Conditionalization and Jeffrey Conditionalization cannot simultaneously satisfy two widely held desiderata on rules for empirical learning. The first desideratum is confirmational holism, which says that the evidential import of an experience is always sensitive to our background assumptions. The second desideratum is commutativity, which says that the order in which one acquires evidence shouldn't affect what conclusions one draws, provided the same total evidence is gathered in the end. (Jeffrey) Conditionalization cannot satisfy either of these desiderata without violating the (...) other. This is a surprising problem, and I offer a diagnosis of its source. I argue that (Jeffrey) Conditionalization is inherently anti-holistic in a way that is just exacerbated by the requirement of commutativity. The dilemma is thus a superficial manifestation of (Jeffrey) Conditionalization's fundamentally anti-holistic nature. (shrink)

We introduce proof nets and sequent calculus for the multiplicative fragment of non-commutative logic, which is an extension of both linear logic and cyclic linear logic. The two main technical novelties are a third switching position for the non-commutative disjunction, and the structure of order variety.

In this paper we review some properties of fuzzy observables, mainly as realized by commutative positive operator valued measures. In this context we discuss two representation theorems for commutative positive operator valued measures in terms of projection valued measures and describe, in some detail, the general notion of fuzzification. We also make some related observations on joint measurements.

Two operations, e.g. measurements, successively applied to the state of a system are said to be non-commutative if the sequence of their application makes a difference for the final result. Non-commuting operations play a crucial role in quantum theory, where they are intimately related to concepts as central as those of complementarity and entanglement. However, their significance is not restricted to the small dimensions of the microworld. For reasons easy to understand, non-commuting operations must be expected to be the rule (...) rather than the exception for operations on mental systems. We demonstrate this with a few examples from our own work, and hope this may contribute to the increasing attention that this exciting new area in consciousness studies is currently attracting worldwide. (shrink)

Conciliatory views of peer disagreement hold that when an agent encounters peer disagreement she should conciliate by adjusting her doxastic attitude towards that of her peer. In this paper I distinguish different ways conciliation can be understood and argue that the way conciliationism is typically understood violates the principle of commutativity of evidence. Commutativity of evidence holds that the order in which evidence is acquired should not influence what it is reasonable to believe based on that evidence. I (...) argue that when an agent encounters more than one peer, and applies the process of conciliation serially, the order she encounters the peers influences the resulting credence. I argue this is a problem for conciliatory views of disagreement, and suggest some responses available to advocates of conciliation. (shrink)

The notions of a commutative (∈, ∈)-neutrosophic ideal and a commutative falling neutrosophic ideal are introduced, and several properties are investigated. Characterizations of a commutative (∈, ∈)-neutrosophic ideal are obtained. Relations between commutative (∈, ∈)-neutrosophic ideal and (∈, ∈)-neutrosophic ideal are discussed. Conditions for an (∈, ∈)-neutrosophic ideal to be a commutative (∈, ∈)-neutrosophic ideal are established. Relations between commutative (∈, ∈)-neutrosophic ideal, falling neutrosophic ideal and commutative falling neutrosophic ideal are considered. Conditions for a falling neutrosophic ideal to be (...) commutative are provided. (shrink)

In this work we introduce a class of commutative rings whose defining condition is that its lattice of ideals, augmented with the ideal product, the semi-ring of ideals, is isomorphic to an MV-algebra. This class of rings coincides with the class of commutative rings which are direct sums of local Artinian chain rings with unit.

We characterize the generalized quantifiers Q which satisfy the scheme $QxQy\phi \leftrightarrow QyQx\phi$ , the so-called self-commuting quantifiers, or quantifiers with the Fubini property.

