In this paper we give an example of intertranslatability between an ontology of individuals (nominalism), an ontology of properties (realism), and an ontology of facts (factualism). We demonstrate that these three ontologies are dual to each other, meaning that each ontology can be translated into, and recaptured from, each of the others. The aiin of the enterprise is to raise the possibility that, at least in some settings, there may be no need for considerations of ontological primacy. Whether the world (...) is made up of things, or properties, or facts, may be no more than a matter of how we look at it. (shrink)

One of the most expected properties of a logical system is that it can be algebraizable, in the sense that an algebraic counterpart of the deductive machinery could be found. Since the inception of da Costa's paraconsistent calculi, an algebraic equivalent for such systems have been searched. It is known that these systems are non self-extensional (i.e., they do not satisfy the replacement property). More than this, they are not algebraizable in the sense of Blok-Pigozzi. The same negative results hold (...) for several systems of the hierarchy of paraconsistent logics known as Logics of Formal Inconsistency (LFIs). Because of this, these logics are uniquely characterized by semantics of non-deterministic kind. This paper offers a solution for two open problems in the domain of paraconsistency, in particular connected to algebraization of LFIs, by obtaining several LFIs weaker than C1, each of one is algebraizable in the standard Lindenbaum-Tarski's sense by a suitable variety of Boolean algebras extended with operators. This means that such LFIs satisfy the replacement property. The weakest LFI satisfying replacement presented here is called RmbC, which is obtained from the basic LFI called mbC. Some axiomatic extensions of RmbC are also studied, and in addition a neighborhood semantics is defined for such systems. It is shown that RmbC can be defined within the minimal bimodal non-normal logic E+E defined by the fusion of the non-normal modal logic E with itself. Finally, the framework is extended to first-order languages. RQmbC, the quantified extension of RmbC, is shown to be sound and complete w.r.t. BALFI semantics. (shrink)

I show how classical and quantum physics approach the problem of randomness and probability. Contrary to popular opinions, neither we can prove that classical mechanics is a deterministic theory, nor that quantum mechanics is a nondeterministic one. In other words it is not possible to show that randomness in classical mechanics has a purely epistemic character and that of quantum mechanics an ontic one. Nevertheless, recent developments of quantum theory and increasing experimental possibilities to check its predictions call for returning (...) to the problem of comparing possibilities given by classical and quantum physics to accommodate and prove the existence of a `genuine randomness'. Recent results concerning `amplification of randomness' show that, in certain sense, quantum physics is in fact ‘more random’ that classical and outperforms it in producing a `truly random process'. (shrink)

In this paper we use the framework of generalized probabilistic theories to present two sets of basic assumptions, called axioms, for which we show that they lead to the Hilbert space formulation of quantum mechanics. The key results in this derivation are the co-ordinatization of generalized geometries and a theorem of Solér which characterizes Hilbert spaces among the orthomodular spaces. A generalized Wigner theorem is applied to reduce some of the assumptions of Solér’s theorem to the theory of symmetry in (...) quantum mechanics. Since this reduction is only partial we also point out the remaining open questions. (shrink)

In his essay ‘“Wang’s Paradox”’, Crispin Wright proposes a solution to the Sorites Paradox (in particular, the form of it he calls the ‘Paradox of Sharp Boundaries’) that involves adopting intuitionistic logic when reasoning with vague predicates. He does not give a semantic theory which accounts for the validity of intuitionistic logic (and the invalidity of stronger logics) in that area. The present essay tentatively makes good the deficiency. By applying a theorem of Tarski, it shows that intuitionistic logic is (...) the strongest logic that may be applied, given certain semantic assumptions about vague predicates. The essay ends with an inconclusive discussion of whether those semantic assumptions should be accepted. (shrink)

In this paper we show a duality between two approaches to represent quantum structures abstractly and to model the logic and dynamics therein. One approach puts forward a “quantum dynamic frame” :2267–2282, 2005), a labelled transition system whose transition relations are intended to represent projections and unitaries on a Hilbert space. The other approach considers a “Piron lattice”, which characterizes the algebra of closed linear subspaces of a Hilbert space. We define categories of these two sorts of structures and show (...) a duality between them. This result establishes, on one direction of the duality, that quantum dynamic frames represent quantum structures correctly; on the other direction, it gives rise to a representation of dynamics on a Piron lattice. (shrink)

