The basic purpose of this essay, the first of an intended pair, is to interpret standard von Neumann quantum theory in a framework of iterated measure algebraic truth for mathematical (and thus mathematical-physical) assertions — a framework, that is, in which the truth-values for such assertions are elements of iterated boolean measure-algebras (cf. Sections 2.2.9, 5.2.1–5.2.6 and 5.3 below).The essay itself employs constructions of Takeuti's boolean-valued analysis (whose origins lay in work of Scott, Solovay, Krauss and others) to provide a (...) metamathematical interpretation of ideas sometimes considered disparate, heuristic, or simply ill-defined: the collapse of the wave function, for example; Everett's many worlds'-construal of quantum measurement; and a natural product space of contextual (nonlocal) hidden variables. (shrink)

This paper focuses on the relation between logic and ontology. In particular, it demonstrates how classical logical theory can clarify the ontological part of Ludwig Wittgenstein’s Tractatus Logico-Philosophicus. To this end, the work examines the adequacy of a formal system that was devised by the Polish logician, mathematician and philosopher Roman Suszko (1919–1979) as a model for the Tractatus. Following a brief explanation of the Tractarian ontology, the main ideas of Suszko’s system and its philosophical significance will be considered. The (...) latter will be illustrated in the context of central Tractarian concepts. Finally, two implications for a better understanding of the Tractarian ontology will be pointed out. (shrink)

We analyze Zorn's Lemma and some of its consequences for Boolean algebras in a constructive setting. We show that Zorn's Lemma is persistent in the sense that, if it holds in the underlying set theory, in a properly stated form it continues to hold in all intuitionistic type theories of a certain natural kind. (Observe that the axiom of choice cannot be persistent in this sense since it implies the law of excluded middle.) We also establish the persistence of some (...) familiar results in the theory of (complete) Boolean algebras--notably, the proposition that every complete Boolean algebra is an absolute subretract. This (almost) resolves a question of Banaschewski and Bhutani as to whether the Sikorski extension theorem for Boolean algebras is persistent. (shrink)

In this paper we construct the ortholattices arising in quantum logic starting from the phenomenologically plausible idea of a collection of ensembles subject to passing or failing various “tests.” A collection of ensembles forms a certain kind of preordered set with extra structure called anorthospace; we show that complete ortholattices arise as canonical completions of orthospaces in much the same way as arbitrary complete lattices arise as canonical completions of partially ordered sets. We also show that the canonical completion of (...) an orthospace of ensembles is naturally identifiable as the complete lattice of properties of the ensembles, thereby revealing exactlywhy ortholattices arise in the analysis of “tests” or experimental propositions. Finally, we axiomatize the hitherto implicit concept of “test” and show how they may be correlated with properties of ensembles. (shrink)

Contrary to what is stated in Lemma 7.1 of [8], it is shown that some Boolean algebras of finitary logic admit finitely additive probabilities that are not σ-additive. Consequences of Lemma 7.1 are reconsidered. The concept of a C-σ-additive probability on B (where B and C are Boolean algebras, and $\mathscr{B} \subseteq \mathscr{C}$ ) is introduced, and a generalization of Hahn's extension theorem is proved. This and other results are employed to show that every S̄(L)-σ-additive probability on s̄(L) can be (...) extended (uniquely, under some conditions) to a σ-additive probability on S̄(L), where L belongs to a quite extensive family of first order languages, and S̄(L) and s̄(L) are, respectively, the Boolean algebras of sentences and quantifier free sentences of L. (shrink)

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 (i.e., the principle also cannot be validated by any atomic Boolean algebra expansion). It follows from a theorem of David Lewis that our logic is as simple as possible in terms of modal nesting depth (two). We prove the consistency of the logic using a generalization of possible world semantics known as possibility semantics. We also prove the completeness of the logic (and two other relevant logics) with respect to possibility semantics. Finally, we observe that the logic we identify naturally arises in the study of Peano Arithmetic. (shrink)

This book presents an English translation of a classic Russian text on duality theory for Heyting algebras. Written by Georgian mathematician Leo Esakia, the text proved popular among Russian-speaking logicians. This translation helps make the ideas accessible to a wider audience and pays tribute to an influential mind in mathematical logic. The book discusses the theory of Heyting algebras and closure algebras, as well as the corresponding intuitionistic and modal logics. The author introduces the key notion of a hybrid that (...) “crossbreeds” topology and order, resulting in the structures now known as Esakia spaces. The main theorems include a duality between the categories of closure algebras and of hybrids, and a duality between the categories of Heyting algebras and of so-called strict hybrids. Esakia’s book was originally published in 1985. It was the first of a planned two-volume monograph on Heyting algebras. But after the collapse of the Soviet Union, the publishing house closed and the project died with it. Fortunately, this important work now lives on in this accessible translation. The Appendix of the book discusses the planned contents of the lost second volume. (shrink)