This book develops a liberal theory of justice in exchange. It identifies the conditions that market exchanges need to fulfill to be just. It also addresses head-on a consequentialist challenge to existing theories of exchange, namely that, in light of new harms faced at the global level, we need to consider the combined consequences of millions of market exchanges to reach a final judgment about whether some individual exchange is just. The author argues that, even if we accept this challenge, (...) the effect of it is minimal. For different reasons, normatively problematic collective market outcomes like externalities, monopolies, violations of the Lockean proviso, inequality, and commodification do not pose particular problems to the justice of market exchanges. He outlines the various conditions a market exchange needs to fulfill to be considered just from a liberal background and in light of the new harms. Ultimately, he shows, it is not the market which is to blame; if we want to tackle issues like global warming or global economic injustice, we should not blindly follow the intuition that we best restrain and regulate markets. Commutative Justice is unique in its focus on justice in exchange rather than on end-state distributive justice, and the way in which it addresses the new harms we are facing today. It will be of interest to researchers and advanced students in philosophy, politics, and economics who are working on questions of economic justice. (shrink)

Quantum B-algebras, the partially ordered implicational algebras arising as subreducts of quantales, are introduced axiomatically. It is shown that they provide a unified semantic for non-commutative algebraic logic. Specifically, they cover the vast majority of implicational algebras like BCK-algebras, residuated lattices, partially ordered groups, BL- and MV-algebras, effect algebras, and their non-commutative extensions. The opposite of the category of quantum B-algebras is shown to be equivalent to the category of logical quantales, in the way that every quantum B-algebra admits a (...) natural embedding into a logical quantale, the enveloping quantale. Partially defined products of algebras related to effect algebras are handled efficiently in this way. The unit group of the enveloping quantale of a quantum B-algebra X is shown to be always contained in X, which gives a functorial subgroup X× of X. Similar subfunctors are obtained for the non-commutative extensions of BCK-algebras and effect algebras. The results of Galatos, Jónsson, and Tsinakis on the splitting of generalized BL-algebras into a semidirect product of a partially ordered group operating on an integral residuated poset are extended to a characterization of twisted semidirect products of a po-group by a quantum B-algebra. (shrink)

A symmetric residuated lattice is an algebra such that is a commutative integral bounded residuated lattice and the equations x=x and =xy are satisfied. The aim of the paper is to investigate the properties of the unary operation ε defined by the prescription εx=x→0. We give necessary and sufficient conditions for ε being an interior operator. Since these conditions are rather restrictive →0)=1 is satisfied) we consider when an iteration of ε is an interior operator. In particular we consider the (...) chain of varieties of symmetric residuated lattices such that the n iteration of ε is a boolean interior operator. For instance, we show that these varieties are semisimple. When n=1, we obtain the variety of symmetric stonean residuated lattices. We also characterize the subvarieties admitting representations as subdirect products of chains. These results generalize and in many cases also simplify, results existing in the literature. (shrink)

In a recent pair of publications, Richard Bradley has offered two novel no-go theorems involving the principle of Preservation for conditionals, which guarantees that one’s prior conditional beliefs will exhibit a certain degree of inertia in the face of a change in one’s non-conditional beliefs. We first note that Bradley’s original discussions of these results—in which he finds motivation for rejecting Preservation, first in a principle of Commutativity, then in a doxastic analogue of the rule of modus ponens —are (...) problematic in a significant number of respects. We then turn to a recent U-turn on his part, in which he winds up rescinding his commitment to modus ponens, on the grounds of a tension with the rule of Import-Export for conditionals. Here we offer an important positive contribution to the literature, settling the following crucial question that Bradley leaves unanswered: assuming that one gives up on full-blown modus ponens on the grounds of its incompatibility with Import-Export, what weakened version of the principle should one be settling for instead? Our discussion of the issue turns out to unearth an interesting connection between epistemic undermining and the apparent failures of modus ponens in McGee’s famous counterexamples. (shrink)

Commuting conversions were introduced in the natural deduction calculus as ad hoc devices for the purpose of guaranteeing the subformula property in normal proofs. In a well known book, Jean-Yves Girard commented harshly on these conversions, saying that ‘one tends to think that natural deduction should be modified to correct such atrocities.’ We present an embedding of the intuitionistic predicate calculus into a second-order predicative system for which there is no need for commuting conversions. Furthermore, we show that the redex (...) and the conversum of a commuting conversion of the original calculus translate into equivalent derivations by means of a series of bidirectional applications of standard conversions. (shrink)