This paper looks at how the idea of pointless topology itself evolved during its pre-localic phase by analyzing the definitions of the concept of topological space of Menger and Nöbeling. Menger put forward a topology of lumps in order to generalize the definition of the real line. As to Nöbeling, he developed an abstract theory of posets so that a topological space becomes a particular case of topological poset. The analysis emphasizes two points. First, Menger's geometrical perspective was superseded by (...) an algebraic one, a lattice-theoretical one to be precise. Second, Menger's bottom–up approach was replaced by a top–down one. (shrink)

The logic of a physical system consists of the elementary observables of the system. We show that for chaotic systems the logic is not any more the classical Boolean lattice but a kind of fuzzy logic which we characterize for a class of chaotic maps. Among other interesting properties the fuzzy logic of chaos does not allow for infinite combinations of propositions. This fact reflects the instability of dynamics and it is shared also by quantum systems with diagonal singularity. We (...) also generalize the fuzzy implication to a probabilistic implication following the hint of von Neumann. In this way we can evaluate the probability of the validity of the logical inference. (shrink)

Since antiquity well into the beginnings of the 20th century geometry was a central topic for philosophy. Since then, however, most philosophers of science, if they took notice of topology at all, considered it as an abstruse subdiscipline of mathematics lacking philosophical interest. Here it is argued that this neglect of topology by philosophy may be conceived of as the sign of a conceptual sea-change in philosophy of science that expelled geometry, and, more generally, mathematics, from the central position it (...) used to have in philosophy of science and placed logic at center stage in the 20th century philosophy of science. Only in recent decades logic has begun to loose its monopoly and geometry and topology received a new chance to find a place in philosophy of science. (shrink)

In this paper it is shown that Heyting and Co-Heyting mereological systems provide a convenient conceptual framework for spatial reasoning, in which spatial concepts such as connectedness, interior parts, (exterior) contact, and boundary can be defined in a natural and intuitively appealing way. This fact refutes the wide-spread contention that mereology cannot deal with the more advanced aspects of spatial reasoning and therefore has to be enhanced by further non-mereological concepts to overcome its congenital limitations. The allegedly unmereological concept of (...) boundary is treated in detail and shown to be essentially affected by mereological considerations. More precisely, the concept of boundary turns out to be realizable in a variety of different mereologically grounded versions. In particular, every part K of a Heyting algebra H gives rise to a well-behaved K-relative boundary operator. (shrink)

In this paper a solution of Whitehead’s problem is presented: Starting with a purely mereological system of regions a topological space is constructed such that the class of regions is isomorphic to the Boolean lattice of regular open sets of that space. This construction may be considered as a generalized completion in analogy to the well-known Dedekind completion of the rational numbers yielding the real numbers . The argument of the paper relies on the theories of continuous lattices and “pointless” (...) topology.

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A framework for the mind-matter problem in a holistic universe which has no parts is outlined. The conceptual structure of modern quantum theory suggests to use complementary Boolean descriptions as elements for a more comprehensive non-Boolean description of a world without an a priori mind-matter distinction. Such a description in terms of a locally Boolean but globally non-Boolean structure makes allowance for the fact that Boolean descriptions play a privileged role in science. If we accept the insight that there are (...) no ultimate building blocks, the existence of holistic correlations between contextually chosen parts is a natural consequence. The main problem of a genuinely non-Boolean description is to find an appropriate partition of the universe of discourse. If we adopt the idea that all fundamental laws of physics are invariant under time translations, then we can consider a partition of the world into a tenseless and a tensed domain.In the sense of a regulative principle, the material domain is defined as the tenseless domain with its homogeneous time. The tensed domain contains the mental domain with a tensed time characterized by a privileged position, the Now. Since this partition refers to two complementary descriptions which are not given a priori we have to expect correlations between these two domains. In physics it corresponds to Newton 's separation of universal laws of nature and contingent initial conditions Both descriptions have a non-Boolean structure and can be encompassed into a single non-Boolean description. Tensed and tenseless time can be synchronized by holistic correlations. (shrink)

