TOPOLOGY AND MODALITY: THE TOPOLOGICAL INTERPRETATION OF FIRST-ORDER MODAL LOGIC

As McKinsey and Tarski showed, the Stone representation theorem for Boolean algebras extends to algebras with operators to give topological semantics for (classical) propositional modal logic, in which the “necessity” operation is modeled by taking the interior of an arbitrary subset of a topological space. In this article, the topological interpretation is extended in a natural way to arbitrary theories of full first-order logic. The resulting system of S4 first-order modal logic is complete with respect to such topological semantics.

It has been known since the work of McKinsey & Tarski (1944) that, by extending the Stone representation theorem for Boolean algebras, topological spaces provide semantics to propositional modal logic. Specifically, a necessity operator obeying the rules of the system S4 can be interpreted by the interior operation in a topological space. This result, however, is limited to propositional modal logic. The aim of this article is to show how the topological interpretation can be extended in a very natural way to first-order modal logic.

Topological semantics for propositional modal logic.
Let us review the topological semantics for propositional S4.

The system S4 of propositional modal logic.
Modal logic is the study of logic in which the words "necessary" and "possible" appear in statements such as • It is necessary that the square of an integer is not negative.
• It is possible that there are more than 8 planets.
The history of modal logic is as old as that of the study of logic in general and can be traced back to the time of Aristotle. The contemporary study of modal logic typically treats modal expressions as sentential operators, in the same way as ¬ is treated. That is, for each formula ϕ of propositional logic, the following are again formulas: 2ϕ "It is necessary that ϕ." 3ϕ "It is possible that ϕ." Formulas are recursively generated from propositional letters p, q, r , . . . using the propositional operators , ⊥, ∧, ∨, →, ¬ as usual, in addition to 2 and 3. Hence, the formulas of the language include ones such as 2(2 p → 3(2q ∧ ¬r )).
Among various axiom systems providing inference rules for modal operators, the system S4 of propositional modal logic consists of the rules listed below, in addition to those of classical propositional logic. 1 Here ϕ, ψ are any sentences and is a propositional constant standing for truth (or it stands for any theorem of propositional logic if the language is not assumed to have the propositional constant). Also, define 3ϕ = ¬2¬ϕ.
2ϕ ϕ 2ϕ 22ϕ 2ϕ ∧ 2ψ 2(ϕ ∧ ψ) 2 ϕ 2ϕ ψ 2ψ 1.2 Topology. The S4 rules in subsection 1.1 have been known, since McKinsey & Tarski (1944), to be exactly the rules of the interior operation on topological spaces. Given a set X , recall that a subset O(X ) ⊆ P(X ) of its powerset P(X ) is said to be a topology on X if it satisfies the following: • If U i ∈ O(X ) for all i ∈ I , then i∈I U i ∈ O(X ) for any index set I . Such a pair (X, O(X )), or often X itself with O(X ) in mind, is called a topological space. The U ⊆ X lying in O(X ) are called open sets of X , and an open set U such that a ∈ U is called a neighborhood of a. On the other hand, F ⊆ X such that X −F = {x ∈ X | x / ∈ F } is an open set is called a closed set. Now, given a topological space (X, O(X )), define an interior operation int on P(X ) as follows: for any subset A ⊆ X ,

Note that int(A) is open because the union of open sets is open. Thus, int(A) is the largest of all open sets U contained in A.
It follows that any open set U is a fixed point of int and can be written as an interior, that is, U = int(U ). Moreover, int obeys the following rules. For any A, B ⊆ X ,

A ⊆ B ⇒ int(A) ⊆ int(B).
Here, if we read A, B for sentences and replace X , ∩, ⊆ with , ∧, , we can see that these rules are just the rules of S4. In a similar manner, the closure cl(A) = X − int(X − A) of A, that is, the smallest closed set containing A, obeys the corresponding S4 rules of 3.

Topological semantics for propositional S4.
Let us now formally define how a language of propositional modal logic is interpreted in a topological space. Suppose we are given a language L of propositional modal logic and a topological space (X, O(X )).

