Maximality of logic without identity

Lindstr\"om theorem obviously fails as a characterization of $\mathcal{L}_{\omega \omega}^{-} $, first-order logic without identity. In this note we provide a fix: we show that $\mathcal{L}_{\omega \omega}^{-} $ is \emph{maximal} among abstract logics satisfying a weak form of the isomorphism property (suitable for identity-free languages and studied in \cite{Casa}), the L\"owenheim--Skolem property, and compactness. Furthermore, we show that compactness can be replaced by being recursively enumerable for validity under certain conditions. In the proofs we use a form of strong upwards L\"owenheim--Skolem theorem not available in the framework with identity.


INTRODUCTION
In the 1960s, Per Lindström [22] showed that first-order logic is maximal (in terms of expressive power) among its extensions satisfying certain combinations of model-theoretic results.The best known of these combinations are: Löwenheim-Skolem theorem + Compactness Löwenheim-Skolem theorem + Recursively enumerable set of validities This list is by no means exhaustive though (the reader can consult the encyclopaedic monograph [3] for a thorough treatment of this topic).Philosophically, these results have been interpreted as providing a case for first-order logic being the "right" logic in contrast to higher-order, infinitary, or logics with generalized quantifiers, which can be argued to be more mathematical beasts (see [19,25]).An implicit assumption of Lindström's work is that identity (=) is a most basic notion and belongs in the base logic.
The classical Lindström theorems clearly fail for first-order logic without identity (L − ωω ) since first-order logic with identity (L ωω ) is a proper extension of L − ωω .In fact, there are continuum-many logics between the former and the latter satisfying the compactness and Löwenheim-Skolem properties, and with recursively enumerable sets of validities (see Example 1 below).
In this article we aim at finding a way to amend Lindström's two central theorems so that they apply in the identity-free context. 1 Our proofs make heavy use of a property that is not available in the context with identity, namely, that we have an unrestricted upwards Löwenheim-Skolem theorem applying even to finite models.We also observe other maximality results: a very simple one for the monadic version of the logic (i.e.restricted to vocabularies that only have unary predicates), L 1− ωω , as well as results for both L − ∞ω and L − ωω in terms of a suitable variant of the Karp property.A simple byproduct will be a preservation theorem characterizing the identity-free fragment of first-order logic (we essentially obtain [10,Cor. 2.10] by a rather different method).
L − ωω has attracted mathematical attention in other works such as [18] where the problem of categoricity of theories in that logic is studied.Moreover, the results in the present paper may provide new insight on the philosophical discussion whether L − ωω is suitable as 1 Recall that any criteria for first-order axiomatizability in terms of closure of a class of structures under certain algebraic operations can be recast as a Lindström-style theorem.In this way, [10,Thm. 3.4] can be seen as a Lindström-style result already in the literature for logic without identity.
a contender for the title of the "right logic" against L ωω .After all, the logicality of the = predicate is not obvious (cf.[15]).So, if the criteria were to involve only indisputably logical operators (so no more than what L ωω already involves), be reasonably expressive (quite a bit can be formalized already in L − ωω , including set theory), and satisfy a neat Lindström-style characterization, L − ωω would appear to be as good an option as any.However, we will not pursue those issues here.
We use the notion of an abstract logic from [3, Def.II.1.1.1]which presents logics as model-theoretic languages [14] (see also [2,16,22]), not as consequence relations or collections of theorems.Furthermore, we assume logics to have the basic closure properties (including for the Boolean connectives) from [3, Def.II.1.2.1], except that in the atom property we use L − ωω as the base logic.For greater generality, we do not require the relativization property.As usual, if L and L ′ are logics, we write L ≤ L ′ if, for any vocabulary τ and any formula ϕ ∈ L(τ ), we can find an equivalent formula ϕ ′ ∈ L ′ (τ ).
