Logical operations and invariance
Enrique Casanovas
University of Barcelona
July 28, 2004. Revised September 8, 2005
Abstract
I present a notion of invariance under arbitrary surjective mappings
for operators on a relational finite type hierarchy generalizing the so called
Tarski-Sher criterion for logicality and I characterize the invariant operators as definable in a fragment of the first-order language. These results
are compared with those obtained by Feferman and it is argued that further clarification of the notion of invariance is needed if one wants to use
it to characterize logicality.
1
Background
The semantical analysis of logical constants stems from the early work of Tarski,
Mautner and Mostowski, and has received a lot of attention recently mainly
due to the contributions of Sher, McGee, van Benthem, and Feferman. Its main
point is the proposal that logicality may be understood as invariance under the
most general transformations. In [13] Tarski develops the idea of identifying
logical notions with those notions that are invariant under all permutations
of the universe of discourse. A similar point of view had been previously but
independently maintained by Mautner in [7]. Lindenbaum and Tarski proved
in [5] that all logical notions from Principia Mathematica are invariant in this
sense.
Mostowski in [9] addressed the problem of characterizing quantifiers. He
distinguishes between limited and unlimited quantifiers. A limited quantifier
is defined on a specific set and it is invariant under all permutations of this
set. An unlimited quantifier determines a limited quantifier on each particular
set and it is required to be invariant under all bijections between sets. This
distinction is used by Gómez-Torrente in [4] to justify the difference between
the Tarski criterion, namely invariance under permutations of the universe, and
the Mostowski criterion, which is invariance under bijections across universes.
Note that in order to be able to apply Mostowski’s criterion we have to assume
that the notions and operations we are discussing are defined on every set.
Sher [10] requires that a logical constant be defined over all models, not
only over all sets. Her criterion for logicality includes, among other conditions,
1
invariance under isomorphisms, which in fact is a weaker condition than mere
invariance under bijections. This difference between Sher’s and other previous
accounts of logical constancy is frequently neglected. For instance, McGee in [8]
and Feferman in [3] identify Sher’s position with the requirement of invariance
under bijections for logical operations, which is Mostowski’s criterion according
to Gómez-Torrente’s terminology. For a concise presentation of Sher’s views
see [11].
The proposal of identifying logical operations and notions with those invariant under bijections across domains has received severe criticism. It is generally
acknowledged that invariance is a necessary condition for logicality, but not a
sufficient one. One reason for this is the observation that all notions formally defined in terms of cardinalities turn out to be invariant under bijections, whereas
their logical character is more than dubious in many cases. Moreover bijections
connect only sets of the same size, and consequently invariance under bijections does not exclude an arbitrary behavior on different cardinalities. In words
of Machover in his review of Sher’s book in [6]: “ For example, let C be the
class whose members are all natural numbers belonging to some non-analytic
set, as well as all infinite cardinals α (regarded as initial ordinals) such that
2ℵα = ℵα+1 . Now let Q be a particle of the same syntactic type as the familiar standard quantifiers ∃ and ∀, interpreted as follows: in any domain whose
cardinality belongs to C, Q is interpreted as ∃, while in any other domain Q
is interpreted as ∀. Clearly, Q under its given interpretation satisfies Sher’s
criterion, and hence she would have us regard it as a logical particle.”
The defense of the criterion of bijection-invariance has been undertaken
mainly by McGee. In [8] he proves, roughly stated, that any bijection-invariant
operation can be defined in an infinitary logical language and maintains that
“this confirms Tarski’s thesis that an operation invariant under permutations
is a logical operation” ([8], page 568). To be more precise, McGee shows that
if F is a bijection-invariant operation, then for each cardinal number κ there is
a formula ϕk in the infinitary language L∞∞ which characterizes F on sets of
cardinality κ. Therefore McGee’s result does not provide a single formula for
F , but a different formula ϕκ for each cardinal number κ.
McGee’s work seems to Feferman “faultless in its execution” but “blatantly
implausible” ([3] page 32). He presents three basic criticisms of McGee’s approach. In his own words ([3] page 37):
1. The thesis assimilates logic to mathematics, more specifically to set theory.
2. The set-theoretical notions involved in explaining the semantics of L∞∞
are not robust.
3. No natural explanation is given by it of what constitutes the same logical
operation over arbitrary basic domains.
Point 3 is for him the main reason for rejecting McGee’s proposal. As mentioned above, this was also one of the objections of Machover to Sher’s criterion
for logicality. Feferman proceeds then to present his own ideas, which try, according to him ([3] page 39), to give an explanation of logical operations “which
2
shows how an operation behaves when applied over one domain M0 connects
naturally with how it behaves over any other domain M00 ”. The solution is to
investigate invariance under mappings instead of invariance under bijections.
He calls it “homomorphism invariance”, which I think it is an unlucky denomination since there is no structure to be preserved and it is really only a matter
of surjective mappings from pure sets to pure sets. Feferman characterizes the
operations (of monadic type) which are invariant in this sense and he finds
out that they are exactly the ones which are definable in the λ-calculus without equality. In other words, the invariant operations are those which can be
characterized using only pure first-order logic. There are some precedents of
Feferman’s results in Van Benthem [14] but they are of very restricted nature.
In the end Feferman is not completely satisfied with his characterization since
it excludes equality, and after a brief discussion of the problem of the logical
character of equality he says we should give it “a distinguished role in logic
even if it should not turn out to be logical on its own under some cross-domain
invariance criterion, such as homomorphism” ([3] page 44).
2
Invariance
My approximation to the semantic analysis of logical operations has, in its initial
steps, many points in common with the one developed by Feferman. Clearly,
invariance under bijections across universes is not the more general kind of
invariance one can think of. Equality (or better inequality) is invariant under
bijections but not necessarily under arbitrary mappings. In mathematics much
more general kinds of transformations – which do not respect inequality – are
many times considered the natural ones. It is therefore convenient to investigate
more general kinds of transformations and to find out which are the notions and
operations invariant under them.
Unexpectedly, I arrived at very different conclusions than those of Feferman.
Invariant notions and operations in my sense turn out to be only a small fragment of all notions and operations definable in first-order logic. In particular,
negation, arbitrary conjunctions and universal quantification are not invariant.
On the other hand it follows from my results that some particular forms of
equality are invariant. Some part of my work in the last part of this paper
has been devoted to try to understand how we arrive at so distant conclusions
starting with the same assumptions and very close intuitions.
Feferman’s analysis takes place in a finite type hierarchy of universes similar
to the one used first by van Benthem. Just to simplify the proofs, Feferman
replaces relational types by functional types and replaces relations by their
characteristic functions. At first sight one can think that his replacement is
innocuous, but in fact it has serious consequences. As I show in section 7,
Feferman’s invariance notion does not correspond to invariance under surjective
mappings when translated back into the relational framework. His definition
of invariance seems to be very reasonable in the functional type hierarchy but
it is fairly unnatural when restated in the relational setting. As pointed out
3
in section 7, it is in fact a different notion of invariance than the one here
investigated. For the first levels of the type hierarchy it coincides with invariance
under preimages of arbitrary mappings. This is a capital point to explain the
difference in conclusions between Feferman’s analysis and mine. If one does not
make the simplification of replacing relations by their characteristic functions,
one is forced to deal with the notion of invariance I am discussing here.
The fact that invariance splits into several notions, some stronger than others, and that accordingly different criteria for logicality appear, should perhaps
be taken as a sign that the logical character of notions and operations from a
semantical point of view is a matter of perspective and a matter of degree. Even
outside the semantical framework, the idea that classifying a notion as a logical
notion is a question of more or less has been considered by some authors. For instance, Byrd discusses in [1] the position that the conditional and the universal
quantifier play a prominent role among logical concepts and Warmbrōd in [15]
proposes a distinction between a secure logical theory with a minimal set of
notions and an extended setting where a great variety of notions is allowed. According to my notion of invariance, existential quantification and disjunction are
particularly strong whereas negation, conjunction and universal quantification
are not. I realize that it is not easy to accept than universal quantification and
conjunction are less logical that existencial quantification and disjunction. But
according to my results there is a natural sense in which they are less robust.
