Positive logics

Lindström’s Theorem characterizes first order logic as the maximal logic satisfying the Compactness Theorem and the Downward Löwenheim-Skolem Theorem. If we do not assume that logics are closed under negation, there is an obvious extension of first order logic with the two model theoretic properties mentioned, namely existential second order logic. We show that existential second order logic has a whole family of proper extensions satisfying the Compactness Theorem and the Downward Löwenheim-Skolem Theorem. Furthermore, we show that in the context of negation-less logics, positive logics, as we call them, there is no strongest extension of first order logic with the Compactness Theorem and the Downward Löwenheim-Skolem Theorem.


Introduction
Our motivating question in this paper is whether we can generalize Lindström's Theorem from first order logic to 1 1 , that is, existential second order logic. In the case of first order logic Lindström's Theorem says that first order logic is maximal with the Compactness Theorem 1 and the Downward Löwenheim-Skolem Theorem 2 among logics satisfying some minimal closure conditions [10]. One of the assumed closure conditions is closure under negation. What happens if we drop this assumption? It seems that this question was first explicitly raised in [7]. The Compactness Theorem and the Downward Löwenheim-Skolem Theorem make perfect sense, whether we have negation or not. These two conditions make no reference to negation.
In earlier related work ( [14]) we showed that a strong form of Lindström's Theorem fails for extensions of L κω and L κκ : For weakly compact κ there is no strongest extension of L κω with the (κ, κ)-compactness property and the Löwenheim-Skolem Theorem down to κ. With an additional set-theoretic assumption, there is no strongest extension of L κκ with the (κ, κ)-Compactness Theorem and the Löwenheim-Skolem theorem down to < κ.
Obviously first order logic itself is not maximal if negation is dropped because existential second order logic 1 1 , and even 1 1,δ (also denoted PC ), i.e. existential second order quantifiers followed by a countable conjunction of first order sentences, which clearly satisfy both the Compactness Theorem and the Downward Löwenheim-Skolem Theorem, also properly extend first order logic.
We are led to the following (interrelated) questions, all in the context of logics where closure under negation is not assumed: Question 1 Is 1 1 (or rather 1 1,δ ) maximal among logics satisfying the Compactness Theorem and the Downward Löwenheim-Skolem Theorem? Question 2 Is there an extension of 1 1 (or 1 1,δ ) which is maximal among logics satisfying the Compactness Theorem and the Downward Löwenheim-Skolem Theorem? Question 3 Is there a characterization of 1 1 (or 1 1,δ ) as maximal among logics satisfying some model-theoretic conditions? Question 4 Is there an extension of 1 1 (or 1 1,δ ) which is maximal (or even strongest) among logics satisfying some model-theoretic conditions?
In this paper we formulate Questions 1 and 2 in exact terms. We answer Question 1 negatively. As to Question 2 we show that there is no strongest 3 extension of 1 1 satisfying the Compactness Theorem and the Downward Löwenheim-Skolem Theorem. The existence of a maximal one (which has no proper such extension) remains open. 1 The (κ, λ)-Compactness Theorem says: Every theory of size ≤ κ, every subset of size < λ of which has a model, has a model. Compactness Theorem means (ω, ω)-Compactness Theorem. 2 The Downward Löwenheim-Skolem Theorem down to κ says: Every sentence in a countable vocabulary, which has a model, has a model of size ≤ κ. "Down to < κ" means "has a model of size < κ". The Downward Löwenheim-Skolem Theorem means the Löwenheim-Skolem Theorem down to ℵ 0 . 3 By strongest extension we mean one which contains every other as a sublogic. Questions 3 and 4 remain completely unanswered. Admittedly, Question 4 is a little vague as both "extension" and "model-theoretical conditions" are left open.
To answer the above Questions 1 and 2 we introduce a family of new generalized quantifiers associated with the very natural and intuitive concept of the density of a set of reals. These quantifiers are defined for the purpose of solving the said questions and may lack wider relevance, although the general study of logics without negation is so undeveloped that it may be too early to say what is relevant and what is not.