L − ωω is, properly speaking, a fragment of L ωω containing the guarded fragments corresponding to basic modal logics.In this setting, the most fruitful approach has been to use bisimulations as a modal analogue of potential isomorphisms in first-order logic [5].In the present context all we require is the notion of weak (partial) isomorphism introduced in [10], which is stronger than bisimulation. 2 Interestingly, the presence of identity can make a substantial difference regarding compactness.For example, monadic first-order logic with the Henkin quantifier, L 1 ωω (Q H ), is not compact and not contained in (monadic) first-order logic with identity for it can express the quantifier "there are at least ℵ 0 -many elements"; however, the identity-free fragment of the very same logic admits the effective elimination of the quantifier Q H and, hence, it is compact [21, Thm.1.5]. 3  The paper is arranged as follows: in §2 we start with the preliminary observation that there is a continuum of abstract logics between L − ωω and L ωω , and we recall the definitions of the properties of abstract logics employed in the paper, while referring to the literature for some particular technical notions.In §3 we present our main new results, that is, Lindström-style characterizations of the identity-free first-order logic and its monadic fragment, together with instrumental observations regarding the logical relations of the involved properties and a useful form of upwards Löwenheim-Skolem theorem.In §4 we examine a few interesting particular extensions of L − ωω that help us understand the role of compactness and the Löwenheim-Skolem property in our characterizations.Finally, in §5, we collect some open problems that arise from this investigation.

PRELIMINARIES
We begin this section by noting that there are continuum-many pairwise non-equivalent abstract logics between L − ωω and L ωω (actually, already between their monadic fragments).Example 1.Consider quantifiers ∃ ≥n with semantics A |= ∃ ≥n x ϕ iff there are at least n elements a such that A |= ϕ[a].For each non-empty X ω \ {0, 1}, we can prove that the logic L − ωω ({∃ ≥n | n ∈ X}) indeed lies properly between L − ωω and L ωω in terms of expressive power and, moreover, there is a continuum of such intermediate abstract logics.
For distinct X, Y ⊆ ω \ {0, 1}, the corresponding logics L − ωω ({∃ ≥n | n ∈ X}) and L − ωω ({∃ ≥n | n ∈ Y }) are also distinct.To see this, it suffices to focus our attention on a monadic vocabulary τ = {P }.Suppose, without loss of generality, that we have an element r ∈ X \ Y .We abbreviate, for n < m, ∃ ≥n x θ ∧ ¬∃ ≥m x θ as ∃ [n,m) x θ, and, for each n, ∃ ≥n x θ as ∃ [n,∞) x θ.Then, using results from [8], any sentence ϕ from the logic 2 This notion has incidentally proven useful in recent philosophical debates on the logicality of quantifiers and other operators [6,11]. 3In contrast, the logic obtained from (monadic) identity-free first-order logic by adding the quantifier "there are at least ℵ 0 elements" does not satisfy compactness [29,Thm. 8].
L − ωω ({∃ ≥n | n ∈ Y }) over the vocabulary τ is equivalent to a disjunction θ 1 ∨ • • • ∨ θ q involving only quantifiers from ϕ where each θ i is of one the three following forms: • ∃ [ni,mi) x P (x) ∧ ∃ [ri,si) x ¬P (x) • ∃ [ni,mi) x P (x) • ∃ [ri,si) x ¬P (x) where n i ≤ m i and r i ≤ s i belong to Y ∪ {1, ∞}.Thus, ϕ just describes an array of possible cardinalities for the interpretations of P and its complement, and clearly, ∃ ≥r x P (x) is equivalent to this disjunction if and only if [r, ∞) = ni<mi [n i , m i ), or r = n i for the least n i , which is impossible as r ∈ Y ∪ {1}.
We use the definitions from [10]: A ∼ B means that there is a relativeness correspondence between the structures [10, Def.2.5] (we prefer to call this a weak isomorphism); A ∼ p B means that there is a back-and-forth system I of partial relativeness correspondences between the models [10,Def. 4.7] (we can say that these structures are partially weakly isomorphic); and we denote by ∼ n the finite approximation of ∼ p [10,Def. 4.2].In the setting of first-order logic without identity, the relation ∼ behaves like a weak notion of isomorphism [10], which motivates the name for the second property defined below. 4he properties of abstract logics that we consider in this article are: • Boundedness property: any sentence ϕ(<, . . . ) which for arbitrary large ordinal type α has a model where the interpretation of < is an irreflexive and transitive binary relation containing a chain of order type α has a model where the interpretation of < contains an infinite descending chain.All these properties, with the exception of Karp − and weak isomorphism, hold in L ωω .