It is not my purpose here to support any particular point of view concerning
the logical character of abstract operations. My analysis can be used to show
that the semantic characterization based on invariance under all possible transformations leads to absurdities. It can also be used to support the view that in
some respect some operations are more logical than others. I think that there
is still a need for exploration. Several notions of invariance have appeared and
some other may still be relevant. My research shows that the notion of invariance is not as clear as one could have thought. Moreover the previous proposals
have been only partially developed. Feferman has been able to characterize invariant operations only of a restricted class. They are operations transforming
a tuple consisting of n subsets of a set M on a m-ary relation on M . It would be
interesting to see whether his description generalizes to more general operations
transforming tuples of mi -ary relations (for 1 ≤ i ≤ n) on m-ary relations, which
is the most reasonable setting. In this article all the mapping-invariant (in my
sense) operations of these more general kinds are investigated. I describe with
some detail these operations and I show they are just the operations definable
in a specific fragment of first-order logic. I do not claim that they constitute
the answer to the question of which are the logical operations.
My approach to invariance is influenced by the fact that in algebra the
most general kind of transformation usually considered is not isomorphism but
homomorphism. On the contrary to what happens with the relation of being an
isomorphic image, the relation of being an homomorphic image is not symmetric
and transitive. Thus, one need to construct its symmetric and transitive closure
in order to obtain an equivalence relation analogous to isomorphism. These
topics are studied in [2] with the main motivation of developing the basic model4
theoretic aspects of first-order logic without equality. The equivalence relation
among structures obtained as the symmetric and transitive closure of being an
homomorphic image is called relativeness in [2]. “Being relative” is the natural
substitute for “being isomorphic” when one does not care of preserving equalities
and inequalities. In the case we are interested here, only pure sets are present
–we do not deal with arbitrary models– and therefore one has to work with
bijections in place of isomorphisms and with surjective mappings in place of
homomorphisms. It is a particular case of the general situation studied in [2],
namely the case in which all structures considered have empty similarity type.
Feferman in [3] is also concerned with the equivalence relation characterized
as the symmetric and transitive closure of being an image under an arbitrary
surjective mapping and he chooses the name similarity to refer to it. Thus, a
similarity is a finite composition of surjective mappings and inverses of surjective
mappings and one says that two sets are similar when there is a similarity
between them. Of course, any two nonempty sets are similar in this sense, but
the important tool is not the similarity relation itself but invariance under every
particular similarity. Similarity is just relativeness particularized to the case of
structures which are only pure sets. Here we will follow Feferman’s terminology.
Of course, we can apply to the similarity relation all results obtained in [2] for
the relativeness relation. This has been very helpful for my understanding of
the situation.
I now describe with some detail the content of the paper. We will look at
notions and operations defined simultaneously over any nonempty set. Over any
such set we will consider the relational and the functional finite type hierarchy
of universes and we will show how different transformations of the basic sets
can be extended to upper levels. In section 3, after fixing terminology, I define
mapping-invariance and I clarify some points concerning invariance under mappings and under similarities. In section 4, I characterize the mapping-invariant
objects at the lower levels of the relational type hierarchy. In sections 5 and 6,
I present my main results: the characterization of all mapping-invariant operators that transform a tuple of relations among individuals of different arities
on a relation among individuals. The particular case of operators transforming
m-ary relations on n-ary relations is studied in detail in section 5. Then in
section 6 I need only to indicate how to generalize the results from section 5 to
the broader setting.
Sections 7 and 8 are devoted to compare my approach with that of Feferman. In section 7, I introduce the finite functional type hierarchy and I prove
the equivalence between similarity-invariance and homomorphism-invariance in
the sense of Feferman [3]. In section 8, Feferman’s proposal is presented, and I
translated it back into the relational setting. In the finite relational type hierarchy I define what I call Feferman invariance and I show that an operator is
Feferman invariant if and only if it is invariant in the sense of Feferman when
translated into the finite functional type framework. Then a last notion of invariance is introduced, the notion of preimage-invariance and I show that it is
the same as Feferman invariance at the lower levels. This explains why every
first-order definable operation is invariant according to Feferman but it is not
5
according to my investigation: his transformations of the universe are the result of taking preimages and as is well-known they preserve intersections and
complements while taking images do not.
The notation is mainly standard. P(X) denotes the power set of X. If
R ⊆ M n and 1 ≤ j ≤ n we will use the notation
fieldj (R) = {a ∈ M : (a1 , . . . , an ) ∈ R for some a1 , . . . , an such that aj = a}.
Thus, for example, if R is a binary relation, then n = 2, field1 (R) = dom (R),
and field2 (R) = rng (R). The union of fieldj (R) for all j is field (R). R−1 is
used for the inverse of a relation R even in the case that R happens to be a non
necessarily one-to-one mapping. If R and S are relations then their relational
product or composition will be
R ◦ S = {(a, b) : for some c, R(a, c) and S(c, b)}.
If R or S is a mapping we still use this notation, identifying the mapping with
its graph. Thus f ◦ g is not defined by f ◦ g(x) = f (g(x)).
3
Mapping-invariance
The relational finite types are generated from the basic type 0 by the rule: if
τ1 , . . . , τn are relational finite types, then (τ1 , . . . , τn ) is a relational finite type.
We will call them in short just relational types. If M is a nonempty set and τ a
relational type, the τ -universe of M is the set Mτ defined recursively as follows:
1. M0 = M .
2. M(τ1 ,...,τn ) = P(Mτ1 × . . . × Mτn ).
When using the notation Mτ , we will always assume that M is nonempty. The
type τ = (τ1 , . . . , τn ) where τi = 0 for every i = 1, . . . , n will be denoted by 0n .
Hence M0n = P(M n ).
Let f : M → N be a mapping and let τ be a relational type. The mapping
f can be extended in a very natural manner to a mapping from the τ -universe
Mτ in the τ -universe Nτ . The mappings fτ : Mτ → Nτ are defined recursively
by the following rules:
1. f0 = f .
2. If τ = (τ1 , . . . , τn ), the mapping fτ : Mτ → Nτ is given by
fτ (a) = {(fτ1 (a1 ), . . . , fτn (an )) : (a1 , . . . , an ) ∈ a}
for all a ⊆ Mτ1 × . . . × Mτn .
6
Observe that each fτ is onto if f is onto, and that fτ is one-to-one if f is oneto-one. Sometimes we will write f instead of fτ when the context does not lead
to confusion.
Compositions of surjective mappings in every possible direction produce the
notion of similarity. A relation π ⊆ M × N is a similarity relation between M
and N if and only if for some n ≥ 2 there are sets M1 , . . . , Mn and mappings
f1 , . . . , fn−1 such that M = M1 , N = Mn and for every i = 1, . . . , n, fi is a
mapping from Mi onto Mi+1 or it is a mapping from Mi+1 onto Mi , and π is
the relational product or composition R1 ◦ . . . ◦ Rn−1 where Ri = fi if fi is from
Mi onto Mi+1 and Ri = fi−1 if it is from Mi+1 onto Mi .
The following is easily proven by induction on the length n of the sequence
of sets M1 , . . . , Mn in the definition of similarity. Moreover it follows from the
proof of Proposition 2.6 in [2] since a similarity is just a relativeness relation for
the case of the empty language.
Fact 3.1. A binary relation π ⊆ M × N is a similarity relation between M
and N if and only if for every a ∈ M there is some b ∈ N such that π(a, b)
and for every b ∈ N there is some a ∈ M such that π(a, b). In other words, if
dom π = M and rng π = N .
A similarity relation π between M and N is also naturally extensible to a
similarity relation πτ between the τ -universes Mτ and Nτ . Assume that there
are sets M1 , . . . , Mn and mappings f1 , . . . , fn−1 such that M = M1 , N = Mn
and for every i = 1, . . . , n, fi is a mapping from Mi onto Mi+1 or it is a mapping
from Mi+1 onto Mi , and π is the relational product R1 ◦. . .◦Rn−1 where Ri = fi
if fi is from Mi onto Mi+1 and Ri = fi−1 if it is from Mi+1 onto Mi . Then πτ
is the relational product R1τ ◦ . . . ◦ Rn−1 τ where Riτ = fi τ if fi is from Mi onto
Mi+1 and Riτ = fi −1
τ if it is from Mi+1 onto Mi .
Remark 3.2. Let π a similarity relation between M and N . The induced
similarity πτ ⊆ Mτ × Nτ can also be defined inductively starting from π0 = π
by the rule that if τ = (τ1 , . . . , τn ) is a relational type, then the similarity πτ
between Mτ and Nτ is given by: πτ (a, b) if and only if
1. for each (a1 , . . . , an ) ∈ a there is a tuple (b1 , . . . , bn ) ∈ b such that πτi (ai , bi )
for each i = 1, . . . , n, and
2. for each (b1 , . . . , bn ) ∈ b there is a tuple (a1 , . . . , an ) ∈ a such that πτi (ai , bi )
for each i = 1, . . . , n.