Notation:
We use M and N to denote structures, and M and N to denote their universes, respectively. For finite sequences s and sets a, we use sˆ a to denote the extension of s by the set a. For sequences of length ≤ ω, s s means that s is an initial segment of s . The empty sequence is denoted ∅. A subset A of 2 ω is said to be dense if for all s ∈ 2 <ω there is s ∈ A such that s s . We use P(ω) to denote the power-set of ω.

Positive logics
We define the concept of a positive logic, meaning a logic without negation, except in front of atomic (and first order) formulas. We have to be careful about substitution in this context. If we are too lax about the substitution 4 of formulas into atomic formulas we end up having a logic which is closed under negation, which is not what we want. Substitution is very natural, but it is not needed in Lindström's characterization of first order logic.
One may ask whether a logic deserves to be called a logic if it is not closed under negation? We do not try to answer this question, but merely point out that there are several logics that do not have a negation in the sense that we have in mind, i.e. in the sense of classical logic. Take, for example, constructive logic. Although it has a negation, it does not have the Law of Excluded Middle, so its negation does not function in the way we mean when we ask whether a logic is closed under negation. In our sense constructive logic is not closed under negation. Another example is continuous logic [3] and the related positive logic of [4]. We have already mentioned existential second order logic 1 1 and its stronger form, 1 1,δ . In the same category as 1 1 are Dependence logic [15] and Independence Friendly Logic [11]. Transfinite game quantifiers yield infinitary logics which are not closed under negation, due to non-determinacy [8]. In the finite context there is the complexity class non-deterministic polynomial time NP, which is equivalent to existential second order logic on finite models, of which it is not known whether it is closed under negation. In this paper we introduce new examples of logics without negation.

Definition 1
A positive logic is an abstract logic 5 in the sense of [10] (see also [6]) which contains first order logic and is closed under disjunction, conjunction, and first order quantifiers ∃ and ∀. We do not require closure under negation, nor closure under substitution.
Example 2 1. First order logic is a positive logic. 2. 1 1 and 1 1,δ are positive logics. 3. If L is a positive logic, then so is 1 1 (L), the closure of L under existential second order quantification.

A class of new quantifiers
In the tradition of [9] we define our new generalized quantifiers by first specifying a class of structures, closed under isomorphisms.

Example 3 A canonical example of a τ d -structure is the model
where A ⊆ 2 ω and Definition 4 For n < ω and η ∈ 2 n we define ψ η (x) as: If η ∈ (M, a), we say that a represents η in M. We also say that M represents the set (M).
One element a can represent several η, but later in Sect. 7 we impose a further restriction to the effect that representation is unique.
Note that if M is a τ d -model, then the property " (M) is dense" is a 1 1 -property of M. Since we aim at a logic which goes beyond existential second order logic, we have to sharpen the requirement of density. The property of τ d -models we are interested in is the property that " (M) \ A is dense" for some preassigned set A ⊆ 2 ω of reals.
Definition 5 Let A ⊆ 2 ω . We define the Lindström quantifier Q A as follows. Suppose M is a model andc ∈ M k . Then we define We define the positive logic L d A as the closure of first order logic under conjunction, disjunction, first order quantifiers ∃ and ∀, the existential second order quantifier ∃R, where R is a relation symbol, and the generalized quantifier Q A . We denote by L d,ω A the extension of L d A obtained by allowing countable conjunctions as a logical operation. Finally, the proper class L d,∞ A denotes the extension of L d A obtained by allowing arbitrary set-size conjunctions as a logical operation.
With the obvious definition of what it means for a positive logic to be a sublogic of another, we can immediately observe that 1 1 is a sublogic of L d A , and 1 1,δ is a sublogic of L d,ω A whatever A is.
For future reference we make the following observation: If η ∈ 2 n ,ȳ = (y 0 , . . . , y n−1 ),ψ as in Definition 5 is a 5-tuple of formulas of L d

then (1) is equivalent to
For every σ ∈ 2 <ω there are η ∈ 2 ω \ A extending σ and a ∈ M such that for some function n → b n 0 , . . . , b n We proceed to proving that the logic L d A , for suitably chosen A ⊆ 2 ω , satisfies the Compactness Theorem and the Downward Löwenheim-Skolem Theorem, and also properly extends 1 1 .