Given a structure A, we denote by A * the reduction of A [10, Def.2.4], i.e., the quotient structure A Ω(A) obtained from the Leibniz congruence relation.

Proposition 2 ( [10]
).Let A and B be structures.Then: Thanks to Proposition 2, the weak isomorphism property can be equivalently formulated as follows: for any structures A and B, A

MAXIMALITY RESULTS
We start this section by showing a form of upwards Löwenheim-Skolem theorem, which will be heavily used in the arguments below: Lemma 3. Let L be an abstract logic with the weak isomorphism property.Then, a theory T ⊆ L(τ ) has a model A of cardinality λ only if, for any κ > λ, there is a model B of T with cardinality κ and a surjective strict homomorphism (in the sense of [10,Def. 2.1]), and hence a weak isomorphism, from B onto A.
Proof.It follows by inspection of the proof of [7,Lem. 2.24] or [1, Ch.IV, §1] (which is only formulated for relational languages but can be easily generalized to languages with function symbols).For any structure A of cardinality λ, in that proof one builds a model B of size κ and a mapping B −→ A which is, in fact, a surjective strict homomorphism.Remark 4. Lemma 3 allows us to see that a plethora of logics do not have the weak isomorphism property, e.g. the logics in Example 1.Interestingly, the usual Lindström quantifiers destroy the property, in particular in the logics L − ωω (Q α ).However, as we will see in Example 16, all of these logics have counterparts which do have the weak isomorphism property.On the other hand, as we will see below, the Henkin quantifier Q H is a curious case of a natural Lindström quantifier that has the weak isomorphism property.Now we can provide an analogue of (1) from [3, Thm.III.1.1.1].
Lemma 5. Let L be an abstract logic such that L − ωω ≤ L. If L has the compactness and weak isomorphism properties, then it also has the finite weak dependence property.
Proof.Given a vocabulary τ, let τ ′ be a disjoint copy and consider the theory Φ(τ, R): For any ϕ ∈ L(τ ), let ϕ ′ denote its renaming in the type τ ′ .Then, Φ(τ, R) |= ϕ ↔ ϕ ′ by closure of the logic L under weak isomorphisms, and by compactness Therefore, Mod τ (ϕ) cannot be a union of these classes (it would be equivalent to a finite disjunction of sentences in L − ωω (τ 0 )) and it must cut some equivalence class in two nonemtpy pieces.In other words, there are τ -structures A n and B n such that By Lemma 3, we may assume that A n and B n have the same infinite power and share the same domain A n .
By [10, Lem.4.4 and Prop.4.5], A n ↾ τ 0 ∼ n B n ↾ τ 0 ; that is, there are sets I 0 , . . ., I n of weak finite τ 0 -partial isomorphisms from A n to B n such that I n = ∅ and for all p ∈ I j+1 a, ∈ A n , b ∈ B n there are q, q ′ ∈ I j such that q, q ′ ⊇ r and a ∈ dom(q), b ∈ rg(q ′ ), and further extension properties guaranteeing that constants and functions are eventually preserved.The set of all finite weak τ 0 -partial isomorphisms has the same power as D, so we may enumerate them as {R p | p ∈ A n }; moreover, we may assume {0, . . ., n} ⊆ A n .Then, renaming B n as B ′ n on the vocabulary τ ′ , as in the proof of Lemma 5, and defining in A n : , where c 0 is a constant, < and I are binary relations, and G is a ternary relation (all fresh symbols; moreover, for each formula ψ ∈ L(τ 0 ), we denote by ψ ′ its renaming in the vocabulary τ ′ 0 ): The second sentence states that I n is non-empty.Sentences 3-6 describe a sequence I 0 , . . ., I n of sets of weak τ 0 -partial isomorphisms in the sense of [10,Def. 4.2].