A collection of papers presented at the First International Summer Institute in Cognitive Science, University at Buffalo, July 1994, including the following papers: ** Topological Foundations of Cognitive Science, Barry Smith ** The Bounds of Axiomatisation, Graham White ** Rethinking Boundaries, Wojciech Zelaniec ** Sheaf Mereology and Space Cognition, Jean Petitot ** A Mereotopological Definition of 'Point', Carola Eschenbach ** Discreteness, Finiteness, and the Structure of Topological Spaces, Christopher Habel ** Mass Reference and the Geometry of Solids, Almerindo E. Ojeda (...) ** Defining a 'Doughnut' Made Difficult, N .M. Gotts ** A Theory of Spatial Regions with Indeterminate Boundaries, A.G. Cohn and N.M. Gotts ** Mereotopological Construction of Time from Events, Fabio Pianesi and Achille C. Varzi ** Computational Mereology: A Study of Part-of Relations for Multi-media Indexing, Wlodek Zadrozny and Michelle Kim. (shrink)

The lattices of the title generalize the concept of a De Morgan lattice. A representation in terms of ordered topological spaces is described. This topological duality is applied to describe homomorphisms, congruences, and subdirectly irreducible and free lattices in the category. In addition, certain equational subclasses are described in detail.

The Uniformly Reflexive Structure was introduced by E. G. Wagner who showed that the theory of such structures generalized much of recursive function theory. In this paper Uniformly Reflexive Structures are constructed as factor algebras of Free nonassociative algebras. Wagner's question about the existence of a model with no computable splinter ("successor set") is answered in the affirmative by the construction of a model whose only computable sets are the finite sets and their complements. Finally, for each countable Boolean algebra (...) R of subsets of a countable set which contains the finite subsets, a model is constructed with R as its family of computable sets. (shrink)

The question of how the probabilistic opinions of different individuals should be aggregated to form a group opinion is controversial. But one assumption seems to be pretty much common ground: for a group of Bayesians, the representation of group opinion should itself be a unique probability distribution, 410–414, [45]; Bordley Management Science, 28, 1137–1148, [5]; Genest et al. The Annals of Statistics, 487–501, [21]; Genest and Zidek Statistical Science, 114–135, [23]; Mongin Journal of Economic Theory, 66, 313–351, [46]; Clemen and (...) Winkler Risk Analysis, 19, 187–203, [7]; Dietrich and List [14]; Herzberg Theory and Decision, 1–19, [28]). We argue that this assumption is not always in order. We show how to extend the canonical mathematical framework for pooling to cover pooling with imprecise probabilities by employing set-valued pooling functions and generalizing common pooling axioms accordingly. As a proof of concept, we then show that one IP construction satisfies a number of central pooling axioms that are not jointly satisfied by any of the standard pooling recipes on pain of triviality. Following Levi, 3–11, [39]), we also argue that IP models admit of a much better philosophical motivation as a model of rational consensus. (shrink)

We say that a ring admits elimination of quantifiers, if in the language of rings, {0, 1, +, ·}, the complete theory of R admits elimination of quantifiers. Theorem 1. Let D be a division ring. Then D admits elimination of quantifiers if and only if D is an algebraically closed or finite field. A ring is prime if it satisfies the sentence: ∀ x ∀ y ∃ z (x = 0 ∨ y = 0 ∨ xzy ≠ 0). Theorem (...) 2. If R is a prime ring with an infinite center and R admits elimination of quantifiers, then R is an algebraically closed field. Let A be the class of finite fields. Let B be the class of 2 × 2 matrix rings over a field with a prime number of elements. Let C be the class of rings of the form $GF(p^n) \bigoplus GF(p^k)$ such that either n = k or g.c.d. (n, k) = 1. Let D be the set of ordered pairs (f, Q) where Q is a finite set of primes and f: Q → A ∪ B ∪ C such that the characteristic of the ring f(q) is q. Finally, let E be the class of rings of the form $\bigoplus_{q \in Q}f(q)$ for some (f, Q) in D. Theorem 3. Let R be a finite ring without nonzero trivial ideals. Then R admits elimination of quantifiers if and only if R belongs to E. Theorem 4. Let R be a ring with the descending chain condition of left ideals and without nonzero trivial ideals. Then R admits elimination of quantifiers if and only if R is an algebraically closed field or R belongs to E. In contrast to Theorems 2 and 4, we have Theorem 5. If R is an atomless p-ring, then R is finite, commutative, has no nonzero trivial ideals and admits elimination of quantifiers, but is not prime and does not have the descending chain condition. We also generalize Theorems 1, 2 and 4 to alternative rings. (shrink)

