In [1] it was shown that if κ is a strong partition cardinal, then every function from [κ ]κ to [κ ]κ is continuous almost everywhere. In this investigation, we explore whether such functions are differentiable or integrable in any sense. Some of them are.

Classical physics and quantum physics suggest two meta-physical types of reality: the classical notion of a objectively definite reality with properties "all the way down," and the quantum notion of an objectively indefinite type of reality. The problem of interpreting quantum mechanics (QM) is essentially the problem of making sense out of an objectively indefinite reality. These two types of reality can be respectively associated with the two mathematical concepts of subsets and quotient sets (or partitions) which are category-theoretically dual (...) to one another and which are developed in two mathematical logics, the usual Boolean logic of subsets and the more recent logic of partitions. Our sense-making strategy is "follow the math" by showing how the logic and mathematics of set partitions can be transported in a natural way to Hilbert spaces where it yields the mathematical machinery of QM--which shows that the mathematical framework of QM is a type of logical system over ℂ. And then we show how the machinery of QM can be transported the other way down to the set-like vector spaces over ℤ₂ showing how the classical logical finite probability calculus (in a "non-commutative" version) is a type of "quantum mechanics" over ℤ₂, i.e., over sets. In this way, we try to make sense out of objective indefiniteness and thus to interpret quantum mechanics. (shrink)

The work broadens – to a considerable extent – Z. Pawlak’s original method (1982, 1992) of approximation of sets. The approximation of sets included in a universum U goes on in the contextual approximation space CAS which consists of: 1) a sequence of Pawlak’s approximation spaces (U,Ci), where indexes i from set I are linearly ordered degrees of contexts (I, <), and Ci is the universum partition U, 2) a sequence of binary relations on sets included in U, relations (...) called context relations indexed with degrees of contexts. The introduction of relations of contexts of sets was inspired by the work of W. Ziarnko (1993). Intuitively, set X is the context of set Y to the degree i, when on the basis of knowledge of the elements of set X we can obtain knowledge about the belonging of these elements to set Y to the degree i which is a degree of uncertainty or ambiguity, or inaccuracy. The operations of approximation of sets Ci-, BNi, Ci+, called lower approximation of the degree i, boundary of the degree i, upper approximation of the degree i, are formulated in a way which is analogous with the systems of Pawlak’s approximation, by expanding their formulas accordingly. In the next part of the work, it is shown (in an analogous way to Bryniarski’s work (1989)) that one can define the following operations on contextual rough sets: the union of contextual rough sets, the intersection of contextual rough sets and the complement of contextual rough set. It is proved that the algebra of contextual rough sets is a ditributive lattice with a unit and with a zero. On the basis of the results presented above the problem of defining operations on contextual rough sets by membership relations (Bryniarski 1989) may be formulated analogously to the classical operations on sets. (shrink)

We prove that the strong polarized relation $\left( {\mu _\mu ^ + } \right) \to \left( {\mu _\mu ^ + } \right)_2^{1.1}$ is consistent with ZFC, for a singular ì which is a limit of measurable cardinals.

James Earl Baumgartner (March 23, 1943–December 28, 2011) came of age mathematically during the emergence of forcing as a fundamental technique of set theory, and his seminal research changed the way set theory is done. He made fundamental contributions to the development of forcing, to our understanding of uncountable orders, to the partition calculus, and to large cardinals and their ideals. He promulgated the use of logic such as absoluteness and elementary submodels to solve problems in set theory, (...) he applied his knowledge of set theory to a variety of areas in collaboration with other mathematicians, and he encouraged a community of mathematicians with engaging survey talks, enthusiastic discussions of open problems, and friendly mathematical conversations. (shrink)