We will be concerned with the invariance of objects and operators. Let τ
be a relational type. An object of type τ is a function a which associates with
every nonempty set M a corresponding element aM ∈ Mτ . We will say that the
object a is mapping-invariant if and only if for every M, N and every surjective
mapping f : M → N , f (aM ) = aN . Equivalently, a is mapping-invariant if
and only if for every M, N , for every similarity relation π between M and N ,
π(aM , aN ). We call the object a bijection-invariant if and only if for every M, N
and every bijection f : M → N , fτ (aM ) = aN .
7
Let τ1 , . . . , τn , τ be relational types. A (τ1 , . . . , τn ; τ )-ary operator is a function F which for each nonempty set M gives a mapping
FM : Mτ1 × . . . × Mτn → Mτ .
Sometimes we omit the subscript in FM to simplify notation if this is not
misleading. We call F mapping-invariant if for all M, N and every mapping
f : M → N from M onto N , for all a1 ∈ Mτ1 , . . . , an ∈ Mτn ,
fτ (FM (a1 , . . . , an )) = FN (fτ1 (a1 ), . . . , fτn (an )).
It is easy to check that a (τ1 , . . . , τn ; τ )-ary operator F is mapping-invariant
if and only if for all M, N and every similarity relation π between M and N ,
for all a1 ∈ Mτ1 , . . . , an ∈ Mτn , and for all b1 ∈ Nτ1 , . . . , bn ∈ Nτn , if πτi (ai , bi )
for all i = 1, . . . , n, then πτ (FM (a1 , . . . , an ), FN (b1 , . . . , bn )).
Following more customary terminology, a (0m1 , . . . , 0mr ; 0n )-ary operator
will be also called (m1 , . . . , mr , n)-ary.
As mentioned in the introduction, Tarski in [13] talks of invariance under
permutations of a fixed universe and not of invariance under bijections across
universes. We may say that the element b ∈ Mτ is permutation-invariant if
and only if for every permutation f of M , fτ (b) = b. Clearly, if a τ -object a is
bijection-invariant, then for every set M , the corresponding element aM ∈ Mτ
is permutation-invariant. On the other hand it is easy to show that for each
set M and each permutation-invariant element b ∈ Mτ we can find a bijectioninvariant τ -object a such that aM = b. Hence both criteria of invariance coincide
when applied to a particular universe.
4
Mapping-invariant objects
It is clear that there are no bijection-invariant objects of type 0. This was
observed by Tarski in [13] for permutation invariance. He also noticed that
the only permutation-invariant objects of type (0) are the empty set and the
universe and more generally, that there are only finitely many permutationinvariant objects of type 0n . For instance, the permutation-invariant objects of
type 02 = (0, 0) are the empty relation, the universal relation, the identity and
its complement. The same happens for bijection-invariance. It is not difficult
to see that the bijection-invariant objects of type 0n are the n-ary relations
first-order definable in the language of pure equality. But the defining formula
might be not uniform for all cardinalities.
Fact 4.1. An object a of type 0n is bijection-invariant if and only if for each
cardinal κ there is a first-order formula ϕκ = ϕκ (x1 , . . . , xn ) in the language of
pure equality such that for each set M of cardinality κ, aM = {(a1 , . . . , an ) ∈
M n : M |= ϕ(a1 , . . . , an )}.
Mapping-invariance is a more restrictive condition. Of course, there is no
mapping-invariant object of type 0. Concerning objects of type 0n , the basic
8
difference with respect to bijection-invariance is that intersections and complements of mapping-invariant relations are not necessarily mapping-invariant.
Equality is mapping-invariant, but inequality is not.
Theorem 4.2. An object a of type 0n is mapping-invariant if and only if it
is the empty object (aM = ∅ for every set M ) or the universe (aM = M n for
every M ) or it is uniformly a union of objects dI for I ⊆ {1, . . . , n}, where
dIM = {(a1 , . . . , an ) ∈ M n : ai = aj for all i, j ∈ I} for every set M .
Proof. It is clear that all these objects are mapping-invariant. Let a be a
mapping-invariant object. We will show that a is of the prescribed form. If
aM = ∅ then aN = ∅ for any other set N , since if π is a similarity relation, π(∅, aN ) implies aN = ∅. I claim that if dIM ∩ aM 6= ∅ for a set M
such that |M | > n, then for any set N , dIN ⊆ aN . From this it will follow that a is a union of objects dI . This includes the case aM = M n , which
corresponds to |I| ≤ 1. Assume |M | > n and let (a1 , . . . , an ) ∈ dIM ∩ aM
and (b1 , . . . , bn ) ∈ dIN . We now show that (b1 , . . . , bn ) ∈ aN . Notice that
π = {(ai , bi ) : i = 1, . . . , n} ∪ ((M r {a1 , . . . , an }) × N ) is a similarity relation
between M and N . By mapping invariance of a, there is a tuple (c1 , . . . , cn ) ∈
aN such that π((a1 , . . . , an ), (c1 , . . . , cn )). By construction of π we see that
(c1 , . . . , cn ) = (b1 , . . . , bn ). Hence (b1 , . . . , bn ) ∈ aN .
Invariant objects of type ((0)) are closely related to quantifiers. Tarski observed that the permutation-invariant objects in this type are the “properties
concerning the number of elements of these classes”. Cardinality also plays a
role for the mapping-invariant objects of type ((0)), but essentially as upper
bound.
Theorem 4.3. An object a of type ((0)) is mapping-invariant if and only if a
is the empty object or it is uniformly a union of the following objects:
1. the object e such that for every set M , eM = {∅},
2. the object u such that for every set M , uM = {M },
3. the object p such that for every set M , pM = P(M ) r {∅},
4. for some cardinal number κ, the object bκ such that for every set M ,
bκM = {A ⊆ M : A 6= ∅ and |A| < κ}.
Proof. It is easy to check that all the described objects are mapping-invariant.
We prove now that all the mapping-invariant objects are of this kind. If aM = ∅
then for any other set N , aN = ∅. Similarly, if ∅ ∈ aM then ∅ ∈ aN for every
N . Now assume aM has nonempty elements different from M and let κ be a
cardinal such that for each cardinal µ smaller than κ, aM has such elements of
cardinality µ. We prove that in this case, for any set N , bκN ⊆ aN . Let B ⊆ N
be nonempty and of cardinality strictly less than κ. Choose a set A ∈ aM of
cardinality ≥ |B| and such that A 6= M , choose a surjective mapping f : A → B
such that f (A) = B, and put π = f ∪ ((M r A) × N ). It is a similarity
9
relation between M and N and hence, by mapping-invariance of a, π(A, C) for
some C ∈ aN . By choice of π it follows that C = B. Thus B ∈ aN . If for
arbitrarily large κ there are sets N such that bκN ⊆ aN , then clearly for every
N , aN = P(N ) or for every N , aN = P(N ) r {∅}. Otherwise, let κ be the least
cardinal such that for some set N , bκN 6⊆ aN . Then for every µ < κ, for every
set N , bµN ⊆ aN . Obviously, κ is a successor cardinal number, say κ = λ+ .
Moreover A 6∈ aN if A is a subset of N such that A 6= N and |A| ≥ λ. If there
is not a set M such that |M | ≥ λ and M ∈ aM then either aN = bλN for every
N or aN = bλN ∪ {∅} for every N .
Assume now that there is a set M such that |M | ≥ λ and M ∈ aM . We prove
that N ∈ aN for every N . From this it will follow that either aN = bλN ∪ {N }
for every N or aN = bλN ∪ {∅} ∪ {N } for every N . Consider an arbitrary set N .
If |N | < λ, then clearly N ∈ aN . In case |M | = |N | ≥ λ, there is a bijection f
between M and N and we put π = f . If M , N have different cardinality ≥ λ
we may assume that |M | > |N | ≥ λ, we choose A ⊆ M and B ⊆ N of the same
cardinality ≥ λ such that A 6= M , we choose a bijection f between A and B,
and we put π = f ∪ ((M r A) × N ). In either case π is a similarity relation
between M and N and there is a set C ∈ aN such that π(M, C) and it follows
that N = C. Hence N ∈ aN .
The last case to be considered is when there is no set M such that aM has
a nonempty element A 6= M . Then aM ⊆ {∅, M } for every M . It is easy to see
that either aM = ∅ for every M , or aM = {M } for every M , or aM = {∅, M }
for every M .
