topology onG is thetopology induced by the embedding :G SP().

Lemma 6. The setG ={ g |g G } is a closed subgroup ofSP().

Proof. Since respects composition G is a group. To see that it is closedlet h S be in the closure ofG, i.e., for all A = A0, . . . , An there is g Gsuch that hAi = gAi for i n. Define f G by f a h { a }, this uniquelydetermines f since the set h { a } is a singleton set: There is g G such thath { a }= g { a }= { ga }.

Assume now that h /G, then there is A such that hA =fA. We mayassume that there is a hA \ fA. For otherwiseh(Ac) \ f(Ac)= since thecomplement (fA)c offA is f(Ac) and also that h(Ac) = (hA)c since there isg G such that gA= hA and g(Ac) = h(Ac) (take A= A, Ac).

Let g G be such that gA = hA and g { f1a } = h { f1a } (take A =A, { f1(a) }). Thus gf1a = f f1a = a and so f1a A contradicting thata /fA.

From this lemma it follows that a set A G is closed in the second-ordertopology iffA is closed in SP().

By using Proposition2 and considering a second-order monadic structure onthe domain as a first-order structure on P(), we can prove the characteriza-tion of automorphism groups of monadic second-order structures.

Proposition7. LetH be a subgroup ofG. The following are equivalent:

1. There is a second order monadic structure q with domain whose auto-morphism group isH.

2. H is the automorphism group of the second-order structure

M = (, OH

R )R(P())k .

3. His closed in the second-order topology.

Proof. The implication2 1 is trivial.To prove that1 3 let qbe a second-order structure with Aut(q) = H, let

M be the first-order part of q and q

the second-order part. By Proposition2 Aut(M) G is closed. Now, q can be seen as a first-order structure onthe domain P(), and by applying Proposition 2 to this structure on P()we get that AutP()(q

) SP() is closed.8 By the comment after Lemma 6

8 AutP()(q

) is the group of all permutations ofP() respecting the quantifiers in q.

Compare this to Aut(q), which is the set of all permutations of respecting the quantifiers

in q .


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Aut(q) G is closed since

Aut(q) =G AutP()(q),

and both of these subgroups are closed. To conclude the argument it is enoughto observe that Aut(q) = Aut(M) Aut(q) and that both of these groups areclosed.

For the last implication 3 2 it is enough to observe that H SP() isclosed and thus, by Proposition2, H is the automorphism group of

(P(), OH

R )R(P())k

and that g G respects Q iffg SP() respects Q, as a first-order quantifieronP(). Thus, His the automorphism group ofM.

To conclude the discussion of automorphism groups of monadic second-orderstructures we prove that not all subgroups ofG are closed in the second-order

topology, implying that not all subgroups are automorphism groups of suchstructures which proves that the second part of Theorem 4can not hold if werestrict Inv() to only include monadic quantifiers. The following example ofnon-closed subgroup comes from Stoller [20].

Assume that is countable and well-orders . Say that a permutationp of is a piece-wise order isomorphism if there are partitions (Ai)ik and (Bi)ikof such thatp restricted to Ai is an order isomorphism from Ai toBi. LetHbe the set of all such piece-wise order isomorphisms.

Lemma 8 ([20]). H is a proper dense subgroup ofG.

Proof. For simplicity assume that = N and that the ordering coincideswith the usual ordering of the natural numbers.

It is rather straight-forward to check that H is a subgroup. That it is properfollows from the following argument. Let g(m) be (n + 1)2 (m + 1 n2) wheren2 m


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Let us now prove the second half of Theorem 4 by observing that the gener-alization of the second-order topology on G where we regard the permuations ofGas permuting binary relations on is the discrete topology and thus that anysubgroup is closed in this topology:

Proposition9. The topology onG given by the basic open sets

GR,S={ g G | g(Ri) =Si }

whereR = R0, . . . , Rn andS = S0, . . . , S n are subsets of 2 is the discrete

topology.