As the above holds for any n, we have models for any finite part of the infinite theory with additional constants c 1 , c 2 , ...: By compactness, we have a model C, < C , I C , G C , c C j j∈ω of this theory.By the axioms, each p ∈ C encodes a weak τ 0 -partial isomorphism 0 , and the sequence . .has the back-and-forth extension property with respect to increasing subindexes: if R p ∈ I j and c ∈ C, then there is a R q ∈ I j+1 such that c ∈ dom(R p ), etc.Hence, K C = j I j has the unrestricted extension property and becomes a Karp system of weak τ 0 -isomorphisms.This is expressible by the finite theory Φ(τ 0 , K, G) which results of changing the backand-forth axioms of Ψ(τ 0 , <, c 0 , I, G) to ∞ω is maximal among its extensions in having the boundedness, and Karp − properties.In fact, all we need from the boundedness property is that it will give us a model where < is not well founded.
Comparing the proof of Theorem 6 with that of its classical counterpart with identity, the reader should note that our approach makes a substantial use of the strong upwards Löwenheim-Skolem theorem given by Lemma 3.This allows us to deal with cardinality situations that in the classical context are dealt with the expressive power of identity.
One may wonder whether we can obtain a Lindström-style characterization for identityfree monadic first-order logic, L 1− ωω , analogous to Tharp's result [25, Thm.1] for monadic first-order logic.The answer is yes and the result does not require, surprisingly, any form of the Löwenheim-Skolem theorem (not even the other two properties if we assume the finite weak dependence property; see Remark 10).Each map δ : I → {0, 1} determines a type A type t δ is consistent with A if for each finite J ⊆ I, A |= ∃x ∧ (t δ (x) ↾ J).Clearly, A and B above have the same consistent types and, if t δ is not consistent with A, there is a witness η δ of the form ¬∃x ∧ (t δ↾J δ (x)), J δ ⊆ f in I, true in both A and B.
Consider the following theory on the vocabulary For each t δ consistent with A and each finite J ⊆ I: δ↾J (a)} for J = {i | P i or P ′ i occur in Σ}.By compactness, there is a model A + B ′ , P A δ , P ′B ′ δ δ∈2 J of the full theory.Then, A and B realize exactly the same types t δ (those originally consistent) and thus A ∼ B, defining aRb iff a and b realize the same type t δ .This contradicts the weak isomorphism property since A |= ϕ and B |= ¬ϕ.
Remark 10.If L has the finite weak dependence property, then the compactness and weak isomorphism properties are not needed in the previous theorem.Indeed, if ϕ depends on finite τ 0 ⊆ τ , the first step of the proof A ↾ τ 0 ≡ − 1 B ↾ τ 0 , A |= ϕ, B |= ¬ϕ, yields already a contradiction, since A and B realize trivially the same t δ types based on τ 0 , and thus A ↾ τ 0 ∼ B ↾ τ 0 .
Since L ωω and L 1 ωω have both the compactness and the Löwenheim-Skolem properties, then we can obtain the following preservation result from Theorems 6 and 9 (which is essentially [10, Cor.2.10]5 proved by a rather different method): is the fragment of L ωω (resp.L 1 ωω ) preserved under weak isomorphisms.
We proceed now to obtain an analogue of the second Lindström theorem from [22].First, we need the following lemma: Proof.Assume ϕ ∈ L(τ ) \ L − ωω (τ ) and ϕ depends on finite τ 0 ⊆ τ .Let τ ′ 0 be a disjoint copy of τ 0 , and set σ = τ 0 ∪ τ ′ 0 ∪ {<, c 0 , I, G, U, E}, which results of adding to the vocabulary in the proof of Theorem 6 a unary predicate symbol U and a binary predicated symbol E. Next, let ρ ⊇ α be finite and consider the sentence θ ∈ L(ρ) which is the conjunction of the theory Ψ introduced in the proof of Theorem 6 plus the following new sentences: 7. ∀x(U x ↔ ∃y(x < y ∨ y < x) "U is the field of <" This says that E satisfies the finite list of axioms of equality for the vocabulary ρ, and guarantees that E is the Leibniz congruence relation (this is enough by [20,§73 Thm. 41]) with respect to ρ. 9. ∀x¬(x < x), ∀xyz(x < y ∧ y < z → x < z), These axioms say, with E replacing =: "< is a strict linear order of U with last element c 0 and immediate predecesor for non minimal elements" Using [10,Lem. 4.4 and Prop. 4.5] and Lemma 3 as in the proof of Theorem 6, for each n < ω, we get a model and E A is true equality.
