On Sahlqvist Formulas in Relevant Logic

This paper defines a Sahlqvist fragment for relevant logic and establishes that each class of frames in the Routley-Meyer semantics which is definable by a Sahlqvist formula is also elementary, that is, it coincides with the class of structures satisfying a given first order property calculable by a Sahlqvist-van Benthem algorithm. Furthermore, we show that some classes of Routley-Meyer frames definable by a relevant formula are not elementary.


Introduction
The Sahlqvist Correspondence Theorem is a celebrated result in modal logic (see [3,6] for slightly different expositions). It tells us that when a modal formula φ has certain syntactic form, we can always compute a first order formula ψ in the signature of Kripke frames such that a frame validates φ iff it satisfies condition ψ, so the class of frames definable by φ is elementary. The proof relies on the so called "Sahlqvist-van Benthem algorithm" for transforming second order frame correspondents of some modal formulas into first order properties.
The present note is a contribution to the correspondence theory of relevant logic in the Routley-Meyer semantic framework (cf. [4,8,16]). 1 The framework has been considerably generalized and investigated in relation to other semantics in [1].
We will show that, mutatis mutandis, the argument for the Sahlqvist Correspondence Theorem (as presented in [3]) can be adapted to the context of relevant logic to prove an analogous result in the Routley-Meyer semantics. This result improves our understanding of the first order properties of Routley-Meyer frames which are definable in relevant logic. It also begins to solve Problem 8.4.18 from [1]. Moreover, we will show that some relevant formulas actually define non-elementary properties of frames, so the problem of elementarity is not trivial.
Correspondence theory for the broader setting of substructural logics with frame semantics was briefly explored by Restall in [14] (pp. 263-265) although the subject appears to have been discussed for the first time in the relevant logic literature in [13]. 2 Recently, Suzuki [18] provided a rather general Sahlqvist result for substructural logics with respect to what he calls bi-approximation semantics (which unfortunately is much more complicated than the Routley-Meyer framework). Other settings without boolean negation where Sahlqvist theorems have been obtained are positive modal logic [5] and relevant modal logic [17]. In particular, the work in [17] is rather close to ours, but the concern there is still modal logic.
In Section 2, we will introduce Routley-Meyer frames and some basic propositions on frame correspondence. In Section 3, we will discuss elementary classes of Routley-Meyer frames and prove that not all classes of Routley-Meyer frames definable by a relevant formula are elementary. In Section 4, we will present a Sahlqvist fragment of relevant languages and establish a Sahlqvist correspondence theorem for the Routley-Meyer semantics. Finally, in Section 5, we will sum up our work.