Propositional S4
( ] of L is a mapping from the set of sentences of L to P(X ). It assigns an arbitrary subset [[ p]] of X to each atomic sentence p and moreover satisfies the conditions below for connectives and operators. Here ϕ, ψ, are the same as before, while ⊥ is either the propositional constant for falsity or any sentence whose negation is provable in propositional logic.
With this notion of interpretation, the correspondence between the rules of Boolean operations on sets and those of the propositional connectives, and the rules of the interior operation and the S4 rules, immediately gives us soundness.
Theorem 1. For any pair of sentences ϕ, ψ of L, In particular, The usual converse statement of completeness can be derived as a corollary of the following even stronger result. , 1944). For any (consistent) theory T in L containing S4, there exist a topological space X and an interpretation [[·]] such that any pair of sentences ϕ, ψ of L satisfies the following:

Theorem 2 (McKinsey & Tarski
Corollary 1. For any pair ϕ, ψ of sentences of L, In particular, 2. Semantics for first-order logic. The goal of this article is to extend the topological semantics in the foregoing section to first-order modal logic. In this section, we introduce some notation for the standard semantics of (nonmodal) first-order logic, which will be convenient for our purposes.
2.1 Denotational interpretation. Suppose we are given a language L in first-order logic. L has primitive relation symbols R i (i ∈ I ), function symbols f j ( j ∈ J ), and constants c k (k ∈ K ). Then, as usual, a structure M = D, R i M , f j M , c k M i∈I, j∈J,k∈K for L consists of the following: Given such a structure and elements a 1 , . . . , a n ∈ D, for any formula ϕ(x 1 , . . . , x n ) with at most the displayed variables x 1 , . . . , x n free, the relation M ϕ[a 1 , . . . , a n ] of modeling a formula is recursively defined as usual. Now we extend the "denotational" point of view to first-order languages. Whereas we gave an interpretation [[ϕ]] to sentences ϕ in subsection 1.3, here for first-order logic we give an interpretation also to formulas containing free variables; so we extend the notation to include interpretations of all formulas. Here it is presupposed that no free variables appear in the formula ϕ except x, y but not that x, y actually appear. To a sentence σ with no free variables, we give [[σ ]] as we did before. We also give interpretation [[x | t ]] to a term t (x) built up from function symbols, constants, and variables.

First-order logic
The interpretation of a formula ϕ is essentially the subset of the model M defined by ϕ: That is, the set of individuals satisfying ϕ. Then, the following properties are easily derived: These properties could also be used as conditions to define the interpretation recursively, skipping altogether. Then, we would need to define [[x, y | ϕ(x) ]] ⊆ D n+1 also for a formula ϕ(x) which does not contain the free variable y, which can be done simply by Similarly, when a term t (x) has n arguments, its interpretation [[x | t ]] is the function f : D n → D recursively defined from f M , c M in the expected way.
The definition of interpretation of formulas can be naturally extended to the case of n = 0 for D 0 = { * }, any 1-element set. That is, while a subset [[x | ϕ ]] of D n is given for a formula ϕ, the interpretation of a sentence σ is in a similar manner given as a subset [[σ ]] of D 0 (a "truth value") as follows: Note that as in (1) we then have, for any formula ϕ with at mostx free, Now, in terms of [[·]], the usual soundness and completeness of first-order logic are expressed as follows.
Theorem 3. Given a language L of first-order logic, for any pair of formulas ϕ, ψ of L with at mostx free, In particular, ϕ is provable ⇐⇒ every interpretation M has M ϕ.

Interpretation and mappings.
Some of the conditions that recursively define interpretation can be considered in terms of images of mappings. We sum up this fact in this subsection because it will be useful shortly. First, let us introduce some notation for images. Given a mapping f : X → Y and subsets A ⊆ X and B ⊆ Y , the direct image of A and the inverse image of B under f shall be written, respectively, as follows: Next, we define, for each n, the projection p n : D n+1 → D n to be (ā, b) →ā. In particular, p 0 : D → D 0 = { * } has p 0 (b) = * for all b ∈ D. Then, we have which corresponds to the rule ∃yϕ ψ ⇐⇒ ϕ ψ of first-order logic. Here the "eigenvariable" condition that y does not occur freely in ψ is expressed by [[x | ψ ]] making sense. 2 Moreover, the substitution of terms can also be expressed by inverse images. Given a formula ϕ(z) and term t (ȳ), with the obvious notation for substitution one has

Topological semantics for first-order modal logic.
We now extend the topological semantics reviewed in subsection 1.3 to first-order logic. To do so, we require the notion of a sheaf over a topological space, which combines the topological semantics of propositional modal logic with the set-valued semantics of first-order logic in section 2. and gives a very natural semantics for first-order modal logic.