Proof. Let < well-order and let g G be some permutation of . LetR = { g(


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the same role that isomorphisms play for first-order logic with equality, see [6].10

The present extension of Krasners correspondence pushes the analogy further.Our extension comes with a twist. In L, on a fixed domain and with

parameters, all subsets of the domain are definable. This is not the case in L.Generalized quantifiers may add expressive power that remains ineffective evenwhen parameters are allowed. As a consequence, the proof method used in theproof of Theorem4breaks down. It relied on describing from below the actionof invariant quantifiers by describing elementwise the sets and relations theyapply to. Such a strategy is no longer available.

To see this, let us first define invariance under similarity. A relation R k

is invariant under the similarity relation on if for all a b we have a Riff b R. Given a quantifier Q over the domain we say that Q is invariantunder if for all relationsR1, . . . , Rk, S1, . . . , S k on such thatRi Siwe haveR1, . . . Rk, Q iff S1, . . . , S k Q. Consider now, as an extreme case, the

operations q= {, , QE

}, where and are the two trivial unary relations11

defined by = , = and QE = {{0, 2, 4, 6, 8, 10,...}}. One may showthat QO = {{1, 3, 5, 7, 9, 11,...}} is invariant under all similarities under whichoperations in qare invariant. But in order to describe from below the action ofQO, we would need to be able to define the set of odd numbers. As can easilybe proven by induction, this is not possible in a language in which and arethe only relation symbols, even in the presence ofQE.

Our alternative strategy will consist in showing that Krasners correspondenceholdswhen the extra expressive power is discarded. More precisely, we will showthat the desired definability and closure properties hold when attention is limitedto the action of generalized quantifiers on definable sets and relations. Notehowever that even though QO is not definable from below in L(q), it isindeed definable full stop, simply by the formula QEx P x. Therefore, the

question whether a version of Krasners correspondence without a restriction todefinable relations holds is still open.

Given a set of operationsqand a quantifier Q, letQq be the restriction ofQto relations definable in L(q) (with parameters):

Qq ={ R1 . . . Rk Q | Ri is definable in L(q, a)a, 1 i k }

Letq be all the relations inqand all the restrictions Qq of quantifiers inq. Itis easy to see that L(q) and L

(q

) are elementary equivalent. In thatsense, restricting quantifiers to their definable parts does not change the effectiveexpressive power of the logic.

Restriction to definable parts on the side of quantifiers will correspond to asimilar restriction on the side of the invariance condition. Invariance will onlybe required with respect to relations that respect the underlying equivalence

relation generated by the operations or the similarities under scrutiny. A set of10In [6], Casanovas & alii use the term relativeness correspondence, whereas Feferman in [8]

goes for similarity relation. Note also that, instead of similarity relations, one may work withsurjective functions. Any relational composition of surjective maps or inverses of surjectivemaps is a similarity relation, and any similarity relation can be written as the composition

f g1, where f :C B and g : C A are onto, see [6] for more details.11It is useful to assume that there are symbols in the language to express these, in order to

guarantee that L(q) is not empty.


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operations qgives us an equivalence relation q corresponding to definabilitywith parameters in L

,

(q), that is a q b iff

L(q)

x((a,x) (b, x)).(2)

Similarly, a set of similarities generates an equivalence relation by the fol-lowing condition:

a b iff for all c k there exists such that a, c b, c, for a finite k .

Invariance is now parametrized by the equivalence relations we are con-sidering. Given a similarity, a quantifier Q is -invariant under if for anyR, S P(k1) . . . P(kl), all invariant under , ifR S then R Q iffS Q. As before, a relation is-invariant under if it is invariant under .This yields the main definitions of this section:

Definition10.- Let Inv() be the set of all relationsR and quantifiers Q on which are

-invariant under all similarities in .- Sim(q) is the set of similarities such that all relations and quantifiers in

qareq-invariant under .