The Compactness Theorem
We use the well-established method of ultraproducts to prove the Compactness Theorem of L d A .

are models and D is an ultrafilter on a set I . Let
Proof We use induction on φ. The cases corresponding to the atomic formulas, the negated atomic fromulas, conjunction (even infinite conjunction), disjunction, ∃, ∀, and ∃R (see e.g. [5, 4.1.14]) are all standard and well known. In the case of disjunction we use the property of ultrafilters that I 1 ∪ I 2 ∈ D implies I 1 ∈ D or I 2 ∈ D. We are left with the induction step for Q A . Let us denote f 0 (i), . . . , f n−1 (i) byf (i) and f 0 /D, . . . , f n−1 /D byf /D. We assume and Suppose η ∈ 2 ω is arbitrary. Let i ∈ I n+1 \ I n . Because B i is dense, there are, for all n < ω, extensions η i ∈ 2 ω of η n and elements for all n < ω. Let h(i) = a i . For i ∈ I n \ I n+1 and m < n, let g n

Every sentence of L d
A with an infinite model has arbitrarily large models.

The only sentences in L d A that have a negation (in the usual sense) are the first order (equivalent) ones.
Proof The usual argument gives 1: Suppose T is a finitely consistent theory in L d A . Let I be the set of finite subsets of T and for each i Then the family J = {A φ : φ ∈ T } has the finite intersection property. Let D be a non-principal ultrafilter on I extending J . Now if φ ∈ T , then Claim 2 follows immediately from Claim from 1. Claim 3 follows from the ultraproduct characterization of first order model classes (see e.g. [5, 4.1.12]) and the characterization of elementary equivalence in terms of ultrapowers [13].
Proof This proof is not specific to L d A , but is rather a well-known consequence of Łoś Lemma, Theorem 8. Let

The Downward Löwenheim-Skolem property
The Downward Löwenheim-Skolem Property, which says that any sentence (of the logic) which has a model has a countable model, is an important ingredient of the Lindström characterization of first order logic. The main examples of logics with this property, apart from first order logic, are L ω 1 ω and its sublogics L(Q 0 ) (with the quantifier "there exists infinitely many", see e.g. [1, p. 8]) and the weak second order logic L 2 w (with quantifiers for variables that range over finite sets, see e.g. [1, p. 9]). We now prove this property for L d A in a particularly strong form. Because of lack of negation the elementary submodel relation M N splits into two different concepts M + N and M − N : and for all a 1 , . . . , a n in N and all . . . , a n ). and for all a 1 , . . . , a n in N and all (a 1 , . . . , a n ).
Similar definitions can be given for L d,ω A and L d,∞ A , and for fragments (i.e. subsets closed under subformulas) thereof.
The Compactness Theorem implies that every infinite model N has arbitrarily large M such that N +  N | φ(a). We use induction on φ. The claim follows from N ⊆ M * for atomic and negated atomic φ. The claim is clearly preserved under conjunction and disjunction. It is also trivially preserved under universal quantifier, since N ⊆ M * . The induction steps for both first and second order existential quantifiers are trivial because of the expansion we have performed on M * . We are left with the quantifier Q A .
Suppose M * satisfies (1) of Definition 5 withc ∈ N k . Thus (2) holds and we want to prove (2) with M * replaced by N . Note that (2) also holds in K . Suppose σ ∈ 2 n is given. There is η ∈ K ∩ (2 ω \ A) extending σ such that K satisfies We conclude that every sentence of L d A which has an infinite model has a countable model and an uncountable model.
The following examples show that Theorem 12 is in a sense optimal:

Example 13
There is an uncountable model M, namely (P(ω), a, ∈), where a is the element ω of P(ω), such that there is no countable model N with N + Our goal in this section is to show that for many A ⊆ 2 ω the logic L d A properly extends 1 1 and L d,ω A properly extends 1 1,δ . We have a spectrum of results to this effect but nothing as conclusive as being able to explicitly point out such a set A. There are obvious reasons for this. The logic 1 1 is very powerful and any "simple" A ⊆ 2 ω is likely to yield L d A which is equivalent to 1 1 rather than properly extending it. This is even more true with L d,ω A and 1 1,δ . We first establish the basic existence of sets A ⊂ 2 ω with the desired properties. We shall then refine the result with further arguments.

