Recent berkeleylogic_fp items

Compatibility, compossibility, and epistemic modality

Sat, 3 Dec 2022 00:00:00 +0000

We give a theory of epistemic modals in the framework of possibility semantics and axiomatize the corresponding logic, arguing that it aptly characterizes the ways in which reasoning with epistemic modals does, and does not, diverge from classical modal logic.

Compatibility and accessibility: lattice representations for semantics of non-classical and modal logics

Sun, 26 Jun 2022 00:00:00 +0000

In this paper, we study three representations of lattices by means of a set with a binary relation of compatibility in the tradition of Ploscica. The standard representations of complete ortholattices and complete perfect Heyting algebras drop out as special cases of the first representation, while the second covers arbitrary complete lattices, as well as complete lattices equipped with a negation we call a protocomplementation. The third topological representation is a variant of that of Craig, Haviar, and Priestley. We then extend each of the three representations to lattices with a multiplicative unary modality; the representing structures, like so-called graph-based frames, add a second relation of accessibility interacting with compatibility. The three representations generalize possibility semantics for classical modal logics to non-classical modal logics, motivated by a recent application of modal orthologic to natural language semantics.

Possibility Semantics

Thu, 19 May 2022 00:00:00 +0000

In traditional semantics for classical logic and its extensions, such as modal logic, propositions are interpreted as subsets of a set, as in discrete duality, or as clopen sets of a Stone space, as in topological duality. A point in such a set can be viewed as a "possible world," with the key property of a world being primeness—a world makes a disjunction true only if it makes one of the disjuncts true—which classically implies totality—for each proposition, a world either makes the proposition true or makes its negation true. This chapter surveys a more general approach to logical semantics, known as possibility semantics, which replaces possible worlds with possibly partial "possibilities." In classical possibility semantics, propositions are interpreted as regular open sets of a poset, as in set-theoretic forcing, or as compact regular open sets of an upper Vietoris space, as in the recent theory of "choice-free Stone duality." The...

Voting Theory in the Lean Theorem Prover

Tue, 2 Nov 2021 00:00:00 +0000

There is a long tradition of fruitful interaction between logic and social choice theory. In recent years, much of this interaction has focused on computer-aided methods such as SAT solving and interactive theorem proving. In this paper, we report on the development of a framework for formalizing voting theory in the Lean theorem prover, which we have applied to verify properties of a recently studied voting method. While previous applications of interactive theorem proving to social choice (using Isabelle/HOL and Mizar) have focused on the verication of impossibility theorems, we aim to cover a variety of results ranging from impossibility theorems to the verication of properties of specic voting methods (e.g., Condorcet consistency, independence of clones, etc.). In order to formalize voting theoretic axioms concerning adding or removing candidates and voters, we work in a variable-election setting whose formalization makes use of dependent types in Lean.

Three roads to complete lattices: orders, compatibility, polarity

Tue, 12 Jan 2021 00:00:00 +0000

This note aims to clarify the relations between three ways of constructing complete lattices that appear in three different areas: (1) using ordered structures, as in set-theoretic forcing, or doubly ordered structures, as in a recent semantics for intuitionistic logic; (2) using compatibility relations, as in semantics for quantum logic based on ortholattices; (3) using Birkhoff’s polarities, as in formal concept analysis.

Logics of Imprecise Comparative Probability

Mon, 14 Dec 2020 00:00:00 +0000

This paper studies connections between two alternatives to the standard probability calculus for representing and reasoning about uncertainty: imprecise probability andcomparative probability. The goal is to identify complete logics for reasoning about uncertainty in a comparative probabilistic language whose semantics is given in terms of imprecise probability. Comparative probability operators are interpreted as quantifying over a set of probability measures. Modal and dynamic operators are added for reasoning about epistemic possibility and updating sets of probability measures.

Algebraic and topological semantics for inquisitive logic via choice-free duality

Fri, 13 Nov 2020 00:00:00 +0000

We introduce new algebraic and topological semantics for inquisitive logic. The algebraic semantics is based on special Heyting algebras, which we call inquisitive algebras, with propositional valuations ranging over only the ¬¬-fixpoints of the algebra. We show how inquisitive algebras arise from Boolean algebras: for a given Boolean algebra B, we define its inquisitive extension H(B) and prove that H(B) is the unique inquisitive algebra having B as its algebra of ¬¬-fixpoints. We also show that inquisitive algebras determine Medvedev’s logic of finite problems. In addition to the algebraic characterization of H(B), we give a topological characterization of H(B) in terms of the recently introduced choice-free duality for Boolean algebras using so-called upper Vietoris spaces (UV-spaces). In particular, while a Boolean algebra B is realized as the Boolean algebra of compact regular open elements of a UV-space dual to B, we show that H(B) is realized as the algebra of compact open...