Observe that when is a group Hof permutations then Inv() = Inv(H),where Inv(H) uses the old definition of Inv. This follows directly from the factthat in this case is equality.

In general, Sim(q) is not a group, because the similarities need not be invert-ible. The relevant closure properties for Sim(q) are the following. A set ofsimilarities is amonoid with involutionif it is closed under composition and tak-ing converses.12. Moreover, is full if it is a monoid with involution, ,

and closed under taking subsimilarities, i.e., such that if and is asimilarity then .

We are now ready to state Krasners correspondence for the equality-free case,with the aforementioned restriction to the definable parts of quantifiers:

Theorem11. Letqbe a set of operations and a set of similarities, then

1. Q Inv(Sim(q)) iffQq is definable inL(q), andR Inv(Sim(q)) iffR is definable inL(q).

2. Sim(Inv()) is the smallest full monoid including.

As a test case letqinclude equality. Then q is equality and Sim(q) = Aut(q).Theorem11says that a quantifier Qis in Inv(Aut(q)) iff it is definable in L(q)which is the same as being definable in L(q). Thus, part one of Theorem4follows easily from Theorem11. Also part two follows since ifH is a subgroupof the symmetric group of then His nothing but equality and Aut(Inv(H))is just Sim(Inv(H)) which is Hby (2) of Theorem11.

In order to prove Krasners correspondence for definable relations, we shallneed some technical machinery. We define two mappings: / and whichwill operate on several different domains, but in a similar way. Let be an

12The converse R1 ofR is such that a R1 b iffb R a.


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equivalence relation on , [a] = { b| a b } be the equivalence class ofa, and/ the set of all equivalence classes. Given R k we define the relationR/ on / by

R/= { [a1], . . . , [ak] (/)k | a1, . . . , ak R } .

IfR (/)k then

R= { a1, . . . , ak k | [a1], . . . , [ak] R } .

Similarily, given a quantifier Q on , the quantifier Q/on / is defined by

Q/= { R1, . . . , Rk | R1, . . . , Rk Q } ,

and given a quantifier Q on /, the quantifierQ on is defined by

Q= { R1, . . . , Rk | R1, . . . , Rk Q } .

Thus, given a set qof operations on and an equivalence relation on , weget a setq/of operations on /. Similarly we can go from a set of operationson / to a set of operations on by the operation.

Since there are at most 22||

many non-equivalent formulas with || free vari-ables the conjuction in (2) above can be bounded by that number, and thus (2)is a formula ofL(q). We now get the following easy lemma.

Lemma 12. For anya andb in, aq b iff

L(q)

x((a, x) (b, x)),

where the conjuntion is only overs with two free variables.

Proof. Follows directly by setting(x, y) to be the formulax q y and x tobe a. Then we have

aq a a q b,and thus thata q b.

Proposition13. Letqbe a set of operations on.

1. R k is definable inL(q, a)a iffR is invariant underq.2. Qq =(Q/q)

Proof. (1). The left to right implication is immediate. For the other impli-cation assumeR is invariant under q. LetR(x1,...,xn) be the formula

a1,...,anR

1in

aiq xi

defining R. I f a R, , q (a) since q is reflexive. If , q (a), there isb R such that aq b. Implying that a R sinceR is invariant under q.

(2). By (1) R k is definable in L(q, a)a iff it is of the form R,whereR (/q)

k. This holds iffR = (R/q).Assume that R Qq, then Ri is definable in L(q, a)a. Thus Ri =

(Ri/ q), proving that R (Q/ q). On the other hand ifR (Q/ q) then each Ri satisfies Ri = (Ri/ q) and thus they are all definable inL(q, a)a. From the definition it also follows that (Q/q) Q and thusR Q.


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Lemma 14. Letq be a set of operators and suppose Sim(q). Then

1. /q is a permutation of/q,2. ifR is invariant underq thenR S, whereS= (R), and3. if a, b , a b and(x) is a formula inL(q) then, q |= (a)

(b).

