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Generalized Topological Semantics for First-Order Modal Logic 205 VI.1 Generalized Topological Semantics for First-Order Modal Logic1 Kohei Kishida 1Draft of November 14, 2010. Draft of November 14, 2010 Abstract. This dissertation provides a new semantics for first-order modal logic. It is philosophi- cally motivated by the epistemic reading of modal operators and, in particular, three desiderata in the analysis of epistemic modalities. (i) The semantic modelling of epistemic modalities, in particular verifiability and falsifiability, cannot be properly achieved by Kripke’s relational notion of accessibility. It requires instead a more general, topological notion of accessibility. (ii) Also, the epistemic reading of modal operators seems to require that we combine modal logic with fully classical first-order logic. For this purpose, however, Kripke’s semantics for quanti- fied modal logic is inadequate; its logic is free logic as opposed to classical logic. (iii) More importantly, Kripke’s semantics comes with a restriction that is too strong to let us se- mantically express, for instance, that the identity of Hesperus and Phosphorus, even if meta- physically necessary, can still be a matter of epistemic discovery. To provide a semantics that accommodates the three desiderata, I show, on the one hand, how the desideratum (i) can be achieved with topological semantics, and more generally neighborhood se- mantics, for propositional modal logic. On the other hand, to achieve (ii) and (iii), it turns out that David Lewis’s counterpart theory is helpful at least technically. Even though Lewis’s own formulation is too liberal—in contrast to Kripke’s being too restrictive—to achieve our goals, this dissertation provides a unification of the two frameworks, Kripke’s and Lewis’s. Through a series of both formal and conceptual comparisons of their ontologies and semantic ideas, it is shown that structures called sheaves are needed to unify the ideas and achieve the desiderata (ii) and (iii). In the end, I define a category of sheaves over a neighborhood frame with certain properties, and show that it provides a semantics that naturally unifies neighborhood semantics for propositional modal logic, on the one hand, and semantics for first-order logic on the other. Completeness theorems are proved. 3 Contents Chapter I. Mathematical Introduction 9 I.1. Neighborhood Semantics for Propositional Modal Logic 9 I.1.1. Basic Definition 9 I.1.2. Some Conditions on Neighborhood Frames 11 I.2. Semantics for First-Order Logic 13 I.2.1. Denotational Interpretation 13 I.2.2. Interpretation and Images 15 I.3. Topological Semantics for First-Order Modal Logic 18 I.3.1. Domain of Possible Individuals 18 I.3.2. Interpreting First-Order Logic 21 I.3.3. Sheaves over a Topological Space 23 I.3.4. Topological-Sheaf Semantics for First-Order Modal Logic 25 I.3.5. First-Order Modal Logic FOS4 27 I.3.6. An Example of Interpretation 29 I.4. Neighborhood Semantics for First-Order Modal Logic 31 I.4.1. Why Sheaves are Needed 31 I.4.2. Sheaves over a Neighborhood Frame 35 I.4.3. Neighborhood-Sheaf Semantics for First-Order Modal Logic 38 Chapter II. Philosophical Introduction 41 II.1. Questions that this Dissertation Tries to Answer 41 II.1.1. Epistemic Logic and Topological Semantics 41 Chapter III. Semantics for First-Order Logic Revisited 51 III.1. More General Languages of First-Order Logic 51 III.1.1. Standard Semantics for Classical First-Order Logic 51 III.1.2. The Forgotten Trio 57 5 Draft of November 14, 2010 III.1.3. What If the Language is not Pure 64 III.2. Operational Semantics for First-Order Free Logic 79 III.2.1. Existence and Two Notions of Domain 79 III.2.2. Operational Semantics: A First Step 85 III.2.3. A Bit Categorical Preliminary 92 III.2.4. Autonomy of Domain of Quantification 98 Chapter IV. Kripkean Semantics for Quantified Modal Logic 111 IV.1. Kripke Semantics for Quantified Modal Logic 111 IV.1.1. Kripke’s Ontology and Semantics 111 IV.1.2. Separation of Modal and Classical 118 IV.2. Autonomous Domains of Quantification for the Kripkean Setting 124 IV.2.1. Operational Form of Kripkean Semantics: A First Step 124 IV.2.2. Kripke’s Operations 132 IV.2.3. Autonomy of Kripkean Domains of Quantification 139 IV.2.4. Autonomy of Domains and Converse Barcan Formula 145 IV.3. Operational Form of Kripkean Semantics: A Second Step 151 IV.3.1. Free-Variable-Sensitive Interpretation of Operators 152 IV.3.2. Preservation of Local Determination Generalized 155 IV.3.3. DoQ-Restrictability Generalized 158 Chapter V. Accessibility and Counterparts 169 V.1. David Lewis’s Counterpart Theory 169 V.1.1. Disjoint Ontology of Possible Individuals and the Notion of Counterparts 169 V.1.2. Counterpart Translation of a Modal Language 173 V.2. Counterpart-Theoretic Semantics 182 