The well known Bell experiment with two actors Alice and Bob is considered. First the simple deduction leading to the CHSH inequality under local realism is reviewed, and some arguments from the literature are recapitulated. Then I take up certain background themes before I enter a discussion of Alice’s analysis of the situation. An important point is that her mind is limited by the fact that her Hilbert space in this context is two-dimensional. General statements about a mind’s limitation during (...) a decision process are derived from recent results on the reconstruction of quantum theory from conceptual variables. These results apply to any decision situation. Let all the data from the Bell experiment be handed over to a new actor Charlie, who performs a data analysis. But his mind is also limited: He has a four-dimensional Hilbert space in the context determined by the experiment. I show that this implies that neither Alice nor Charlie can have the argument leading to the CHSH inequality as a background for making decisions related to the experiment. Charlie may be any data analyst, and he may communicate with any person. It is argued that no rational person can be convinced by the CHSH argument when making empirical decisions on the Bell situation. (shrink)

This paper offers the Boolean many-valued solution to the Sorites Paradox. According to the precisification-based Boolean many-valued theory, from which this solution arises, sentences have not only two truth values, truth (or 1) and falsity (or 0), but many Boolean values between 0 and 1. The Boolean value of a sentence is identified with the set of precisifications in which the sentence is true. Unlike degrees fuzzy logic assigns to sentences, Boolean many values are not linearly but only partially ordered; (...) so there are values that are incomparable. Despite this fact, the sentences in a sorites series can be taken as having values in a linear order, and losing (or gaining) value as we move from sentence to sentence in the series. So there is no sharp borderline between 0 and 1. Assigning Boolean many values instead of the two truth values to sentences for their vagueness values is analogous to assigning propositions instead of the truth values to sentences for their semantic values, as propositions are, from our viewpoint, also Boolean many values. (shrink)

We present a manuscript of Paul Lorenzen that provides a proof of consistency for elementary number theory as an application of the construction of the free countably complete pseudocomplemented semilattice over a preordered set. This manuscript rests in the Oskar-Becker-Nachlass at the Philosophisches Archiv of Universität Konstanz, file OB 5-3b-5. It has probably been written between March and May 1944. We also compare this proof to Gentzen's and Novikov's, and provide a translation of the manuscript.

Quantum set theory and topos quantum theory are two long running projects in the mathematical foundations of quantum mechanics that share a great deal of conceptual and technical affinity. Most pertinently, both approaches attempt to resolve some of the conceptual difficulties surrounding QM by reformulating parts of the theory inside of nonclassical mathematical universes, albeit with very different internal logics. We call such mathematical universes, together with those mathematical and logical structures within them that are pertinent to the physical interpretation, (...) ‘Q-worlds’. Here, we provide a unifying framework that allows us to better understand the relationship between different Q-worlds, and define a general method for transferring concepts and results between TQT and QST, thereby significantly increasing the expressive power of both approaches. Along the way, we develop a novel connection to paraconsistent logic and introduce a new class of structures that have significant implications for recent work on paraconsistent set theory. (shrink)

The use of probability theory is widespread in our daily life as well as in scientific theories. In virtually all cases, calculations can be carried out within the framework of classical probability theory. A special exception is given by quantum mechanics, which gives rise to a new probability theory: quantum probability theory. This dissertation deals with the question of how this formalism can be understood from a philosophical and physical perspective. The dissertation is divided into three parts. In the first (...) part, the formalism of quantum probability theory and its relation to quantum mechanics is presented. The second part considers the possibility of reformulating quantum probability theory to embed it within classical probability. Mathematically, this possibility overlaps with the possibility of the existence of hidden variables theories of quantum mechanics: theories that attempt to solve fundamental problems in quantum mechanics by introducing additional physical properties of systems. The conclusion of this part is that, although classical reformulations of quantum probability theory are formally possible, they offer little insight into the formalism of quantum probability theory itself. The third part regards a more direct investigation of quantum probability theory. A reformulation of quantum probability theory is obtained by constructing a quantum logic on the basis of empirical non-probabilistic predictions of quantum mechanics. In this reformulation quantum probability functions are conditional probability functions on an algebra of propositions about measurements and measurement outcomes. (shrink)