Proof. Let beq.(1) To prove that / is a function we prove that for all a, a and all

b, b such that a a, a b and a b we have b b. This is provedby induction on the formulas in L(q). In fact we prove, by induction, theslightly stronger statement that if(x, c) is a formula ofL(q, ) (observethat c can be an infinite string) then , q|=(b, c) iff , q|=(b, c).

For the base case take R qand suppose R(b, c). Let d be such that d c,then R(a, d) sincea, d b, c. NowR(a, d) sincea a and then we haveR(b, c)

sincea

,d b

, c.For the induction steps negation, disjunction and existential quantificationare easy. Let us thus take Q q and suppose , q |= Qx(b, c, x). Then(b, c, x)x Q and by the induction hypothesis (b

, c, x)x = (b, c, x)x

and thus , q|=Qx(b, c, x).The proof of the bijective property is very similar and left to the reader.(2) From (1) we have that if Sim(q) and R respects then R S iff

S= (R). This is because if a band b (R) then there is a R such thata band so by (1) a a and since R respects, a R.

(3) The proof is by induction on . For the base case we have a b, where Sim(q), andR q. Thus R is invariant under andRa iffRb.

For the induction step negation and (infinite) conjuntion is trivial. For the casewith (an infinite string of) existential quantifers assume that , q|=x(a, x) is

true and thus there are csuch that , q|=(a, c). Since is a similarity relationthere are d such that a, c b, d and thus by the induction hypothesis we have, q|=(b, d) and thus , q|=x(b, x).

We are left with the case of a generalized quantifier Q q. Suppose , q |=Qx(a, x), i.e., thatR = (a, x)x Q. Since Sim(q),R is invariant under and by (2) R S where S = (R), we have S Q. Now, (R) = S is(b, x)x and thus , q|=Qx(b, x).

Next we prove that satisfies similar properties:

Lemma 15. Let be a monoid of similarites with involution and . Then

1. / is a permutation of/,2. ifR is invariant under thenR S, whereS= (R), and

3. if also

,

a,b

, a

b

and(x)

is a formula inL(

Inv())then, Inv()|=(a) (b).

Proof. Let be.(1) To prove that / is a function we prove that for all a, a and all

b, b such thata a, a b and a b we have b b. This follows directlyfrom being closed under composition and taking inverses. The proof of thebijective property is similar and left to the reader.


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(2) If a b and b (R) then there is a R such that a b and so by (1)a a and since R is invariant under , a R.

(3) The proof is by induction on . For the base case we have a b, where , and R Inv(). Thus R is invariant under andRaiffRb.

For the induction step negation and (infinite) conjuntion is trivial. For thecase with (an infinite string of) existential quantifiers assume that , Inv() |=x(a, x) is true and thus there are c such that , Inv()|= (a, c). Since isa similarity relation there are d such that a, c b, d and thus by the inductionhypothesis we have , Inv()|=(b, d) and , Inv(P i)|=x(b, x).

We are left with the case of a generalized quantifier Q q. SupposeQx(a, x),i.e., that R = (a, x)x Q. IfR is invariant under we are finished sincethen by (2)R SwhereS= (R), we haveS Q. Now,(R) =Sis (b, x)xand thus , Inv()|=Qx(b,x). To see that R is invariant under take c Rand d such that c d. There is such that c d and then since ,

= and a, c

a,d, and thus , Inv() |= (a, c) implies that, Inv()|=(a, d) by the induction hypothesis and we have d R.

In fact and q are very much related as the next proposition shows.

Proposition16. Letqbe a set of operators and a monoid with involution,then

1. if , then is the same relation asInv(), and2. q is the same asSim(q).