All that is left to show is that if for a countably infinite structure A we have A |= θ, then U A * is finite and non-empty.The first thing to notice is that < A * is a strict linear ordering with last element [c 0 ] and immediate predecesors for non miminal elements, because E collapses to true identity in A * .Now, suppose that U A * is infinite, then we have an infinite descending sequence . But then we have the sequence Proof.Assume for a contradiction that L ≤ L − ωω .Using Vaught's generalization of Trakhtenbrot theorem to L − ωω [28], we obtain a finite purely relational vocabulary τ ′ such that the set V fin ⊆ L − ωω (τ ′ ) of sentences valid on finite models is not recursively enumerable.Let θ ∈ L − ωω (σ ∪ τ ′ ) where σ and θ are given by Lemma 12 (we may obviously assume σ ∩ τ ′ = ∅).Now we may observe that where ψ U is the relativization in L − ωω of ψ to the unary predicate U (which is possible since L − ωω has the relativization property).If ψ ∈ V fin , then whenever A is a countably infinite σ ∪ τ ′ -structure such that A |= θ we must have that U A * is finite and non-empty by Lemma 12, thus A * |= ψ U , and by the weak isomorphism property, A |= ψ U as desired (given that A ∼ A * ).But for any sentence χ of L, χ iff χ is valid on countably infinite structures: if χ, a countably infinite countermodel for χ can be found by either applying the Löwenheim-Skolem property or Lemma 3 as needed. 7On the other hand, if θ → ψ U and A is a τ ′ -model of size n, say, we may assume (since τ ′ ∩ σ = ∅) that A ∼ = (A ′ |U ) ↾ τ ′ for a model A ′ that comes from extending and expanding A to a τ ′ ∪ σ-model of θ given by (1) in a suitable way.Hence, A ′ |= ψ U and thus A |= ψ.Since, by hypothesis, L is effectively regular and recursively enumerable for validity, we must have then that V fin is recursively enumerable after all, which is a contradiction.

Remark 14. Proper extensions of L −
ωω which are recursively enumerable for validity and have the weak isomorphism property are given in Examples 16 and 17 below.Notice that an analogous theorem for the monadic case is trivial because, in the presence of the weak isomorphism property, the effectivity of the logic implies the finite weak dependence property.
Remark 15.Other maximality results can be obtained by similar methods to those in this paper.For example, L − ωω is the maximal logic with the weak isomorphism property, compactness and the so called Tarski union property.This can be seen by adapting the argument of [3, Thm.III.2.2.1] for L ωω to the context without identity with the help of [13,Prop. 2.8].We conjecture that the λ-omitting types theorem also provides a characterization of the maximality of L − ωω (cf.[23]).

EXTENSIONS OF L − ωω
In this section, we collect a number of interesting examples of identity-free logics that help answer some questions posed by our results, e.g. is there a proper extension of L − ωω satisfying both the compactness and weak isomorphism properties? 8Notice that the infinitary logic L − ω1ω is an example of an abstract logic with the weak isomorphism and Löwenheim-Skolem properties, but without compactness.
Our examples will rely on the addition of suitable Lindström quantifiers which conveniently differ from usual definitions found in the literature.Indeed, adding a Lindström quantifier to L − ωω usually destroys the weak isomorphism property, as is the case with cardinality and cofinality quantifiers.However, each quantifier has a natural version closed under weak isomorphisms.