This paper uses a non-distributive system of Boolean fractions (a|b), where a and b are 2-valued propositions or events, to express uncertain conditional propositions and conditional events. These Boolean fractions, 'a if b' or 'a given b', ordered pairs of events, which did not exist for the founders of quantum logic, can better represent uncertain conditional information just as integer fractions can better represent partial distances on a number line. Since the indeterminacy of some pairs of quantum events is due (...) to the mutual inconsistency of their experimental conditions, this algebra of conditionals can express indeterminacy. In fact, this system is able to express the crucial quantum concepts of orthogonality, simultaneous verifiability, compatibility, and the superposition of quantum events, all without resorting to Hilbert space. A conditional (a|b) is said to be "inapplicable" (or "undefined") in those instances or models for which b is false. Otherwise the conditional takes the truth-value of proposition a. Thus the system is technically 3-valued, but the 3rd value has nothing to do with a state of ignorance, nor to some half-truth. People already routinely put statements into three categories: true, false, or inapplicable. As such, this system applies to macroscopic as well as microscopic events. Two conditional propositions turn out to be simultaneously verifiable just in case the truth of one implies the applicability of the other. Furthermore, two conditional propositions (a|b) and (c|d) reside in a common Boolean sub-algebra of the non-distributive system of conditional propositions just in case b = d, their conditions are equivalent. Since all aspects of quantum mechanics can be represented with this near classical logic, there is no need to adopt Hilbert space logic as ordinary logic, just a need perhaps to adopt propositional fractions to do logic, just as we long ago adopted integer fractions to do arithmetic. The algebra of Boolean fractions is a natural, near-Boolean extension of Boolean algebra adequate to express quantum logic. While this paper explains one group of quantum anomalies, it nevertheless leaves no less mysterious the 'influence-at-a-distance', quantum entanglement phenomena. A quantum realist must still embrace non-local influences to hold that "hidden variables" are the measured properties of particles. But that seems easier than imaging wave-particle duality and instant collapse, as offered by proponents of the standard interpretation of quantum mechanics. (shrink)

We show that for finite n⩾3n⩾3, every first-order axiomatisation of the varieties of representable n-dimensional cylindric algebras, diagonal-free cylindric algebras, polyadic algebras, and polyadic equality algebras contains an infinite number of non-canonical formulas. We also show that the class of structures for each of these varieties is non-elementary. The proofs employ algebras derived from random graphs.

We formulate a restricted version of the Tukey-Teichmüller Theorem that we denote by (rTT). We then prove that (rTT) and (BPI) are equivalent in ZF and that (rTT) applies rather naturally to several equivalent forms of (BPI): Alexander Subbase Theorem, Stone Representation Theorem, Model Existence and Compactness Theorems for propositional and first-order logic. We also give two variations of (rTT) that we denote by (rTT)+ and (rTT)++; each is equivalent to (rTT) in ZF. The variation (rTT)++ applies rather naturally to (...) various Selection Lemmas due to Cowen, Engeler, and Rado. (shrink)

We review the fact that an MV-algebra is the same thing as a lattice-ordered effect algebra in which disjoint elements are orthogonal. An HMV-algebra is an MV-effect algebra that is also a Heyting algebra and in which the Heyting center and the effect-algebra center coincide. We show that every effect algebra with the generalized comparability property is an HMV-algebra. We prove that, for an MV-effect algebra E, the following conditions are mutually equivalent: (i) E is HMV, (ii) E has a (...) center valued pseudocomplementation, (iii) E admits a central cover mapping γ such that, for all p, q∈E, p∧q=0⇒γ(p)∧q=0. (shrink)

Using the mathematical notion of an entity to represent states in quantum and classical mechanics, we show that, in a strict sense, proper superpositions are possible in classical mechanics.

The two concepts of probability used in physics are analyzed from the formal and the material points of view. The standard theory corresponds toprob 1 (probability of the coexistence of two properties). A general logicomathematical theory ofprob 2 (probability of transition between states) is presented in axiomatic form. The underlying state algebra is neither Boolean nor Birkhoff-von Neumann but partial Boolean. In the Boolean subalgebras,prob 1 theory holds. The theory presented contains the logicomathematical foundations of quantum mechanics and, as degenerate (...) cases, the theories of stochastic games and of Markov chains. (shrink)

We offer a proof of the duality theorem for finitely presented MV-algebras and rational polyhedra, a folklore and yet fundamental result. Our approach develops first a general dual adjunction between MV-algebras and subspaces of Tychonoff cubes, endowed with the transformations that are definable in the language of MV-algebras. We then show that this dual adjunction restricts to a duality between semisimple MV-algebras and closed subspaces of Tychonoff cubes. The duality theorem for finitely presented objects is obtained by a further specialisation. (...) Our treatment is aimed at showing exactly which parts of the basic theory of MV-algebras are needed in order to establish these results, with an eye towards future generalisations. (shrink)