Proof. (1) First, Inv(). Assumea b. For any c , there is a such thatac bc, hence, by (3) in Lemma15and since is full, for everyformula (x, y) in L(Inv()), we have , Inv() (a, c) iff , Inv() (b, c). By Lemma12, this suffices for a Inv() b.

Second, Inv(). Assume a Inv() b. For any c k, define a relation

Hac of arityk +1 byHac = {d

k+1 |there is such thatac d}. CheckthatHac is invariant under . Assume that d d

for some . Ifd Hac,there is a with a, c d. Since is closed under composition, ,and then a, c d implies d Hac,. Ifd Hac, there is a

witha, c d. Since is closed under composition and taking converses, ,and then a, c d implies d Hac. ThereforeH

ac is the interpretation of

a predicate symbolPin the language L(Inv()). By definition, , Inv() P(a, c). Sincea Inv() b, this implies , Inv() P(b, c), hence there is a such thatac bc.

(2) First, qSim(q). Assumea Sim(q) b. For any c , there is a such that ac bc, hence, by (3) in Lemma 14, for every formula (x, y) inL(q), we have , q(a, c) iff , q(b, c). By Lemma12, this suffices for

aInv()b.Second, qSim(q). Assumeaq b. Let c

k, we need to find a Sim(q)

withac bc. It is sufficient to show thatHSim(q)ac is definable in L

(q) by some

formula , because then , q (a, c) and a q b implies , q (b, c), givingthe needed .

HSim(q)ac is invariant under Sim(q) since Sim(q) is closed under compositions.

By Proposition 13, HSim(q)ac is definable in L

(q, ) if H

Sim(q)ac is invariant


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under q. Thus, it is sufficient to show thatq Sim(q). Let R be an n-aryrelation in q, d1...dn R and d1...dn q d

1

...dn

. d1...dn R and d1 q d1

,henced1d2...dn R, and by repeating this reasoning, d

1...d

n R. Let Q be a

quantifier in q, R and Ssequences of relations on invariant under q and ofa type appropriate for Q. We need that ifRq S, then R Q iffSQ. It issufficient to note that ifR is invariant under q and R q S, then R= S. Letd R and e such that eq d. e R sinceR is invariant under q, hence sinceRq S, d S. Similarly, let d Sand esuch that eq d. SinceR q S, e R,and then eq d and the fact that R is invariant under q implies d R.

Lemma 17. Sim(q) is a full monoid.

Proof. Let = Sim(q). To check that is closed under taking conversesis trivial, and by using (2) of Lemma14 we can easily see that is also closedunder compositions.

We prove that if and is a similarity relation then . Itis trivial to check that every relation ofq is invariant under . For the case ofquantifiers letR Sbe such that R and Sare invariant under q. ThenR Sbecause if not we may assume, without loss of generality, there are a b, a Rand b / S. There are c such that a c and thus, since both a b and a cwe have by (1) in Lemma 14 that bq ccontradicting thatSis invariant underq.

Given a set of operations q we now know that q is the same as Sim(q)so when working with Inv(Sim(q)) we can use q instead ofSim(q); this willimplicity be used in the following proposition.

Proposition18. Letqbe a set of operators and beq. We have

1. Sim(q)/ is Aut(q/), and2. Inv(Sim(q))is the set of all quantifersQsuch thatQ/ is in Inv(Aut(q/)).

Proof. (1) For the left-to-right inclusion let Sim(q). By Lemma 14,f = / is a bijection on / and if Q q then R Q/ iff R Q iff(R) Q ifff(R) Q ifff(R) Q/.

For the other inclusion let f Aut(q/). Definea b ifff([a]) = [b]. Then is a similarity relation on and f = /. Let Q q and R is invariantunder then R Q iffR/ Q/ ifff(R/) Q/ ifff(R/) Q. Butf(R/) = (R). We have proved that R Q iff (R) Q, and thus Q isinvariant under.