In this paper we present an equivalence between the category of commutative regular rings and the category of Boolean-valued fields, i.e., Boolean-valued sets for which the field axioms are true. The author used this equivalence in [12] to develop a Galois theory for commutative regular rings. Here we apply the equivalence to give an alternative construction of an algebraic closure for any commutative regular ring.Boolean-valued sets were developed in 1965 by Scott and Solovay [10] to simplify independence proofs in set (...) theory. They later were applied by Takeuti [13] to obtain results on Hilbert and Banach spaces. Ellentuck [3] and Weispfenning [14] considered Boolean-valued rings which consisted of rings and associated Boolean-valued relations. To the author's knowledge, the present work is the first to employ the Boolean-valued sets of Scott and Solovay to obtain results in algebra.The idea that commutative regular rings can be studied by examining the properties of related fields is not new. For several years algebraists and logicians have investigated commutative regular rings by representing a commutative regular ring as a subdirect product of fields or as the ring of global sections of a sheaf of fields over a Boolean space. These representations depend, as does the work presented here, on the fact that the set of central idempotents of any ring with identity forms a Boolean algebra. The advantage of the Boolean-valued set approach is that the axioms of classical logic and set theory are true in the Boolean universe. Therefore, if the axioms for a field are true for a Boolean-valued set, then other properties of the set can be deduced immediately from field theory. (shrink)

The non-commutative counterpart of the well-known Łukasiewicz propositional logic is developed, in strong connection with the algebraic theory of psMV-algebras. An extension by a new unary logical connective is also considered and a stronger completeness result is proved for this system.

Lange (2000) famously argues that although Jeffrey Conditionalization is non-commutative over evidence, it’s not defective in virtue of this feature. Since reversing the order of the evidence in a sequence of updates that don’t commute does not reverse the order of the experiences that underwrite these revisions, the conditions required to generate commutativity failure at the level of experience will fail to hold in cases where we get commutativity failure at the level of evidence. If our interest in (...) commutativity is, fundamentally, an interest in the order-invariance of information, an updating sequence that does not violate such a principle at the more fundamental level of experiential information should not be deemed defective. This paper claims that Lange’s argument fails as a general defense of the Jeffrey framework. Lange’s argument entails that the inputs to the Jeffrey framework differ from those of classical Bayesian Conditionalization in a way that makes them defective. Therefore, either the Jeffrey framework is defective in virtue of not commuting its inputs, or else it is defective in virtue of commuting the wrong kinds of ones. (shrink)

A large-scale field experiment in Rotterdam, Netherlands, tested whether nudging could increase public transport use. During one work week, 4000 commuters on six bus lines, received a free travel card holder. On the three bus lines in the experimental condition, the card holders displayed a social label that branded bus passengers as sustainable travelers because of their bus use. On the three bus lines in the control condition, there was no such message on the card holders. Analysis of the number (...) of rides per hour showed that the intervention led to a change from pre-intervention to post-intervention period that was estimated to be 1.18 rides per day greater on experimental lines than on control lines. This experiment shows that public transport operators can increase public transport use by incorporating messages that positively label passengers as sustainable travelers in their communication strategies. (shrink)

Several investigations in probability theory and the theory of expert systems show that it is important to search for some reasonable generalizations of fuzzy logics (e.g. Łukasiewicz, Gödel or product logic) having a non-associative conjunction. In the present paper, we offer a non-associative fuzzy logic L CBA having as an equivalent algebraic semantics lattices with section antitone involutions satisfying the contraposition law, so-called commutative basic algebras. The class (variety) CBA of commutative basic algebras was intensively studied in several recent papers (...) and includes the class of MV-algebras. We show that the logic L CBA is very close to the Łukasiewicz one, both having the same finite models, and can be understood as its non-associative generalization. (shrink)

In this paper, I sketch out the way our bodies are engaged while commuting in order to elucidate several key aspects of the bodily experience of “in-between-ness.” I discover that within the rhythm and movement of the in-between, our bodies can open to a specific kind of conceptual creativity—an insight that I unfold in reference to the unanticipated innovation and transformation that accompanies other bodily experiences of in-between-ness more generally. This sketch, however, also demands that I reflect on phenomenological methodology, (...) and our ontological presuppositions about the “things” that we intend to “go back to.” I conclude by suggesting that while Deleuze and Guattari’s rhizomatic ontology helps us explain the findings of my phenomenological sketch of commuting, Merleau-Ponty’s embodied phenomenology can help to flesh out (quite literally) Deleuze and Guattari’s suggestion that all the work happens in “the logic of the AND” ( A Thousand Plateaus 25). While the method I call on in this paper is decidedly phenomenological, the conclusions I reach invite me to explore further the promise of a more “rhizomatic phenomenological” practice. (shrink)

How should we respond to cases of disagreement where two epistemic agents have the same evidence but come to different conclusions? Adam Elga has provided a Bayesian framework for addressing this question. In this paper, I shall highlight two unfortunate consequences of this framework, which Elga does not anticipate. Both problems derive from a failure of commutativity between application of the equal weight view and updating in the light of other evidence.