After Theorem 4.3 one might think that all quantifiers ∃<κ xϕ(x) are at our
disposal in constructing mapping-invariant operators and in defining mappinginvariant objects. But this is not the case. It is misleading to restrict the
notion of a quantifier over a set M to a specification of a collection of subsets
of M . This might be enough to quantify formulas with just one free variable.
Quantification of a formula with more free variables involves transformation of
n + 1-ary relations on n-ary relations. In the next section I will characterize all
mapping-invariant transformations of this kind. The quantifier ∃≤1 is clearly
not among them.
A similar situation arises with respect to the logical connectives. In a footnote Corcoran, the editor of [13], explains that during the Buffalo lecture that
originated the paper [13] Tarski indicated that truth-functions can be accommodated in the relational type hierarchy over the universe M by identifying the
truth values T and F with the universe M and the empty set. With this identification every binary truth-function is an element of M((0),(0),(0)) and hence we can
discuss its invariance. It is easy to check that all truth-function are mappinginvariant objects in this sense. However this is again misleading. One thing
is conjunction acting on sentences and a completely different thing is conjunction acting on formulas with free variables. Conjunction of formulas with free
variables involves the operation of intersection, which is not mapping-invariant.
10
5
Characterizing mapping-invariant (m, n)-ary operators
This section is devoted to the characterization of all (m, n)-ary mapping-invariant
operators. I start fixing some basic operators and fixing also some generating
operations to obtain more complex operators. I will show that all mappinginvariant operators are generated with these operations from the basic ones.
The (m, n)-ary operators are the following ones:
m
m
1. The constant (m, 1)-ary operators C>
and C⊥
whose actions on R ⊆ M m
are given by
m
(a) C>
(R) = M
m
(b) C⊥
(R) = ∅
m
2. The (m, n · m)-ary diagonal operator ∆m
n such that for any R ⊆ M ,
∆m
n (R) = {n × a : a ∈ R}
where n × a is the n-fold concatenation of the tuple a, that is, n × a =
(b1 , . . . , bn·m ) if a = (a1 , . . . , am ) and bk·m+i = ai for 1 ≤ i ≤ m.
3. The (m, m−1)-ary i-projection operator Πm
i (where m ≥ 2 and 1 ≤ i ≤ m)
such that for any R ⊆ M m ,
Πm
i (R) = {(a1 , . . . , ai−1 , ai+1 , . . . , am ) : (a1 , . . . , am ) ∈ R}.
4. For any σ ∈ Sym{1, . . . , m}, the (m, m)-ary permutation operator Pσ such
that for any R ⊆ M m ,
Pσ (R) = {(aσ(1) , . . . , aσ(n) ) : (a1 , . . . , an ) ∈ R}.
The generating operations for operators are the following ones:
1. Product. If F is (m, n1 )-ary and G is (m, n2 )-ary, the product F × G is
the (m, n1 + n2 )-operator such that for any R ⊆ M m ,
F × G(R) = F (R) × G(R).
2. Sum. If F and G are (m, n)-ary operators, the sum F ∪G is the (m, n)-ary
operator such that for any R ⊆ M m ,
F ∪ G(R) = F (R) ∪ G(R).
3. Composition. If F is (m, n)-ary and G is (n, k)-ary, the composition of F
and G is the (m, k)-ary operator G ◦ F such that for any R ⊆ M m ,
G ◦ F (R) = G(F (R)).
11
Operators generated from projections, diagonals and permutations by composition will be called intern.
Remark 5.1. A (m, n)-ary operator F is intern if and only if there is a map
σ : {1, . . . , n} → {1, . . . , m} such that for any R ⊆ M m ,
F (R) = {(a1 , . . . , an ) : for some (a01 . . . , a0m ) ∈ R, ai = a0σ(i) for all i = 1, . . . , n}
For the next lemma we need the technical notion of free system. Let R ⊆
M m . We call (M, R) a free system if R is infinite, for all a ∈ M there is at
most one tuple (a1 , . . . , am ) ∈ R such that a = ai for some i and, finally, for all
(a1 , . . . , am ) ∈ R, ai 6= aj if i 6= j.
Lemma 5.2. Let F be a (m, n)-ary mapping-invariant operator. Let R ⊆ M m
and assume (M, R) is a free system. For any (a1 , . . . , an ) ∈ FM (R) there are a
decomposition
˙ k
{1, . . . , n} = I1 ∪˙ . . . ∪I
and operators F1 , . . . , Fk such that
1. Fl is (m, nl )-ary, where nl = |Il |.
2. If i ∈ Il and ai = aj , then j ∈ Il .
3. Fl is intern if {ai : i ∈ Il } has more than one element or if it has just one
element a and a ∈ fieldj (R) for some j.
4. If {ai : i ∈ Il } has only one element a and a 6∈ field (R), then Fl =
m
∆1nl ◦ C>
, that is Fl (S) = {nl × a : a ∈ N } for all S ⊆ N m .
5. (ai : i ∈ Il ) ∈ Fl (R).
6. There is a σ ∈ Sym{1, . . . , m} such that for all S ⊆ N m , if (N, S) is a
free system, then
Pσ (F1 (S) × . . . × Fk (S)) ⊆ F (S).
Proof. If a1 ∈ field (R), we choose as I1 a maximal subset of {1, . . . , m} for
which 1 ∈ I1 and there is an intern (m, n1 )-ary operator G such that (aj : j ∈
I1 ) ∈ G(R), and we put F1 = G. If a1 6∈ field (R), we take I1 = {i : a1 = ai }
m
and we put F1 = ∆1n1 ◦ C>
. In either case (ai : i ∈ I1 ) ∈ F1 (R) and also
j ∈ I1 whenever aj = ai for some i ∈ I1 . Now let l be the first element of
{1, . . . , m} r I1 . Again, if al ∈ field (R) we choose as I2 a maximal subset of
{1, . . . , m} r I1 for which l ∈ I2 and there is an intern (m, n2 )-ary operator G
such that (ai : i ∈ I2 ) ∈ G(R), and we put F2 = G. In case al 6∈ field (R)
m
we set as before I2 = {i : al = ai } and F2 = ∆1n2 ◦ C>
. This procedure
eventually ends and the decomposition is finished. We will show that it verifies
6. Assume S ⊆ N m , (N, S) is a free system, and (b1 , . . . , bn ) is a tuple such that
(bi : i ∈ Il ) ∈ Fl (S) for all l = 1, . . . , k. We will prove that (b1 , . . . , bn ) ∈ F (S).
a
a
The permutation σ will be the natural enumeration of the sequence sa
1 s2 . . . sk
12
where every sl is the enumeration of Il in increasing order. Let J be the set of
all l ∈ {1, . . . , k} which do not fall under case 4, that is, all l for which Fl is
intern. If l ∈ J, we can find a mapping σl : Il → {1, . . . , m} such that for all
T ⊆ K m,
Fl (T ) = {(ci : i ∈ Il ) : for some (c01 , . . . , c0m ) ∈ T, ci = c0σl (i) for all i ∈ Il }
and we can find (al1 , . . . , aln ) ∈ R and (bl1 , . . . , bln ) ∈ S such that ai = alσl (i)
and bi = blσl (i) for all i ∈ Il . For l 6∈ J we choose al and bl such that ai = al
and bi = bl for all i ∈ Il . Let L be the set of all l ∈ {1, . . . , k} r J such that
bl ∈ field (S). For l ∈ L choose now a tuple (bl1 , . . . , blm ) ∈ S such that bl = bli for
some i and choose arbitrary (al1 , . . . , alm ) ∈ R with the only requirement that
these tuples are all different and also different from the chosen for l ∈ J. This
is possible because R is infinite.
Let A = M r ({ali : 1 ≤ i ≤ m and l ∈ J ∪ L} ∪ {al : l 6∈ J}) and
B = N r ({bli : 1 ≤ i ≤ m and l ∈ J ∪ L} ∪ {bl : l 6∈ L}). Since R is infinite, A
and B are nonempty sets. Let
π = (A × B) ∪ {(ali , bli ) : 1 ≤ i ≤ m and l ∈ J ∪ L} ∪ {(al , bl ) : l ∈ {1, . . . , k} r J}.
It is a similarity relation between M and N and moreover π(R, S). By mappinginvariance of F , π(F (R), F (S)). Hence (c1 , . . . , cn ) ∈ F (S) for some c1 , . . . , cn
0
such that π(ai , ci ) for all i = 1, . . . , n. By construction, ali 6= ali0 for (l, i) 6= (l0 , i0 )
0
0
and also ali 6= al for all i, l, l0 and al 6= al for all l 6= l0 . Hence (c1 , . . . , cn ) =
(b1 , . . . , bn ).