Routley-Meyer Frames
In this paper, a relevant language L will contain a countable list PROP of propositional variables p, q, r . . . and the logical symbols: ∼ (negation), ∧ (conjunction), ∨ (disjunction), • (fusion), → (implication), t (the Ackermann constant). Formulas are constructed in the usual way: In this paper, a Routley-Meyer frame for L is a structure F = W, R, * , O , where W is a non-empty set, ∅ = O ⊆ W , * is an operation * : W −→ W , and R ⊆ W × 1 The Routley-Meyer framework remains today the most prominent non-algebraic approach to the semantics of propositional relevant languages (cf. [4,8]). Recent applications (e.g., [15,19]) are easy to find. The Routley-Meyer semantics has also been given an appropriate philosophical justification in [2] and much more recently in [9]. If anything comes close to being the "standard" relational semantics for propositional relevant logics, it surely is the Routley-Meyer approach. 2 Many thanks to an anonymous referee who pointed this out.
W ×W satisfies p1-p5 below. In the standard way, we will abbreviate ∃z(Oz∧Rzxy) by x ≤ y.
These frames are essentially the same as what are called B •t -frames in [4] (p. 80) except that we have left out the condition that * is an involution. Hence, our starting basic system of relevant logic is slightly weaker than B •t from [4]. The reason for this is that we are interested in finding the exact first order frame correspondents of the double negation laws ∼∼ p → p and p →∼∼ p using our Sahlqvist-van Benthem algorithm, but this task becomes trivial if * is an involution. Indeed, the system is just the Hilbert calculus B •t presented in [4] (p. 74) minus the axiom ∼∼ p → p, whose axioms and rules we review here for the sake of completeness (a substitution rule will be tacitly assumed): is a function such that for any p ∈ PROP, V (p) is upward closed under the ≤ relation, that is, x ∈ V (p) and x ≤ y implies that y ∈ V (p). We define satisfaction at w in M recursively as follows: When w is an arbitrary world of F, we also write F, w φ to mean that for every model M based on F, M, w φ. Validity and this latter semantic notions will be the focus of the present study. We will be interested in what the languages defined above can say at the level of frames, not so much of models. Proof This is traditionally proven by induction on formula complexity (see [8]). The case φ = p follows simply because V (p) is upward closed under ≤, while the case φ = t follows because ≤ is a transitive relation.
Next let φ =∼ ψ. By p5, we must have that y * ≤ x * given our assumption that x ≤ y. Now, if M, x ∼ ψ then M, x * ψ, and by inductive hypothesis, it follows that M, y * ψ, so M, y ∼ ψ as desired.
Let φ = ψ → χ . If M, x ψ → χ , then for every a, b such that Rxab, if M, a ψ then M, b ψ. Now suppose that M, y ψ → χ , so there are a , b such that Rya b while M, a ψ and M, b χ . However, by p2, it must be that Rxa b , which is impossible given the assumption that M, x ψ → χ . Hence, actually M, y ψ → χ .
Let φ = ψ • χ . Similar to the previous case but appealing to p4 instead. The remaining cases are trivial.
Consider a monadic second order language that comes with one function symbol * , a unary predicate O, a distinguished three place relation symbol R, and a unary predicate variable P for each p ∈ PROP. Following the tradition in modal logic, we might call this a correspondence language L corr for L (cf. [3]). Now we can read a model M as a model for L corr in a straightforward way: W is taken as the domain of the structure, V specifies the denotation of each of the predicates P , Q, . . . , the collection O is the object assigned to the predicate O, while * is the denotation of the function symbol * of L corr and R the denotation of the relation R of L corr .
Where t is a term in the correspondence language, we write φ t/x for the result of replacing x with t everywhere in the formula φ. As expected, it is easy to specify a translation from the formulas of the relevant language into formulas of first order logic with one free variable as follows: The symbols ¬ and ⊃ represent boolean negation and material implication in classical logic (which should not be confused with the relevant ∼ and →).
Next we prove a proposition to the effect that our proposed translation is adequate. While stands for satisfaction as defined for relevant languages, will be the usual Tarskian satisfaction relation from classical logic.

Proposition 2 For any w, M, w φ if and only if M T x (φ)[w].
Hence, without loss of generality, we can switch (when convenient) to the following presentation of L as a fragment of the classical language L corr : Working with this presentation makes life easier in the next definition.
Definition 1 A relevant formula φ is said to be positive if T x (φ) is a formula built up from atomic formulas involving only unary predicates and first order formulas where the only non-logical symbols are R and O, using the connectives ∃, ∀, ∧ and ∨. On the other hand φ is negative if T x (φ) is a formula built up from boolean negations of atomic formulas involving only unary predicates and first order formulas where the only non-logical symbols are R and T, using the connectives ∃, ∀, ∧ and ∨.
A remark seems in order here: the translation of a formula of the form φ → ψ is in general not positive as it involves ⊃ with its usual definition in terms of ¬ and ∨. Proof This follows from Theorem 10.3.3 (a) in [11].
We say that a formula φ of L corr corresponds to (or is a correspondent of) a relevant formula φ if for any Routley-Meyer frame F, F φ iff F φ . A formula ψ of L corr locally corresponds to a relevant formula φ if for any Routley-Meyer frame F and world w of If we abreviate the formula of L corr which expresses that the value of a given predicate P is upward closed under ≤ by Up ≤ (P ), the next proposition is easily established using Proposition 2.

