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We develop a probabilistic semantics for modal logic, which was introduced in recent years by Dana Scott. This semantics is intimately related to an older,.
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A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Philosophy in the Graduate Division of the University of California, Berkeley
Committee in Charge:
Paolo Mancosu (Co-Chair) Barry Stroud (Co-Chair) Christos Papadimitriou
To Dana and Grisha, for turning a philosopher into a mathematician,
and to Barry and Paolo, for turning her back again.
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Acknowledgements
This dissertation had its beginnings in the Colorado mountains. I went there in a break between Summer Session 2009 and the beginning of the fall semester to visit a friend, Darko Sarenac. At the time, I was having serious doubts about fnishing my degree, and was exploring the possibility of dropping out to become a photographer in the more remote parts of Southwestern New Mexico. Darko and I discussed the different things I might photograph, and even, as I remember, made planstotraveltogetherwithmycameratoWyomingandtheSouth. Soon enough, though, we got to talking about logic. On a hike up a mountain as a storm set in, I learned that there was a deep connection between topology and modal logic—indeed, that there was a whole feld called topological modal logic that had been quite active at Stanford and elsewhere in the last several years. In those 48 hours, Darko taught me the basics. By the time he dropped me off at the airport in Denver, we had plans to write a paper together. (This paper forms the frst chapter of the dissertation.) Thank you, Darko. Without you, I would be somewhere in New Mexico. I want to thank, most of all, the people who have worked with me throughout my graduate career at Berkeley. I was incredibly fortunate to have Barry Stroud as an advisor from very early on. Our conversations shaped the way that I think about so many things, and his way of doing philosophy has been a great infuence on me. Although the topic I eventually chose for my dissertation was quite far afeld from Barry’s own interests, he was the frst to encourage it. More than anyone else, Barry was witness to the many ups and downs of my career at Berkeley, and I always felt his staunch support and confdence. I was also very fortunate to have Paolo Mancosu as an advisor and mentor. Paolo was the frst professor to take me on as a Graduate Student Instructor. I learned from him how logic could be taught in a way that was clear, engaging, and philosophically rich. When it came time to my writing a dissertation in modal logic, Paolo was aware of the many professional challenges that lay ahead and despite this, was fully supportive of the work I was doing and the project I had chosen. He provided me with invaluable advice and help at critical moments. In the frst few days of working together, Darko surreptitiously sent an e-mail to Grigori Mints at Stanford, encouraging him to be in contact with me. I still remember seeingGrisha forthe frsttimeaftermany years at atalk byDana Scottin the Berkeley Logic Colloquium. At the talk, Dana introduced a new, probabilistic semantics for modal logic—a semantics about which very little was known at the time. Some days after the talk, Grisha approached me. “Tamar,” he said, “Vai you
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a spatial operator, which picks out the interior of a region of topological space. Tarskiand McKinsey’s work in the 1930’s and 1940’s led to what is now called the topological semantics for modal logic. Their elegant completeness results pre- date Kripke semantics by more than a decade, but in the years after the introduc- tion of the Kripke framework, the topological semantics was largely forgotten. The fexibility of the Kripke framework—the fact that it can be used to model not just S 4 , but many different propositional and predicate modal logics—as well as its intuitive appeal are perhaps jointly responsible for the near-oblivion into which the topological semantics fell. In the last ffteen years or so, however, things have changed. Modal logicians, familiar with the many advances in temporal logics (or modal logics used to describe time, and temporal processes) started asking, ‘What about a modal logic of space?’ Tarski’s work on the topological semantics came to be seen as the foundation stone of a much broader project: using modal logic to describe, make distinctions between, and systematize our reasoning about space and spatial structures. This research program has produced many new and interesting results in recent years: logicians have simplifed and refned Tarski and McKinsey’s original completeness results; extended Tarski’s topological semantics to more complex, multi-modal languages; and proved new results concerning the model theory and complexity of these extensions. In the pages that follow, we take Tarski’s topological semantics as our starting point. This is not to say that we ignore Kripke’s relational semantics—far from it. Interesting relationships between the two will be developed throughout the text. But the primary aim of this work is not, in fact, to develop either Tarski’s topolog- ical semantics, or Kripke semantics. Rather, it is to introduce the reader to a new way of interpreting modal languages—one that can be developed quite naturally, as we’ll see, from Tarski’s topological semantics, but which differs in important ways from any of the well-known semantics for modal logics to date.^2 Those semanticsall share the following feature. In a given modal model (or formal interpretation ofthe modal language), each formula is either true or false. In Kripke semantics, we say that a formula is true in a model if it’s true at every (accessible) possible world in the model. In the topological semantics, we say that a formula is true in a model if it’s true at every point in the relevant topological space. What if instead we inter- preted modal languages probabilistically? What if, in other words, each formula in a given model gotassignednotjusta truth value, buta probability valuebetween0 and 1? The idea for a probabilistic semantics was introduced by Dana Scott in the last several years, in talks given at Stanford and Berkeley. As Scott said, the semantics “provides rich ingredients for building many kinds of structures having (^2) For a probabilistic semantics for classical logic, see K. Popper’s (31) and H. Field’s (12). See
also Keisler’s (16) and (17).
