A generalization of the Routley-Meyer semantic framework. Nick Thomas January 8, 2014 Abstract We develop an axiomatic theory of “generalized Routley-Meyer (GRM) logics.” These are first-order logics which are can be characterized by model theories in a certain generalization of Routley-Meyer semantics. We show that all GRM logics are subclassical, have recursively enumerable consequence relations, satisfy the compactness theorem, and satisfy the standard structural rules and conjunction and disjunction introduction/elimination rules.

There are many different axiomatic approaches to the theory of systems of logic; see, e.g., Beziau [2007] and Gabbay [1994]. All of these approaches have in common that they wish to allow us to study infinitely many different systems of logic, by proving theorems which quantify over systems of logic. The most extensively studied approach to the theory of logics is algebraic logic; see, e.g., Dunn and Hardegree [2001] and Font and Jansana [2009]. In algebraic logic we consider logics which are definable in terms of algebraic semantics over a class of algebras, most often definable in terms of algebraic equations. Shifting our focus away from universal logic, perhaps the most widely studied semantic paradigm other than algebraic semantics is Kripke-style semantics, and its generalization into Routley-Meyer semantics (see §48 of Anderson et al. [1992]). It seems natural to explore a theory of universal logic based on Kripkestyle semantics, and more generally Routley-Meyer style semantics. In such a theory we would consider logics definable in terms of semantics over a class of Routley-Meyer frames. Here we will develop such a framework. Let us demarcate more precisely the class of logics we will target with our framework. We are interested in first-order logics defined over the usual firstorder syntax, with the familiar connectives and quantifiers, as listed previously. We want our logics to be subclassical, to have recursively enumerable consequence relations, and to satisfy the compactness theorem. We want them to satisfy the standard structural rules; that is, we are not here interested in substructural logics. We also want them to satisfy the standard introduction and elimination rules for conjunction and disjunction. Let us say some words about the latter criterion. The subclassical logics used in real life vary greatly in their treatment of negation and implication. 1

There is almost nothing that can be said in general about how negation and implication behave in real world subclassical logics. On the other hand, conjunction and disjunction tend, in almost all cases, to be governed by the standard introduction and elimination rules. Negation and implication are the “problematic” connectives; conjunction and disjunction are the “unproblematic” ones. So we feel comfortable taking on board the standard rules for conjunction and disjunction. We have demarcated the space of logics we are interested in, and we intend to study logics in the class of interest which can be defined in terms of some Routley-Meyer style semantics. The main task of the paper is to spell out what we mean by a “Routley-Meyer style semantics.” The main other task is to show that all logics definable by the type of semantics we identify are in fact in the class of interest: they are subclassical, compact, have recursively enumerable consequence relations, etc. Some generalization of the Routley-Meyer framework will be required in order to capture as many logics as possible. Ideally we want all prominent real world logics in the class of interest to fall under our theory. So far as the author has explored, this is true of our theory. The most complicated test case we look at is Kit Fine’s semantics for firstorder relevant logics, as given in §53 of Anderson et al. [1992]. As Fine shows, relevant logics in general require fairly elaborate control over the domains of the points. Meeting the demands of Fine’s semantics will in the end motivate us to make the domains of the points part of the frame itself, in departure from common practice. We also make other slight generalizations from standard Routley-Meyer semantics, most notably replacing the star operator ? with a binary relation. We will think of frames as themselves being models in first-order classical logic. The properties of frames (that the accessibility relation is a preorder, etc.) are expressed as first-order statements which the frames are required to satisfy. This point of view becomes useful when we turn to defining the concept of a logic, where we find ready at hand the notion of a first-order theory. The basic differences between logics in Routley-Meyer style semantics are the constraints they put on frames. We can therefore think of logics as first-order theories which frames may be required to satisfy. We say that a given frame is a valid frame for a given logic just in case the frame (which is a model) satisfies the logic (which is a theory). This strategy is responsible for making it true that our logics are compact and have recursively enumerable consequence relations. Both of these facts follow from the fact that our logics are characterized by first-order theories in classical logic. Because of this fact, the class of models of a given signature in a given logic is described by a first-order theory. The condition of a model satisfying a given statement is also first-order expressible. These facts let us transform questions about entailment in a given logic of our theory into questions about entailment in classical first-order logic. Compactness and recursive enumerability follow.