Proposition 4
For any relevant formula φ(p 1 , . . . , p n ), Routley-Meyer frame F and world w of F, the following holds: A class (or, informally, a property) K of Routley-Meyer frames will be said to be definable by a relevant formula φ if K is exactly the class of all frames validating φ. So, at the level of frames and validity, relevant languages can be seen as fragments of monadic second order logic as opposed to fragments of first order logic at the level of models. To be precise, the "relevant fragment" of monadic second order logic can be taken to be any formula equivalent to a formula in the set

Elementary Classes of Frames
A class of structures K is said to be elementary if there is a first order formula φ such that K is identical to the class of all models of φ. Since frames are clearly structures, then it makes sense to talk about elementary classes of frames. Note that it is easy to prove that a class of Routley-Meyer frames which is definable by a formula of relevant logic is elementary iff it is closed under ultraproducts. Goldblatt [10] observed that this was so for modal logic and the same quick little argument applies here. Recall that a class of structures is elementary iff both the class and its complement are closed under ultraproducts ([7], Corollary 6.1.16 (ii) 3 ). Now, the complement of a class of Routley-Meyer frames definable by a relevant formula φ is always closed under ultraproducts. For it is the class of all models of a 1 1 sentence ∃P 1 , . . . , P n (Up ≤ (P 1 )∧· · ·∧Up ≤ (P n )∧¬T x (φ) T /x ) and 1 1 formulas are preserved under ultraproducts ( [7], Corollary 4.1.14). Hence, we have proven the observation. Furthermore, following a similar argument to that in [21] for modal logic, we can actually show that ultrapowers suffice for a characterization of elementarity.
As an example of a relevant formula which defines an elementary class of frames we have p 0 ∨ (p 0 → p 1 ) ∨ · · · ∨ 0≤i≤n−1 p i → p n . The intuitionistic counterpart of this formula on intuitionistic Kripke frames corresponds to a cardinality claim ( [6], Proposition 2.40). The first order correspondent of the formula on Routley-Meyer frames is ∀x 1 , y 1 , . . . , x n , y n n i=1 x i ≤ y i ⊃ 1≤j<i≤n x i ≤ y j . This can be seen as follows. Let F be an arbitrary Routley-Meyer frame. Suppose first that F satisfies our first order condition and that for some model M based on F, M, Without loss of generality, we may assume that for some T ∈ O, we have The example in the above paragraph also serves as a point of comparison between the expressive power of relevant formulas on Routley-Meyer frames and intuitionistic formulas on Kripke frames. More illustrations of relevant formulas defining elementary classes will follow in Section 4, as applications of our Sahlqvist correspondence theorem.
One question that arises immediately is whether all relevant formulas have first order correspondents, i.e., define elementary classes. The next results show that this is not the case (though Section 4 will give a positive result). The formula witnessing this fact in the following theorem is roughly the analogue of the famous McKinsey axiom from modal logic (which was shown not to be elementary simultaneously by van Benthem [20] and Goldblatt [10]). We use a techinique due to van Benthem [20].