(2) TakeQ Inv(Sim(q)) and f Aut(q/). By (1) choose Sim(q) suchthat f =/. ThenR Q/ iffR Q iff(R) Q iff(R)/ Q/,but(R)/= f(R).

On the other hand takeQ such thatQ/ Inv(Aut(q/)),R invariant under and Sim(q), then f = / is in Aut(q/). ThusR Q iffR/ Q/ifff(R/) Q/ifff(R/) Q, butf(R/) = (R).

From this proposition we get what was announced as the main theorem of thissection:

Theorem 11. Letqbe a set of operations and a set of similarities, then


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1. Q Inv(Sim(q)) iffQq is definable in L(q), andR Inv(Sim(q)) iffRis definable in L

(q).2. Sim(Inv()) is the smallest full monoid including .

Proof. (1) By Proposition 18, Q Inv(Sim(q)) iff Q/ Inv(Aut(q/)).Thus, by Theorem4,Q/is definable in L(q/) by a sentence . Let bethe translation of into L(q) where = is replaced by .

By induction of formulas we see that for all a and all(x) in L(q/):

/ |=([a]) iff |=(a),

where, again, is the translation of into L(q).It follows that defines(Q/) and since is definable in L(q) we have

what we wanted.(2) Follows by the same argument as in (1).(3) We prove that for a full monoid we have that Sim(Inv()) = . The

general theorem then follows from the fact that every Sim(q) is a full monoid.The inclusion Sim(Inv()) is trivial. For the other includion let H =

/. It is a group of permutations by Lemma 15. By straightforward checkingwe see that

Inv()/= Inv(H).

From (1) in Proposition18we get that Sim(Inv())/= Aut(Inv()/) andthus that this is equal to Aut(Inv(H)), which by Theorem4is nothing else thanH. Thus,

Sim(Inv())/= /

and by the assumption that is full we get that Sim(Inv()) = : If Sim(Inv()) then there is such that /= /, and then ( ), and thus .

5. Concluding remarks.

5.1. Invariance under permutation as a logicality criterion. Let usfirst go back to languages with equality and groups of permutations. The mainbenefit of a Krasnerian approach to McGees result is to show that the correspon-dence between purportedly logical operations on a domain and the symmetricgroup on that set is a special case of the general duality which holds betweensets of relations and groups of permutations, when sets of relations consist infirst-order and second-order relations closed under definability in L. Thelight thus shed on McGees result is ambiguous, depending on how the closurecondition is looked at. One might insist that groups of permutations are con-sequently shown to be a sure guide to characterizing sets of relations. WhenL provides the logical context, anyrelevant structure may be characterized

in terms of invariance under permutation, and therefore the structure consistingof all and only logical relations on a given domain, in particular, is thus char-acterizable. Now which group of permutations should be the one giving us all(and only the) logical relations? Since logical notions have generality as theirdistinguishing feature, we should go for the largest possible group, and take allpermutations: this is the gist of the generality argument. Since logic is formaland does not make differences between objects, we should require insensitivity


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to any possible way of switching objects, and take again all permutations: this isthe gist of the formality argument. Of course, several steps are in need of further

justification. Maybe identifying being a logical notion and being a general no-tion is too quick, because generality remains underspecified. Or maybe beingformal and not making distinctions between objects are not quite the same see MacFarlane [16] for a thorough discussion of these issues. But our goal hereis not to challenge the relevance of the conceptual analysis in terms of generalityor indifference to objects. Our point is rather to suggest that these argumentsmay be deemed to be on the right track only if we know that logical relationsare characterizable in terms of invariants of a group of permutations. Once thischaracterizability is granted, both arguments go through, and nicely converge.