It is introduced a new algebra$$(A, \otimes, \oplus, *, \rightharpoonup, 0, 1)$$(A,⊗,⊕,∗,⇀,0,1)called$$L_PG$$LPG-algebra if$$(A, \otimes, \oplus, *, 0, 1)$$(A,⊗,⊕,∗,0,1)is$$L_P$$LP-algebra (i.e. an algebra from the variety generated by perfectMV-algebras) and$$(A,\rightharpoonup, 0, 1)$$(A,⇀,0,1)is a Gödel algebra (i.e. Heyting algebra satisfying the identity$$(x \rightharpoonup y ) \vee (y \rightharpoonup x ) =1)$$(x⇀y)∨(y⇀x)=1). The lattice of congruences of an$$L_PG$$LPG-algebra$$(A, \otimes, \oplus, *, \rightharpoonup, 0, 1)$$(A,⊗,⊕,∗,⇀,0,1)is isomorphic to the lattice of Skolem filters (i.e. special type ofMV-filters) of theMV-algebra$$(A, \otimes, \oplus, *, 0, 1)$$(A,⊗,⊕,∗,0,1). The variety$$\mathbf {L_PG}$$LPGof$$L_PG$$LPG-algebras (...) is generated by the algebras$$(C, \otimes, \oplus, *, \rightharpoonup, 0, 1)$$(C,⊗,⊕,∗,⇀,0,1)where$$(C, \otimes, \oplus, *, 0, 1)$$(C,⊗,⊕,∗,0,1)is ChangMV-algebra. Any$$L_PG$$LPG-algebra is bi-Heyting algebra. The set of theorems of the logic$$L_PG$$LPGis recursively enumerable. Moreover, we describe finitely generated free$$L_PG$$LPG-algebras. (shrink)

This paper introduces the category of b-frames as a new tool in the study of complete lattices. B-frames can be seen as a generalization of posets, which play an important role in the representation theory of Heyting algebras, but also in the study of complete Boolean algebras in forcing. This paper combines ideas from the two traditions in order to generalize some techniques and results to the wider context of complete lattices. In particular, we lift a representation theorem of Allwein (...) and MacCaull to a duality between complete lattices and b-frames, and we derive alternative characterizations of several classes of complete lattices from this duality. This framework is then used to obtain new results in the theory of complete Heyting algebras and the semantics of intuitionistic propositional logic. (shrink)

The aim of this text is to reply to criticisms of the logics of evidence and truth and the epistemic approach to paraconsistency advanced by Barrio [2018], and Lo Guercio and Szmuc [2018]. We also clarify the notion of evidence that underlies the intended interpretation of these logics and is a central point of Barrio’s and Lo Guercio & Szmuc’s criticisms.

There are many ontologies of the world or of specific phenomena such as time, matter, space, and quantum mechanics1. However, ontologies of information are rather rare. One of the reasons behind this is that information is most frequently associated with communication and computing, and not with ‘the furniture of the world’. But what would be the nature of an ontology of information? For it to be of significant import it should be amenable to formalization in a logico-grammatical formalism. A candidate (...) ontology satisfying such a requirement can be found in some of the ideas of K. Turek, presented in this paper. Turek outlines the ontology of information conceived of as a part of nature, and provides the ‘missing link’ to the Z axiomatic set theory, offering a proposal for developing a formal ontology of information both in its philosophical and logicogrammatical representations. (shrink)

The problem of how to rationally aggregate probability measures occurs in particular when a group of agents, each holding probabilistic beliefs, needs to rationalise a collective decision on the basis of a single ‘aggregate belief system’ and when an individual whose belief system is compatible with several probability measures wishes to evaluate her options on the basis of a single aggregate prior via classical expected utility theory. We investigate this problem by first recalling some negative results from preference and judgment (...) aggregation theory which show that the aggregate of several probability measures should not be conceived as the probability measure induced by the aggregate of the corresponding expected utility preferences. We describe how McConway’s :410–414, 1981) theory of probabilistic opinion pooling can be generalised to cover the case of the aggregation of infinite profiles of finitely additive probability measures, too; we prove the existence of aggregation functionals satisfying responsiveness axioms à la McConway plus additional desiderata even for infinite electorates. On the basis of the theory of propositional-attitude aggregation, we argue that this is the most natural aggregation theory for probability measures. Our aggregation functionals for the case of infinite electorates are neither oligarchic nor integral-based and satisfy a weak anonymity condition. The delicate set-theoretic status of integral-based aggregation functionals for infinite electorates is discussed. (shrink)