Theorem 5 The class of Routley-Meyer frames defined by the formula
is not elementary. In other words, the above formula has no first order correspondent.
and * is the identity function. F 0 is trivially a Routley-Meyer frame, since in F 0 , a ≤ b iff a = b. Note that any subset of W is upward closed with respect to ≤ due to this. It is easily seen that F 0 , w t∨ ∼ t for any w ∈ W by our definition of * , which basically makes ∼ collapse to boolean negation in F 0 . Clearly, F 0 has uncountable cardinality. Also, we observe that over frames where * is the identity (such as F 0 ), a formula like (M), of the form ∼ φ ∨ ψ, is essentially a material implication.
. By the latter, for each q n there must be some q n,i such that . Take r f to be such that F 0 , V , q n,f (n) p for any n ∈ ω. Then since Rsr f q Now suppose (M) has a first order correspondent φ. By the above paragraph, F 0 φ. By the downward Löwenheim-Skolem theorem, there is a countable elementary 1}} as a subset of its domain. We will show that (M) is not valid in F 0 . This will produce a contradiction since F 0 φ, given that F 0 and F 0 are elementarily equivalent.
Being countable, F 0 must leave out an element z g (g ∈ {0, 1} ω ) of W . Consider a valuation V on F 0 such that V (p) = {q n,g(n) : n ∈ ω}. We note that F 0 , V , q p, so F 0 , V , q ∼ p. Now, for any q n , since Rq n,g(n) q n,g(n) q n , F 0 , V , q n p • (t∨ ∼ t). Similarly if w = q n (n ∈ ω), F 0 , V , w p • (t∨ ∼ t). Hence, for all w in the domain of F 0 , F 0 , V , w p • (t∨ ∼ t), which implies that F 0 , V , q t∨ ∼ t → (p • (t∨ ∼ t)). Finally, we want to show that F 0 , V , q (t∨ ∼ t) • (t∨ ∼ t → p). If Rxyq either (1) x = y = q or (2) x = y = q n,i for some n ∈ ω, i ∈ {0, 1}, or (3) x = s and y = z f for some z f in the domain of F 0 . It suffices to show that in all three cases F 0 , V , y t∨ ∼ t → p. If (1), since F 0 , V , q p and given that Rqqq, F 0 , V , y t∨ ∼ t → p. If (2), since Rq n,i q n,h(g(n)) q n,h(g(n)) (where h : {0, 1} −→ {0, 1} is the function such that h(0) = 1 and h(1) = 0) and F 0 , V , q n,h(g(n)) p, we have that F 0 , V , y t∨ ∼ t → p. If (3), f = g, so they differ in their value for some n, hence, q n,f (n) = q n,g(n) , which implies that F 0 , V , q n,f (n) p. Given that Rr f q n,f (n) q n,f (n) , we see that F 0 , V , y t∨ ∼ t → p, as desired.
Next we provide a much easily graspable non-elementary class of Routley-Meyer frames which is definable in the language of relevant logic. For the conjunction of conditions (i) and (ii) in Proposition 6 is not expressible in first order logic as will be seen through a compactness argument. This time we build a relevant logic analogue of the well-known Löb axiom.
Put R # xy = df ∃z(Rxyz ∨ Rxzy) and p = df (p∨ ∼ p → p) ∧ (∼ p → p∧ ∼ p). Observe that for any frame F, and world x ∈ W , R # T x holds, so F ∀x(R # T x). In any frame F where ∀x(x * ≤ x ∧ x ≤ x * ), using the Hereditary Lemma, we see that a formula of the form ∼ φ ∨ ψ behaves essentially as a material implication in a classical language at the level of models based on F. Also, for any valuation V in any such frame F, F, V , w p iff for all x, y such that Rwxy, F, V , x p and F, V , y p iff for all x such that R # wx, F, V , x p. Now, for any frame F, . . . and (iii) for any x, y, R # T x and R # xy implies that R # T y.
Proof Let F be an arbitrary Routley-Meyer frame. We have that if (i) holds, the validity of ( p ⊃ p) ∧ p ⊃ p implies (ii). For suppose (ii) fails, then there is an infinite sequence of worlds T = s 0 The latter means that v ≤ s i for some 0<i<ω, however since F, V , s i+1 p and R # s i s i+1 , it must be that F, V , s i p, and by the Hered- Similarly, if (iii) fails, we have some x, y such that R # T x and R # xy but not R # T y. Just consider a valuation V such that V (p) = {w : w x, y}. V (p) turns out to be upwards closed under ≤ due to the transitivity of ≤. This concludes the left to right direction of the proposition.
For the converse suppose F, V , T ( p ⊃ p) ∧ p ⊃ p. If (i) holds, one can build the desired sequence to falsify (ii) by taking x such that R # T x while F, V , x p and applying F, V , T ( p ⊃ p) in conjunction with (iii).
To see that the conjunction of conditions (i) and (ii) in Proposition 6 is not a first order property, let us suppose it is to derive a contradiction. Say φ is the first order formula expressing (i) and (ii). Expand the correspondence language by adding a constant c i for each i>0. Let be the collection of first order formulas axiomatizing our class of Routley-Meyer frames and the set of first order formulas {O(T ), R # T c 1 , R # c i c i+1 : 0<i<ω}. For each n>0, take the frame F n where W = {k : k ≤ n}, R = { 0, i, i : i ≤ n} ∪ { j, j + 1, j + 1 : j<n}, T is simply the number 0 while O = {0} and * the identity. Each finite subset of {φ} ∪ ∪ has a model in F n for sufficiently large n. By compactness, the whole thing must have a model, which is impossible since φ forbids from being true by assumption.
So we have seen that at the level of frames, the language of relevant logic can go beyond the expressive power of first order logic. More precisely, the fragment of monadic second order logic corresponding to the language of relevant logic over frames can express some non-first order concepts. Does it contain first order logic, though? We will give next an easy argument that it does not (though there was never a reason to believe that it did). This shows that the "relevant fragment" of monadic second order logic is indeed incomparable with first order logic in terms of expressive power. Proof Routine induction on formula complexity.