Theorem 14
There are sets A ⊆ 2 ω such that 2 ω \ A is dense and Q A is not definable in 1 1 , nor in 1 1,δ , nor in L ω 1 ω 1 .
Proof Let A α , α < 2 ω , be disjoint dense subsets of 2 ω . For any X ⊆ 2 ω , let The following result merely improves the previous result: . Let M be resplendent (see [2]) such that M B ≺ M. Let M + be an expansion of M such that every 1 1 -sentence true in M + has a witness in the vocabulary (countable).
Hence all the first order consequences of φ are true in M B . Since M is resplendent, M | φ. Since φ has a witness in M + , N + | φ. Hence N + | ψ A whence (N + ) \ A is dense. This is a contradiction, as (N + ) \ A = ∅.
Theorem 16 Let P be the poset of finite partial functions (ω 1 + ω 1 ) × ω → 2 i.e. the forcing for adding ω 1 +ω 1 Cohen reals. Let G be P-generic and η α ∈ 2 ω , α < ω 1 +ω 1 , the Cohen reals added by G. Let A be the set of η such that η = η α (mod finite) for some α < ω 1 Proof Let B be the set of η such that η = η α (mod finite) for some α < ω 1 + ω 1 . Then The function f induces an complete embeddingf of P into P. The mappingf induces a mapping τ → τf between Pterms. Let N be the image of M B under this mapping. Now N | ψ A . However, N | φ, whence N | ψ A , a contradiction. Let φ be a 1 1 sentence ∃Rφ 0 such that ψ A and φ are logically equivalent, contradicting our desired conclusion. Let D α : α < ω 1 be a sequence of disjoint countable dense subsets of D ∩U . Let N α be a countable model representing the set D α , whence it satisfies ψ A , hence φ, and there is an expansion N * α of N α to a model of φ 0 . Let N = N * α : α < ω 1 . Let B = (H θ , ∈, <), for a large enough cardinal θ and for a well-ordering < of H θ . We choose a countable elementary submodel By Theorem IV.5.19 of [12] there is a sequence B α : α < 2 ω of countable elementary extensions of B * such that for every α < β < 2 ω : (a) B α has standard ω. (b) B α has a (possibly non-standard) member c α of (ω 1 ) B * . (c) If an element of ω 2 is definable in both B α and B β , then it is in B * .
Let N + η be the c η 'th member of the sequence N * α : α < ω 1 as interpreted in B η . So necessarily N + η is a model of φ 0 and hence its reduct N + η τ d is a model of φ, and further of ψ A . We have continuum many models N + η τ d of ψ A . However, we will now show that the number of η for which the model N + By the disjointness clause (c) above we get the claimed contradiction.
We now finish the proof of Theorem 17: Suppose η is such that N + η | ψ A . This is a contradiction because N + η | φ.

No strongest extension
We show that there is no strongest extension among positive logics of first order logic, or 1 1 , or 1 1,δ , with the Compactness Theorem and the Downward Löwenheim-Skolem Theorem.
Intuitively, TL says that R 2 is a tree-like partial order extending R 0 and R 1 . For example, the model M A of Example 3 always satisfies TL . If M | TL , then one element a of M can represent only one η, i.e. η, η ∈ (M, a) implies η = η . (5) Definition 18 We define the Lindström quantifier Q A as follows. Suppose M is a model andc ∈ M k . Then we define that M satisfies if and only if Mψ | TL and (Mψ ) where Mψ is as in Definition 5 and (Mψ ) is as in Definition 4.

