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Recent eScholarship items from Working PapersMon, 19 Aug 2019 12:23:35 +0000A note on the consistency operator
https://escholarship.org/uc/item/40j0v0hb
A note on the consistency operatorhttps://escholarship.org/uc/item/40j0v0hbThu, 9 May 2019 00:00:00 +0000Choice-free Stone duality
https://escholarship.org/uc/item/00p6t2v4
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...https://escholarship.org/uc/item/00p6t2v4Fri, 7 Sep 2018 00:00:00 +0000On the Logics with Propositional Quantifiers Extending S5Π
https://escholarship.org/uc/item/1bf3g4fn
Scroggs's theorem on the extensions of S5 is an early landmark in the modern mathematical studies of modal logics. From it, we know that the lattice of normal extensions of S5 is isomorphic to the inverse order of the natural numbers with infinity and that all extensions of S5 are in fact normal. In this paper, we consider extending Scroggs's theorem to modal logics with propositional quantifiers governed by the axioms and rules analogous to the usual ones for ordinary quantifiers. We call them Π-logics. Taking S5Π, the smallest normal Π-logic extending S5, as the natural counterpart to S5 in Scroggs's theorem, we show that all normal Π-logics extending S5Π are complete with respect to their complete simple S5 algebras, that they form a lattice that is isomorphic to the lattice of the open sets of the disjoint union of two copies of the one-point compactification of N, that they have arbitrarily high Turing-degrees, and that there are non-normal Π-logics extending S5Π.https://escholarship.org/uc/item/1bf3g4fnMon, 20 Aug 2018 00:00:00 +0000The Logic of Comparative Cardinality
https://escholarship.org/uc/item/2nn3c35x
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.https://escholarship.org/uc/item/2nn3c35xTue, 7 Aug 2018 00:00:00 +0000Operationalism Meets Modal Logic
https://escholarship.org/uc/item/3dg565gq
<p>Guided by a desire to eliminate language that refers to unobservable structure from mechanics, Ernst Mach proposed a definition of mass in terms of more directly observable data. A great deal of literature surrounds the question of whether this proposed definition accomplishes its stated goal, or even whether it constitutes a definition. In this talk we aim to bring clarity to this debate by using methods from model theory and from modal logic to classify, reconstruct, and evaluate these arguments. In particular, we exhibit a general construction of first-order modal frames for appropriately presented scientific theories with epistemic constraints. These frames allow us to characterize which properties are “modally definable” in the sense of Bressan.</p>https://escholarship.org/uc/item/3dg565gqSat, 23 Jun 2018 00:00:00 +0000Reflection ranks and ordinal analysis
https://escholarship.org/uc/item/1159j6ck
It is well-known that natural axiomatic theories are well-ordered by consistency strength. However, it is possible to construct descending chains of artificial theories with respect to consistency strength. We provide an explanation of this well-orderness phenomenon by studying a coarsening of the consistency strength order, namely, the $\Pi^1_1$ reflection strength order. We prove that there are no descending sequences of $\Pi^1_1$ sound extensions of $\mathsf{ACA}_0$ in this order. Accordingly, we can attach a rank in this order, which we call reflection rank, to any $\Pi^1_1$ sound extension of $\mathsf{ACA}_0$. We prove that for any $\Pi^1_1$ sound theory $T$ extending $\mathsf{ACA}_0^+$, the reflection rank of $T$ equals the proof-theoretic ordinal of $T$. We also prove that the proof-theoretic ordinal of $\alpha$ iterated $\Pi^1_1$ reflection is $\varepsilon_\alpha$. Finally, we use our results to provide straightforward well-foundedness proofs of ordinal notation systems...https://escholarship.org/uc/item/1159j6ckSat, 23 Jun 2018 00:00:00 +0000One Modal Logic to Rule Them All? (Extended Technical Report)
https://escholarship.org/uc/item/07v9360j
One Modal Logic to Rule Them All? (Extended Technical Report)https://escholarship.org/uc/item/07v9360jMon, 26 Mar 2018 00:00:00 +0000Fine's canonicity theorem for some classes of neighborhood frames
https://escholarship.org/uc/item/1w71d5g8
Fine's canonicity theorem for some classes of neighborhood frameshttps://escholarship.org/uc/item/1w71d5g8Thu, 22 Feb 2018 00:00:00 +0000Modal Correspondence Theory for Possibility Semantics
https://escholarship.org/uc/item/7t12914n
New version: <a href="/uc/item/2kf0p9bg">https://escholarship.org/uc/item/2kf0p9bg</a>https://escholarship.org/uc/item/7t12914nWed, 21 Feb 2018 00:00:00 +0000Possibility Frames and Forcing for Modal Logic (February 2018)
https://escholarship.org/uc/item/0tm6b30q