This volume contains the proceedings of the Logic at Harvard conference in honor of W. Hugh Woodin's 60th birthday, held March 27–29, 2015, at Harvard University. It presents a collection of papers related to the work of Woodin, who has been one of the leading figures in set theory since the early 1980s. The topics cover many of the areas central to Woodin's work, including large cardinals, determinacy, descriptive set theory and the continuum problem, as well as connections between set (...) theory and Banach spaces, recursion theory, and philosophy, each reflecting a period of Woodin's career. Other topics covered are forcing axioms, inner model theory, the partition calculus, and the theory of ultrafilters. This volume should make a suitable introduction to Woodin's work and the concerns which motivate it. The papers should be of interest to graduate students and researchers in both mathematics and philosophy of mathematics, particularly in set theory, foundations and related areas. (shrink)

Reviewed Works:Edwin W. Miller, On a Property of Families of Sets.Ben Dushnik, E. W. Miller, Partially Ordered Sets.P. Erdos, Some Set-theoretical Properties of Graphs.G. Fodor, Proof of a Conjecture of P. Erdos.P. Erdos, R. Rado, A Partition Calculus in Set Theory.P. Erdos, R. Rado, Intersection Theorems for Systems of Sets.A. Hajnal, Some Results and Problems on Set Theory.P. Erdos, A. Hajnal, On a Property of Families of Sets.A. Hajnal, Proof of a Conjecture of S. Ruziewicz.P. Erdos, A. Hajnal, (...) R. Rado, Partition Relations for Cardinal Numbers.P. Erdos, A. Hajnal, On a Problem of B. Jonsson.P. Erdos, A. Hajnal, On Chromatic Number of Graphs and Set-Systems. (shrink)

In this study, we analyse the notion of “differential heterogenesis” proposed by Deleuze and Guattari on a morphogenetic perspective. We propose a mathematical framework to envisage the emergence of singular forms from the assemblages of heterogeneous operators. In opposition to the kind of differential calculus that is usually adopted in mathematical-physical modelling, which tends to assume a homogeneous differential equation applied to an entire homogeneous region, heterogenesis allows differential constraints of qualitatively different kinds in different points of space and (...) time. These constraints can then change in time, opening the possibility for new kinds of differential dynamics and the emergence of distinct entities and forms. Formally, we show that operators with different phase spaces can be assembled on the basis of a result of Rothschild & Stein (1976. Hypoelliptic differential operators and nilpotent groups. Acta Mathematica 137. 247–320). Furthermore, operators with different dynamics can be assembled by means of a partition of the unit.After stating the concept of differential heterogenesis in terms of contemporary mathematics, we show that this construction sheds light on the constitution of the semiotic function. In fact, both the Merleau-Pontian and the Deleuzian approaches share a common conceptualisation of the semiotic function and its emergence in terms of amorphodynamics of heterogeneous assemblages with a divergent actualisation. This divergent actualisation allows the co-constitution of various expression and content planes. Finally, we show that the divergent actualisation can be interpreted as the directions of principal eigenvectors of the actualized flow. (shrink)

In a previous paper, a statistical method of constructing quantum models of classical properties has been described. The present paper concludes the description by turning to classical mechanics. The quantum states that maximize entropy for given averages and variances of coordinates and momenta are called ME packets. They generalize the Gaussian wave packets. A non-trivial extension of the partition-function method of probability calculus to quantum mechanics is given. Non-commutativity of quantum variables limits its usefulness. Still, the general form (...) of the state operators of ME packets is obtained with its help. The diagonal representation of the operators is found. A general way of calculating averages that can replace the partition function method is described. Classical mechanics is reinterpreted as a statistical theory. Classical trajectories are replaced by classical ME packets. Quantum states approximate classical ones if the product of the coordinate and momentum variances is much larger than Planck constant. Thus, ME packets with large variances follow their classical counterparts better than Gaussian wave packets. (shrink)