Seeing McGees result as a special case of a more encompassing Krasneriancorrespondence reveals L to be a built-in feature of the framework. Defin-ability in Lis the closure condition on sets of relations which corresponds to

working with groups of permutations, rather than with some other kind of trans-formations. As soon as we choose to explicate logicality (either qua generality orqua formality) in terms of groups of permutation, we let infinitary syntax sneakin. Now the generality and formality arguments tell us which group of permuta-tions we should pick, if we are to single out logical relations by picking one, butthey do not tell us that we should single out logical relations by picking a groupof permutations to start with. Thus, the fact that the infinitary logic Lendsup being a full-blown logic on this account of logicality might be deemed anartificial effect of the invariance under permutation framework. McGee himselftook his result to be an argument in favor of invariance for permutation as a log-icality criterion, inasmuch as every operation on the list is intuitively logical([17], p.567), the list being the list of basic operations ofL. But this presup-poses that having reasons to count conjunction as logical automatically counts as

having reasons to count infinitary conjunction, and similarly for quantification.Blaming invariance under permutation for its infinitary upshot is no new crit-

icism (see [8], [3] and [7]), but, in the light of the generalized Krasnerian corre-spondence, this infinitary burden is seen to be the very consequence of the use ofpermutations in order to demarcate the extensions that are admissible for logicalconstants.

5.2. Invariance under similarity as a logicality criterion. Invarianceunder similarities has been put forward as an alternative logicality criterion byFeferman [8].13 The shift from permutations to similarities makes for greatergenerality, which is crucial for the generality argument. It also provides a wayto escape the commitment of permutations to cardinalities. Every cardinalityquantifier, no matter how wild (there are at least 17 many) is permutationinvariant. Shifting to invariance under similarities expells these wild creatures

from the paradise of logical quantifiers.14

13Feferman now favors mixed approches in which invariance only provides necessary condi-

tions for logicality, see [7] and [9].14Feferman had another reason to favor similarities, namely the fact that they can relate do-

mains of different sizes, and hence force a greater homogeneity for the extensions of quantifiersacross domains. We do not deal with across-domain invariance here.


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Feferman shows that an operation is definable in first-order logic withoutequality just in case it is definable in the -calculus from homomorphism invari-ant15 monadicquantifiers and asks whether there is a natural characterizationof the homomorphims invariant propositional operations in general, in terms oflogics extending the predicate calculus ([8], p. 47). The question is all the morepressing that the syntactic restriction to monadic quantifiers lacks a proper justi-fication. With reference to an unpublished proof given in [2], it has been claimedthat invariance under homomorphism corresponds to definability in pure L(see [3] and [7]). The proof, however, was faulty, breaking down because of unde-finable subsets. Theorem11 provides a correct generalization. However, it doesnot answer Fefermans exact question, since the correspondence we eventuallyget only holds with respect to the action of quantifiers on definable sets. Whatwe know by Theorem 11 is that invariance under homomorphism correspondsto definability in pure L when quantifiers are restricted to their action on

definable subsets of the domain. Whether the original claim was correct is stillunknown.The restriction to definable sets notwithstanding, Theorem 11shows that the

shift from permutations to similarities or homorphisms does not change the pic-ture when it comes to the infinitary explosion of invariance criteria: infinitarysyntax is every bit as much hardwired in invariance under similarities as it is ininvariance under permutations. Moreover, invariance criteria do not seem wellsuited to help us adjudicate the status of equality as a logical constant or not.As highlighted by the close parallelism between Theorem4and Theorem11, al-lowing identity at level of relations exactly corresponds to banning non-injectivefunctions at the level of invariance conditions. Whether there are better groundson which to adjudicate the issue, or whether it should be considered as primarilya matter of convention remains to be seen.

5.3. Further perspective. To conclude, we would like to mention two linesof further research. The first one is somewhat tangential to the main project.We have seen that automorphism groups of monadic second order structures arecharacterized as closed groups in a way similar to the first-order case. What dothese closed groups tell us about the structures they come from? The questionis whether remarkable properties of monadic second order structures can bereduced to properties of automorphism groups a famous example in the first-order case is the Ryll-Nardzewski theorem which says that being 0 categoricalis such a property.