A topological description of space is given, based on the relation of connection among regions and the property of being limited. A minimal set of 10 constraints is shown to permit definitions of points and of open and closed sets of points and to be characteristic of locally compact T2 spaces. The effect of adding further constraints is investigated, especially those that characterise continua. Finally, the properties of mappings in region-based topology are studied. Not all such mappings correspond to point (...) functions and those that do correspond to continuous functions. (shrink)

The main result established in this paper is the following: If the base normed spaceV of completely additive weights is a norm-determining subspace of the space of finitely additive weights V acting on the order unit space spanning the operational logic, thenV has the ε-Jordan-Hahn property iff V has the approximate Jordan-Hahn property. Several examples illustrating the theory are given.

From early work of N. Goodman to recent approaches by H. Field and D. Lewis, there have been attempts to combine 2nd order languages with calculi of individuals. This paper is a contribution, containing basic definitions and distinctions and some metatheorems, to the development of a general metatheory of such theories.

A bounded ultrasheaf is a nonstandard universe constructed from a superstructure in a Boolean valued model of set theory. We consider the bounded elementary embeddings between bounded ultrasheaves. Then the standardization principle is true if and only if the ultrafilters are comparable by the Rudin-Frolik order. The base concept is that the bounded elementary embeddings correspond to the complete Boolean homomorphisms. We represent this by the Rudin-Keisler order of ultrafilters of Boolean algebras.

Our results concern the existence of a countable extension U of the countable atomless Boolean algebra B such that U is a "nonconstructive" extension of B. It is known that for any fixed admissible indexing φ of B there is a countable nonconstructive extension U of B (relative to φ). The main theorem here shows that there exists an extension U of B such that for any admissible indexing φ of B, U is nonconstructive (relative to φ). Thus, in this (...) sense U is a countable totally nonconstructive extension of B. (shrink)

Employing the theory of Birkhoff polarities as a model of model theory yields an inductively defined dual structure which is a formalization of semantics and which allows for simple proofs of some new results for model theory.

This paper is a sequel to the joint publication of Scott and Krauss in which the first aspects of a mathematical theory are developed which might be called "First Order Probability Logic". No attempt will be made to present this additional material in a self-contained form. We will use the same notation and terminology as introduced and explained in Scott and Krauss, and we will frequently refer to the theorems stated and proved in the preceding paper. The main objective of (...) this study is to show that the probability of symmetric probability systems may be represented as a "weighted average" of what might be called "product probabilities". We then discuss some applications of our results to Carnap's "Principle of instantial relevance", which plays an important role in his system of inductive logic. (shrink)

In this note, we outline a definition of propositional manifold and logical cohomology. An application is also considered for mathematics: two Boole algebras of mathematical propositions are non equivalent if their two cohomologies are not isomorphic.

We develop a series of small infinitary epistemic logics to study deductive inference involving intra-/interpersonal beliefs/knowledge such as common knowledge, common beliefs, and infinite regress of beliefs. Specifically, propositional epistemic logics GL are presented for ordinal α up to a given αo so that GL is finitary KDn with n agents and GL allows conjunctions of certain countably infinite formulae. GL is small in that the language is countable and can be constructive. The set of formulae Lα is increasing up (...) to α = ω but stops at ω We present Kripke-completeness for GL for each α ≤ ω, which is proved using the Rasiowa–Sikorski lemma and Tanaka–Ono lemma. GL has a sufficient expressive power to discuss intra-/interpersonal beliefs with infinite lengths. As applications, we discuss the explicit definability of Axioms T, 4, 5, and of common knowledge in GL Also, we discuss the rationalizability concept in game theory in our framework. We evaluate where these discussions are done in the series GL, α ≤ ω. (shrink)

In the second installment to Gruszczyński and Pietruszczak we carry out an analysis of spaces of points of Grzegorczyk structures. At the outset we introduce notions of a concentric and \-concentric topological space and we recollect some facts proven in the first part which are important for the sequel. Theorem 2.9 is a strengthening of Theorem 5.13, as we obtain stronger conclusion weakening Tychonoff separation axiom to mere regularity. This leads to a stronger version of Theorem 6.10. Further, we show (...) that Grzegorczyk points are maximal contracting filters in the sense of De Vries, but the converse inclusion is not necessarily true. We also compare the notions of a Grzegorczyk point and an ultrafilter, and establish several properties of topological spaces based on Grzegorczyk structures. The main results of the paper are representation and completion theorems for G-structures. We prove both set-theoretical and topological representation theorems for various classes of G-structures. We also present topological object duality theorem for the class of complete G-structures and the class of concentric spaces, both restricted to structures which satisfy countable chain condition. We conclude the paper with proving equivalence of the original Grzegorczyk axiom with the one accepted by us as axiom. (shrink)

A comprehensive formal system is developed that amalgamates the operational and the realistic approaches to quantum mechanics. In this formalism, for example, a sharp distinction is made between events, operational propositions, and the properties of physical systems.