Another Problem in Possible World Semantics

Fri, 3 Jul 2020 00:00:00 +0000

In "A Problem in Possible-World Semantics," David Kaplan presented a consistent and intelligible modal principle that cannot be validated by any possible world frame (in the terminology of modal logic, any neighborhood frame). However, Kaplan's problem is tempered by the fact that his principle is stated in a language with propositional quantification, so possible world semantics for the basic modal language without propositional quantifiers is not directly affected, and the fact that on careful inspection his principle does not target the world part of possible world semantics---the atomicity of the algebra of propositions---but rather the idea of propositional quantification over a complete Boolean algebra of propositions. By contrast, in this paper we present a simple and intelligible modal principle, without propositional quantifiers, that cannot be validated by any possible world frame precisely because of their assumption of atomicity...

Inquisitive Intuitionistic Logic

Sun, 28 Jun 2020 00:00:00 +0000

Inquisitive logic is a research program seeking to expand the purview of logic beyond declarative sentences to include the logic of questions. To this end, inquisitive propositional logic extends classical propositional logic for declarative sentences with principles governing a new binary connective of inquisitive disjunction, which allows the formation of questions. Recently inquisitive logicians have considered what happens if the logic of declarative sentences is assumed to be intuitionistic rather than classical. In short, what should inquisitive logic be on an intuitionistic base? In this paper, we provide an answer to this question from the perspective of nuclear semantics, an approach to classical and intuitionistic semantics pursued in our previous work. In particular, we show how Beth semantics for intuitionistic logic naturally extends to a semantics for inquisitive intuitionistic logic. In addition, we show how an explicit view of inquisitive...

A note on Murakami’s theorems and incomplete social choice without the Pareto principle

Wed, 15 Jan 2020 00:00:00 +0000

In Arrovian social choice theory assuming the independence of irrelevant alternatives, Murakami (1968) proved two theorems about complete and transitive collective choice rules that satisfy strict non-imposition (citizens’ sovereignty), one being a dichotomy theorem about Paretian or anti-Paretian rules and the other a dictator-or-inverse-dictator impossibility theorem without the Pareto principle. It has been claimed in the later literature that a theorem of Malawski and Zhou (1994) is a generalization of Murakami’s dichotomy theorem and that Wilson’s (1972) impossibility theorem is stronger than Murakami’s impossibility theorem, both by virtue of replacing Murakami’s assumption of strict non-imposition with the assumptions of non-imposition and non-nullness. In this note, we first point out that these claims are incorrect: non-imposition and non-nullness are together equivalent to strict non-imposition for all transitive collective choice rules. We then generalize Murakami’s...

A Semantic Hierarchy for Intuitionistic Logic

Sat, 3 Nov 2018 00:00:00 +0000

Brouwer's views on the foundations of mathematics have inspired the study of intuitionistic logic, including the study of the intuitionistic propositional calculus and its extensions. The theory of these systems has become an independent branch of logic with connections to lattice theory, topology, modal logic and other areas. This paper aims to present a modern account of semantics for intuitionistic propositional systems. The guiding idea is that of a hierarchy of semantics, organized by increasing generality: from the least general Kripke semantics on through Beth semantics, topological semantics, Dragalin semantics, and finally to the most general algebraic semantics. While the Kripke, topological, and algebraic semantics have been extensively studied, the Beth and Dragalin semantics have received less attention. We bring Beth and Dragalin semantics to the fore, relating them to the concept of a nucleus from pointfree topology, which provides a unifying perspective on the...

Complete Additivity and Modal Incompleteness

Wed, 19 Sep 2018 00:00:00 +0000

In this paper, we tell a story about incompleteness in modal logic. The story weaves together a paper of van Benthem [1979], “Syntactic aspects of modal incompleteness theorems,” and a longstanding open question: whether every normal modal logic can be characterized by a class of completely ad- ditive modal algebras, or as we call them, V-BAOs. Using a first-order reformulation of the property of complete additivity, we prove that the modal logic that starred in van Benthem’s paper resolves the open question in the negative. In addition, for the case of bimodal logic, we show that there is a naturally occurring logic that is incomplete with respect to V-BAOs, namely the provability logic GLB [Japaridze, 1988, Boolos, 1993]. We also show that even logics that are unsound with respect to such algebras do not have to be more complex than the classical propositional calculus. On the other hand, we observe that it is undecidable whether a syntactically defined logic is V-complete....