Given a basic conditional of the form, If A then C, individuals usually list three cases as possible: A and C, not‐A and not‐C, not‐A and C. This result corroborates the theory of mental models. By contrast, individuals often judge that the conditional is true only in the case of A and C, and that cases of not‐A are irrelevant to its truth or falsity. This result corroborates other theories of conditionals. To resolve the discrepancy, we devised two new tasks: (...) the “collective” truth task, in which participants judged whether sets of assertions about a specific individual, such as: If A then C, not‐A, C, could all be true at the same time; and one in which participants judged the truth of conditional predictions about specific future events. The results consistently matched the three possibilities, thereby corroborating the model theory. They also showed a massive violation of the probability calculus in estimates of the probabilities of the four cases in the partition of conditionals (A and C, A and not‐C, not‐A and C, and not‐A and not‐C), which summed to over 200%. (shrink)

Drawing upon rational choice and investor attention theories, we examine how accusations of corporate bribery and subsequent investigations shape market reactions. Using event study methodology to measure loss in firm value for public firms facing bribery investigations from 1978 to 2010, we found that total market penalties amounted to $60.61 billion. We ran moderated multiple regression analysis to examine further the degree to which the unique characteristics of bribery explain variations in market penalties. Companies committing bribery in less corrupt host (...) countries and with the involvement of compromised executives experienced greater market penalties than did other companies. After partitioning share value losses into components for regulatory penalties, class action settlements, and loss to reputation, we found that reputational penalties account for 81.8¢ of every dollar of share value loss. Omission of reputational penalties in rational choice calculus underestimates bribery costs by 4.5 times. The results suggest that firms should not underestimate the importance of market-imposed reputational penalties by merely considering regulator-imposed fines and sanctions. (shrink)

A broad class of inductive logics that includes the probability calculus is defined by the conditions that the inductive strengths [A|B] are defined fully in terms of deductive relations in preferred partitions and that they are asymptotically stable. Inductive independence is shown to be generic for propositions in such logics; a notion of a scale-free inductive logic is identified; and a limit theorem is derived. If the presence of preferred partitions is not presumed, no inductive logic is definable. This (...) no-go result precludes many possible inductive logics, including versions of hypothetico-deductivism. (shrink)

A broad class of inductive logics that includes the probability calculus is defined by the conditions that the inductive strengths [A|B] are defined fully in terms of deductive relations in preferred partitions and that they are asymptotically stable. Inductive independence is shown to be generic for propositions in such logics; a notion of a scale-free inductive logic is identified; and a limit theorem is derived. If the presence of preferred partitions is not presumed, no inductive logic is definable. This (...) no-go result precludes many possible inductive logics, including versions of hypothetico-deductivism. (shrink)

In a recent paper, we have shown that the class of Boolean contact algebras (BCAs) has the hereditary property, the joint embedding property and the amalgamation property. By Fraïssé’s theorem, this shows that there is a unique countable homogeneous BCA. This paper investigates this algebra and the relation algebra generated by its contact relation. We first show that the algebra can be partitioned into four sets {0}, {1}, K, and L, which are the only orbits of the group of base (...) automorphisms of the algebra, and then show that the contact relation algebra of this algebra is finite, which is the first non-trivial extensional BCA we know which has this property. (shrink)

An MV-algebra A=(A,0,¬,⊕) is an abelian monoid (A,0,⊕) equipped with a unary operation ¬ such that ¬¬x=x,x⊕¬0=¬0, and y⊕¬(y⊕¬x)=x⊕¬(x⊕¬y). Chang proved that the equational class of MV-algebras is generated by the real unit interval [0,1] equipped with the operations ¬x=1−x and x⊕y=min(1,x+y). Therefore, the free n-generated MV-algebra Free n is the algebra of [0,1]-valued functions over the n-cube [0,1] n generated by the coordinate functions ξ i ,i=1, . . . ,n, with pointwise operations. Any such function f is a (...) McNaughton function, i.e., f is continuous, piecewise linear, and each piece has integer coefficients. Conversely, McNaughton proved that all McNaughton functions f: [0,1] n →[0,1] are in Free n . The elements of Free n are logical equivalence classes of n-variable formulas in the infinite-valued calculus of Łukasiewicz. The aim of this paper is to provide an alternative, representation-free, characterization of Free n. (shrink)