Sheaves. First, recall that a map
Definition 1. A sheaf over a topological space X consists of a topological space F and a local homeomorphism π : F → X , meaning that every point a of F has some neighborhood U a such that π(U ) is open and the restriction π | U : U → π(U ) of π to U is a homeomorphism. 3 F is called the total space, and π is called the projection from F to X.
Taking a concrete example, R (with its usual topology) and π : R → S 1 such that π(a) = e i2πa = (cos 2πa, sin 2πa) form a sheaf over the circle S 1 (with the subspace topology in R 2 ). We may say that R draws a spiral over S 1 , so that, for every a ∈ R, a neighborhood U small enough is homeomorphic to its image π(U ).
One of the properties of sheaves important for the goal of this article is that a local homeomorphism π : F → X is not only continuous but also an open map, which means that π(U ) ⊆ X is an open subset of X for every open U ⊆ F.
It is also important that we can consider sheaves from the following viewpoint. Given a sheaf π : F → X , take any point p of X and define the "stalk" F p ⊆ F at p as follows: F p is also called the fiber of F over p; it is shown in the figure above to be a single line over p. Because fibers do not intersect each other, F is partitioned into fibers, so that the underlying set |F| of the space F can be recovered by taking the disjoint union of all fibers. That is, we can write indicates that the union is disjoint. By the local homeomorphism condition, each fiber F p forms a discrete subspace of F. In the context of semantics for first-order modal logic, we may think of the fibers as "possible worlds" which "change continuously" over the space X .
Let us mention maps of sheaves as well. A map f from a sheaf (F, π F ) to another (G, π G ) is simply a continuous map f : F → G such that π G • f = π F , that is, such that the following diagram commutes.
Thus, f respects the fibers; that is, the underlying map f can be written as a bundle of maps f p : F p → G p from fibers to fibers: It is an important fact that maps of sheaves are necessarily also local homeomorphisms and hence are open maps.
Last, for a sheaf π : F → X , the diagonal map : F → F × X F defined to be a → (a, a) is a map of sheaves and hence is an open map. 4 Therefore, in particular, the image 3.2 Topological semantics in terms of sheaves. Speaking figuratively, the extension of topological semantics using sheaves corresponds to taking the "product" of topological semantics for propositional modal logic and denotational semantics for first-order logic. The topology, or the "horizontal axis," on a space X and a sheaf F gives interpretation to the modal operator 2, and each fiber, along the "vertical axis," plays the role of a "possible world," a set providing the first-order interpretation.
Consider a language L gained by adding the modal operator 2 to a language of firstorder logic. Here, in defining formulas recursively, the usual conditions coming from firstorder logic do not discriminate formulas containing modality from ones not (e.g., in the same way that (¬ϕ)[t/z], which is gained by substituting the term t for the free variable z in ¬ϕ, and ¬(ϕ[t/z]), by applying ¬ to ϕ[t/z], are the same formula, we identify (2ϕ)[t/z] and 2(ϕ[t/z]) as the same formula 2ϕ[t/z]). Then, in a similar manner to subsection 2.1, we define a structure to interpret formulas of L as consisting of the following: 5 • A topological space X and a sheaf π : D → X over it.
Decomposing this structure into fibers, we can see that, for each point p ∈ X , the fiber D p gets a standard L-structure Taking a sentence ∃yϕ, for example, its interpretation is As can be seen in this example, the interpretation of a sentence σ with no free variables is given as a subset of D 0 = X , the "worlds" p ∈ X at which σ is true.
Finally, we of course use the topology of X and D to interpret the modal operator 2, that is, Since sentences are interpreted by subsets of X , we define in a similar manner to (1) and (2) of subsections 1.3 and 2.1, respectively, as follows.

Definition 2. A formula ϕ is true in an interpretation
Note that this specification does indeed agree with the "classical" one of McKinsey and Tarski at the level of propositional modal logic.