The deductive system in Boole's Laws of Thought (LT) involves both an algebra, which we call proto-Boolean, and a "general method in Logic" making use of that algebra. Our object is to elucidate these two components of Boole's system, to prove his principal results, and to draw some conclusions not explicit in LT. We also discuss some examples of incoherence in LT; these mask the genius of Boole's design and account for much of the puzzled and disparaging commentary LT has (...) received. Our evaluation of Boole's logical system does not differ substantially from that advanced in Hailperin's exhaustive study, Boole's Logic and Probability. Unlike the latter work, however, we make direct use of the polynomials native to LT rather than appealing to formalisms such as multisets and rings. (shrink)

. Relational semantics for nonclassical logics lead straightforwardly to topological representation theorems of their algebras. Ortholattices and De Morgan lattices are reducts of the algebras of various nonclassical logics. We define three new classes of topological spaces so that the lattice categories and the corresponding categories of topological spaces turn out to be dually isomorphic. A key feature of all these topological spaces is that they are ordered relational or ordered product topologies.

The principle of set theory known as the Axiom of Choice has been hailed as “probably the most interesting and, in spite of its late appearance, the most discussed axiom of mathematics, second only to Euclid's axiom of parallels which was introduced more than two thousand years ago” (Fraenkel, Bar-Hillel & Levy 1973, §II.4). The fulsomeness of this description might lead those unfamiliar with the axiom to expect it to be as startling as, say, the Principle of the Constancy of (...) the Velocity of Light or the Heisenberg Uncertainty Principle. But in fact the Axiom of Choice as it is usually stated appears humdrum, even self-evident. For it amounts to nothing more than the claim that, given any collection of mutually disjoint nonempty sets, it is possible to assemble a new set — a transversal or choice set — containing exactly one element from each member of the given collection. Nevertheless, this seemingly innocuous principle has far-reaching mathematical consequences — many indispensable, some startling — and has come to figure prominently in discussions on the foundations of mathematics. It (or its equivalents) have been employed in countless mathematical papers, and a number of monographs have been exclusively devoted to it. (shrink)

Gelfand quantales are complete unital quantales with an involution, *, satisfying the property that for any element a, if a b a for all b, then a a* a = a. A Hilbert-style axiom system is given for a propositional logic, called Gelfand Logic, which is sound and complete with respect to Gelfand quantales. A Kripke semantics is presented for which the soundness and completeness of Gelfand logic is shown. The completeness theorem relies on a Stone style representation theorem for (...) complete lattices. A Rasiowa/Sikorski style semantic tableau system is also presented with the property that if all branches of a tableau are closed, then the formula in question is a theorem of Gelfand Logic. An open branch in a completed tableaux guarantees the existence of an Kripke model in which the formula is not valid; hence it is not a theorem of Gelfand Logic. (shrink)

Both early analytic philosophy and the branch of mathematics now known as topology were gestated and born in the early part of the 20th century. It is not well recognized that there was early interaction between the communities practicing and developing these fields. We trace the history of how topological ideas entered into analytic philosophy through two migrations, an earlier one conceiving of topology geometrically and a later one conceiving of topology algebraically. This allows us to reassess the influence and (...) significance of topological methods for philosophy, including the possible fruitfulness of a third conception of topology as a structure determining similarity. (shrink)

I argue that the construction of representation theorems is a powerful tool for creating novel objects and theories in mathematics, as the construction of a new representation introduces new pieces of information in a very specific way that enables a solution for a problem and a proof of a new theorem. In more detail I show how the work behind the proof of a representation theorem transforms a mathematical problem in a way that makes it tractable and introduces information into (...) it that it did not contain at the beginning of the process. (shrink)