Arrow's Decisive Coalitions

Fri, 7 Sep 2018 00:00:00 +0000

In his classic monograph, Social Choice and Individual Values, Arrow introduced the notion of a decisive coalition of voters as part of his mathematical framework for social choice theory. The subsequent literature on Arrow’s Impossibility Theorem has shown the importance for social choice theory of reasoning about coalitions of voters with different grades of decisiveness. The goal of this paper is a fine-grained analysis of reasoning about decisive coalitions, formalizing how the concept of a decisive coalition gives rise to a social choice theoretic language and logic all of its own. We show that given Arrow’s axioms of the Independence of Irrelevant Alternatives and Universal Domain, rationality postulates for social preference correspond to strong axioms about decisive coalitions. We demonstrate this correspondence with results of a kind familiar in economics—representation theorems—as well as results of a kind coming from mathematical logic—completeness theorems....

Choice-free Stone duality

Fri, 7 Sep 2018 00:00:00 +0000

The standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this paper, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski’s observation that the regular open sets of any topological space form a Boolean algebra. We prove without choice principles that any Boolean algebra arises from a special spectral space X via the compact regular open sets of X; these sets may also be described as those that are both compact open in X and regular open in the upset topology of the specialization order of X, allowing one to apply to an arbitrary Boolean algebra simple reasoning about regular opens of a separative poset. Our representation is therefore a mix of Stone...

The Logic of Comparative Cardinality

Tue, 7 Aug 2018 00:00:00 +0000

This paper investigates the principles that one must add to Boolean algebra to capture reasoning not only about intersection, union, and omplementation of sets, but also about the relative size of sets. We completely axiomatize such reasoning under the Cantorian definition of relative size in terms of injections.

One Modal Logic to Rule Them All?

Tue, 26 Jun 2018 00:00:00 +0000

One Modal Logic to Rule Them All?

Indicative Conditionals and Dynamic Epistemic Logic

Sat, 5 Aug 2017 00:00:00 +0000

Recent ideas about epistemic modals and indicative conditionals in formal semantics have significant overlap with ideas in modal logic and dynamic epistemic logic. The purpose of this paper is to show how greater interaction between formal semantics and dynamic epistemic logic in this area can be of mutual benefit. In one direction, we show how concepts and tools from modal logic and dynamic epistemic logic can be used to give a simple, complete axiomatization of Yalcin's [16] semantic consequence relation for a language with epistemic modals and indicative conditionals. In the other direction, the formal semantics for indicative conditionals due to Kolodny and MacFarlane [9] gives rise to a new dynamic operator that is very natural from the point of view of dynamic epistemic logic, allowing succinct expression of dependence (as in dependence logic) or supervenience statements. We prove decidability for the logic with epistemic modals and Kolodny and MacFarlane's indicative conditional...

Axiomatization in the Meaning Sciences

Fri, 7 Apr 2017 00:00:00 +0000

While much of semantic theorizing is based on intuitions about logical phenomena associated with linguistic constructions—phenomena such as consistency and entailment—it is rare to see axiomatic treatments of linguistic fragments. Given a fragment interpreted in some class of formally specified models, it is often possible to ask for a characterization of the reasoning patterns validated by the class of models. Axiomatizations provide such a characterization, often in a perspicuous and efficient manner. In this paper, we highlight some of the benefits of providing axiomatizations for the purpose of semantic theorizing. We illustrate some of these benefits using three examples from the study of modality.

A Note on Algebraic Semantics for S5 with Propositional Quantifiers

Mon, 13 Mar 2017 00:00:00 +0000

In two of the earliest papers on extending modal logic with propositional quantifiers, R. A. Bull and K. Fine studied a modal logic S5Π extending S5 with axioms and rules for propositional quantification. Surprisingly, there seems to have been no proof in the literature of the completeness of S5Π with respect to its most natural algebraic semantics, with propositional quantifiers interpreted by meets and joins over all elements in a complete Boolean algebra. In this note, we give such a proof. This result raises the question: for which normal modal logics L can one axiomatize the quantified propositional modal logic determined by the complete modal algebras for L?

Preferential Structures for Comparative Probabilistic Reasoning

Sat, 11 Feb 2017 00:00:00 +0000

Qualitative and quantitative approaches to reasoning about uncertainty can lead to different logical systems for formalizing such reasoning, even when the language for expressing uncertainty is the same. In the case of reasoning about relative likelihood, with statements of the form φ ≥ ψ expressing that φ is at least as likely as ψ, a standard qualitative approach using preordered preferential structures yields a dramatically different logical system than a quantitative ap- proach using probability measures. In fact, the standard pref- erential approach validates principles of reasoning that are incorrect from a probabilistic point of view. However, in this paper we show that a natural modification of the preferential approach yields exactly the same logical system as a probabilistic approach—not using single probability measures, but rather sets of probability measures. Thus, the same preferential structures used in the study of non-monotonic logics and belief...