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1

Definition of generalized Routley-Meyer logics

As stated, we wish to think of frames as models in first-order classical logic. Recall also that we want our frames, in departure from the usual way of doing things, to include both points and the domains of the points. We will begin with frames that just have points, as it will be useful at times to isolate just the “point set” part of a frame. We will call this structure a “point frame,” and again these will be for us a kind of first-order model. The signature of point frames is ΣP = {R, N, ≤∀ , ≤∃ , ω}, where R is a ternary relation symbol, N, ≤∀ , and ≤∃ are binary relation symbols, and ω is a constant symbol. (Note that we will always denote points by Greek letters, because we wish to reserve Roman letters for the objects of domains.) N generalizes the Routley star; it is like the Routley star, but not required to be a function. In frames where for every α there is a unique β such that αN β, we continue to denote that unique β by α? . In general for α to satisfy ¬φ, we will require that φ be false at all points β such that αN β. Our generalization from ? to N is required to accommodate intuitionistic logic under its usual semantics, and also to accommodate relevant logics under Fine’s semantics. ≤∀ and ≤∃ say what worlds we look at to evaluate quantified statements; for instance, in intuitionistic logic we must look at all worlds later in the accessibility order to evaluate universal quantifiers. ω is the “real world,” usually called 0 in relevance logic literature. We use the Greek letter ω in order to follow our convention that points are denoted by Greek letters. Definition 1. A “point frame” P is a (first-order, classical) model in the signature ΣP . We require it to satisfy the following first-order statements: 1. (Negation-worlds exist.) ∀α∃β(αN β). 2. (Counterfactual worlds exist.) ∀α∃β, γ(Rαβγ). 3. (Quantifiers range over something.) ∀α(∃β(α ≤∀ β) ∧ ∃β(α ≤∃ β)). The reason for axioms (1)-(3) is that in their absence, we could construct logics which were not subclassical, as we shall see later. Now we shall define a notion of homomorphisms of point frames. Our homomorphisms are like homomorphisms of models which have some of the “invertibility” properties of isomorphisms, without being bijections. The reader may also observe that our homomorphisms are somewhat like bisimulations (Definition 2.16 in Blackburn et al. [2002]), except they are required to be functions. The motivation for this notion of homomorphism is that makes it true (as we will see later) that if f : P → Q is a homomorphism, then P defines a weaker logic than Q. 3

Definition 2. A homomorphism of point frames is a function f : P → Q (with P and Q point frames) such that: 1. f (ω P ) = ω Q . 2. For all binary relation symbols S ∈ {N, ≤∀ , ≤∃ }: (a) If α, β ∈ P and P |= αSβ, then Q |= f (α)Sf (β). (b) If α ∈ P, β ∈ Q and Q |= f (α)Sβ, then there is γ ∈ P such that f (β) = γ and P |= αSγ. 3. If α, β, γ ∈ P and P |= Rαβγ, then Q |= Rf (α)f (β)f (γ). 4. If α ∈ P, β, γ ∈ Q and Q |= Rf (α)βγ, then there are δ,  ∈ P such that f (δ) = β, f () = γ, and P |= Rαδ. An “isomorphism” of point frames is a bijective homomorphism. Observe that every homomorphism of point frames is a homomorphism of models, and that the isomorphisms of point frames are exactly the isomorphisms of models (between point frames). Now let us turn to full frames, which consist of points and the domains of the points. For the language of frames we will use two-sorted first-order logic, with the two sorts being called “points” and “objects.” For variables denoting points we will use Greek letters α, β, ..., and for variables denoting objects we will use Roman letters a, b, .... The signature of frames is ΣF = ΣP ∪ {∈, 7→}, where: 1. The relation symbols in ΣP range over points, and ω ∈ ΣP is a point constant symbol. 2. ∈ is a binary relation symbol taking an object on the LHS and a point on the RHS. 3. 7→ is a quaternary relation symbol taking an object, a point, an object, and a point, in that order. We write (a, α) 7→ (b, β) as syntactic sugar for 7→(a, α, b, β). a ∈ α is meant to be read as, “the object a is part of the domain of α.” The statement (a, α) 7→ (b, β) means that object a at world α is a “modal ancestor” of object b at world β. All atomic statements true about a at α become true of b at world β. This is a more fine-grained version of the usual monotonicity relation ≤ between points. This more fine-grained version is used in relevant logics under Fine’s semantics. Definition 3. A “frame” F is a two-sorted model of (possibly an expansion of) ΣF . We require, furthermore, that F satisfy the following statements: 4

1. (Domains are nonempty.) ∀α∃a(a ∈ α). 2. (7→ is reflexive.) ∀α∀a((a, α) 7→ (a, α)). If F is a frame, we let Fpt denote F ’s set of points, and Fob denote F ’s set of objects. Given α ∈ Fpt , we define Fob (α) = {a ∈ Fob : F |= a ∈ α}. Now we define homomorphisms of frames, and some useful special cases of frame homomorphisms. Again our motivation for this definition of homomorphism is that it makes it true that if f : F → G is a homomorphism, then F defines a weaker logic than G, as we shall see later. Definition 4. Let F, G be frames. A homomorphism of frames is a map f : Fpt ∪ Fob → Gpt ∪ Gob such that f (Fpt ) ⊆ Gpt and f (Fob ) ⊆ Gob . We define def

def

fpt = f |Fpt , fob = f |Fob . We require that: 1. fpt : Fpt → Gpt is a homomorphism of point frames (when we view Fpt and Gpt as the point frames which inherit their interpretations of the symbols in ΣP from F and G). 2. For all α, f (Fob (α)) = Gob (f (α)). 3. For all α, β ∈ Fpt , a, b ∈ Fob , if F |= (a, α) 7→ (b, β) then G |= (f (a), f (α)) 7→ (f (b), f (β)). We say that f is a “domain contraction” if fpt is a bijection, and for all α, β ∈ Fpt , a, b ∈ Fob , if G |= (f (a), f (α)) 7→ (f (b), f (β)) then F |= (a, α) 7→ (b, β). We say that f is an “isomorphism” if f is a domain contraction and fob is a bijection. Observe that this coincides with the standard notion of isomorphism for two-sorted models. Definition 5. Let F be a frame, and Σ a first-order signature, which may contain relation and constant symbols. An “F, Σ-model” is a tuple M = (F, I, J), where: 1. I is a function defined on pairs (α, R) of points in F and relation symbols in Σ, such that for each α ∈ Fpt and each n-ary relation R ∈ Σ, I(α, R) ⊆ (Fob )n . 2. J is a function defined on the constant symbols in Σ such that for each c ∈ Σ, J(c) ∈ Fob . Furthermore, we require that: 1. For all constant symbols c ∈ Σ and all α ∈ Fpt , F |= J(c) ∈ α. 2. For all α, β ∈ Fpt , all n-ary relation symbols R ∈ Σ, and all a1 , ..., an , b1 , ..., bn ∈ Fob , if for each 1 ≤ i ≤ n, F |= ai ∈ α, bi ∈ β, (ai , α) 7→ (bi , β), then if (a1 , ..., an ) ∈ I(α, R) then (b1 , ..., bn ) ∈ I(β, R). 5