Definition 2 Let
When one omits condition (i) in Definition 2, we might speak of a bounded morphism from a frame F into a frame F .

Proposition 8 If there is a bounded morphism f from a frame
) and x ≤ y in F, which by conditions (ii) and (v) in Definition 2 and our assumption, implies that f (x) ≤ f (y). Hence, f (y) ∈ V (p), which means that y ∈ V (p). So f is now a bounded morphism from the model F, V into the model F , V , which, using Proposition 8, means that F φ.
Finally, let F 1 be the frame W, R, * , {s, t} where W = {s, t}, R = W × W × W and * = { t, t , s, t }. On the other hand, let F 2 be the frame W , R , * , {s} where W = {s}, R = W × W × W and * is the identity. There is only one function f : W −→ W . The function f is a bounded morphism from F 1 onto F 2 as it is easy to check. The first order property ∃x(Ox ∧ x = x * ) (where the denotation of T is s in both F 1 and F 2 ) holds at F 1 but fails at F 2 , so by Proposition 8, ∃x(Ox ∧ x = x * ) is not definable by a relevant formula.

A Sahlqvist Correspondence Theorem
In this section, we prove the result promised in Section 1 using the groundwork layed out in Section 2. In the proof we use lambda terms as in [3], which are to be understood as predicate constants (i.e., the denotation of the lambda term λu.(φ) is fixed by the set of worlds satisfying φ).

Lemma 9
Let φ, θ, ψ and χ be relevant formulas such that θ contains no propositional variable and φ and χ have no propositional variable in common. Suppose φ, ψ and χ have first order frame correspondents. Then θ → φ, φ ∧ ψ and φ ∨ χ have first order correspondents.
Proof Suppose φ , ψ , χ are the first order local correspondents of φ, ψ and χ respectively.
For (θ → φ), supposing that all its propositional variables appear in the list p 1 , . . . , p n , we have that For (φ∧ψ), supposing that all its propositional variables appear in the list p 1 , . . . , p n , we have that Right to left is obvious. For the converse suppose that F, w φ ∨ χ , F, w φ and F, w χ . Then there are V 1 , V 2 such that F, V 1 , w φ and F, V 2 , w χ . It is easy to show by induction on formula complexity that for any two valuations V , V on a frame F such that V (p) = V (p) for all propositional variables p appearing in a formula φ, F, V , w φ iff F, V , w φ for any world w of F. Say p 1 , . . . , p n and q 1 , . . . , q m are the propositional variables appearing in φ, χ respectively. Take a valuation V 3 such that V 3 (p i ) = V 1 (p i ) while V 3 (q j ) = V 2 (q j ). It follows that F, V 3 , w φ and F, V 3 , w χ , contradicting our assumption that F, w φ ∨ χ .
Definition 3 A formula φ → ψ is called a relevant Sahlqvist implication if ψ is positive while φ is a formula built up from propositional atoms, double negated atoms (i.e., formulas of the form ∼∼ p), negative formulas, the constant t and implications of the form t → p (for any propositional variable p) using only the connectives ∨, ∧ and •.