non-standard random elements.” At the time, however, many fundamental ques- tions about the semantics—particularly those relating to completeness—were still unanswered. In the chapters that follow, we answer some of these questions, and show that the probabilistic semantics can be elegantly extended to more complex, multi-modal languages.
In embarking on the work that follows, the question naturally arises: Why defne a new semantics for modal logic in the frst place? Isn’t the standard Kripke seman- tics good enough? There are two ways to respond. On the one hand, we may start out from an interest in existing modal languages (or existing axiomatic modal systems), and be interested in what the different semantics for these languages are. Here, of course, the probabilistic semantics will have quite different features from the stan- dard Kripke semantics and even from the topological semantics for S 4. Formulas, as we noted, acquire not just truth values in probabilistic models, but probability values. Someone interested in the various uses to which probability theory has been put in the more formal areas of philosophy might take interest in this new seman- tics for this reason. But secondly, one might start out from an interest in certain mathematical objects themselves—topological spaces, say, or topological spaces together with Borel measures in the present case. Then one will want to know: to what extent can modal languages describe, make distinctions between, and help us reason about these structures? From this point of view, the fexibility of Kripke semantics—the fact that it can be used to interpret not just S 4 , but many different modal logics—is not essential. What we want to know is what modal logics the mathematical objects we’re interested in give rise to, and what distinctions between such objects can be made within the confnes of different modal languages.^3 As the reader moves forward through the work of the next chapters, she is invited to keep these two perspectives in mind. The new semantics presented here is not meant as a rival for Kripke (or relational) semantics. Rather, the hope is thattheprobabilisticsemanticscantakeitsplacealongside thoseothersemantics,
(^3) J. Van Benthem makes this point in connection with Tarski’s topological semantics:
Some modal logicians see topological models as a means of providing new semantics for existing modal languages, mostly for logic-internal purposes. This can be motivated a bit more profoundly by thinking of topologies as models for information, making this interest close to central logical concerns. But someone primarily interested in Space as such will not worry about the semantics of modal languages. She will rather be interested in spatial structures by themselves,andspatiallogicswillbejudgedby how welltheyanalyze oldstructures,discover new ones, and help in reasoning about them. (40, p. 11)
fact that in classical logic a false proposition implies (in the algebraic sense) any proposition, and a true proposition is implied by any proposition. Insymbols,
¬ P → ( P → Q );
P → ( Q → P )
Under the ordinary meaning of implication, Lewis thought that ‘P implies Q’ means something like, ‘Q can be legitimately inferred from P.’ But one cannot legitimately infer any proposition from a false proposition. The paradoxes of the material conditional highlighted the way in which the material conditional of clas- sical logic failed to capture the ordinary meaning of “implies”—a connective which Lewis thought stood at the foundations of fundamental notions in logic. “Unless ‘implies’ has some ‘proper’ meaning, there is no criterion of validity, no possibil- ity even of arguing the question whether there is one or not,” Lewis claimed. “And yet the question, What is the ‘proper’ meaning of ‘implies’? remains peculiarly diffcult.” (24, p. 325) What system of logic, if not the classical one, could formalize the ordinary meaning of “implies”? The proposition expressed by ‘A implies B’ was, accord- ing to Lewis, equivalent to the proposition expressed by ‘Either not-A or B.’ But Lewis distinguished between what he called an extensional and intensional read- ing of “or.” On the extensional reading, “or” is the truth-functional disjunction of classical logic. This yields the algebraic meaning of “implies” as a material conditional. But on the intensional reading of “or,” Lewis claimed that “at least one of the disjuncts is necessarily true.” Using this intensional reading to under- stand the ordinary meaning of implies, ‘A implies B’ is equivalent to ‘Necessarily not-A or B.” To understand the ordinary “implies,” Lewis was moved to appeal to modal vocabulary—vocabulary that he thought functioned differently from any of the truth-functional connectives of classical logic. Lewis came at modal logics from a syntactic, or axiomatic, point of view. His aim was to identify axioms and rules of inference in a new, modally-enriched language—ones that would be appropriate to what he took to be ordinary impli- cation. In an appendix to their 1932 volume, Symbolic Logic , Lewis and Langford defned fve different axiomatic modal systems, S 1 - S 5. In these systems ‘ 3 ’ is taken as a modal primitive, with the intended interpretation “possibly” or “it is possible that.” The strict conditional—which was meant to formalize ordinary implication—is then defned in terms of this modal primitive as follows:
P ⇒ Q ≡ ¬ 3 ( P & ¬ Q )
In words: ‘P implies Q’ is equivalent to ‘It is not possible that P and not-Q.’ (Al- though Lewis did not himself introduce a separate “necessity” operator, D, it can
be defned in terms of 3 and¬ in the usual way: D≡ φ (^) ¬ (^3) (^) ¬ φ. In words: ‘Nec- essarily P’ is equivalent to ‘It is not possible that not-P.’) The systems, S 1 - S 5 , were the frst modern axiomatic systems of modal logic.
Already at the beginning, there was a range of views about what modalities the symbols ‘D’ and ‘ 3 ’ naturally expressed. While Lewis took them to symbolize necessity and possibility, respectively, Go¨ del saw in the new language a way of talking about provability within a formal system. He interpreted ‘D’ as the senten- tial operator ‘It is provable that...,’ and argued that on this interpretation, S 4 was the correct axiomatic system. McKinsey and Tarski, meanwhile, noticed the deep connection between Lewis’s axioms for S 4 and Kuratowski’s axioms for a topo- logical interior operator. They interpreted ‘D’ spatially, as picking out the interior of a region of topological space (more on this below). Finally, Prior interpreted modal languages temporally, and took ‘D’ and ‘ 3 ’ to symbolize the temporal sen- tential operators, ‘Henceforth... ’ and ‘At some point in the future... ’ (or ‘Until now... ’ and ‘Atsome point in the past... ’). These differing viewpoints struck, in some sense, at the heart of the modal logic program: What was modal logic about? What modalities did itseekto formalize? Although Lewis did not concern himself with the problem of giving a for- mal semantics for modal languages, interest in the subject was quickly growing. Broadly speaking, there were two competing traditions that developed more or less simultaneously: the algebraic tradition, in which modal languages are interpreted in Boolean algebras with operators, and the relational tradition, which culminated in Kripke’s possible world semantics. We focus in this section on the latter, in view of it’s present-day prominence. An early precursor to Kripke’s possible worlds semantics was proposed by Carnap.^5 According to Carnap, “necessity” was to be interpreted as logical truth, or analyticity (truth in virtue of meaning alone). Infuenced by Leibnitz’s analysis of necessity as that which holds in all possible worlds, Carnap introduced the notion of a state description. A state description for a propositional language, L , is a collection of sentences in which for every propositional variable P in L , either P or P ¬ is in the collection, but not both—and nothing else is in the collection. Each state description is a total specifcation of truth for the propositional variables in L. We can think of a state description as picking out some possible world, or possible state of affairs, as described by the language L. The collection of all state descriptions for L is, then, the collection of all possible worlds or states of affair
(^5) See (8) and (9).