This work presents a computational interpretation of the construction process for cyclic linear logic and non-commutative logic sequential proofs. We assume a proof construction paradigm, based on a normalisation procedure known as focussing, which efficiently manages the non-determinism of the construction. Similarly to the commutative case, a new formulation of focussing for NL is used to introduce a general constraint-based technique in order to dealwith partial information during proof construction. In particular, the procedure develops through construction steps propagating constraints in (...) intermediate objects called abstract proofs. (shrink)

It is shown that all the assumptions for symmetric monoidal categories follow from a unifying principle involving natural isomorphisms of the type →, called medial commutativity. Medial commutativity in the presence of the unit object enables us to define associativity and commutativity natural isomorphisms. In particular, Mac Lane’s pentagonal and hexagonal coherence conditions for associativity and commutativity are derived from the preservation up to a natural isomorphism of medial commutativity by the biendofunctor . This preservation (...) boils down to an isomorphic representation of the Yang–Baxter equation of symmetric and braid groups. The assumptions of monoidal categories, and in particular Mac Lane’s pentagonal coherence condition, are explained in the absence of commutativity, and also of the unit object, by a similar preservation of associativity by the biendofunctor . In the final section one finds coherence conditions for medial commutativity in the absence of the unit object. These conditions are obtained by taking the direct product of the symmetric groups for 0≤i≤n. (shrink)

A simple rule of probability revision ensures that the final result of a sequence of probability revisions is undisturbed by an alteration in the temporal order of the learning prompting those revisions. This Uniformity Rule dictates that identical learning be reflected in identical ratios of certain new-to-old odds, and is grounded in the old Bayesian idea that such ratios represent what is learned from new experience alone, with prior probabilities factored out. The main theorem of this paper includes as special (...) cases Field's theorem on commuting probability-kinematical revisions and the equivalence of two strategies for generalizing Jeffrey 's solution to the old evidence problem to the case of uncertain old evidence and probabilistic new explanation. (shrink)

In [9] it is proved the categorical isomorphism of two varieties: bounded commutative BCK-algebras and MV -algebras. The class of MV -algebras is the algebraic counterpart of the infinite valued propositional calculus L of Lukasiewicz . The main objective of the present paper is to study that isomorphism from the perspective of logic. The B-C-K logic is algebraizable and the quasivariety of BCKalgebras is the equivalent algebraic semantics for that logic . We call commutative B-C-K logic, briefly cBCK, to the (...) extension of B-C-K logic associated to the variety of commutative BCK–algebras. Moreover, we present the extension Boc of cBCK obtained by adding the axiom of “boundness”. We prove that the deductive system Boc is equivalent to L. We observe that cBCK admits two interesting extensions: the logic Boc, treated in this paper, which is equivalent to the system L of Lukasiewicz, and the logic Co that is naturally associated to the system Balo of `-groups . This constructions establish a link between L and Balo , that would be a logical approach to the categorical relationship between MV–algebras and `-groups. (shrink)

The Convention on Biological Diversity and its Nagoya Protocol established an Access and Benefit Sharing system between utilizers and providers of genetic resources. ABS is understood as a tool that should promote commutative justice between the involved parties. This essay discusses what exactly it is that is being exchanged in the ABS process. It critically analyses moral claims to compensation that are implied by the ABS system for genetic resources. It argues that with the exception of cases in which traditional (...) knowledge is involved, states are not automatically entitled to compensation in return for the utilization of genetic resources growing within their territory. However, biodiversity-rich states that make an effort to protect biodiversity must be compensated for complying with the requests set out in the CBD. Although it acknowledges that the NP is a step towards recognizing this claim, this essay argues that ABS is not the appropriate method to compensate for biodivers... (shrink)

The notions of a C-energetic subset and permeable C-value in BCK-algebras are introduced, and related properties are investigated. Conditions for an element t in [0, 1] to be an permeable C-value are provided. Also conditions for a subset to be a C-energetic subset are discussed. We decompose BCK-algebra by a partition which consists of a C-energetic subset and a commutative ideal.