Definition 19
We define L d A as the closure of first order logic under ∧, ∨, ∃, ∀, ∃R and Q A . The fragment, where Q A is applied to first order formulasψ only is denoted Proof We follow the proof of Theorem 8. The only point that requires attention is the induction step for Q A . We assume

Theorem 20 (Łoś Lemma for L d
and demonstrate M | Q A x 0 x 1 ψ 0 (x 0 , x 1 ,f /D) . . . ψ 4 (x 0 ,f /D). As in the proof of Theorem 8, it can be shown that the set B of η ∈ 2 ω such that there is a ∈ M such that for some b n 0 , . . . , b n n−1 in i M i /D we have M | n,k ψ,η n (b n 0 , . . . , b n n−1 , a,f /D) for all n < ω, is the full set 2 ω . It follows that 2 ω ∩ A ω 1 = A ω 1 and hence that 2 ω ∩ A ω 1 ∈ A, as claimed.
Proof The ultrafilter we used in the proof of Corollary 9 was regular, hence ω 1incomplete.
Proof Let A be as above but A α = β<α A β for all limit α. Let S, S ⊆ ω 1 be disjoint stationary sets. Note that the set of elements of S that are limits of elements of S is stationary, because it contains the intersection of S with the closed unbounded set of limits of elements of S. Similarly, the set of elements of S that are limits of elements of S is stationary. Let A = A α : α ∈ S ˆ A ω 1 and A = A α : α ∈ S ˆ A ω 1 . Now both {α ∈ S : A α = β∈α∩S A β } and {α ∈ S : A α = β∈α∩S A β } are stationary. Let and similarly ψ A . Let φ be the sentence ψ A ∧ ψ A ∧ T L . This sentence has a model, namely M A ω 1 . Suppose it has a countable model N .
be the existential second order formula R 6 (x) ∧ ∃F(F is a one-one function from R 7 onto {y : Q 2 (x, y)}).
Despite the negative result of Theorem 25, Theorem 22 still holds for the fragment of L d A obtained by dropping existential second order quantifiers.

Definition 26 Let L d0
A be defined as L d A (Definition 19) except that existential second order quantification is not allowed. Let L d1 A be defined as the extension of L d A by adding negation to the logical operations.

Clearly, L d0
A is a positive logic and it satisfies the Compactness Theorem because even L d A does. The logic L d1 A is an abstract logic in the sense of [10]. Unlike our positive logics, it is closed under negation and also closed under substitution. Note that L d0 A ≤ L d1 A .

Theorem 27 (Downward Löwenheim-Skolem-Tarski Theorem) Suppose M is a model for a countable vocabulary and X ⊆ M is countable. Then there is N L d1
A M such that X ⊆ N and |N | ≤ ℵ 0 . In particular, N L d0 Proof This is as in the proof of Theorem 22. We first expand M as follows: For every L d1 A -formula φ(z), wherez = z 0 , . . . , z k−1 , there is a predicate symbol R φ of arity k such that M | ∀z(φ(z) ↔ R φ (z)). Let τ be the original vocabulary of M and τ * the vocabulary of the expansion. For any atomic formulasψ of the vocabulary τ * let g(n, k,ψ, η) be the function which maps n, k,ψ and η ∈ 2 n to n,k ψ,η (y 0 , . . . , y n , x,z). Let K ≺ H θ , where θ ≥ (2 ω ) + such that M ⊆ H θ , |K | = ℵ 0 , {A, ω 1 , τ * , M, X , g} ∪ ω 1 ∪ τ ∪ X ⊆ K , and δ = K ∩ ω 1 ∈ S. Let N be the restriction of M to K , i.e. the universe N of N is M ∩ K and the constants, relations and functions of M are relativized to N .
As in the proof of Theorem 12, N is closed under the interpretations of function symbols of the vocabulary of M.
Claim: If φ(x) is a τ * -formula in L d1 A and c ∈ N , then N | φ(c) ↔ R φ (c). The proof of this claim is as in the proof of Theorem 22. Since N ⊆ M in the vocabulary τ * , the claim implies N L d1 A M.

The logic L d1
A is closed under negation and satisfies the Downward Löwenheim-Skolem Theorem. Thus it cannot satisfy the Compactness Theorem, although its sublogic L d0 A does.
Funding Open Access funding provided by University of Helsinki including Helsinki University Central Hospital.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.