. With an overall objective of establishing association between air pollutants and incidence of respiratory diseases, the environmental professionals and medical practitioners have made significant contribution, using statistical mechanics in modelling epidemiological data, population characteristics, and pollution parameters. Broadly speaking, the studies have shown that the increase in vehicular traffic has been one of the causes of respiratory diseases. However, the WHO Centre for Environment and Health, Europe in its 2005 document states: “There is little evidence for a causal relationship (...) between asthma prevalence/incidence and air pollution in general, though the evidence is suggestive of a causal association between the prevalence/incidence of asthma symptoms and living in close proximity to traffic”. Decision making process in a real world is invariably based on perceptions which are expressed in words or may be in numeric terms and not in probability terms. In the paper, we made an attempt to model the perceptions of experienced pulmonologists in arriving at their combined degree of belief/plausibility/ignorance for all the possible combinations of identified respiratory diseases, using evidence theory and fuzzy relational calculus without collecting sizeable parametric data accumulated over a period of years. Tightening pollution norms by the regulatory authorities is an overall objective of the global efforts on greenhouse gases reduction in general, and air pollution mitigation in particular. The concept of solar battery operated electric vehicles for road transport is advocated, initially for two/three wheelers, and extending it to four wheelers, especially in the developing countries. (shrink)

This paper presents detailed formalizations of ontological arguments in a simple modal natural deduction calculus. The first formal proof closely follows the hints in Scott’s manuscript about Gödel’s argument and fills in the gaps, thus verifying its correctness. The second formal proof improves the first one, by relying on the weaker modal logic KB instead of S5 and by avoiding the equality relation. The second proof is also technically shorter than the first one, because it eliminates unnecessary detours and (...) uses Axiom 1 for the positivity of properties only once. The third and fourth proofs formalize, respectively, Anderson’s and Bjørdal’s variants of the ontological argument, which are known to be immune to modal collapse. (shrink)

This article investigates the proof theory of the Quantified Argument Calculus as developed and systematically studied by Hanoch Ben-Yami [3, 4]. Ben-Yami makes use of natural deduction, we, however, have chosen a sequent calculus presentation, which allows for the proofs of a multitude of significant meta-theoretic results with minor modifications to the Gentzen’s original framework, i.e., LK. As will be made clear in course of the article LK-Quarc will enjoy cut elimination and its corollaries.

In this paper we develop what we can describe as a “dual two-sided” cut-free sequent calculus system for the non-classical logics of truth lp, k3, stt and a non-reflexive logic ts which is, arguably, more elegant than the three-sided sequent calculus developed by Ripley for the same logics. Its elegance stems from how it employs more or less the standard sequent calculus rules for the various connectives and truth, and the fact that it offers a rather neat (...) connection between derivable sequents and validity in comparison to the calculus developed by Ripley. (shrink)

A new justification of probabilism is developed that pays close attention to the structure of the underlying space of possibilities. Its central assumption is that rational numerical degrees of bel...

A three-valued propositional logic is presented, within which the three values are read as ?true?, ?false? and ?nonsense?. A three-valued extended functional calculus, unrestricted by the theory of types, is then developed. Within the latter system, Bochvar analyzes the Russell paradox and the Grelling-Weyl paradox, formally demonstrating the meaninglessness of both.

In this paper, we present a simple sequent calculus for the modal propositional logic S5. We prove that this sequent calculus is theoremwise equivalent to the Hilbert-style system S5, that it is contraction-free and cut-free, and finally that it is decidable. All results are proved in a purely syntactic way.

The first example of an intuitively meaningful propositional logic without the finite model property, and still the simplest one in the literature. The question of its decidability appears still to be open.