The system FOS4 of first-order modal logic.
The topological semantics given in the previous subsection is a very natural extension of the topological semantics for the system S4 of propositional modal logic to first-order logic, which can be seen from the fact that a system that is sound and complete with respect to it can be gained by simply taking the union of the axioms and rules of first-order logic and S4.
Definition 3. System FOS4 consists of the following axioms and rules: 1. All axioms and rules of (classical) first-order logic. In applying schemes, formulas containing the modal operator and ones not are not distinguished. Especially in the following axiom of identity, ϕ may contain the modal operator.
2. The rules of S4 propositional modal logic. That is, for any formulas ϕ, ψ and for as before, Listing some theorems of FOS4, not only do we have 2∃y2ϕ ∃y2ϕ, but also the following proof is available.
2ϕ ∃y2ϕ 22ϕ 2∃y2ϕ 2ϕ 2∃y2ϕ ∃y2ϕ 2∃y2ϕ The last step satisfies the eigenvariable condition that y does not occur freely in the right formula. Similarly, continuity is required to model substitution so that, for any formula ϕ(z) and term t (ȳ), we will have the required equality indicated by ! below: Also, by substituting 2x = z for ϕ(z) in the first-order axiom x = y ϕ(x) → ϕ(y) of identity, we have while 2x = x is gained by the S4 rule from another axiom of identity, namely, x = x. Therefore, x = y 2x = y is provable. Thus, the diagonal which interprets identity, has to be open, and therefore the diagonal map has to be an open map. This, together with the necessity of projections being open continuous maps, shows that the soundness of FOS4 for topological semantics actually requires the use of sheaves. Indeed, we have the following.
Theorem 4. For any formulas ϕ and ψ, Moreover, we also have completeness in the strong form of section 1.
Theorem 5. For any (consistent) theory T of L containing FOS4, there exists a topological interpretation M = (π : D → X, [[·]]) such that, for any pair of formulas ϕ, ψ of L with no free variables exceptx, the following holds.
In particular, for any sentence σ , Corollary 2. For any pair of formulas ϕ, ψ of L with no free variables exceptx, Moreover, for any sentence σ , The proof of Theorem 5 is beyond the scope of this article, but we provide a sketch as an appendix for the curious reader.

Examples of the interpretation.
To help understand how the combination of topology and quantification works in this semantics, let us take an example of a concrete interpretation.

Necessary properties of individuals.
Let us recall the example of a sheaf given in subsection 3.1, that is, the infinite helix over the circle with projection π : R + → S 1 such that π(a) = (cos 2πa, sin 2πa), except that we now take D = R + = {a ∈ R | 0 < a }, the positive reals instead of R. Thus, we have a spiral infinitely continuing upward but with an open downward end at 0; this is also a sheaf. So let M = (π, [[·]]) interpret the binary relation symbol by the "no greater than" relation of real numbers on this sheaf as follows: That is, in each fiber R + p , the order is just the usual one on the reals. Then, consider the truth of the following sentences under this interpretation: ∃x∀y.x y "There exists x such that x is the least." ∃x2∀y.x y "There exists x such that x is necessarily the least." So [[ ∃x2∀y.
In this way, 1 ∈ R + is "actually the least" in its fiber (or "possible world") R + (1,0) = {1, 2, 3, . . .} but not "necessarily the least." Intuitively speaking, 1 is the least in the world R + (1,0) , but any neighborhood of this world, no matter how small a one we take, contains some world ({ε, 1 + ε, 2 + ε, 3 + ε, . . .} for ε > 0) in which 1 is no longer the least. Note that here we used the notion "1 in worlds near by" for explanation. Even though 1 only exists in R + (1,0) , this notion still makes sense because the local homeomorphism property of the sheaf allows us to find an associated point in any other world in a sufficiently small neighborhood.
Finally That is, M is also a countermodel for the Barcan formula of the form "∀2 → 2∀." (In contrast, "converse Barcan" "2∀ → ∀2" and "∃2 → 2∃" are provable in FOS4 in a similar manner to the proof in p. 156 and are valid in the topological semantics.)