Let M = (F, I, J) be an F, Σ-model. We define the satisfaction relation |=M (or just |= where clear) between elements α of Fpt and formulas φ in Σ without free variables, which may also mention elements of Fob . It is defined as follows: 1. α 2 ⊥. def

2. α |= R(t1 , ..., tn ) iff (δM (t1 ), ..., δM (tn )) ∈ I(α, R), where δM (ti ) = J(ti ) def

if ti is a constant symbol in Σ, and δM (ti ) = ti if ti ∈ Fob . 3. α |= φ ∧ ψ iff α |= φ and α |= ψ. 4. α |= φ ∨ ψ iff α |= φ or α |= ψ. 5. α |= ¬φ iff for all β ∈ Fpt such that F |= αN β, β 2 φ. 6. α |= φ → ψ iff for all β, γ ∈ Fpt such that F |= Rαβγ, if β |= φ then γ |= ψ. 7. α |= ∀x(φ(x)) iff for all β ∈ Fpt such that F |= α ≤∀ β and all a ∈ Fob such that F |= a ∈ β, β |= φ(a). 8. α |= ∃x(φ(x)) iff for some β ∈ Fpt such that F |= α ≤∃ β and some a ∈ Fob such that F |= a ∈ β, β |= φ(a). We say that M |= φ iff ω F |= φ. Definition 6. A “generalized Routley-Meyer (GRM) logic” L is a recursively enumerable theory of two-sorted first-order logic in some signature Σ ⊇ ΣF . We require, furthermore, that: 1. (Nontriviality.) There is a frame F such that F |= L. 2. (Domain expansion property.) If f : F → G is a domain contraction and G |= L, then F |= L. Let L be a generalized Routley-Meyer logic, and Σ a first-order signature. An “L, Σ-model” is any F, Σ-model where F |= L. Now let T be a first-order theory in Σ, and φ a formula in Σ (all without free variables). We say that T |=L φ iff for all L, Σ-models M , if M |= ψ for all ψ ∈ T then M |= φ. Let us now explain the two axioms we have given for GRM logics. Both axioms are required to make it true that GRM logics are subclassical. Nontriviality is an obvious requirement; a logic with no frames would be the trivial logic, in which anything follows from anything. The domain expansion property is meant to rule out certain logics which are non-subclassical in virtue of exerting overly tight control over the domains of their frames. When the domains of all points in F are infinite the domain expansion property holds for L¨owenheim-Skolem type reasons, but when some points in F have finite domains, it is a nontrivial statement. Without the domain expansion property, we could for example construct a GRM logic which was identical to classical logic except that it required the 6

domains of points to contain no more than five elements. Then a (classically consistent) theory which required six or more distinct objects to exist would be trivial in this theory, violating subclassicality. However, such a logic would not have the domain expansion property, because we can always make a domain contraction which maps a frame with more than five elements in some points onto a frame with no more than five elements in any of its points.

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Examples

We give some examples of axiomatizations of logics as generalized RoutleyMeyer logics. For this purpose we define: def

α ⊆ β = ∀a(a ∈ α → a ∈ β). def

α ≈ β = α ⊆ β ∧ β ⊆ α. These relations will give us the only form of domain control we will usually need. We also define def α ≤ β = ∀a((a, α) 7→ (a, β)). This is the usual monotonicity relation, and the only use of 7→ we will need for the examples other than Kit Fine’s semantics. As a further notational convention, in logics where ∀α, β, γ(Rαβγ → β = γ), we treat R as a binary relation, writing αRβ for Rαββ. Similarly, recall that in logics where ∀α∃!β(αN β), we denote the unique β such that αN β by α? . Example 1 (Classical logic). ∀α, β(α = β). This works because the axioms on point frames will require that the unique point in a frame satisfying this axiom be related to itself by all of the relations in ΣP , and this makes the truth definition for GRM logics coincide with the classical truth definition. Example 2 (Intuitionistic logic). ∀α, β, γ(Rαβγ → β = γ). R (as a binary relation) is a preorder. ∀α, β(αRβ ↔ αN β ↔ α ≤ β ↔ α ⊆ β ↔ α ≤∀ β). ∀α, β(α ≤∃ β ↔ α = β). Example 3 (FDE). ∀α∃!β(αN β). ∀α(α?? = α). ∀α(α? ≈ α). ∀α, β, γ(Rαβγ ↔ (β = α? ∧ γ = α)). ∀α, β(α ≤∀ β ↔ α ≤∃ β ↔ α = β). 7