table for a given sentence, ‘ φ ,’ is an assignment of truth values to the propositional variables occurring in ‘ φ ’. Again, we can think of these rows as possible worlds in some attenuated sense. In Kripke’s early conception, a model for a formula ‘ φ ’ in the standard propositional modal language is a pairh G, K i , where K is some collection of truth assignments for the propositional variables occurring in ‘ φ ,’ and G is a member of K. (Thus, a model is a partial truth table in which one row is highlighted.) Each truth assignment in K assigns a truth value to every subformula occurring in ‘ φ ’ according to the usual recursive clauses for Boolean connectives, as well as the following rule for the ‘D’-modality:
D ψ is true just in case ψ is true in every member of K
Thus, to say that ‘ φ ’ is true in a model is to say that ‘ φ ’ is true throughout all truth assignments in that model. In this semantics, we say that a formula is valid if it is true in every such model. Notice that under these rules, neither ‘ 3 P ’ nor ‘ 3 ( P P ∨ ¬ )’ is valid! Indeed, if we select only rows of the truth table where ‘ P ’ is false, then in this model, ‘ 3 P (^) ≡ ¬ ¬D P ’ is false. So ‘ 3 P ’ is not valid. More generally, depending on K — or our selection of rows of the truth table—‘ 3 P ’ is true in some models and false in others (and the same for ‘D P ,’or boxed formulas generally). The ability to restrict the collection of possible worlds, K , that matter for the truth of modal formulas is what allows us to do away with the problems faced by Carnap. Kripke showed that the partial truth table semantics is sound and complete for Lewis’s S 5 —the strongestoftheaxiomaticsystemsintroducedinLewisandLangford(1932). But what about weaker propositional logics? Consider, for example, the for- mula ‘ P (^) →D 3 P .’ This formula is not a theorem of S 4 , and so should not come out valid in any (complete) semantics for that logic. But the formula is satisfed in every partial truth table. Indeed, if there is some row of the truth table in which P is true, then 3 P is true in every row, and so D 3 P is true in every row as well. If, on the other hand, there is no row where P is true, then the formula comes out true in every row in virtue of the fact that the antecedent is false. The simple partial truth tables semantics, while suitable for S 5 (where this formula is a theorem), did not suggest a semantics for the full range of propositional modal logics. In order to give a proper semantics for these systems, a full-blown relational structure had to be developed. (Notice that such structure was implicit—in hindsight—in the sim- ple partial truth tables. If we think of each row in a truth table as a possible world, then a partial truth table consists of some collection of possible worlds, each of which is related to every other.) Such structures had been considered in some form byHintikka,Kanger,andPriorbutitwasn’tuntilKripke’sworkintheearly1960’s
w 0
w 1 w 2
w 3
Figure 1: A Kripke frame, F = h w 0 , W, R i, where W = { w 0 , w 1 , w 2 , w 3 } and
R = {h w 0 , w 1 i , h w 0 , w 2 i , h w 2 , w 3 i}.
that a fully fexible and workable version was articulated.^10
1.3 Kripke semantics
By now, several of the ideas that appear in the mature version of Kripke semantics [Kripke, 1963] are familiar. The semantics interprets modal formulas in relational structures (or frames), which consist of some set of possible worlds, together with a binary ‘accessibility’ relation on worlds. Pictorially, we can think of a Kripke frame as a graph consisting of some collection of nodes together with arrows point- ing from some nodes to others. (See Figure 1.) To say that ‘D φ ’ is true at a par- ticular world, w , is to say that ‘ φ ’ is true throughout the worlds that are accessible from w. More informally: It is to say that from the point of view of w, φ is true as far as the eye can see. (Similarly, to say that ‘ 3 φ ’ is true at w is to say that ‘ φ ’ is true at some possible worldaccessible from w .) Formally, a Kripke frame is a triple F = w h 0 , W, R (^) i, where W is a set of possible worlds, w 0 is a member of W (the actual world), and R is a binary relation on worlds. A Kripke model is a pair F, V h , wherei F is a frame, and V : W P× →
{>^ ,^ ⊥}is a^ valuation function , assigning to each world and propositional variable a truth value (the truth value of that proposition in the given world). We extend the valuation function to the set of all formulas in the language in the way one would expect. In words, ‘ φ ∨ ψ ’ is true at a world w just in case ‘ φ ’ is true at w or ‘ ψ ’ is true at w ; ‘ φ ∧ ψ ’ is true at w just in case ‘ φ ’ is true at w and ‘ ψ ’ is true at w ; and ‘¬ φ ’ is true at w just in case φ is not true at w. But what about the modal symbol, ‘D’? The formula ‘D φ ’ is true at w just in case ‘ φ ’ is true at each world w^0 such
(^10) For a very thorough account of this history, see (13).