Defining functions and names.
In first-order logic, when a structure M satisfies ∀x∃!yϕ(x, y) ("eachx has a unique y such that ϕ(x, y)"), a new function symbol f ϕ can be introduced into the language and interpreted in M so that M ∀x∀y( f ϕ (x) = y ↔ ϕ(x, y)). Does a corresponding fact hold in FOS4?
Consider the "codiscrete" topological space consisting of 2 points, that is, X = {p, q}, O(X ) = {X, ∅}. Moreover, consider the sheaf over X consisting of 2 copies of X , that is, with π : D → X defined as (u, i) → u. On this sheaf, let us set the interpretation of an (n + 1)-ary relation symbol R so that y)). This is implied by the fact that M does not satisfy the consequent of the theorem ∀x∀y2( f (x) = y ↔ R(x, y)) ∀x∃!y2R(x, y) of FOS4, where ∀x∃!y2R(x, y) is short for ∀x∃y∀z(y = z ↔ 2R(x, z)). The same thing can be expressed in terms of the interpretation as follows. The interpretation would not be continuous and hence not a map of sheaves. The same thing can be said about names with n = 0. That is, even when M ∃!yϕ(y) holds, a name c such that M ∀y(c = y ↔ ϕ(y)) cannot be defined in general. For example, M in the previous subsection has ∃!x∀y.x y true but cannot have a name for such x.
On the other hand, not only in this sheaf but in any interpretation M, a function symbol f ϕ can be defined so that M ∀x∀y( f ϕ (x) = y ↔ 2ϕ(x, y)) if M ∀x∃!y2ϕ(x, y). To sum up, in FOS4, a necessary description defines a name, which then has a continuous denotation, whereas a contingent description need not have a corresponding denotation.

Comparison to other semantics for modal logic.
Let us compare the topological semantics to other preceding semantics for quantified S4. To prepare ourselves for the comparison, it is helpful to first review the relation between the following 3 semantics for propositional S4:  Tarski (1944) showed that any topo-Boolean algebra can occur as a subalgebra of the algebra of a space. 6 Several ideas have been proposed to extend the semantics above to quantified modal logic. One is to extend (iii) by completing the algebra so that it is equipped with arbitrary meets (for ∀) and joins (for ∃) to interpret the quantifiers. This completion was shown by Rasiowa & Sikorski (1963) to give a semantics with respect to which first-order S4 is complete. 7 Another idea is to extend (i) or (ii) by equipping each possible world with a domain of individuals. The current notion of a Kripke sheaf derives from early work in topos theory (Lawvere, 1969(Lawvere, , 1970 and is defined to be a presheaf over a preorder (W, R) (S4 Kripke frame), namely, a functor from (W, R) to the category Sets of sets. This means that a Kripke sheaf D over an S4 Kripke frame (W, R) assigns a set D(x), "domain of individuals," to each world x ∈ W and functorially provides a mapping D xy : D(x) → D(y) for each x, y ∈ W such that x Ry; then for a ∈ D(x), we can read D xy (a) ∈ D(y) to be "a in the world y." 8 Such a fibration of preorders can be equivalently written as follows: 9 a Kripke sheaf consists of 2 S4 Kripke frames (W, R) and (D, ρ) and a p-morphism 10 π : (D, ρ) → (W, R) satisfying Then, π −1 (x) ⊆ D corresponds to D(x). π(a) is "the world where the individual a lives," and b in ( * ) is D π(a)x (a). QS4 = (quantified S4 with equality) is known to be complete with respect to Kripke sheaves (see, e.g., Shehtman & Skvortsov, 1990). The topological semantics of this article is the extension of (ii) analogous to Kripke sheaves extending (i). In other words, the relation between (i) and (ii) is preserved in the relation between Kripke sheaves and topological semantics: any Kripke sheaf π : (D, ρ) → (W, R) becomes a local homeomorphism by taking the Alexandroff topology 6 McKinsey & Tarski (1944) showed the dual result for closure algebras. 7 As Rasiowa & Sikorski (1963) showed, the completeness result does not require all meets and joins. 8 Such a functorial (presheaf) definition of Kripke sheaves is found in Ghilardi (1989), Ghilardi & Meloni (1988), and Goldblatt (1979). Note that D xy need not be an injection, whereas each D xy is an inclusion map in a conventional Kripke frame with a domain of individual. 9 See Shehtman & Skvortsov (1990). 10 A map π : (D, ρ) → (W, R) of Kripke frames is called a p-morphism when and are satisfied. both at (W, R) and at (D, ρ). 11 More precisely, indeed, the category of Kripke sheaves over a preorder P and monotone maps respecting fibers is exactly the topos of all sheaves over the space P with the Alexandroff topology. The approach of this article also extends the Kripke sheaf approach by extending the interpretation to functions and names, which have been ignored in the existing semantics in terms of Kripke sheaves; 12 in this sense, the semantics is for first-order, but not just quantified, modal logic.