Two axiomatizations of the nonassociative and commutative Lambek syntactic calculus are given and their equivalence is proved. The first axiomatization employs Permutation as the only structural rule, the second one, with no Permutation rule, employs only unidirectional types. It is also shown that in the case of the Ajdukiewicz calculus an analogous equivalence is valid only in the case of a restricted set of formulas. Unidirectional axiomatizations are employed in order to establish the generative power of categorial grammars based on (...) the nonassociative and commutative Lambek calculus with product. Those grammars produce CF-languages of finite degree generated by CF-grammars closed with respect to permutations. (shrink)

The paper presents a generalization of pregroup, by which a freely-generated pregroup is augmented with a finite set of commuting inequations, allowing limited commutativity and cancelability. It is shown that grammars based on the commutation-augmented pregroups generate mildly context-sensitive languages. A version of Lambek’s switching lemma is established for these pregroups. Polynomial parsability and semilinearity are shown for languages generated by these grammars.

Journal of Political Philosophy, Volume 29, Issue 4, Page 480-495, December 2021.

The relationship between q-spaces (c.f. [9]) and quantum spaces (c.f. [5]) is studied, proving that both models coincide in the case of Spec A, the spectrum of a non-commutative C*-algebra A. It is shown that a sober T 1 quantum space is a classical topological space. This difficulty is circumvented through a new definition of point in a quantale. With this new definition, it is proved that Lid A has enough points. A notion of orthogonality in quantum spaces is introduced, (...) which permits us to express the usual topological properties of separation. The notion of stalks of sheaves over quantales is introduced, and some results in categorial model theory are obtained. (shrink)

I will argue that Kochen-Specker arguments do not provide an algebraic proof for quantum contextuality since, for the argument to be effective, operators must be uniquely associated with measur...

We show that a stable groupG satisfying certain commutator conditions is nilpotent. Furthermore, a soluble stable group with generically splitting automorphism of prime order is nilpotent-by-finite. In particular, a soluble stable group with a generic element of prime order is nilpotent-by-finite.

The so-called "non-commutativity" of probability kinematics has caused much unjustified concern. When identical learning is properly represented, namely, by identical Bayes factors rather than identical posterior probabilities, then sequential probability-kinematical revisions behave just as they should. Our analysis is based on a variant of Field's reformulation of probability kinematics, divested of its (inessential) physicalist gloss.

A theory of the joint measurement of quantum mechanical observables is generalized in order to make it applicable to the measurement of the local observables of field theory. Subsequently, the property of local commutativity, which is usually introduced as a postulate, is derived by means of the theory of measurement from a requirement of mutual nondisturbance, which, for local observables performed at a spacelike distance from each other, is interpreted as a requirement of macrocausality. Alternative attempts at establishing a (...) deductive relationship between relativistic causality and local commutativity are reviewed, but found wanting, either because of the assumption of an unwarranted objectivity of the object system (algebraic approach) or because of the use of a projection postulate (operational approach). Finally, the quantum mechanical nonobjectivity is related to certain features of nonlocality which are present in the formalism of quantum mechanics. (shrink)

Mathematical notation includes a vast array of signs. Most mathematical signs appear to be symbolic, in the sense that their meaning is arbitrarily related to their visual appearance. We explored the hypothesis that mathematical signs with iconic aspects—those which visually resemble in some way the concepts they represent—offer a cognitive advantage over those which are purely symbolic. An early formulation of this hypothesis was made by Christine Ladd in 1883 who suggested that symmetrical signs should be used to convey commutative (...) relations, because they visually resemble the mathematical concept they represent. Two controlled experiments provide the first empirical test of, and evidence for, Ladd's hypothesis. In Experiment 1 we find that participants are more likely to attribute commutativity to operations denoted by symmetric signs. In Experiment 2 we further show that using symmetric signs as notation for commutative operations can increase mathematical performance. (shrink)