Lemma 5.3. Let F be a (m, n)-ary mapping-invariant operator. Assume M is
infinite. For any (a1 , . . . , an ) ∈ FM (∅) there are a decomposition
˙ k
{1, . . . , n} = I1 ∪˙ . . . ∪I
and operators F1 , . . . , Fk such that
m
1. Fl = ∆1nl ◦ C>
, where nl = |Il |.
2. If i ∈ Il and ai = aj , then j ∈ Il .
3. (ai : i ∈ Il ) ∈ Fl (∅).
4. There is a σ ∈ Sym{1, . . . , m} such that for any infinite set N ,
Pσ (F1N (∅) × . . . × FkN (∅)) ⊆ FN (∅).
Proof. It is basically the same reasoning as in the proof of Lemma 5.2 except
that intern operators are not needed. Given (b1 , . . . , bn ) such that (bi : i ∈
Il ) ∈ FlN (∅) for each l = 1, . . . , k we have to show that (b1 , . . . , bn ) ∈ FN (∅).
Once al and bl are chosen, define A = M r {al : l = 1, . . . , k}, B = N r {bl :
l = 1, . . . , k} and π = (A × B) ∪ {(al , bl ) : l = 1, . . . , k}. Since M and N
13
are infinite, A and B are nonempty sets and π is a similarity relation between M and N . Clearly π(∅, ∅), and by mapping-invariance π(FM (∅), FN (∅)).
Then π((a1 , . . . , an ), (c1 , . . . , cn )) for some (c1 , . . . , cn ) ∈ FN (∅). It follows that
(c1 , . . . , cn ) = (b1 , . . . , bn ).
Lemma 5.4. For all R, M such that R ⊆ M m and R 6= ∅, there are S, N, f
such that S ⊆ N m , (N, S) is a free system and f : N → M is a mapping from
N onto M such that f (S) = R.
Proof. Choose an enumeration {(ai1 , . . . , aim ) : i ∈ I} of R whose index set I
is infinite and choose an enumeration (bi : i ∈ J) of M r field (R) such that
J ∩ (I × {1, . . . , m}) = ∅. Let N = J ∪ (I × {1, . . . , m}) and let f : N → M
be the mapping defined by f (i) = bi for i ∈ J and f ((i, j)) = aij for (i, j) ∈
I × {1, . . . , m}. If S = f −1 (R), then (N, S) is a free system, f is surjective and
f (S) = R.
Lemma 5.5. Let F be an (m, n)-ary mapping-invariant operator. Let R ⊆ M m ,
S ⊆ N m and assume f : N → M is a surjective mapping such that f (S) = R.
Then
1. f (FN (S)) = FM (R).
2. If G is also an (m, n)-ary mapping-invariant operator and FN (S) = GN (S),
then FM (R) = GM (R).
Proof. 1 is clear, since f is a similarity relation connecting R and S. 2 follows
from 1 since FM (R) = f (FN (S)) = f (GN (S)) = GM (R).
Theorem 5.6. An (m, n)-ary operator F is mapping-invariant if and only if
there are two (m, n)-ary operators G, H generated from the basic operators by
sum, product and composition and such that:
1. For all M , FM (∅) = GM (∅).
2. For all M and all nonempty R ⊆ M m , FM (R) = HM (R).
Proof. Let us consider first the case of nonempty arguments. By lemmas 5.4
and 5.5 we may restrict our attention to free systems. By Lemma 5.2 if (M, R)
is a free system, then FM (R) is a union of components, where a component is a
m
permutation of products of intern operators and operators of the form ∆1nl ◦C>
.
Since there are only finitely many choices for components, it is in fact a finite
union of them. If we take all components for F arising in all free (M, R), by
point 6 of Lemma 5.2 we see that in fact their union works for obtaining the
value of F in all free (M, R). The case of the empty set is similar but using
Lemma 5.3 and observing that for any M there are N, f such that N is infinite
and f : N → M is onto.
14
Definition 5.1. Let P be a m-ary relation symbol and let ϕ = ϕ(x1 , . . . , xn ) be a
first-order formula having P as its only extralogical symbol. The (m, n)-operator
Gϕ attached to ϕ is defined as follows: for any nonempty M and R ⊆ M m ,
n
Gϕ
M (R) = {(a1 , . . . , an ) ∈ M : (M, R) |= ϕ(a1 , . . . , an )}.
Definition 5.2. Consider the first-order language L whose only extralogical
symbol is the m-ary relation symbol P . For convenience we assume that > and
⊥ are formulas. The mapping-invariant formulas are all L-formulas generated
with the following rules:
1. > and ⊥ are mapping-invariant.
2. P (y1 , . . . , ym ) is mapping-invariant for any distinct variables y1 , . . . , ym .
3. ¬∃y1 . . . ym P (y1 , . . . , ym ) is mapping-invariant for any distinct variables
y1 , . . . , y m .
4. If ϕ, ψ are mapping-invariant, then (ϕ ∨ ψ) is mapping-invariant.
5. If ϕ, ψ are mapping-invariant and have no common free variable, then
(ϕ ∧ ψ) is mapping-invariant.
6. If ϕ = ϕ(x1 , . . . , xn ) is mapping-invariant and y is different from all the
variables x1 , . . . , xn , then for all i = 1, . . . , n, (ϕ(x1 , . . . , xn ) ∧ xi = y) is
mapping-invariant
7. If ϕ is mapping-invariant, then ∃xϕ is mapping-invariant.
Observe that > and ⊥ are dispensable since we can replace them respectively by the mapping-invariant formulas (ψ ∨ ¬ψ) and (ψ ∧ ¬ψ) where ψ =
∃y1 . . . ym P (y1 , . . . , ym ).
Theorem 5.7. An (m, n)-ary operator F is mapping-invariant if and only if
F = Gϕ for some mapping-invariant formula ϕ = ϕ(x1 , . . . , xn ) in a language
whose only extralogical symbol is the m-ary relation symbol P .
Proof. It is easy to check that by induction on ϕ that all operators Gϕ for
mapping-invariant ϕ are mapping-invariant. For the rest we first need to show
that any operator generated by sum, product and composition from the basic
ones is definable by a mapping-invariant formula. This can be easily done by
induction on the length of the generating sequence of the operator. Now using
Theorem 5.6 we see that there are two mapping-invariant formulas ψ(x1 , . . . , xn )
and χ(x1 , . . . , xn ) such that for all M , FM (∅) = Gψ
M (∅) and for all M and all
nonempty R ⊆ M m , FM (R) = GχM (R). Let ϕ(x1 , . . . , xn ) be the formula
(ψ(x1 , . . . , xn ) ∧ ¬∃yP (y)) ∨ (χ(x1 , . . . , xn ) ∧ ∃yP (y))
where y = y1 , . . . , ym is a tuple of distinct variables. It is a mapping-invariant
ψ
ϕ
χ
formula and clearly Gϕ
M (∅) = GM (∅) while for nonempty R, GM (R) = GM (R).
ϕ
Hence F = G .
15
6
More general operators
Recall that an (m1 , . . . , mr , n)-ary operator is an operator F which for every
nonempty set M gives a mapping
FM : P(M m1 ) × . . . × P(M mr ) → P(M n ).
Sometimes I will call operators with these arities general operators. Here I
characterize the mapping-invariant general operators. The methods are closely
parallel to the ones used in the previous section. Therefore I will state the
results and give only brief indications about the proofs.
The basic general operators are the basic operators introduced in the previous section and
N
1. The product. It is the (m, n, m +
n)-ary operator
such that for all M ,
N
and all R ⊆ M m and S ⊆ M n , M (R, S) = R × S.
S
2. The sum. It is the (m, m,
m)-ary operator
such that for all M , and all
S
R ⊆ M m and S ⊆ M m , M (R, S) = R ∪ S.
The general composition is the operation which for any (m1 , . . . , ms , n)-ary operator F and any (mi1 , . . . , miri , mi )-ary operators Fi (where i = 1, . . . , s), gives an
(m11 , . . . , m1r1 , m21 , . . . , m2r2 , . . . , ms1 , . . . , msrs , n)-ary operator compF1 ,...,Fs ,F such
i
that for every set M and all subsets Rji ⊆ M mj (i = 1, . . . , s and j = 1, . . . , ri ),
1
s
compF1 ,...,Fs ,F (R11 , . . . , Rrss ) = F (F1 (R11 , . . . , Rm
), . . . , Fs (R1s , . . . , Rm
))
r1
rs
A general operator is intern if it is generated from projections, diagonals, and
permutations by general composition.