Categorical and topos-theoretic formulations.
Local homeomorphisms over a topological space X (as in (ii)) form the category LH/ X , which is well known to be categorically equivalent to the category Sh(X ) of functorial sheaves over X , that is functors from the complete Heyting algebra O(X ) (an instance of (iii)) to Sets satisfying certain conditions. By virtue of this fact, the semantics of this article in terms of local homeomorphisms can also be formulated in terms of functorial sheaves to be a version of functorial semantics.
Moreover, since LH/ X and Sh(X ) as well as their underlying (discrete) structures Sets/|X | and Sh(|X |) are elementary topoi, a topos-theoretic formulation is also available for the semantics of this article by considering the forgetful functor id * : Sh(X ) → Sets/|X |. In this formulation, we take a sheaf F and the Boolean algebra Sub Sets/|X | (id * F) ∼ = P(|F|) of subsets of F, which is equipped with the interior operation int coming from the topology of F to interpret the modal operator 2. 13 id * and its right adjoint id * constitute the geometric morphism id : Sets/|X | → Sh(X ), which can also be viewed as induced by the (continuous) identity map id : |X | → X . This morphism helps us push forward with the topos-theoretic point of view, because int can be obtained from the comonad id * • id * , as in: Although the point-set topological formulation presented in this article is more elementary and perspicuous, the topos-theoretic one is more useful for generalizations. For example, we see from it that any geometric morphism of topoi (not just id * id * ) induces 11 The parallelism is even deeper than mentioned here. With the condition ( * ) dropped, any p-morphism π : (D, ρ) → (W, R) is called a Kripke bundle (see Shehtman & Skvortsov (1990)); topologically speaking, it is an open bundle (open continuous map) with the Alexandroff topology. If semantics includes not only Kripke sheaves but also Kripke bundles, the substitution of terms is lost. In parallel to this, the substitution is lost if topological semantics includes not only sheaves (local homeomorphisms) but also open bundles, and the discussion in pp. 12f. illuminates why. 12 In the Kripke framework, Dragalin's (1979) semantics dealt with functions and names, but for intuitionistic first-order logic. This logic does not require the general sheaf structure (which FOS4 or even QS4 = does); instead Dragalin used Kripke frames with increasing domains (with which FOS4 and QS4 = are incomplete). In such a semantics, the identity of individuals across worlds is given, or in other words, we need not (and Dragalin did not) make explicit the fact that functions and names have to be interpreted by maps of sheaves or monotone maps. 13 Indeed, the authors originally presented (Awodey & Kishida, 2005) the topological interpretation of this paper under this topos-theoretic formulation; this paper has served to reformulate it purely in terms of elementary (point-set) topology. a modality on its domain. 14 This immediately suggests natural models for intuitionistic modal logic, typed modal logic, and higher order modal logic. One conceptual difference between the local-homeomorphism formulation and the functorial one is that, in the former, 2 is interpreted by topological interior, as it was originally in McKinsey & Tarski (1944). In this sense, the local-homeomorphism semantics can be properly called the extension of McKinsey and Tarski's topological semantics. In the same way that (ii) connects the 3 approaches (i)-(iii), the topological semantics of this article (extending (ii)) subsumes Kripke sheaf semantics (the extension of (i)) on one hand and can be seen to categorically subsume the algebraic topological semantics (Rasiowa & Sikorski (1963), extending (iii)) on the other hand, 15 thereby giving unification to these three approaches to first-order modal logic.
Historically, extending (iii) by functorial sheaves is already suggested in Shehtman & Skvortsov (1990). 16 Also, Hilken & Rydeheard (1999) formulated the sheaf extension of (ii) and stated its completeness as an open problem. The completeness of first-order S4 with respect to the topological semantics is first shown by the authors of this article (Awodey & Kishida, in preparation) but in the strong form of Theorem 5, that is, the existence of a canonical model for every theory containing FOS4.

Acknowledgments.
A grateful acknowledgment goes to inspiring discussions with and helpful comments by Horacio Arló-Costa, Nuel Belnap, Johan van Benthem, Mark Hinchliff, Paul Hovda, Ken Manders, Eric Pacuit, Rohit Parikh, Dana Scott, and especially Guram Bezhanishvili, Silvio Ghilardi, and Rob Goldblatt as well as an anonymous referee for their accurate suggestions, which improved section 5. We also thank the organizers, Aldo Antonelli, Alasdair Urquhart, and Richard Zach, of the Banff Workshop "Mathematical Methods in Philosophy" for the opportunity to present this research.