Example 4 (LP, K3, and classical logic). The axioms of FDE, plus: 1. (LP.) ∀α(α? ≤ α). 2. (K3.) ∀α(α ≤ α? ). 3. (Classical logic.) ∀α(α ≤ α? ∧ α? ≤ α). Example 5 (RM). We imitate the binary relational semantics (as in §49 of Anderson et al. [1992]), using the Routley star. ∀α, β, γ(Rαβγ → β = γ). R (as a binary relation) is a linear order. ∀α∃!β(αN β). ∀α(α? Rα). ∀α, β(αRβ ↔ β ? Rα? ). ∀α, β(α ≈ β). ∀α, β(αRβ ↔ α ≤ β). ∀α, β(α = β ↔ α ≤∀ β ↔ α ≤∃ β). Example 6 (B). We refer the reader again to §53 of Anderson et al. [1992] for an exposition of Kit Fine’s variable domain semantics. We will make implicit reference to that exposition, rather than reproducing the details of Fine’s semantics here. Our frames use the signature Σ = ΣF ∪ {S, l, ◦, −, , ↑, ↓, →}. S is a unary predicate on points. l is a unary function from points to points. − is a binary relation on points. ◦, ↑, and ↓ are ternary relations on points. → is a quaternary relation on a point, two objects, and a point, in that order.  is a binary relation on points, corresponding to Fine’s ≤, but confusingly not the same as our ≤. When Fine uses functions defined on D, we replace these with functions defined on points, and require them to take the same value on points with matching domains. This clearly does no harm. Similarly, we implement partial functions as relations, in the usual way. Fine’s clause (iv) in the definition of possible models, characterizing l, translates as ∀α, β(α ≈ β → l(α) = l(β)). ∀α(l(α) ≈ α). Clause (v) translates as ∀α, β(∃γ(◦(α, β, γ)) ↔ α ≈ β)). 8

∀α, β(α ≈ β → ∃!γ(◦(α, β, γ) ∧ α ≈ γ ≈ β)). For clause (vi), we need ∀α(Sα ↔ ∃β(−(α, β)). ∀α(Sα → ∃!β(−(α, β) ∧ α ≈ β)). Clause (vii) should be easy, and for clause (viii) we need ∀α, β(∃γ(↑(α, β, γ) ↔ α ⊆ β)). ∀α, β(α ⊆ β → ∃!γ(↑(α, β, γ) ∧ γ ≈ β). ∀α, β, γ, δ, (↑(α, β, γ) ∧ ↑(α, δ, ) ∧ β ≈ δ → γ = ). Clause (ix) follows the same pattern, and clause (x) is ∀α, a, b(∃β(→(α, a, b, β)) ↔ (a, b ∈ α ∧ a 6= b))). ∀α, a, b((a, b ∈ α ∧ a 6= b) → ∃!β(→(α, a, b, β) ∧ α ≈ β)). Now we turn to axiomatizing Fine’s axioms on possible models. In section (I), axioms (i)-(viii) are straightforward to write down, and (ix) becomes ∀α, β(α  β → α ≤ β). The axioms of (II), on levels, need attention because we are not able to talk about levels directly. Instead we talk about the domains of specific points in order to say things indirectly about levels. A confusing notational conflict is that for Fine, Greek letters denote levels, whereas for us they denote points. Our translations of his axioms read: (i) ∀α∃β(α ⊆ β ∧ ∃a(a ∈ / α ∧ a ∈ β)). (ii) ∀α, β∃γ(α ⊆ γ ∧ β ⊆ γ). (iii) ∀α, β, γ((α ⊆ β ⊆ γ) → ∃δ∀a(a ∈ δ ↔ (a ∈ α ∨ (a ∈ γ ∧ a ∈ / β)))). In section (III), axioms (i)-(iii) are straightforward to write, and axiom (iv) reads ∀α, β, γ(↓ (α, β, γ) → γ ≤ α). The axioms of section (IV) present no difficulties, and neither do axioms (i)-(viii) in section (V). Axiom (ix) in that section translates as ∀α, a, b(∃β(→(β, a, b, α)) → ((a, α) 7→ (b, α) ∧ (b, α) 7→ (a, α))). Finally, in order to get the correct truth conditions, we need to postulate ∀α, β(αN β ↔ ∃γ(α  γ ∧ Sγ ∧ β = −γ)). ∀α, β, γ(Rαβγ ↔ β = α ∧ γ = α ◦ β). 9

∀α, β(α ≤∀ β ↔ ∃γ, δ(↑ (α, γ, δ) ∧ δ  β)). ∀α, β(α ≤∃ β ↔ ∃γ, δ(↓ (β, γ, δ) ∧ δ  α)). The conditions for the universal and existential quantifier, which we take as axiomatic, are theorems for Fine; they are, respectively, condition (v’) and Lemma 10. Note also that for Fine, φ ∨ ψ is taken to abbreviate ¬(¬φ ∧ ¬ψ), so that its truth conditions are defined differently; but since ¬(¬φ ∧ ¬φ) is in fact equivalent to disjunction in this setting, the definitions are equivalent.

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Basic properties of GRM logics

Here we give some basic inferences and meta-inferences valid in all GRM logics. These are the “obvious” inferences, and it is interesting to ask whether there are other inferences which are valid in all GRM logics but do not follow from the ones given. Throughout we let L be a GRM logic. Conjunction. φ, ψ |=L φ ∧ ψ.

φ ∧ ψ |=L φ

φ ∧ ψ |=L ψ.

Disjunction. φ |=L φ ∨ ψ T |=L φ ∨ ψ

ψ |=L φ ∨ ψ.