Lemma 11 Every relevant Sahlqvist implication has a local first order correspondent on Routley-Meyer frames.
Proof Let φ → ψ be a relevant Sahlqvist implication. We consider the local frame correspondent Renaming variables we can make sure that no two quantifiers bind the same variable and that x remains free. One can abbreviate T x (ψ) z/x , which is a positive formula, as POS.
Here, REL is a conjunction of atomic first order expressions involving only the ternary predicate R, (DN)AT is a conjunction of translations of (double negated) propositional variables, IMP is a conjunction of translations of formulas of the form t → p i , that is, formulas of the form ∀y, z(Rxyz ∧ ∃b(Ob ∧ b ≤ y) ⊃ P i z), while NEG is a conjunction of negative formulas. Our purpose will be to eliminate all the second order quantifiers in (1).
Given that any unary predicate appearing in POS also appears in the antecedent of (4), (5) is a first order formula in the signature of Routley-Meyer frames, which contains R, * , and O as the only non-logical symbols. We have seen that (1) implies (5). All that is left is to show that (5) implies (1). Recall that (1) is equivalent to (3). Thus it suffices to prove that (5) implies (3). Assume (5), let P 1 , . . . , P n be arbitrary and suppose further that so, by (5), we obtain that The procedure in the above proof is better understood by working out some examples, which we will do next for the benefit of the reader.
Example 12 By the Sahlqvist-van Benthem algorithm, the following list of equivalences must hold: It is easy to check that this correspondence is accurate. First, if F ∀y, z(Rxyz ⊃ y ≤ y * ∨ z z) [w] and M is any model based on F, if Rwyz and M, y p∧ ∼ p either z z or M, y p and M, y p (contraposing the Hereditary Lemma). So, Take a valuation V on F such that V (p) = {x : y ≤ x} and V (q) = {x : x z} (note that these two sets are upward closed under ≤ by transitivity of ≤). V suffices to show that F, w p∧ ∼ p → q. Note that from this correspondence, we obtain that F p∧ ∼ p → q iff F ∀y, z(y ≤ z ⊃ y ≤ y * ) iff F ∀y(y ≤ y * ).
Example 13 By the Sahlqvist-van Benthem algorithm, Next we establish that the correspondence is correct. Suppose that F ∀y, z(Rxyz ⊃ y * * ≤ z) [w], so, using the Hereditary Lemma, F, w ∼∼ p → p. On the other hand if F ∀y, z(Rxyz ⊃ y * * ≤ z)[w], i.e., F ∃y, z(Rxyz ∧ y * * z) [w]. Then any valuation V such that V (p) = {x : x z} guarantees that F, V , z p and F, V , y * * p, so F, w ∼∼ p → p.
The above condition reduces to ∀x(x * * ≤ x ∧ x ≤ x * * ) when we consider validity with respect to all the worlds in O of the frame. This is not the same in general as ∀x(x = x * * ) which is the condition usually required to validate ∼∼ p → p. However, it is certainly the case that, using the construction from Theorem 5 [12], by restricting our attention to just the frames where ≤ is antisymmetric, we get exactly the same set of validities as in the general case.