I generalize now the technical notion of free system to make it useful for
general operators. Let R1 ⊆ M m1 , . . . , Rr ⊆ M mr . We say that (M, R1 , . . . , Rr )
is a general free system if each (M, Ri ) is a free system and the Ri have disjoint
fields.
Lemma 6.1. Let F be a (m1 , . . . , mr , n)-ary mapping-invariant operator. Let
R1 ⊆ M m1 , . . . , Rr ⊆ M mr , let L ⊆ {1, . . . , r} and assume (M, Ri )i∈L is a
general free system and Ri = ∅ for i 6∈ L. For any (a1 , . . . , an ) ∈ F (R1 , . . . , Rr )
there is a decomposition
˙ k11 ∪˙ . . . ∪I
˙ 1r ∪˙ . . . ∪I
˙ krr
{1, . . . , n} = I11 ∪˙ . . . ∪I
and operators F11 , . . . , Fk11 , . . . , F1r , . . . , Fkrr such that
1. Flp is (mp , nl )-ary, where nl = |Ilp |.
2. If i ∈ Ilp and ai = aj , then j ∈ Il .
3. Flp is intern if {ai : i ∈ Ilp } has more than one element or it has just one
element a and a ∈ fieldj (Rl ) for some j.
16
4. If {ai : i ∈ Ilp } has only one element a and a 6∈ field (R), then Flp =
m
∆1nl ◦ C> p .
5. (ai : i ∈ Ilp ) ∈ Flp (Rl ).
6. There is a σ ∈ Sym{1, . . . , m1 , m1 + 1, . . . , m1 + m2 + . . . + mr } such that
for all S1 ⊆ N m1 , . . . , Sr ⊆ N mr , if (N, Si )i∈L is a general free system
and Si = ∅ for i 6∈ L, then
1
r
Pσ (F11 (S1 )×. . .×Fm
(S1 )×. . .×F1r (Sr )×. . .×Fm
(Sr )) ⊆ F (S1 , . . . , Sr ).
1
r
Proof. The fact that in a general free system (M, Ri )i∈L all the relations Ri have
disjoint fields allows us to adapt the proof of Lemma 5.2 in a straightforward
way to this setting. The details are left to the reader.
Theorem 6.2. Let F be an (m1 , . . . , mr , n)-ary operator. Then F is mappinginvariant if and only if for each L ⊆ {1, . . . , r} there is an (m1 , . . . , mr , n)-ary
operators GL generated from the general basic operators by general composition
and such that for any set M and all R1 ⊆ M m1 , . . . , Rr ⊆ M mr such that
Ri 6= ∅ if and only if i ∈ L,
F (R1 , . . . , Rr ) = GL (R1 , . . . , Rr ).
Proof. General free systems satisfy the corresponding version of Lemma 5.4 and
therefore can be used in the same fashion as free systems were used in the proof
of Theorem 5.6. For the case L = ∅ one has to use an infinite set M instead of
a general free system.
Definition 6.1. Let P1 , . . . , Pr be relation symbols (where Pi is mi -ary) and
let ϕ = ϕ(x1 , . . . , xn ) be a first-order formula having P1 , . . . , Pr as its only
extralogical symbol. The (m1 , . . . , mr , n)-operator Gϕ attached to ϕ is defined
as follows: for any nonempty M and Ri ⊆ M mi for i = 1, . . . , r,
n
Gϕ
M (R1 , . . . , Rr ) = {(a1 , . . . , an ) ∈ M : (M, R1 , . . . , Rr ) |= ϕ(a1 , . . . , an )}.
The mapping-invariant formulas of the first order language whose extralogical
symbols are P1 , . . . , Pr are all formulas generated with the rules described in the
previous section except that instead of rules 2 and 3 we now have:
2 Pi (y1 , . . . , ymi ) is mapping-invariant for all distinct y1 , . . . , ymi for all i =
1, . . . , r.
3 ¬∃y1 . . . ymi Pi (y1 , . . . , ymi ) is mapping-invariant for all distinct y1 , . . . , ymi
for all i = 1, . . . , r.
Theorem 6.3. Let F be an (m1 , . . . , mr , n)-ary operator. Then F is mappinginvariant if and only if F = Gϕ for some mapping-invariant formula ϕ =
ϕ(x1 , . . . , xn ) in a language whose only extralogical symbols are the relational
symbols P1 , . . . , Pr (where Pi is mi -ary and corresponds to Ri ).
17
Proof. As in the proof of Theorem 5.7, we find a mapping-invariant formula
ψL (x1 , . . . , xn ) for each L ⊆ {1, . . . , r} in such a way that FM (R1 , . . . , Rr ) =
L
m1
, . . . , Rr ⊆ M mr and Ri 6= ∅ for i ∈ L. The trick
Gψ
M (R1 , . . . , Rr ) if R1 ⊆ M
for coding all the ψL in only one mapping-invariant formula ϕ(x1 , . . . , xn ) is
the same as the one presented in the proof of Theorem 5.7. We take for ϕ the
disjunction of all formulas of the form
^
^
ψL (x1 , . . . , xn ) ∧
∃yPi (y) ∧
¬∃yPi (y)
i6∈L
i∈L
7
The functional type setting
Let us consider now a modification of the finite type hierarchy. We start with the
basic type 0 of individuals and the boolean type b. The finite functional types
are obtained from the types 0 and b by the rule that whenever τ1 , . . . , τn , µ
are finite functional types, then also (τ1 , . . . , τn → µ) is a finite functional
type. We will call them more briefly functional types. Given a nonempty set
M , we associate to each functional type τ a corresponding τ -universe Mτ over
M . The starting universes are the universe of individuals M0 = M and the
boolean universe Mb = {0, 1}. The remainder universes are obtained recursively
according to the rule that if τ = (τ1 , . . . , τn → µ) then Mτ is the set of all
mappings h : Mτ1 × . . . × Mτn → Mµ .
Relational types, universes, objects, and operators can be represented in the
functional setting by replacing hereditarily every relation by its characteristic
function. For every relational type τ there is a corresponding functional type
τ ∗ defined recursively as follows: 0∗ = 0 and
(τ1 , . . . , τn )∗ = (τ1∗ , . . . , τn∗ → b).
Every element a of Mτ corresponds also to an element a∗ of Mτ ∗ . This correspondence is in fact a bijection between Mτ and Mτ ∗ . For a ∈ M0 we just
take a∗ = a. If τ = (τ1 , . . . , τn ) and a ∈ Mτ , we define a∗ as the mapping from
Mτ1∗ × . . . × Mτn∗ into {0, 1} such that
a∗ (a∗1 , . . . , a∗n ) = χa (a1 , . . . , an )
for all a1 ∈ Mτ1 , . . . , an ∈ Mτn , where χa is the characteristic function of a as a
subset of Mτ1 × . . . × Mτn .
Let τ be a functional type and let f : M → N be a surjective mapping.
In general it is not possible to extend f naturally to a corresponding total
mapping fτ : Mτ → Nτ for any functional type τ as I did for relational types,
but nevertheless we can extend it as a partial surjective mapping. If τ = 0
we put fτ = f and if τ = b we take for fτ the identity on Mb = {0, 1}. If
τ = (τ1 , . . . , τn → µ), and every fτi is surjective, we can define fτ as follows.
18
1. Its domain is the subset of Mτ which consists of all mappings
a : Mτ1 × . . . × Mτn → Mµ
such that
(a) a(a1 , . . . , an ) ∈ dom (fµ ) for all a1 ∈ dom (fτ1 ), . . . , an ∈ dom (fτn )
(b) for all a1 ∈ dom (fτ1 ), . . . , an ∈ dom (fτn ), b1 ∈ dom (fτ1 ), . . . , bn ∈
dom (fτn ), if fτi (ai ) = fτi (bi ) for all i = 1, . . . , n, then
fµ (a(a1 , . . . , an )) = fµ (a(b1 , . . . , bn )).
2. For any such a we define fτ (a) as the mapping from Nτ1 × . . . , ×Nτn into
Nµ such that for all a1 ∈ dom (fτ1 ), . . . , an ∈ dom (fτn ),
fτ (a)(fτ1 (a1 ), . . . , fτn (an )) = fµ (a(a1 , . . . , an )).