T, φ |=L ζ T |=L ζ

T, ψ |=L ζ

.

Structural rules. φ |=L φ

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T |=L φ , if T ⊆ U U |=L φ

T, φ |=L ψ T |=L φ . T |=L ψ

Subclassicality

Next we are interested in establishing that all GRM logics are subclassical. Many of the axioms on GRM logics serve to rule out various pathologies which would violate subclassicality, and we now review these. First, let us explain the reason for the three axioms we make on point frames: that negation worlds exist, that counterfactual worlds exist, and that quantifiers range over something. Suppose we drop these axioms, and consider, for example, the logic where we take classical logic and add the axiom ¬∃α(ωN α). In this logic, as you may verify, every statement of the form ¬φ is derivable from no premises. This violates subclassicality. With the axiom that negation worlds exist, this logic has no frames and therefore is trivial, and, as desired,

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not a GRM logic. Similar non-subclassicalities can be obtained in the absence of the other point frame axioms, in just the same way. The domain expansion property also serves to rule out non-subclassicalities; as we explained previously, it rules out logics which control the domains of points too tightly, for example putting a finite upper bound on the number of objects in a point’s domain. The desire for subclassicality is also the reason we did not include a primitive, semantic equality symbol. Suppose we had an equality relation = defined by the truth condition α |= t1 = t2 iff δ(t1 ) = δ(t2 ) (with δ defined as in the truth definition). Then we define a logic L by the following requirements on frames: 1. There are exactly two points, ω and β. 2. Rωββ, and Rβββ, and these are all the R-relations. 3. N, ≤∀ , and ≤∃ are the identity relation. 4. The domain of β is a proper superset of the domain of ω. Observe that L has the domain expansion property, and so is a GRM logic. Let M be an L, ∅-model. Defining > = ¬⊥, we have M |= ∀x(> → ∃y(y 6= x)). Thus that statement follows from no premises in L, violating subclassicality. Of course we can force that statement to be a tautology in a simpler way, by augmenting classical logic with the requirement that domains have at least two elements. But if necessary we could rule this out as a GRM logic fairly easily, just by requiring a “domain contraction property” whereby L-frames were closed under taking images of domain contractions, as well as preimages. The logic we have given is harder to rule out, because Kit Fine’s semantics for relevant logics actually requires the ability to say that domains are properly expanded in certain contexts. It is not clear therefore what would rule out the logic we have given. This argues against having primitive equality in GRM logics. It seems like the kind of fine-grained control we give our logics over their models’ domains is incompatible with the object theories having the kind of fine-grained access to their models’ domains which primitive equality provides, if we want to hold onto subclassicality. Now we shall prove that all GRM logics are subclassical. This will essentially be a corollary of some facts about the categories of frames and point frames. First, if f : F → G is a frame homomorphism, then any argument valid in all F, Σ-models will be valid in all G, Σ-models, so that F defines a weaker “logic” than G does. Secondly, the one-element point frame which characterizes classical logic is the terminal object in the category of point frames; and more generally, given any classical frame F and any frame G, we can expand F ’s domain in such a way that it is homomorphic into G. These facts suffice to establish subclassicality.

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Definition 7. Let F be a class of frames, Σ a signature, T a theory in Σ, and φ a formula in Σ. We say that T |=F φ iff for all F, Σ-models M with F ∈ F, if M |= ψ for all ψ ∈ T then M |= φ. Similarly, given a frame F , we say that T |=F φ iff T |={F } φ. Theorem 1. Let F, G be frames, and f : F → G a homomorphism. Let Σ be a signature, T a theory in Σ, and φ a formula in Σ. If T |=F φ, then T |=G φ. Proof. Suppose T |=F φ. Let M = (G, I, J) be a G, Σ-model. Suppose M |= ψ for all ψ ∈ T . We construct an F, Σ-model M 0 = (F, I 0 , J 0 ) as follows: 1. For all α ∈ F , all n-ary relations R ∈ Σ, and all a1 , ...an ∈ Fob , we say I 0 (α, R) = {(a1 , ..., an ) ∈ (Fob )n : (f (a1 ), ..., f (an )) ∈ I(f (α), R)}. 2. Let c ∈ Σ be a constant symbol. J(c) ∈ ω G , and by condition (2) in the definition of frame homomorphisms there is a ∈ Fob such that f (a) = J(c) and a ∈ ω F . Let α ∈ Fpt . Since denotations of constants are rigid, J(c) ∈ Gob (f (α)) = f (Fob (α)). Therefore a ∈ Fob (α). Summarizing, f (a) = J(c) and a ∈ Fob (α) for all α ∈ Fpt . The latter fact means that we can assign J 0 (c) = a, and we have f (J 0 (c)) = J(c). We also need to check the remaining condition (2) on being a model. Let α, β ∈ Fpt , R ∈ Σ an n-ary relation symbol, a1 , ..., an , b1 , ..., bn ∈ Fob . Suppose for all 1 ≤ i ≤ n, F |= ai ∈ α, bi ∈ β, and F |= (ai , α) 7→ (bi , β). Suppose (a1 , ..., an ) ∈ I 0 (α, R). By construction, we know that (f (a1 ), ..., f (an )) ∈ I(f (α), R). By homomorphism, for each 1 ≤ i ≤ n, G |= (f (ai ), f (α)) 7→ (f (bi ), f (β)). Because M is a model, (f (b1 ), ..., f (bn )) ∈ I(f (β), R). By construction of M 0 , (b1 , ..., bn ) ∈ I 0 (β, R). This gives us condition (2). So M 0 is a model. If φ is a formula in Σ with no free variables but maybe mentioning elements of Fob , we let f (φ) denote the result of replacing instances of any given a ∈ Fob with f (a) ∈ Gob wherever they occur. Our main claim is that if α ∈ Fpt and φ is a formula of the type described, then α |=M 0 φ iff f (α) |=M f (φ). We prove this by induction on φ. Base cases. For ⊥ this is obvious. Now consider an atomic formula R(t1 , ..., tn ). If ti is a constant symbol, then δM 0 (ti ) = J 0 (c), and δM (ti ) = J(c) = f (J 0 (c)). Then f (δM 0 (ti )) = J(c) = δM (ti ) = δM (f (ti )). If ti is an object in Fob , then f (δM 0 (ti )) = f (ti ) = δM (f (ti )). So for any term ti , f (δM 0 (ti )) = δM (f (ti )). Let α ∈ Fpt . We have that α |=M 0 R(t1 , ..., tn ) iff (δM 0 (t1 ), ..., δM 0 (tn ))) ∈ I 0 (α, R) iff (f (δM 0 (t1 )), ..., fM (δM 0 (tn ))) ∈ I(f (α), R) iff 12