Example 16 (The logic L − ωω (Q − α )).Consider the Lindström quantifier Q − α defined as: The satisfaction condition for this operator then is The quantifier Q α may be recovered by letting E be the real identity relation =.
The first observation we wish to make is that Q − 1 (seen as a Lindström quantifier) is closed under weak isomorphisms, i.e To see this, suppose that A, M, E ∈ Q − 1 and R is a weak isomorphism from A, M, E onto A ′ , M ′ , E ′ .E ′ is an equivalence relation on A ′ compatible with M ′ because that fact can be expressed as a formula in L − ωω .We wish to show then that R induces a bijection M E −→ M ′ E ′.Consider the relation R ′ defined as [x]R ′ [y] iff xRy.We wish to show that R ′ is in fact a bijection.It is obviously surjective since R is.For functionality: assume that x ∈ M , xRy 1 and xRy 2 , then, since xEx, we must have that y 1 E ′ y 2 , which then means that if Injectivity is obtained by an analogous argument in reverse.Hence, | M ′ E ′| ≥ ω 1 as desired.
L − ωω (Q − 1 ) is clearly more expressive than L − ωω since the latter has the Löwenheim-Skolem property but the former does not (thus, the quantifier Q − 1 is not definable in L − ωω ).Recall that a logic L is said to be congruence closed [24] if, for any ϕ ∈ L(τ ), there is a sentence ϕ E ∈ L(τ ∪ {E}) (where E is a new binary predicate) such that for any structure A and any equivalence relation E on A. We will follow the notation of [9] in using qL to denote the congruence closure of a given logic L, obtained by adjoining to L the sentences defined by ( * ) as new quantifiers (see [24]).Then it is not difficult to observe that the logic L − ωω (Q − 1 ) is contained in the logic (with identity) qL ωω (Q 1 ).By the definition above, can be expressed by the relativized sentence ((Q 1 x(x = x)) θ ) {x|ϕ(x)} .Recall a logic is (κ, λ)-compact if every set of sentences of cardinality ≤ κ which has models for each of its subsets of cardinality < λ, has itself a model.By [24,Prop. 3.2], for any L, if L is (κ, λ)-compact, so is qL, and hence qL ωω (Q 1 ) is (ω, ω)-compact since L ωω (Q 1 ) is, which means that L − ωω (Q − 1 ) also inherits this property.Once more, by [24,Prop. 3.2], since L ωω (Q 1 ) is recursively enumerable for validity, qL ωω (Q 1 ) is too, and hence, so is the logic L − ωω (Q − 1 ).

Then, we have that
• A |= ∀xy (ϕ(x, y) ∨ ϕ(y, x) ∨ θ(x, y)), and Once more, the quantifier Q cfω can be defined as above by letting E be the true identity relation =.
We can show that the quantifier ) is recursively enumerable for validity.Moreover, we can observe that L − ωω (Q cfω− ) does not have a Löwenheim-Skolem theorem.For example, the sentence in the signature {E, <} with two binary relation symbols, has no countable models since it produces in the quotient model an infinite linear order without last element with cofinality = ω, and hence ≥ ω 1 .
Interestingly enough, some known quantifiers can be shown to preserve the weak isomorphism property: Example 18 (The logic L − ωω (Q H )). Recall the Henkin quantifier Q H which is defined as follows:

Then, we have that
First, we must show that Q H is closed under weak isomorphisms.Assume then that A, M, E ∈ Q H and R is a weak isomorphism from A, M onto A ′ , M ′ .Then M ⊆ A 4 , M ⊇ f ×g for some f, g : A −→ A. All we need to do now is define f ′ , g ′ : A ′ −→ A ′ such that M ′ ⊇ f ′ × g ′ .Define f ′ as follows: take any a 1 ∈ A ′ , we know then that Ra 0 a 1 for some a 0 ∈ A, so let f ′ (a 1 ) be some b 1 ∈ A ′ such that Rf (a 0 )b 1 .Do a similar thing for g ′ .Now, for any a , and since R is a weak isomorphism and a 0 , f (a 0 ), b 0 , g(b 0 ) ∈ M by hypothesis, a 1 , f ′ (a 1 ), b 1 , g ′ (b 1 ) ∈ M ′ , as desired.