The second line of research concerns further inquiries into the duality betweensets of transformations and sets of relations closed with respect to definabilityin a given logic. Theorem11 leaves as an open question whether the restrictionto the definable parts of quantifiers is mandatory. More generally, this duality

could be extended to other kinds of transformations and less demanding closureconditions. In [1], Barwise provides such a result for first-order relations withrespect to semi-automorphisms (automorphisms between substructures whichare part of a back and forth system) and closure under definability in L. Thissuggests generalizing again to second-order relations. Another tempting move

15Homomorphism invariance is equivalent to invariance under similarities.


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would consist in getting to L by suitably weakening the conditions on semi-automorphisms. Finally, the connection between properties of the logical syntaxand properties of the transformations still remains to be fully understood. Itwould be nice to achieve even greater generality by connecting explicitly abstractproperties of transformations (e.g. the depth of Ehrenfeucht-Frasse games) andproperties of the syntax for the logic providing the closure condition.

REFERENCES

[1] K. J. Barwise,Back and forth through infinitary logic,Studies in model theory, Math-

ematical Association of America, 1973, pp. 534.[2] Denis Bonnay, Quest-ce quune constante logique ?, Ph.D. thesis, Universite Paris I,

Paris, France, November 2006.[3] , Logicality and invariance, The Bulletin of Symbolic Logic, vol. 14 (2008),

no. 1, pp. 2968.

[4] Peter J. Cameron, Oligomorphic permutation groups, Cambridge University Press,1990.

[5] E. Casanovas, Logical operations and invariance, Journal of Philosophical Logic,vol. 36 (2007), no. 1, pp. 3360.

[6] E. Casanovas, P. Dellunde, and R. Jansana,On elementary equivalence for equality-free logic, Notre Dame Journal of Formal Logic, vol. 37 (1996), no. 3, pp. 506522.

[7] Solomon Feferman, Set-theoretical invariance criteria for logicality, Notre DameJournal of Formal Logic, vol. 51 (10), no. 1, pp. 320.

[8] , Logic, logics, and logicism, Notre Dame Journal of Formal Logic, vol. 40

(1999), no. 1, pp. 3154.

[9] , Which quantifiers are logical?, Synthese, (to appear).[10] Carol Karp, Languages with expressions of infinite length, North-Holland, Ams-

terdam, 1964.[11] Felix Klein, Vergleichende betrachtungen uber neuere geometrische forschungen,

Mathematische Annalen, vol. 43 (1893), pp. 63100.

[12] Marc Krasner,Une generalisation de la notion de corps,Journal de mathematiques

pures et appliquees, vol. 17 (1938), no. 3-4, pp. 367385.

[13] , Generalisation abstraite de la theorie de galois , Colloque dalgebre et de

theorie des nombres, Editions du Centre National de la Recherche Scientifique, 1950, pp. 163

168.[14] Serge Lang, Algebra, third ed., Graduate Texts in Mathematics, vol. 211, Springer-

Verlag, New York, 2002.[15] Per Lindstrom, First-order predicate logic with generalized quantifie rs, Theoria, vol.

32 (1966), pp. 165171.

[16] John MacFarlane, What does it mean to say that logic is formal, Ph.D. thesis,University of Pittsburgh, 2000.

[17] Vann McGee, Logical operations, Journal of Philosophical Logic, vol. 25 (1996),pp. 567580.

[18] Bruno Poizat, La theorie de galois abstraite, Elefteria, vol. 3 (1985), pp. 99107.

[19] Gila Sher, The bounds of logic, MIT Press, Cambridge, Mass., 1991.[20] Gerald Stoller, Example of a proper subgroup of S which has a set-transitivity

property, Bull. Amer. Math. Soc., vol. 69 (1963), pp. 220221.[21] Alfred Tarski, What are logical notions?, History and Philosophy of Logic, vol. 7

(1986), pp. 143154.


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