We check now that fτ is surjective. Let b ∈ Nτ . For any c ∈ Nµ choose
a corresponding c0 ∈ dom (fµ ) such that fµ (c0 ) = c. Define a ∈ Mτ as the
mapping a : Mτ1 × . . . × Mτn → Mµ such that for all a1 ∈ Mτ1 , . . . , an ∈ Mτn ,
a(a1 , . . . , an ) = (b(fτ1 (a1 ), . . . , fτn (an )))0 . Observe that a ∈ dom fτ . Now,
given b1 ∈ Nτ1 , . . . , bn ∈ Nτn take a1 ∈ Mτ1 , . . . , an ∈ Mτn with fτi (ai ) = bi
for all i = 1, . . . , n, and note that b(b1 , . . . , bn ) = fµ (a(a1 , . . . , an )). Therefore
b = fτ (a).
We can also think of extending a similarity relation π between M and N to
every τ -universe for any functional type τ . The extension πτ will be a similarity
relation between its domain dom (πτ ) ⊆ Mτ and its range rng (πτ ) ⊆ Nτ , so
not necessarily a similarity between Mτ and Nτ . We start by taking π0 = π
and by taking as πb the identity in the boolean universe Mb = Nb = {0, 1}. Let
τ = (τ1 , . . . , τn → µ). We define πτ ⊆ Mτ × Nτ by stipulating that for a ∈ Mτ
an b ∈ Nτ ,
(∗) πτ (a, b) if and only if for all a1 ∈ Mτ1 , . . . , an ∈ Mτn , b1 ∈ Nτ1 , . . . , bn ∈ Nτn
if πτi (ai , bi ) for i = 1, . . . , n, then πµ (a(a1 , . . . , an ), b(b1 , . . . , bn )).
This last proposal of extensions seems very natural when working with similarities. For any given similarity π between M and N there is a unique system
(πτ )τ a functional type of relations πτ ⊆ Mτ × Nτ which satisfy the defining condition (∗) stated above. But if π is a surjective mapping, we obtain the same
extension πτ if we consider it only as a similarity relation. Moreover, if π is a relational composition of mappings and inverses of mappings, then the extensions
πτ are nothing more than the corresponding composition of extensions of mappings. This has not been noticed by Feferman in [3] and therefore he distinguises
between what he calls homomorphism-invariance and similarity-invariance. Before proving the equivalence we need to show an easy result according to with
every chain of surjective mappings and inverses of them can be shortened to
length two.
19
Lemma 7.1. If π is a similarity relation between M and N , there is a set K
an surjective mappings f : K → M and g : K → N such that π = f −1 ◦ g.
Proof. Let K be a set of indexes big enough to enumerate π as a set of pairs in
the form
π = {(ai , bi ) : i ∈ K}
and define f : K → M by f (i) = ai and g : K → N by g(i) = bi .
Proposition 7.2. Let τ be a functional type.
1. If f : M → N is a surjective mapping, then it is a similarity relation
between M and N and its extension fτ as a mapping coincides with its
extension as a similarity relation.
2. Assume that M1 , . . . , Mn are sets and f1 , . . . , fn−1 are mappings such that
M = M1 , N = Mn and for every i = 1, . . . , n, fi is from Mi onto Mi+1
or it is from Mi+1 onto Mi , and π is the relational product R1 ◦ . . . ◦ Rn−1
where Ri = fi if fi : Mi → Mi+1 and Ri = fi−1 if fi : Mi+1 → Mi . Then
πτ is the relational product R1τ ◦ . . . ◦ Rn−1 τ , where Riτ is the partial
mapping fi τ if fi : Mi → Mi+1 and it is its inverse if fi : Mi+1 → Mi .
Proof. 1. It is easy to check by induction on the complexity of τ that the
system of partial mappings (fτ )τ a functional type satisfies condition (∗). 2. This
is proven by induction on n, but using Lemma 7.1 we see that it will be enough
to prove the result in three particular cases:
Case 1. f : K → M , g : K → N are surjective and π = f −1 ◦ g. We
need only to check that the system (fτ−1 ◦ gτ )τ a functional type satisfies condition
(∗) and this is done by induction on the complexity of τ . Consider the case
τ = (τ1 , . . . , τn → µ). Let a ∈ Mτ and b ∈ Nτ . Let R = fτ−1 ◦ gτ . If R(a, b)
then for some c ∈ dom (fτ ) ∩ dom (gτ ) we have fτ (c) = a and gτ (c) = b. Let
Si = fτ−1
◦ gτi and assume Si (ai , bi ) for all i = 1, . . . , n. Then for each i there is
i
some ci ∈ dom (fτi ) ∩ dom (gτi ) such that fτi (ci ) = ai and gτi (ci ) = bi . Clearly
c(c1 , . . . , cn ) ∈ dom (fµ ) ∩ dom (gµ ) and fµ (c(c1 , . . . , cn )) = a(a1 , . . . , an ), and
gµ (c(c1 , . . . , cn )) = b(b1 , . . . , bn ). Hence T (a(a1 , . . . , an ), b(b1 , . . . , bn )) for T =
fµ−1 ◦ gµ . Now we assume the righthand side of (∗) holds for a, b and we show
that R(a, b). We have to find some c ∈ dom (fτ ) ∩ dom (gτ ) with fτ (c) = a and
gτ (c) = b. Let ci ∈ Kτi for each i = 1, . . . , n. If some ci is not in dom (fτi ) ∪
dom (gτi ), we choose for c(c1 , . . . , cn ) some arbitrary element of Kµ . If all ci ∈
dom (fτi ) but some ci 6∈ dom (gτi ) we observe that a(fτ1 (c1 ), . . . , fτn (cn )) ∈ Mµ
and since fµ is surjective we can define c(c1 , . . . , cn ) as an element d ∈ dom (fµ )
such that fµ (d) = a(fτ1 (c1 ), . . . , fτn (cn )). The procedure is similar if all ci ∈
dom (gτi ) but some ci 6∈ dom (fτi ). Assume now ci ∈ dom (fτi )∩dom (gτi ) for all
i = 1, . . . , n. Let Si = fτ−1
◦ gτi . Then Si (fτi (ci ), gτi (ci )) for all i and we can use
i
the righthand side of (∗) to make sure that there is some d ∈ dom (fµ )∩dom (gµ )
such that fµ (d) = a(fτ1 (c1 ), . . . , fτn (cn )) and gµ (d) = b(gτ1 (c1 ), . . . , gτn (cn )).
We take as c(c1 , . . . , cn ) such a d in this last case. It follows that c ∈ dom (fτ ) ∩
dom (gτ ), fτ (c) = a and gτ (c) = b. Hence R(a, b).
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Case 2. f : M → N , g : K → N , h : K → P are surjective and π = f ◦g −1 ◦h.
Similar to the previous case. Given a ∈ Mτ and b ∈ Pτ which satisfy the
righthand side of (∗) we check that a ∈ dom (fτ ) and we define in a similar way
as before some c ∈ dom (gτ ) ∩ dom (hτ ) such that fτ (a) = gτ (c) and hτ (c) = b.
Case 3. f : N → M , g : K → N , h : K → P are surjective and π =
f −1 ◦ g −1 ◦ h. Also similar to the other cases. Now the main point is to start
with a ∈ Mτ and b ∈ Pτ for which the righthand side of (∗) holds and to
find c ∈ dom (gτ ) ∩ dom (hτ ) such that gτ (c) ∈ dom (fτ ), fτ (gτ (c)) = a and
hτ (c) = b
In the functional type setting, every operator can be identified with an
object. Therefore it would be enough to consider only objects for all invariance purposes. Nevertheless, it is convenient to have both notions (object and
operator) to compare them with objects and operators of the relational type
setting. The definitions of invariance are similar to the ones given in the relational case, but types are functional now. Let us say that an object a of
functional type τ is mapping-invariant if for all M, N and every surjective mapping f : M → N , aM ∈ dom (fτ ) and fτ (aM ) = aN . Feferman in [3] calls this
notion homomorphism-invariance although the sets M, N are pure sets, without
any added structure. He calls an object a similarity-invariant if for all M, N
and every similarity relation π between M and N , πτ (aM , aN ). This notion of
invariance can be easily defined also for (τ1 , . . . , τn ; µ)-ary operators as I did in
the relational case. The details are left to the reader.
Corollary 7.3. In the functional type hierarchy an object or an operator is
mapping-invariant (i.e., homomorphism-invariant in the sense of Feferman) if
and only if it is similarity-invariant.
Proof. By Proposition 7.2.