(δM (f (t1 )), ..., δM (f (tn )) ∈ I(f (α), R)iff f (α) |=M R(f (t1 ), ..., f (tn )) iff f (α) |=M f (R(t1 , ..., tn )). So we are done with this case. Conjunction and disjunction. Easy. Negation. Suppose α |=M 0 ¬φ. Let β ∈ Gpt be such that G |= f (α)N β. We need to show that β 2 f (φ). By homomorphism there is γ ∈ Fpt such that f (γ) = β and F |= αN γ. Then γ 2 φ. By induction, β 2 f (φ). Therefore f (α) |= f (¬φ). Now suppose f (α) |=M f (¬φ). Let β ∈ Fpt be such that F |= αN β. By homomorphism, G |= f (α)N f (β). Therefore f (β) 2 f (φ). By induction, β 2 φ. Therefore α |= ¬φ. Implication. Suppose α |=M 0 φ → ψ. Let β, γ ∈ Gpt be such that G |= Rf (α)βγ. Suppose β |= f (φ). By homomorphism, there are δ,  ∈ Fpt such that F |= Rαδ and f (δ) = β, f () = γ. Since f (δ) = β and β |= f (φ), by induction δ |= φ. Since α |= φ → ψ,  |= ψ. Since f () = γ, by induction γ |= f (ψ). This shows that f (α) |= f (φ) → f (ψ). Rewriting, f (α) |= f (φ → ψ). Now the other direction. Suppose f (α) |=M f (φ → ψ). Rewriting, f (α) |= f (φ) → f (ψ). Let β, γ ∈ Fpt be such that F |= Rαβγ. By homomorphism, G |= Rf (α)f (β)f (γ). Suppose β |= φ. By induction f (β) |= f (φ). Therefore f (γ) |= f (ψ). By induction γ |= ψ. This shows that α |= φ → ψ. Universal quantification. Suppose α |=M 0 ∀x(φ(x)). Let β ∈ Gpt be such that G |= f (α) ≤∀ β. Let b ∈ Gob (β). By homomorphism, there is γ ∈ Fpt such that f (γ) = β and F |= α ≤∀ γ, and there is g ∈ Fob (γ) such that f (g) = b. We have γ |= φ(g). By induction, β |= f (φ(g)). Observe that f (φ(g)) = f (φ)(f (g)) = f (φ)(b). So γ |= f (φ)(b). This shows that f (α) |= ∀x(f (φ)(x)). Rewriting, f (α) |= f (∀x(φ(x))). Now suppose f (α) |=M f (∀x(φ(x))). Put differently, f (α) |= ∀x(f (φ)(x)). Let β ∈ Fpt be such that F |= α ≤∀ β. Let b ∈ Fob (β). By homomorphism, G |= f (α) ≤∀ f (β), and f (b) ∈ Gob (f (β)). Therefore f (β) |= f (φ)(f (b)). Rewriting, f (β) |= f (φ(b)). By induction, β |= φ(b). This shows that α |= ∀x(φ(x)). Existential quantification. Exercise.