Example 14 By the Sahlqvist-van Benthem algorithm,
Now we show that the correspondence is indeed correct. First, let F ∀y, z, u 1 , . Suppose M is an arbitrary model based on F, then if Rwyz ∃u 1 u 2 (Ru 1 u 2 y ∧ P u 1 ∧ Qu 2 ) both hold, u 1 ≤ z and u 2 ≤ z, and, by the Hereditary Lemma, P z and Qz as desired. Hence, F, w p • q → p ∧ q. On the other hand, suppose F ∀y, z, u 1 Observe that in fact the condition simplifies to ∀y, z(Rxyz ⊃ x ≤ z ∧ y ≤ z) by some manipulations.
Example 15 By the Sahlqvist-van Benthem algorithm, The latter condition can be written as ∀x, y, z(Ox ∧ Rxyz ⊃ ∃u 2 u 3 (Ryu 2 u 3 ∧ ∃b(Ob ∧ b ≤ u 2 ) ∧ u 3 ≤ z)) when we consider correspondence with respect to the worlds in O of a given frame. This condition is actually equivalent to the condition ∀x∃b(Ob ∧ Rxbx) corresponding in [4] to (t → p) → p ( [4], p. 80, q6).
All that is left is to verify that F ∀y, Let M be any Routley-Meyer model based on F. Consider arbitrary worlds w 1 , w 2 such that Rww 1 w 2 holds in M and suppose that M, w 1 t → p, which means that ∀u, v(Ryuv ∧ ∃b(Ob ∧ b ≤ u) ⊃ P v) holds in M. Since Rww 1 w 2 , we can conclude that ∃u 2 u 3 (Rw 1 u 2 u 3 ∧ ∃b(Ob ∧ b ≤ u 2 ) ∧ u 3 ≤ w 2 ), but then P u 3 , and, by the Hereditary Lemma, P w 2 as desired. On the other hand, suppose that F ∀y, z(Rxyz ⊃ ∃u 2 u 3 (Ryu 2 u 3 ∧∃b(Ob ∧ b ≤ u 2 ) ∧ u 3 ≤ z))[w], so there are worlds w 1 , w 2 such that Rww 1 w 2 and ∀u 2 u 3 (Rw 1 u 2 u 3 ∧ ∃b(Ob ∧ b ≤ u 2 ) ⊃ u 3 w 2 ). Take any model M based on F such that V (p) = {x ∈ W : x w 2 }. V (p) is upward before, using (the contrapositive of) Lemma 3 and the fact that a negative formula is just the (boolean) negation of a positive formula, we see that (1) implies (1), so they are indeed equivalent.
We also see that (1) is in fact equivalent to where ¬POS ∨ NEG is, of course, a negative formula. Finally, let π 1 (x i1 ), . . . , π k (x ik ) be all the conjuncts of (T)NAT and IMP in the antecedent of (3) in which the unary predicate P i occurs. Then if π j (x ij ) appears in one of the conjuncts in IMP, it must be a formula of the form ∀yz(Rx ij yz ∧ P i y ⊃ ∃b(Ob ∧ b ≤ z)), in which case we define σ (π j (x ij )) = λu.(∀z(Rx ij uz ⊃ ∃b(Ob ∧ b ≤ z))). On the other hand if π j (x ij ) appears is one of the conjuncts in (T)NAT we put σ (π j (x ij )) = λu.(u x * ij ) in case π j (x ij ) = ¬P i x * ij and σ (π j (x ij )) = λu.(u x * * * ij ) in case π j (x ij ) = ¬¬¬P i x * * * ij . Note that σ is a well-defined function and that for any π j (x ij ), if π j (x ij )[w] then, then ∀y(P i y ⊃ σ (π j (x ij ))(y))[w]. Next define δ(P i ) = λu.(σ (π 1 (x i1 ))(u) ∧· · ·∧σ (π k (x ik ))(u)). Now, if (T)NAT[ww 1 . . . w k ] and IMP[ww 1 . . . w k ] then ∀u(P i u ⊃ δ(P i )(u))[ww 1 . . . w k ]. The remainder of the proof is as before but using again the contrapositive formulation of Lemma 3 and noting that the intersection of a collection of upward closed sets under ≤ is also upward closed under ≤.
Definition 5 A relevant Sahlqvist formula is any formula built up from (dual) relevant Sahlqvist implications, propositional variables, and negated propositional variables using ∧, the operations on formulas (for any propositional variable free relevant formula θ ) defined by (φ) = θ → φ, and applications of ∨ where the disjuncts share no propositional variable in common.

Theorem 18 Every relevant Sahlqvist formula has a local first order correspondent on Routley-Meyer frames.
Proof Immediate from Lemma 11, Lemma 16 and Lemma 9. The only thing the reader should note is that F, w p iff F x x[w] and F, w ∼ p iff F x * x * [w].

Conclusion
In this paper we have defined a fragment of relevant languages analogue to the Sahlqvist fragment of modal logic. We then went to establish that every class of Routley-Meyer frames definable by a formula in this fragment is actually elementary. This isolates a modest but remarkable collection of relevant formulas. We also showed that there are properties of Routley-Meyer frames definable by relevant formulas which are not first order axiomatizable.