We show how effect algebras arise in physics and how they can be used to tie together the observables, states and symmetries employed in the study of physical systems. We introduce and study the unifying notion of an effect-observable-state-symmetry-system (EOSS-system) and give both classical and quantum-mechanical examples of EOSS-systems.

Dramatic changes or revolutions in a field of science are often made by outsiders or 'trespassers,' who are not limited by the established, 'expert' approaches. Each essay in this diverse collection shows the fruits of intellectual trespassing and poaching among fields such as economics, Kantian ethics, Platonic philosophy, category theory, double-entry accounting, arbitrage, algebraic logic, series-parallel duality, and financial arithmetic.

This dissertation explores the concepts of naturalness, intrinsicality, and duplication. An intrinsic property is had by an object purely in virtue of the way that object is considered in itself. Duplicate objects are exactly similar, considered as they are in themselves. The perfectly natural properties are the most fundamental properties of the world, upon which the nature of the world depends. In this dissertation I develop a theory of intrinsicality, naturalness, and duplication and explore their philosophical applications. Chapter 1 introduces (...) the notions, gives a preliminary survey of some proposed conceptual connections between the notions, and sketches some of their proposed applications. Chapter 2 gives my background assumptions and introduces notational conventions. In chapter 3 I present a theory of naturalness. Although I take ‘natural’ as a primitive, I clarify this notion by distinguishing and explicating various conceptions of naturalness. In chapter 4 I give a theory of various notions related to naturalness, especially intrinsicality and duplication. I show that ‘intrinsic’ and ‘duplicate’ are interde nable, and then give analyses of these and other notions in terms of naturalness. If, as I think likely, naturalness cannot be analyzed, then what is the proper response? David Lewis suggests: accept naturalness as a primitive. I am sympathetic to this proposal, but not to the form Lewis gives it: chapter 5 contains an argument against Lewis’s theory of naturalness. In chapter 6 I reject the idea that naturalness is analyzable in terms of immanent universals. I focus on the work of D. M. Armstrong. I also criticize Armstrong’s arguments against transcendent universals. (shrink)

The 1981 Dahlem conference was a catalyst for contemporary evolutionary developmental biology (Evo-devo). This introductory chapter rehearses some of the details of the history surrounding the original conference and its associated edited volume, explicates the philosophical problem of conceptual change that provided the rationale for a workshop devoted to evaluating the epistemic revisions and transformations that occurred in the interim, explores conceptual change with respect to the concept of evolutionary novelty, and highlights some of the themes and patterns in the (...) different contributions to the present volume, Conceptual Change in Biology: Scientific and Philosophical Perspectives on Evolution and Development. (shrink)

Evo-Devo exhibits a plurality of scientific “cultures” of practice and theory. When are the cultures acting—individually or collectively—in ways that actually move research forward, empirically, theoretically, and ethically? When do they become imperialistic, in the sense of excluding and subordinating other cultures? This chapter identifies six cultures – three /styles/ (mathematical modeling, mechanism, and history) and three /paradigms/ (adaptationism, structuralism, and cladism). The key assumptions standing behind, under, or within each of these cultures are explored. Characterizing the internal structure of (...) the cultures is necessary for understanding how they collaborate or compete, and how they are fragmented or integrated, in the rich interdisciplinary /trading zone/ (Galison 1997) of Evo-Devo. Evo-Devo is an important example of how science can progress through a radical plurality of perspectives and cultures. (shrink)

mathematicians for over 60 years. Amazingly, the Argonne team's automated theorem-proving program EQP took only 8 days to find a proof of it. Unfortunately, the proof found by EQP is quite complex and difficult to follow. Some of the steps of the EQP proof require highly complex and unintuitive substitution strategies. As a result, it is nearly impossible to reconstruct or verify the computer proof of the Robbins conjecture entirely by hand. This is where the unique symbolic capabilities of Mathematica (...) 3 come in handy. With the help of Mathematica, it is relatively easy to work out and explain each step of the dense EQP proof in detail. In this paper, I use Mathematica to provide a detailed, step-by-step reconstruction of the highly complex EQP proof of the Robbins conjecture. (shrink)