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We have established our main claim, that for all α ∈ Fpt and all φ of the appropriate kind, α |= φ iff f (α) |= f (φ). Let ψ ∈ T . Recall that M |= ψ; i.e., ω G |= ψ. Since ψ does not mention any objects in the models, f (ψ) = ψ; so ω G |= f (ψ). f (ω F ) = ω G , so ω F |= ψ. Since T |=F φ, ω F |= φ. By the claim, ω G |= f (φ), i.e. M |= φ. This shows that T |=G φ. Now we consider classical logic C = {∀α, β(α = β)}. There is a unique-up-to-isomorphism point frame PC which satisfies C. (PC has one object, which must be ω, and, by the point frame axioms, must be related to itself by all relations in ΣP .) Theorem 2. PC is the terminal object in the category of point frames. That is, for any point frame P , there is a unique homomorphism f : P → PC . Proof. Let c be the unique element of PC . Let P be a point frame. (Uniqueness.) If f : P → PC is a homomorphism, then f (p) = c for all p ∈ P . This implies uniqueness. (Existence.) We need only check that the unique map f : P → PC is a homomorphism. Condition (1) holds because f (ω P ) = c = ω PC . Let S ∈ {N, ≤∀ , ≤∃ }. 2(a) holds vacuously because the conclusion is always true. Let α ∈ P, β ∈ PC . Suppose PC |= f (α)Sβ. β = c. By the point frame axioms, there is γ ∈ P such that P |= αSγ. f (γ) = c = β. This satisfies 2(b). (3) holds vacuously because the conclusion is always true. For (4) we argue as for 2(b). Now we give a result which lets us build homomorphisms from a domainexpansion of an arbitrary frame into an arbitrary classical frame. Theorem 3. Let F, G be frames, such that |Gpt | = 1 (so G ∼ = PC as a point frame). Then there is a frame H, a domain contraction g : H → F , and a frame homomorphism h : H → G. Proof. We let c ∈ Gpt be the point in G. We let Gob stand for Gob (c), since we do not care about any other objects in G. Define H as follows. 1. Let Hpt = Fpt , and let H’s interpretations of the symbols in ΣF be inherited from F . 2. Let Hob = Fob × Gob . 3. For each α ∈ Hpt , let Hob (α) = Fob (α) × Gob . 4. Given α, β ∈ Hpt , (a, b), (c, d) ∈ Hob , let H |= ((a, b), α) 7→ ((c, d), β) iff b = d and F |= (a, α) 7→ (c, β).

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Define g by letting gpt be the identity map, and gob = π1 , the projection map. It should be evident that g is a domain contraction; in particular, for condition (2) of being a frame homomorphism, we have g(Hob (α)) = π1 (Hob (α)) = Fob (α) = Fob (g(α)). Define h by letting hpt be the unique point homomorphism into PC , and hob = π2 . For condition (2) of being a frame homomorphism, we have h(Hob (α)) = π2 (Hob (α)) = Gob = Gob (h(α)). For condition (3), suppose α, β ∈ Hpt , (a, b), (c, d) ∈ Hob , and H |= ((a, b), α) 7→ ((c, d), β). By construction b = d. Since h(α) = h(β), G |= (b, h(α)) 7→ (d, h(β)); so we are happy. Theorem 4. (Subclassicality.) Let L be a GRM logic, and let C be classical logic. Let Σ be a signature, T a theory in Σ, and φ a formula in Σ. If T |=L φ, then T |=C φ. Proof. Suppose T |=L φ. Let F be a frame such that F |= C. By nontriviality, there is a frame G such that G |= L. By Theorem 3, there is a frame H, a domain contraction f : H → G, and a frame homomorphism g : H → F . By the domain expansion property, H |= L. Therefore T |=H φ. By Theorem 1, T |=F φ. Since F was an arbitrary frame satisfying C, T |=C φ. The argument just given will work, more generally, to show that if F is any nonempty class of frames with the domain expansion property, then F is subclassical in the sense that if T |=F φ then T |=C φ.

5

Recursively enumerable consequence and compactness

We would like to show that all GRM logics have recursively enumerable consequence relations, and satisfy the compactness theorem. Both of these things are actually true for a fairly simple reason. We can think of GRM models as classical first-order models; the condition of being a valid model for a given logic L is expressible in first-oder logic; and the condition of a point in such a model satisfying a statement φ is also expressible in first-order logic. This lets us transform questions about entailment in any given GRM logic to questions about entailment in classical logic. Let us see the details. We give an alternate definition of GRM models as a type of classical twosorted first-order models, in the following way. Let L be a logic, whose frame signature is Σ. Let ∆ be a first-order signature; we can assume it is disjoint from Σ. We define the signature ∆0 as follows. For each constant symbol c ∈ ∆, we let c ∈ ∆0 . For each n-ary relation symbol R ∈ ∆, we let there be a relation symbol R0 ∈ ∆0 which takes a point followed by n objects. 15

Then we can define an “L, ∆-model” to be a two-sorted model in the signature Σ ∪ ∆0 such that M |= L, M satisfies the first-order statements which all frames are required to satisfy (from the definition of point frames and the definition of frames), and: 1. For each c ∈ ∆, M |= ∀α(c ∈ α). 2. For each n-ary relation R ∈ ∆, M |= ∀α, β∀a1 , ..., an , b1 , ..., bn (

^

(ai ∈ α ∧ bi ∈ β ∧ (ai , α) 7→ (bi , β))

1≤i≤n

→ (R0 (α, a1 , ..., an ) → R0 (β, b1 , ..., bn ))). This definition shows that there is a recursively enumerable first-order theory T which, among models in Σ ∪ ∆0 , is true of exactly the L, ∆-models. The next piece of information we need is that if M is an L, ∆-model and α ∈ Mpt , the statement that α |= φ is expressible in first-order logic. That is, for every formula φ in ∆, without free variables but which may also have constants drawn from Mob , there is a first-order statement [φ](α) (with α a free variable) in Σ ∪ ∆0 , again possibly with constants from Mob , such that for all α ∈ Mpt , M |= [φ](α) iff α |=M φ. This amounts to showing that the truth definition is first-order expressible. We define [φ](α) by induction on formulas: [⊥](α) [R(t1 , ..., tn )](α)

def

=

def

⊥.

=

R(α, t1 , ..., tn ).

[φ ∧ ψ](α)

def

[φ](α) ∧ [ψ](α).