Possibility Frames and Forcing for Modal Logic (February 2018)https://escholarship.org/uc/item/0tm6b30qSat, 17 Feb 2018 00:00:00 +0000A Representation Theorem for Possibility Models
https://escholarship.org/uc/item/881757qn
This paper is about the relation between two kinds of models for propositional modal logic: possibility models in the style of Humberstone and possible world models in the style of Kripke. We show that every countable possibility model <em>M</em> is completed by a Kripke model <em>K</em>, its <em>worldization</em>; every total world of <em>K</em> is the limit of more and more refined possibilities in <em>M</em>, and every possibility in <em>M</em> is realized by some total world of <em>K</em>. In addition, we define a general notion of a <em>possibilization</em> of a Kripke model, which is a possibility model whose possibilities are sets of worlds from the Kripke model. We then characterize the class of possibility models that are isomorphic to the possibilization of some Kripke model. In particular, every possibility model in this class can be represented as a possibilization of one of its worldizations; and every possibility model can be naturally transformed into one in this...https://escholarship.org/uc/item/881757qnSat, 30 Dec 2017 00:00:00 +0000Arrow's Decisive Coalitions
https://escholarship.org/uc/item/5mr296jp
<p>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. We present...https://escholarship.org/uc/item/5mr296jpSun, 8 Oct 2017 00:00:00 +0000On the inevitability of the consistency operator
https://escholarship.org/uc/item/1fm9h9bq
On the inevitability of the consistency operatorhttps://escholarship.org/uc/item/1fm9h9bqSun, 8 Oct 2017 00:00:00 +0000Results in Modal Correspondence Theory for Possibility Semantics
https://escholarship.org/uc/item/2kf0p9bg
Results in Modal Correspondence Theory for Possibility Semanticshttps://escholarship.org/uc/item/2kf0p9bgThu, 5 Oct 2017 00:00:00 +0000Complete Additivity and Modal Incompleteness
https://escholarship.org/uc/item/8pp4d94t
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 <em>completely additive</em> 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....https://escholarship.org/uc/item/8pp4d94tThu, 29 Sep 2016 00:00:00 +0000Possibility Frames and Forcing for Modal Logic (June 2016)
https://escholarship.org/uc/item/9v11r0dq
New version: https://escholarship.org/uc/item/0tm6b30qhttps://escholarship.org/uc/item/9v11r0dqTue, 21 Jun 2016 00:00:00 +0000First-order possibility models and finitary completeness proofs
https://escholarship.org/uc/item/8ht6w3kk
This paper builds on Humberstone's idea of defining models of propositional modal logic where total possible worlds are replaced by partial possibilities. We follow a suggestion of Humberstone by introducing possibility models for quantified modal logic. We show that a simple quantified modal logic is sound and complete for our semantics. Although Holliday showed that for many propositional modal logics, it is possible to give a completeness proof using a canonical model construction where every possibility consists of finitely many formulas, we show that this is impossible to do in the first-order case. However, one can still construct a canonical model where every possibility consists of a computable set of formulas and thus still of finitely much information.https://escholarship.org/uc/item/8ht6w3kkWed, 24 Feb 2016 00:00:00 +0000Scott Ranks of Models of a Theory
https://escholarship.org/uc/item/5q9634jv
<p>The Scott rank of a countable structure is a measure, coming from the proof of Scott's isomorphism theorem, of the complexity of that structure. The Scott spectrum of a theory (by which we mean a sentence of $\mc{L}_{\omega_1 \omega}$) is the set of Scott ranks of countable models of that theory. In $ZFC + PD$ we give a descriptive-set-theoretic classification of the sets of ordinals which are the Scott spectrum of a theory: they are particular $\bfSigma^1_1$ classes of ordinals.</p><p>Our investigation of Scott spectra leads to the resolution (in $ZFC$) of a number of open problems about Scott ranks. We answer a question of Montalb\'an by showing, for each $\alpha < \omega_1$, that there is a $\Pi^{\infi}_2$ theory with no models of Scott rank less than $\alpha$. We also answer a question of Knight and Calvert by showing that there are computable models of high Scott rank which are not computably approximable by models of low Scott rank. Finally, we answer a question of...https://escholarship.org/uc/item/5q9634jvWed, 24 Feb 2016 00:00:00 +0000Possibility Frames and Forcing for Modal Logic
https://escholarship.org/uc/item/5462j5b6
New version: https://escholarship.org/uc/item/0tm6b30qhttps://escholarship.org/uc/item/5462j5b6Fri, 1 Jan 2016 00:00:00 +0000