Ordinary Language Philosophy has largely fallen out of favour, and with it the belief in the primary importance of analyses of ordinary language for philosophical purposes. Still, in their various endeavours, philosophers not only from analytic but also from other backgrounds refer to the use and meaning of terms of interest in ordinary parlance. In doing so, they most commonly appeal to their own linguistic intuitions. Often, the appeal to individual intuitions is supplemented by reference to dictionaries. In recent times, (...) Internet search engine queries for expressions of interest have become quite popular. Apparently, philosophers attempt to surpass the limits of their own linguistic intuitions by appealing to experts or to factual uses of language. I argue that this attempt is commendable but that its execution is wanting. Instead of appealing to dictionaries or Internet queries, philosophers should employ computer-based linguistic corpora in order to confirm or falsify hypotheses about the factual use of language. This approach also has some advantages over methods employed by experimental philosophers. If the importance of ordinary language is stressed, the use of linguistic corpora is hardly avoidable. (shrink)

This paper illustrates how Priestley duality can be used in the transfer of an optimal natural duality from a minimal generating algebra for a quasi-variety to other generating algebras. Detailed calculations are given for the quasi-variety of Kleene algebras and the quasi-varieties n of pseudocomplemented distributive lattices (n 1).

We extend the relational algebra of Chin and Tarski so that it is multisorted or, as we prefer, typed. Each type supports a local Boolean algebra outfitted with a converse operator. From Lyndon, we know that relation algebras cannot be represented as proper relation algebras where a proper relation algebra has binary relations as elements and the algebra is singly-typed. Here, the intensional conjunction, which was to represent relational composition in Chin and Tarski, spans three different local algebras, thus the (...) term distributed in the title. Since we do not rely on proper relation algebras, we are free to re-express the algebras as typed. In doing so, we allow many different intensional conjunction operators. We construct a typed logic over these algebras, also known as heterogeneous algebras of Birkhoff and Lipson. The logic can be seen as a form of relevance logic with a classical negation connective where the Routley-Meyer star operator is reified as a converse connective in the logic. Relevance logic itself is not typed but our work shows how it can be made so. Some of the properties of classical relevance logic are weakened from Routley-Meyer’s version which is too strong for a logic over relation algebras. (shrink)

Stone representation theorems are a central ingredient in the metatheory of philosophical logics and are used to establish modal embedding results in a general but indirect and non-constructive way. Their use in logical embeddings will be reviewed and it will be shown how they can be circumvented in favour of direct and constructive arguments through the methods of analytic proof theory, and how the intensional part of the representation results can be recovered from the syntactic proof of those embeddings. Analytic (...) methods will also be used to establish the embedding of subintuitionistic logics into the corresponding modal logics. Finally, proof-theoretic embeddings will be interpreted as a reduction of classes of word problems. (shrink)

This paper presents the algebraic and Kripke modelsoundness and completeness ofa logic over Boolean monoids. An additional axiom added to thelogic will cause the resulting monoid models to be representable as monoidsof relations. A star operator, interpreted as reflexive, transitiveclosure, is conservatively added to the logic. The star operator isa relative modal operator, i.e., one that is defined in terms ofanother modal operator. A further example, relative possibility,of this type of operator is given. A separate axiom,antilogism, added to the logic (...) causes the Kripke models to support acollection of abstract topological uniformities which become concretewhen the Kripke models are dual to monoids of relations. The machineryfor the star operator is shownto be a recasting of Scott-Montague neighborhood models. An interpretationof the Kripke frames and properties thereof is presented in terms ofcertain CMOS transister networks and some circuit transformation equivalences.The worlds of the Kripke frame are wires and the Kripke relation is a specializedCMOS pass transistor network. (shrink)

We propose a model of an agent’s probability and utility that is a compromise between Savage (The foundations of statistics, Wiley, 1954) and Jeffrey (The Logic of Decision, McGraw Hill, 1965). In Savage’s model the probability–utility pair is associated with preferences over acts which are assignments of consequences to states. The probability is defined on the state space, and the utility function on consequences. Jeffrey’s model has no consequences, and both probability and utility are defined on the same set of (...) propositions. The probability–utility pair is associated with a desirability relation on propositions. Like Savage we assume a set of consequences and a state space. However, we assume that states are comprehensive, that is, each state describes a consequence, as in Aumann (Econometrica 55:1–18, 1987). Like Jeffrey, we assume that the agent has a preference relation, which we call desirability, over events, which by definition involves uncertainty about consequences. For a given probability and utility of consequences, the desirability relation is presented by conditional expected utility, given an event. We axiomatically characterize desirability relations that are represented by a probability–utility pair. We characterize the family of all the probability–utility pairs that represent a given desirability relation. (shrink)