Take now the sentence ϕ inf ∈ L − ωω (Q H )(τ ) where τ = {E} and E is binary: "E is an equivalence relation" ∧ ∃z ∃f, g ∀x, y (¬zEf (x) ∧ (f (x)Ey → g(y)Ex)) Since Q H is closed under weak isomorphisms, A ∼ A * = A E A in the vocabulary τ , and A E A |= ∀x, y (xEy ↔ x = y), we have that A |= ϕ inf only if A E A |= ∃z ∃f, g ∀x, y (z = f (x)∧(f (x) = y → g(y) = x)).The latter sentence says that A E A is infinite.On the other hand, for a τ -structure A, if A |= "E is an equivalence relation" and A E A = A * is infinite, A E A |= ∃z ∃f, g ∀x, y (z = f (x) ∧ (f (x) = y → g(y) = x)), so, reversing the previous reasoning, A |= ϕ inf .
Hence, we might consider the following theory T in the vocabulary τ : Summary of properties of some logics.
and |A| < |B|.By Lemma 3, there is a C of power |B| and a surjective strict homomorphism h : C → A. Thus, r • h is a τ 0 -weak isomorphism from C onto B. Renaming the last structure as B ′ with τ ′ we may put C and B ′ together in a structure C + B ′ sharing the same domain.Then, C + B ′ , r • h |= Φ(τ 0 , R), and hence, C + B ′ , r • h |= ϕ ↔ ϕ ′ this implies: C |= ϕ iff B |= ϕ.But A ∼ C with respect to full τ, then A |= ϕ iff B |= ϕ.If |A| = |B|, we apply the construction directly with A and B. We are now ready to provide the main result of this paper: Theorem 6.Let L be an abstract logic such that L − ωω ≤ L. If L has the weak isomorphism, compactness, and Löwenheim-Skolem properties, then L ≤ L − ωω .Proof.Assume ϕ ∈ L(τ ) \ L − ωω (τ ) and ϕ depends on a finite vocabulary τ 0 ⊆ τ (by compactness and Lemma 5).Notice that there are only finitely many sentences of rank ≤ n in L − ωω (τ 0 ) [10, Lem.4.4], thus the relation A ↾ τ 0 ≡ − n B ↾ τ 0 has finitely many equivalence classes of structures of type τ and the equivalence class of a structure A coincides with Mod τ (Θ A ) for the sentence

Lemma 12 . 6
Let L be an abstract logic such that L − ωω ≤ L satisfying the finite weak dependence and weak isomorphism properties.If L extends properly L − ωω , then there exist a finite vocabulary σ containing at least one unary relation U and, for each finite vocabulary ρ ⊇ σ, a sentence θ ∈ L(ρ) such that (1) for each n ≥ 1, there is a model A |= θ with |U A | = n, and (2) if A |= θ and A is countably infinite, then U A * is finite and non-empty.

•
Compactness property: for any vocabulary τ , Φ ⊆ L(τ ), if every finite subset of Φ has a model then Φ has a model.• Löwenheim-Skolem property: for any vocabulary τ , and sentence ϕ ∈ L(τ ), ϕ has a countable model if it has an infinite model.• Weak isomorphism property: for any structures A and B, A ∼ B only if A ≡ L B. • Finite weak dependence property: for any vocabulary τ and any ϕ ∈ L(τ ), there is a finite τ 0 ⊆ τ s.t. for any τ -structures A and B, if A ↾ τ 0 ∼ B ↾ τ 0 , then A |= ϕ iff B |= ϕ. • Karp − property: for any structures A and B, A ∼ p B only if A ≡ L B.
contradicting the weak isomorphism property.Theorem 13.Let L be an effectively regular abstract logic [3, Def.II.1.2.4] such that L − ωω ≤ L.Then, L has the weak isomorphism property, is recursively enumerable for validity, and has the Löwenheim-Skolem property only if L ≤ L − ωω .