[φ ∨ ψ](α)

def

=

[φ](α) ∨ [ψ](α).

def

=

∀β(αN β → ¬[φ](β)).

def

=

∀β, γ(Rαβγ → ([φ](β) → [ψ](γ)).

def

=

∀β∀a((α ≤∀ β ∧ a ∈ β) → [φ(a)](β)).

def

∃β∃a(α ≤∃ β ∧ a ∈ β ∧ [φ(a)](β).

[¬φ](α) [φ → ψ](α) [∀x(φ(x))](α) [∃x(φ(x))]

=

=

The following statements announce that the definition has the required property: 1. α |= ⊥ iff M |= ⊥. 2. α |= R(t1 , ..., tn ) iff M |= R(α, t1 , ..., tn ). 3. α |= φ ∧ ψ iff M |= [φ](α) ∧ [ψ](α). 4. α |= φ ∨ ψ iff M |= [φ](α) ∨ [ψ](α). 5. α |= ¬φ iff M |= ∀β(αN β → ¬([φ](β)). 6. α |= φ → ψ iff M |= ∀β, γ(Rαβγ → ([φ](β) → [ψ](γ))). 16

7. α |= ∀x(φ(x)) iff M |= ∀β∀a((α ≤∀ β ∧ a ∈ β) → [φ(a)](β)). 8. α |= ∃x(φ(x)) iff M |= ∃β∃a(α ≤∃ β ∧ a ∈ β ∧ [φ(a)](β)). Now we can show that L, our arbitrary GRM logic, has a recursively enumerable consequence relation. Let U be an r.e. first-order theory in ∆. Let [U ] denote {[φ](ω) : φ ∈ U }. By definition, U |=L φ iff for every L, ∆-model M , if ω M |= U then ω M |= φ. We can express the same condition by saying T ∪ [U ] |= [φ](ω). To recursively enumerate the consequences of U in L, we need just recursively enumerate the consequences of T ∪ [U ] in classical logic and look for the consequences of the form [φ](ω). The compactness theorem for L follows similarly. Suppose U |=L φ. Then ˆ ] |= T ∪ [U ] |= [φ](ω). Then by compactness of classical first-order logic, Tˆ ∪ [U ˆ ˆ ˆ ˆ [φ](ω) for some finite T ∪ [U ] ⊆ T ∪ [U ]. Then T ∪ [U ] |= [φ](ω). Then U |=L φ, ˆ is a finite subset of U . and U

6

Open problems

We close by describing two open problems about GRM logics which we think are interesting.

6.1

Giving an external definition of GRM logics

We have given an “internal” definition of GRM logics, as logics which can be presented according to a particular kind of semantics. It would be interesting to give an equivalent “external” definition of GRM logics, as logics whose consequence relations satisfy a specific list of properties. Let us explain. We have given a number of properties possessed by all GRM logics: they validate the inferences in Section 3, they are subclassical, and they are recursively enumerable and compact. Can we extend this list with additional properties true of all GRM logics in such a fashion that any logic with all the listed properties is a GRM logic? If so, this would be informative in a couple of ways. Firstly, it would say, in a paradigm neutral way, exactly how GRM logics fit into the broader scheme of logics. Secondly, it would allow us to take an arbitrary logic satisfying the given properties and study it model-theoretically, in the framework we have given.

6.2

Giving an algebraic characterization of GRM logics

Another open problem is that of giving an algebraic characterization of GRM logics. GRM logics are presumably a special case of algebraic logics, under some suitably broad definition of “algebraic logics.” It would be interesting to find a way of algebraizing arbitrary GRM logics. One obstacle here is that the algebraic theory of first-order logics is not very heavily researched (but see Caliero and Gon¸calves [2007] for recent work on this). Also, it is presumably true that not all GRM logics are characterized 17

by algebraic varieties, because of the difference in expressiveness between the language of algebraic equations and full first-order logic. Therefore some notion of algebraic logic would presumably be needed which was more permissive than “logics characterized by varieties of algebras.” It would be especially interesting to find a class of algebraic first-order logics which are exactly the GRM logics. This would be useful in that it would give us another point of view on the same class of logics. It would also probably shed light on the problem of algebraizing the standard logical connectives, in that we would need to find an algebraic way of requiring each connective to behave somewhat like it does classically, in the sense in which they do in GRM logics.

References Alan Ross Anderson, Nuel D. Belnap, and J.M. Dunn. Entailment: The Logic of Relevance and Necessity, volume 2. Princeton University Press, 1992. Jean-Yves Beziau, editor. Logica Universalis: Towards a General Theory of Logic. Birkh¨ auser, 2nd edition, 2007. Patrick Blackburn, Martin de Rijke, and Yde Venema. Modal Logic. Cambridge University Press, 2002. Carlos Caliero and Ricardo Gon¸calves. On the algebraization of many-sorted logics. In Recents Trends in Algebraic Development Techniques - Selected Papers, volume 4409 of Lecture Notes in Computer Science. Springer, 2007. J. Michael Dunn and Gary M. Hardegree. Algebraic Methods in Philosophical Logic. Oxford University Press, 2001. Josep Maria Font and Ramon Jansana. A General Algebraic Semantics for Sentential Logics, volume 7 of Lecture Notes in Logic. Association for Symbolic Logic, 2nd edition, 2009. Dov M. Gabbay, editor. What is a Logical System?, volume 4 of Studies in Logic and Computation. Oxford University Press, 1994.

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