Lambek categorial grammars is a class of formal grammars based on the Lambek calculus. Pentus proved in 1993 that they generate exactly the class of context-free languages without the empty word. In this paper, we study categorial grammars based on the Lambek calculus with the permutation rule LP. Of particular interest is the product-free fragment of LP called the Lambek-van Benthem calculus LBC. Buszkowski in his 1984 paper conjectured that grammars based on the Lambek-van Benthem calculus (LBC-grammars for short) generate (...) exactly permutation closures of context-free languages. In this paper, we disprove this conjecture by presenting a language generated by an LBC-grammar that is not a permutation closure of any context-free language. Firstly, we introduce an ad-hoc modification of vector addition systems called linearly-restricted branching vector addition systems with states and additional memory (LRBVASSAMs for short) and prove that the latter are equivalent to LBC-grammars. Then we construct an LRBVASSAM that generates a non-semilinear set and thus disprove Buszkowski’s conjecture. Since Buszkowski’s conjecture is false, not so much is known about the languages generated by LBC-grammars or by LP-grammars. The equivalence of LRBVASSAMs and LBC-grammars allows us to establish a number of their properties. We show that LP-grammars generate the same class of languages as LBC-grammars; that is, removing product from LP does not decrease expressive power of corresponding categorial grammars. We also prove that this class of languages is closed under union, intersection, concatenation, and Kleene plus. (shrink)

The notion of a commutator lattice is investigated. It is shown that the class of commutator lattices is coextensive with the class of lattices with residuation.

The purpose of this paper is to present in a uniform way the commutator theory for k-deductive system of arbitrary positive dimension k. We are interested in the logical perspective of the research — an emphasis is put on an analysis of the interconnections holding between the commutator and logic. This research thus qualifies as belonging to abstract algebraic logic, an area of universal algebra that explores to a large extent the methods provided by the general theory of deductive systems. (...) In the paper the new term ‘commutator formula’ is introduced. The paper is concerned with the meanings of the above term in the models provided by the commutator theory and clarifies contexts in which these meanings occur. The work is presented in an abstracted form: main ideas are outlined but proofs are deferred to the second part of the paper. (shrink)

A first-order temporal non-commutative logic TN[l], which has no structural rules and has some l-bounded linear-time temporal operators, is introduced as a Gentzen-type sequent calculus. The logic TN[l] allows us to provide not only time-dependent, resource-sensitive, ordered, but also hierarchical reasoning. Decidability, cut-elimination and completeness (w.r.t. phase semantics) theorems are shown for TN[l]. An advantage of TN[l] is its decidability, because the standard first-order linear-time temporal logic is undecidable. A correspondence theorem between TN[l] and a resource indexed non-commutative logic RN[l] (...) is also shown. This theorem is intended to state that “time” is regarded as a “resource”. (shrink)

Abstract Based on recalling two characteristic features of Bayesian statistical inference in commutative probability theory, a stability property of the inference is pointed out, and it is argued that that stability of the Bayesian statistical inference is an essential property which must be preserved under generalization of Bayesian inference to the non?commutative case. Mathematical no?go theorems are recalled then which show that, in general, the stability can not be preserved in non?commutative context. Two possible interpretations of the impossibility of generalization (...) of Bayesian statistical inference to the non?commutative case are offered, none of which seems to be completely satisfying. (shrink)

A simple rule of probability revision ensures that the final result ofa sequence of probability revisions is undisturbed by an alterationin the temporal order of the learning prompting those revisions.This Uniformity Rule dictates that identical learning be reflectedin identical ratios of certain new-to-old odds, and is grounded in the oldBayesian idea that such ratios represent what is learned from new experiencealone, with prior probabilities factored out. The main theorem of this paperincludes as special cases (i) Field's theorem on commuting probability-kinematical (...) revisions and (ii) the equivalence of two strategiesfor generalizing Jeffrey's solution to the old evidence problem tothe case of uncertain old evidence and probabilistic new explanation. (shrink)

Short-circuit evaluation denotes the semantics of propositional connectives in which the second argument is evaluated only if the first is insufficient to determine the value of the expression. Com...