8
Feferman’s analysis of mapping-invariance
We have seen that every relational type τ can be represented as a functional
type τ ∗ and every a ∈ Mτ has a corresponding element a∗ ∈ Mτ ∗ . In a similar
way, to any operator F in the relational type setting corresponds an operator
F ∗ in the functional type setting. It is defined by
∗
FM
(a∗1 , . . . , a∗n ) = FM (a1 , . . . , an )∗ .
Now I raise the question of what kind of invariance on relational types τ is
the correct translation of mapping-invariance on the corresponding functional
types τ ∗ . We will see that it is closely related to preimage-invariance and that
it is just preimage-invariance at the first levels of the type hierarchy. Before
discussing preimage-invariance I introduce a new version of invariance, which is
exactly the relational counterpart of mapping-invariance on functional types.
Let f : M → N be a surjective mapping. I present another way of extending
f to every universe Mτ for any relational type τ . I will call it the Feferman
21
extension of f to τ and I will use the notation fτF e for it. We start, as usual,
putting f0F e = f and define inductively fτF e for τ = (τ1 , . . . , τn ). In general it is
going to be a partial surjective mapping.
1. Its domain consists of all a ⊆ Mτ1 × . . . × Mτn such that for all a1 ∈
dom (fτF1e ), . . . , an ∈ dom (fτFne ), b1 ∈ dom (fτF1e ), . . . , bn ∈ dom (fτFne ), if
fτFi e (ai ) = fτFi e (bi ) for all i = 1, . . . , n, then (a1 , . . . , an ) ∈ a if and only if
(b1 , . . . , bn ) ∈ a.
2. For any such a, we define fτF e (a) as
{(fτF1e (a1 ), . . . , fτFne (an )) : (a1 , . . . , an ) ∈ a∩(dom (fτF1e )×. . .×dom (fτFne ))}
Lemma 8.1. Let f : M → N be surjective, let τ be a relational type and assume
a ∈ Mτ . Then
1. a ∈ dom (fτF e ) iff a∗ ∈ dom (fτ ∗ )
2. If a ∈ dom (fτF e ), then fτF e (a)∗ = fτ ∗ (a∗ ).
Proof. It is an easy induction on the complexity of τ .
The notion of invariance for objects and operators in the context of these
extensions fτF e is defined as in the other cases. We will call it Feferman invariance. For instance, an object a of type τ is Feferman invariant if and only if
for any surjective mapping f : M → N , aM ∈ dom (fτF e ) and fτF e (aM ) = aN .
Invariance for operators is defined in the obvious way.
Proposition 8.2. An operator F is Feferman invariant if and only if its corresponding operator F ∗ in the functional type setting is mapping-invariant, i.e.,
homomorphism-invariant in the sense of Feferman.
Proof. Let F be a (τ1 , . . . , τn ; µ)-ary operator. Assume F is Feferman invariant.
Let f : M → N be surjective and let a1 ∈ Mτ1 , . . . , an ∈ Mτn be such that a∗i ∈
dom (fτi∗ ) for each i = 1, . . . , n. By Lemma 8.1, ai ∈ dom (fτFi e ) and fτFi e (ai )∗ =
fτi∗ (a∗i ) for every i = 1, . . . , n. By Feferman invariance of F , FM (a1 , . . . , an ) ∈
dom (fµF e ) and
fµF e (FM (a1 , . . . , an )) = FN (fτF1e (a1 ), . . . , fτFne (an )).
∗
(a∗1 , . . . , a∗n ) = FM (a1 , . . . , an )∗ ∈ dom (fµ∗ ) and
By Lemma 8.1 FM
∗
fµ∗ (FM
(a∗1 , . . . , a∗n )) = fµF e (FM (a1 , . . . , an ))∗ .
Hence
∗
fµ∗ (FM
(a∗1 , . . . , a∗n )) = FN (fτF1e (a1 ), . . . , fτFne (an ))∗ = FN∗ (fτ1∗ (a∗1 ), . . . , fτn∗ (a∗n )).
The other direction has a similar proof.
22
Let f : M → N be a surjective mapping. I propose a last way of extending
f to every universe Mτ for any relational type τ . We well call it the preimage
extension of f to τ and we will use the notation fτp for it. The extension fτp
will not be in general a total mapping from Mτ onto Nτ . It is a mapping but
its domain is only a subset of Mτ . As usual, we start with f0p = f . Now let
τ = (τ1 , . . . , τn ).
1. The domain of fτp consists of all relations a ⊆ Mτ1 , × . . . × Mτn for which
there is some b ⊆ Nτ1 × . . . × Nτn such that
a = {(a1 , . . . , an ) ∈ dom (fτp1 )×. . .×dom (fτpn ) : (fτp1 (a1 ), . . . , fτpn (an )) ∈ b}.
2. For each such a ∈ dom (fτp ) we set
fτp (a) = {(fτp1 (a1 ), . . . , fτpn (an )) : (a1 , . . . , an ) ∈ a}.
Let F be an operator of relational type (τ1 , . . . , τn ; τ ). We call F preimageinvariant if for any sets M, N , for any surjective mapping f : M → N ,
if a1 ∈ dom (fτp1 ), . . . , an ∈ dom (fτpn ), then FM (a1 , . . . , an ) ∈ dom (fτp ) and
fτp (FM (a1 , . . . , an )) = FN (fτp1 (a1 ), . . . , fτpn (an )). An object a of relational type
τ is preimage-invariant if for any sets M, N , for any surjective mapping f :
M → N , aM ∈ dom (fτp ) and fτp (aM ) = aN .
We will concentrate the following discussion on operators, but the results
can be easily extended also to objects.
Lemma 8.3. If τ = 0n and f : M → N is surjective, then fτF e = fτp .
Proof. We show that dom (fτF e ) = dom (fτp ) and fτF e (a) = fτp (a) for every a ∈
dom (fτF e ). Let a ∈ dom (fτF e ) and let b = {(f (a1 ), . . . , f (an )) : (a1 , . . . , an ) ∈
a}. It follows that a = {(a1 , . . . , an ) : (f (a1 ), . . . , f (an )) ∈ b} and therefore
a ∈ dom (fτp ). Clearly in this case fτF e (a) = b = fτp (a). On the other hand, if a ∈
dom (fτp ) there is some b ⊆ N n such that a = {(a1 , . . . , an ) : (f (a1 ), . . . , f (an ) ∈
b}. Whenever a1 , . . . , an , b1 , . . . , bn ∈ M , (a1 , . . . , an ) ∈ a, and f (ai ) = f (bi ) for
every i = 1, . . . , n, we have that (f (b1 ), . . . , f (bn )) = (f (a1 ), . . . , f (an )) ∈ b and
therefore (b1 , . . . , bn ) ∈ a. This means that a ∈ dom (fτp ).
Theorem 8.4. An (m1 , . . . , mr , n)-ary operator F is preimage-invariant if and
only if its corresponding operator in the functional type setting F ∗ is mappinginvariant, i.e., homomorphism-invariant in the sense of Feferman.
Proof. By Proposition 8.2 is enough to prove that F is preimage-invariant if and
only if it is Feferman invariant. Let τi = 0mi for every i = 1, . . . , r. Assume F is
preimage-invariant and let f : M → N be surjective. Let R1 ⊆ M m1 , . . . , Rr ⊆
M mr and assume Ri ∈ dom (fτFi e ) for each i = 1, . . . , r. By Lemma 8.3 Ri ∈
dom (fτpi ) and fτpi (Ri ) = fτFi e (Ri ) for every i = 1, . . . , r. By preimage-invariance,
FM (R1 , . . . , Rr ) ∈ dom (f0pn ) and
f0pn (FM (R1 , . . . , Rr )) = FN (fτp1 (R1 ), . . . , fτpr (Rr )).
23
By Lemma 8.3 again, FM (R1 , . . . , Rr ) ∈ dom (f0Fne ) and f0Fne (FM (R1 , . . . , Rr )) =
f0pn (FM (R1 , . . . , Rr )). Therefore
f0Fne (FM (R1 , . . . , Rr )) = FN (fτF1e (R1 ), . . . , fτFre (Rr ))
and F is Feferman invariant. The other direction is proven in a similar way.
ACKNOWLEDGMENT
Work partially supported by grant BFM2002-01034 of Spanish MCYT and grant
2002SGR 00126 of Catalan DURSI. I thank Xavier Caicedo for correcting some
mistaken statements concerning Linsdtröm quantifiers made in a previous version. I also thank Genoveva Martı́ for some comments improving the exposition.
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