w 1 w 2
Figure 2: A refuting model in Kripke semantics for the formula ‘ P → D 3 P .’
w 3...^ w 3
¬ P ¬ P
w 2 w 1 w 4
¬ P P ¬ P
Figure 3: A refuting model for the formula ‘ P → 3 P ’ in an arbitrary, non- refexive Kripke frame. The Kripke frame is non-refexive at the world w 1 , which is where we falsify ‘ P → 3 P .’
1.4 Space and topological semantics
Relational structures provide a natural setting for interpreting modal languages, but let us now shift gears. We said above that some two decades before Kripke intro- duced Kripke frames, Tarski noticed a surprising connection between the axioms of Lewis’s S 4 , and the axioms used to describe topological space. His work led to what is now called the topological semantics for modal logic. Here modalities are interpreted not via a binary accessibility relation between worlds, but via the topo- logical structure of space. To understand the semantics, we need to say something about what a topology , or topological space is.
In our ordinary lives, we have a number of well-entrenched views about space and spatial properties. Any two distinct points bear a precise distance relation to one another. A sequence of points that converges, converges to a single point. No two points are infnitely far away. And so on. From a mathematical point of view, these features of space are not universal. When we think about space mathematically, we think in more general terms: there are many different kinds of space, with different spatial properties. For example, not all spaces come with a notion of ‘distance.’
to the class of refexive Kripke frames. Similar arguments show that other axioms correspond to the class of transitive frames, symmetric frames, and frames in which R is an equivalence relation.
In some spaces, it is impossible to say that one point stands three units away from another. Indeed, spaces that do allow for a notion of distance are rather special: we call them metric spaces, or spaces that have a metric (read: distance) function defned on them. What, then, is space in the fully general, mathematical sense that we are after? A space, as we think of it here, is just a collection of points that are glued together in a certain way. There are two ways to understand this. The frst involves the notion of a neigh- borhood , or as mathematicians say, open set.^13 Think of the city of London. That city is made up of a very large number of different points on the earth that lie inside of its municipal boundaries. These points lie at various distances from one another: the Big Ben is (let us suppose) one mile from the Tate Modern, which is itself an- other half mile from the London Eye. But quite apart from specifc distances, there are also neighborhoods in London: Hampstead, Notting Hill, Chelsea, and so on. Some of these neighborhoods overlap; others are disjoint. Imagine throwing out all information about the relative distances between individual points in the city. London, as you view it now, is a collection of points linked together by a system of neighborhoods. The information about neighborhoods furnishes some sense of how points in this space are related to one another spatially. When we speak of space mathematically, in a completely general way, we view it in this way: as a collection ofpoints togetherwith a systemof neighborhoods, oropen sets. These open sets, or neighborhoods, must satisfy certain conditions if they are to defne a topology on the underlying set of points. In words these conditions state that the entire space and the empty set are open; the intersection of any two open sets is open; and fnally, the union of any collection of open sets is open. More
formally, a topological space is a pair, h X, T i, where X is a set (of ‘points’), and T is a collection of subsets of X that satisfes the following conditions:
i ∈ I Si^ ∈^ T^.
We call the sets inT open. Any collection of subsets of X that satisfes these con- ditions defnes a topology on X. Again, space according to this defnition consists of a collection of points together with a system of open sets, or neighborhoods. A second, less familiar way to think about space is as a set of points together withan interior operator. Thisoperatoridentifes,foranysubsetofpoints,whatthe (^13) Although it is standard to use the expression ‘neighborhood of x ’ to mean any set containing an
open set containing x , we use the term ‘neighborhood’ to mean, simply, open set.