We introduce a cyclic proof system for the two-way alternation-free modal μ-calculus. The system manipulates one-sided Gentzen sequents and locally deals with the backwards modalities by allowing analytic applications of the cut rule. The global effect of backwards modalities on traces is handled by making the semantics relative to a specific strategy of the opponent in the evaluation game. This allows us to augment sequents by so-called trace atoms, describing traces that the proponent can construct against the opponent’s strategy. (...) The idea for trace atoms comes from Vardi’s reduction of alternating two-way automata to deterministic one-way automata. Using the multi-focus annotations introduced earlier by Marti and Venema, we turn this trace-based system into a path-based system. We prove that our system is sound for all sequents and complete for sequents not containing trace atoms. (shrink)

Summary We have considered complete consistent systems in the first‐oder predicate calculus with identity, and have studied the set of the models of such a system by means of the maximal consistent condition‐sets associated with the system. The results may be summarized thus: (a) A complete consistent system is no‐categorical (= categorical in the denumerable domain) if and only if for every n, the number of different conditions in n variables is finite (T10). (b) If a complete consistent system (...) has a model M with finite character (i.e. a model M such that every maximal consistent condition‐set satisfied in M has a finite basis), then this model M is uniquely characterized by the property that every other model is an arithmetic extension of M (T 5). (c) Every complete consistent system, which has only a denumerable number of different associated maximal consistent condition‐sets, has a model with finite character (T8). (shrink)

This paper shows how the classical finite probability theory (with equiprobable outcomes) can be reinterpreted and recast as the quantum probability calculus of a pedagogical or toy model of quantum mechanics over sets (QM/sets). There have been several previous attempts to develop a quantum-like model with the base field of ℂ replaced by ℤ₂. Since there are no inner products on vector spaces over finite fields, the problem is to define the Dirac brackets and the probability calculus. The (...) previous attempts all required the brackets to take values in ℤ₂. But the usual QM brackets <ψ|ϕ> give the "overlap" between states ψ and ϕ, so for subsets S,T⊆U, the natural definition is <S|T>=|S∩T| (taking values in the natural numbers). This allows QM/sets to be developed with a full probability calculus that turns out to be a non-commutative extension of classical Laplace-Boole finite probability theory. The pedagogical model is illustrated by giving simple treatments of the indeterminacy principle, the double-slit experiment, Bell's Theorem, and identical particles in QM/Sets. A more technical appendix explains the mathematics behind carrying some vector space structures between QM over ℂ and QM/Sets over ℤ₂. (shrink)

Inductive characterizations of the sets of terms, the subset of strongly normalizing terms and normal forms are studied in order to reprove weak and strong normalization for the simply-typed λ-calculus and for an extension by sum types with permutative conversions. The analogous treatment of a new system with generalized applications inspired by generalized elimination rules in natural deduction, advocated by von Plato, shows the flexibility of the approach which does not use the strong computability/candidate style à la Tait and (...) Girard. It is also shown that the extension of the system with permutative conversions by η-rules is still strongly normalizing, and likewise for an extension of the system of generalized applications by a rule of ``immediate simplification''. By introducing an infinitely branching inductive rule the method even extends to Gödel's T. (shrink)

We prove a finite model theorem and infinitary completeness result for the propositional -calculus. The construction establishes a link between finite model theorems for propositional program logics and the theory of well-quasi-orders.

One of the most important contributions of A. Church to logic is his invention of the lambda calculus. We present the genesis of this theory and its two major areas of application: the representation of computations and the resulting functional programming languages on the one hand and the representation of reasoning and the resulting systems of computer mathematics on the other hand.

The interpretation of propositions in Lukasiewicz's infinite-valued calculus as answers in Ulam's game with lies--the Boolean case corresponding to the traditional Twenty Questions game--gives added interest to the completeness theorem. The literature contains several different proofs, but they invariably require technical prerequisites from such areas as model-theory, algebraic geometry, or the theory of ordered groups. The aim of this paper is to provide a self-contained proof, only requiring the rudiments of algebra and convexity in finite-dimensional vector spaces.