Inferring Probability Comparisons

Fri, 18 Nov 2016 00:00:00 +0000

The problem of inferring probability comparisons between events from an initial set of comparisons arises in several contexts, ranging from decision theory to artificial intelligence to formal semantics. In this paper, we treat the problem as follows: beginning with a binary relation $\succsim$ on events that does not preclude a probabilistic interpretation, in the sense that $\succsim$ has extensions that are probabilistically representable, we characterize the extension $\succsim^+$ of $\succsim$ that is exactly the intersection of all probabilistically representable extensions of $\succsim$. This extension $\succsim^+$ gives us all the additional comparisons that we are entitled to infer from $\succsim$, based on the assumption that there is some probability measure of which $\succsim$ gives us partial qualitative information. We pay special attention to the problem of extending an order on states to an order on events. In addition to the probabilistic interpretation,...

Complete Additivity and Modal Incompleteness

Thu, 29 Sep 2016 00:00:00 +0000

In this paper, we tell a story about incompleteness in modal logic. The story weaves together a paper of van Benthem [1979], “Syntactic aspects of modal incompleteness theorems,” and a longstanding open question: whether every normal modal logic can be characterized by a class of completely additive modal algebras, or as we call them, V-BAOs. Using a first-order reformulation of the property of complete additivity, we prove that the modal logic that starred in van Benthem’s paper resolves the open question in the negative. In addition, for the case of bimodal logic, we show that there is a naturally occurring logic that is incomplete with respect to V-BAOs, namely the provability logic GLB [Japaridze, 1988, Boolos, 1993]. We also show that even logics that are unsound with respect to such algebras do not have to be more complex than the classical propositional calculus. On the other hand, we observe that it is undecidable whether a syntactically defined logic is V-complete....

On the Modal Logic of Subset and Superset: Tense Logic over Medvedev Frames

Sun, 18 Sep 2016 00:00:00 +0000

On the Modal Logic of Subset and Superset: Tense Logic over Medvedev Frames

A Bimodal Perspective on Possibility Semantics

Sat, 6 Aug 2016 00:00:00 +0000

In this paper we develop a bimodal perspective on possibility semantics, a framework allowing partiality of states that provides an alternative modeling for classical propositional and modal logics [Humberstone 1981, Holliday 2015]. In particular, we define a full and faithful translation of the basic modal logic K over possibility models into a bimodal logic of partial functions over partial orders, and we show how to modulate this analysis by varying across logics and model classes that have independent topological motivations. This relates the two realms under comparison both semantically and syntactically at the level of derivations. Moreover, our analysis clarifies the interplay between the complexity of translations and axiomatizations of the corresponding logics: adding axioms to the target bimodal logic simplifies translations, or vice versa, complex translations can simplify frame conditions. We also investigate a transfer of first-order...

Partiality and Adjointness in Modal Logic

Wed, 13 Jul 2016 00:00:00 +0000

Following a proposal of Humberstone, this paper studies a semantics for modal logic based on partial “possibilities” rather than total “worlds.” There are a number of reasons, philosophical and mathematical, to find this alternative semantics attractive. Here we focus on the construction of possibility models with a finitary flavor. Our main completeness result shows that for a number of standard modal logics, we can build a canonical possibility model, wherein every logically consistent formula is satisfied, by simply taking each individual finite formula (modulo equivalence) to be a possibility, rather than each infinite maximally consistent set of formulas as in the usual canonical world models. Constructing these locally finite canonical models involves solving a problem in general modal logic of independent interest, related to the study of adjoint pairs of modal operators: for a given modal logic L, can we find for every formula φ a formula f(φ) such that...

A computability theoretic equivalent to Vaught's conjecture

Wed, 29 Jun 2016 00:00:00 +0000

A computability theoretic equivalent to Vaught's conjecture

The complexity of computable categoricity

Wed, 29 Jun 2016 00:00:00 +0000

The complexity of computable categoricity

Locales, Nuclei, and Dragalin Frames

Sun, 12 Jun 2016 00:00:00 +0000

It is a classic result in lattice theory that a poset is a complete lattice iff it can be realized as fixpoints of a closure operator on a powerset. Dragalin [9,10] observed that a poset is a locale (complete Heyting algebra) iff it can be realized as fixpoints of a nucleus on the locale of upsets of a poset. He also showed how to generate a nucleus on upsets by adding a structure of “paths” to a poset, forming what we call a Dragalin frame. This allowed Dragalin to introduce a semantics for intuitionistic logic that generalizes Beth and Kripke semantics. He proved that every spatial locale (locale of open sets of a topological space) can be realized as fixpoints of the nucleus generated by a Dragalin frame. In this paper, we strengthen Dragalin’s result and prove that every locale—not only spatial locales—can be realized as fixpoints of the nucleus generated by a Dragalin frame. In fact, we prove the stronger result that for every nucleus on the upsets of a poset, there is a...