A. Products of sheaves.
Here we review the standard definition of (fibered) products of sheaves (cf. Mac Lane & Moerdijk, 1992). We first need to recall some basic definitions in general topology.
Given finitely many topological spaces X 1 , . . . , X n , we can introduce a topology on the cartesian product X 1 × · · · × X n by declaring products U 1 × · · · × U n ⊆ X 1 × · · · × X n of open sets U 1 ⊆ X 1 , . . . , U n ⊆ X n to be basic open sets and thereby defining the union of any number of those basic open sets to be an open set. This topology is called the product topology.
Given a topological space (X, O(X )) and any subset S ⊆ X , we can define another topological space (S, O(S)), called a subspace of (X, O(X )) by setting: Now let us define the product of sheaves. The product of sheaves π F : F → X and π G : G → X is in general not the product space F × G of topological spaces F and G; instead, we take the product "over X ," written F × X G. In the same way that the underlying set of a sheaf is a bundle of fibers, the underlying set of a product of sheaves is given as a bundle of products of fibers. Thus, given This is called a fibered product. Since this set |F × X G| is a subset of F × G, we can then define the topology on F × X G to be the subspace topology of the product topology on F × G. The projection π : F × X G → X (i.e., from the total space to the base space) maps (a, b) ∈ F p × G p to p. One can show that this projection π : F × X G → X is a local homeomorphism if both π F and π G are. We can also consider the projections p F : F × X G → F and p G : F × X G → G (from the product to the components), which map (a, b) ∈ F p × G p to a ∈ F p and b ∈ G p , respectively. Then, of course π = π F • p F = π G • p G . In sum, schematically, we have the situation The n-fold product F × X · · ·× X F of a sheaf π : F → X over X is written π n : F n → X . We write F n p for the fiber (F n ) p = (F p ) n . When n = 0, F 0 is X itself, because the 0-fold product of each fiber F p of F is a singleton F 0 p = { * }: Hence, the projection π 0 : F 0 → X is the identity map.

B. Sketch of a completeness proof.
Here we sketch a proof for Theorem 5, namely, the completeness of FOS4 with respect to the topological semantics. See Awodey & Kishida (in preparation) for the details.
Theorem 5. For any (consistent) theory T in a first-order language L and containing FOS4, there exists a topological interpretation M = (π : D → X, [[·]]) such that any pair of formulas ϕ, ψ of L with no free variables exceptx satisfies the following: To sketch our proof, it is illuminating to first review a proof for the topological completeness of propositional S4, because our proof extends the essential idea of that case.
Theorem 2. For any (consistent) theory T in a propositional language L and containing S4, there exists a topological interpretation (X, [[·]]) such that any pair of sentences ϕ, ψ of L satisfies the following: Proof of Theorem 2 (sketch). Consider the Lindenbaum algebra B of T, which is a Boolean algebra equipped with the operation b : [ϕ] → [2ϕ]. Next, take the set U of ultrafilters in B and the Stone representation · : B → P(U ). That is, The map · is an injective Boolean homomorphism. Note that the topology defined in the proof above coincides with the usual Stone space topology on U if 2 is trivial, that is, ϕ 2ϕ. More importantly, we should note that each ultrafilter u in B can be considered a model of T, that is, u ϕ if T proves ϕ, where we write u ϕ to mean [ϕ] ∈ u. In other words, the essential idea of the proof above is to take the collection of all (propositional) models of T and give it the topology with basic open sets defined by extensions of all 2ϕ. Now, given any consistent theory T in a first-order modal language L, our proof extends this key idea by first taking a sufficiently large set M 0 of first-order models of T in the following way. Consider the nonmodal first-order language L = L ∪ {2ϕ | ϕ is a formula of L } given by adding to L an n-ary basic relation symbol 2ϕ for each formula ϕ of L with exactly n free variables. Then, Gödel's completeness theorem for first-order logic yields a class M = ∅ of structures M for L such that, for any formula ϕ, M 0 , unfortunately, cannot in general be topologized so that π is a sheaf in the required way. To secure the necessary sheaf condition, we need to "label" M 0 so that every a ∈ M∈M 0 |M| has a name in the language. So let us extend the language L to L * = L ∪ {c i | i < λ } by adding λ-many new constant symbols. Then, consider the following collection of structures for L * :