V.2.1. Semantically Rewriting Lewis’s Semantic Ideas 182 V.2.2. Operational Form of Counterpart-Theoretic Semantics 190 V.2.3. Bundle Formulation of Counterpart Theory 197 Chapter VI. Generalized Topological Semantics for First-Order Modal Logic 205 VI.1. Topological Semantics for First-Order Modal Logic 205 VI.1.1. Upshots from the Previous Chapters 205 6 Draft of November 14, 2010 VI.1.2. Classical Semantics in a Category of Sets over a Set 209 VI.1.3. Topological Spaces over a Space 213 VI.1.4. Sheaves over a Topological Space 216 VI.1.5. Topological-Sheaf Semantics for First-Order Modal Logic 219 VI.2. Neighborhood Semantics for First-Order Modal Logic 221 VI.2.1. Basic Definitions for Neighborhood Frames 221 VI.2.2. Products of Neighborhood Frames 225 VI.2.3. Some Subcategories of Neighborhood Frames 232 VI.2.4. Neighborhood Frames over a Frame 235 VI.2.5. Sheaves over a Neighborhood Frame 241 VI.2.6. Neighborhood-Sheaf Semantics for First-Order Modal Logic 246 VI.3. Completeness 247 VI.3.1. Sufficient Set of Models with All Names 248 VI.3.2. Frames of Models with Logical Topology 252 VI.3.3. Products and Logical Topology 258 VI.3.4. Completing the Completeness Proof 259 Bibliography 265 7 CHAPTER I Mathematical Introduction In this short chapter, I briefly lay out, without proofs, the principal mathematical results of this dissertation. The precise, full exposition of them is found in later chapters, mostly in Chapter ??, along with proofs. I.1. Neighborhood Semantics for Propositional Modal Logic To describe it in mathematical terms, the chief result of this dissertation is to extend neighbor- hood semantics for propositional modal logic to first-order modal logic. In this section, we lay out neighborhood semantics for propositional modal logic to prepare ourselves for the extension. I.1.1. Basic Definition. Let us fix a propositional modal language L, that is, a language ob- tained by adding unary sentential operators □ and ^, called modal operators, to any language of classical propositional logic. Neighborhood semantics can be regarded as a kind of possible-world semantics, in the sense that it interprets L with a structure that consists of • a set X , ?, and • a map ~−⟧ : sent(L) !PX, where sent(L) is the set of sentences of L, among other things. We may call points in X possible worlds, and subsets of X propositions, so that we can read w 2 ~'⟧ as meaning that ' is true at w. In a manner coherent to this reading, we define validity in ~−⟧ (or in a suitable tuple such as (X; ~−⟧)) in the following manner. Note that we take binary sequents as units of validity; so, accordingly, we will consider formulations of logic in which a logic or theory proves binary sequents. • A binary sequent ' ` is valid in ~−⟧ if ~'⟧ ⊆ ~ ⟧. By the validity of a sentence ', we mean the validity of > ` ', where ~>⟧ = X—that is, ' is valid in ~−⟧ if ~'⟧ = X. • An inference (Γ; (' ` ))—deriving a sequent ' ` from premises Γ of sequents—is valid in ~−⟧ if it preserves validity, that is, if either ~'i⟧ * ~ i⟧ for some sequent 'i ` i in Γ or ~'⟧ ⊆ ~ ⟧. 9 Draft of November 14, 2010 Propositional logic /o /o /o /o /o /o /o /o /o /o / X ' /o /o /o /o /o /o /o /o /o /o /o /o /o /o /o / ~'⟧ ⊆ X We can extend ~−⟧ to interpret sentential operators, so that, for each n-ary operator ⊗, we have ~⊗⟧ :(PX)n !PX and then ~⊗⟧(~'1⟧;:::; ~'n⟧) = ~⊗('1;:::;'n)⟧: For the logic to have its non-modal base classical, we set ~:⟧ = X n −; so that ~:'⟧ = X n ~'⟧; ~^⟧ = \; so that ~' ^ ⟧ = ~'⟧ \ ~'⟧; ~_⟧ = [; so that ~' _ ⟧ = ~'⟧ [ ~'⟧; ~!⟧ = !; so that ~' ! ⟧ = ~'⟧ ! ~'⟧:1 What is characteristic of neighborhood semantics is to further equip X with • a map N : X ! PPX, called a neighborhood function. Such a map N is mathematically equivalent to • an operation int : PX !PX, called an interior operation, via the correspondence (1) w 2 int(A) () A 2 N(w) for every A ⊆ X and w 2 X. We assume no constraint at all for N or int, though we will consider a few in Subsection I.1.2 (and some turn out essential for the extension of neighborhood semantics to the first-order modal logic). Any pair (X; N) of the type above is called a neighborhood frame. Over such a structure (X; N), the modal operator □ is interpreted by the interior operation int defined by N. That is, ~□⟧ = int; so that ~□'⟧ = int(~'⟧); which means, by (1), that w 2 ~□'⟧ () ~'⟧ 2 N(w); 1The binary operation ! :(PX)2 !PX is such that A ! B = (X n A) [ B. 10 Draft of November 14, 2010 thus, when □ is read as “necessarily”, N(w) amounts to the family of propositions necessarily true at w. The operator ^ is interpreted by a dual of int, the closure operation cl : PX !PX, such that cl(A) = X n int(X n A); that is, ⟦^⟧ = cl; so that ~^'⟧ = X n int(X n ~'⟧): Hence, with : interpreted classically, that is, with ~:⟧ = X n −, ^ can simply be defined as :□:.
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