Recently, new types of independence of a pair of C *- or W *-subalgebras (1,2) of a C *- or W *-algebra have been introduced: operational C *- and W *-independence (Rédei and Summers, http://arxiv.org/abs/0810.5294, 2008) and operational C *- and W *-separability (Rédei and Valente, How local are local operations in local quantum field theory? 2009). In this paper it is shown that operational C *-independence is equivalent to operational C *-separability and that operational W *- (...) class='Hi'>independence is equivalent to operational W *-separability. Specific further sub-types of both operational C *- and W *-separability and operational C *- and W *-independence are defined and the problem of characterization of the logical interdependencies of the independence notions is raised. (shrink)

We formulate several statements in Algebraic Logic that turn out to be independent of ZFC. We relate such statements to Martin's axiom, omitting types for variants of first order logic and topological properties of Baire spaces.

Various notions of independence of observables have been proposed within the algebraic framework of quantum field theory. We discuss relationships between these and the recently introduced notion of logical independence in a general operator-algebraic context. We show that C*-independence implies an analogue of classical independence.

Ekstein has shown that causal independence neither implies nor is implied by commutativity in an infinite-dimensional, reducible construction. DeFacio and Taylor have presented a finite-dimensional irreducible example of Ekstein's proposition. Avishai and Ekstein have shown that the original question regarding locality for algebraic quantum field theories remainsopen. We concur with that claim and offer additional arguments. A new denumerably infinite-dimensional, irreducible example is presented here which shows that a sort of “orthogonality” among operators is involved. Some observations on (...) localC*-andW*-algebras are given. (shrink)

If κω is a cardinal, a complete Boolean algebra is called κ-dependent if for each sequence bβ: β<κ of elements of there exists a partition of the unity, P, such that each pP extends bβ or bβ′, for κ-many βκ. The connection of this property with cardinal functions, distributivity laws, forcing and collapsing of cardinals is considered.

We define a formula φ in a first-order language L , to be an equation in a category of L -structures K if for any H in K , and set p = {φ;i ϵI, a i ϵ H} there is a finite set I 0 ⊂ I such that for any f : H → F in K , ▪. We say that an elementary first-order theory T which has the amalgamation property over substructures is equational if every quantifier-free (...) formula is equivalent in T to a boolean combination of equations in Mod, the category of models of T with embeddings for morphisms. Thus, we develop a theory of independence with respect to equations in general categories of structures, which is similar to the one introduced in stability but which, in our context, has an algebraic character. (shrink)

We prove that the form of conditional independence at play in database theory and independence logic is reducible to the first-order dividing calculus in the theory of atomless Boolean algebras. This establishes interesting connections between independence in database theory and stochastic independence. As indeed, in light of the aforementioned reduction and recent work of Ben-Yaacov :957–1012, 2013), the former case of independence can be seen as the discrete version of the latter.

Recall that a subset X of a Boolean algebra A is independent if for any two finite disjoint subsets F , G of X we have ∏ x∈F x ∏ y∈G −y≠0. The independence of a BA A , denoted by Ind, is the supremum of cardinalities of its independent subsets. We can also consider the maximal independent subsets. The smallest size of an infinite maximal independent subset is the cardinal invariant i , well known in the case A= (...) P / fin . In this article we consider the collection of all cardinalities of infinite maximal independent subsets of a BA A ; we call this set the spectrum of infinite maximal independent subsets , denoted by Spind. Note that infinite maximal independent subsets exist in any BA which is not superatomic. The main result is that any set of infinite cardinals can occur as Spind for some infinite BA A . Beyond this we give results concerning the way that Spind changes under various algebraic operations. However, the basic components of most algebras that we deal with are free algebras. (shrink)

Accounts of logical independence which coincide when applied in the case of classical logic diverge elsewhere, raising the question of what a satisfactory all-purpose account of logical independence might look like. ‘All-purpose’ here means: working satisfactorily as applied across different logics, taken as consequence relations. Principal candidate characterizations of independence relative to a consequence relation are that there the consequence relation concerned is determined by only by classes of valuations providing for all possible truth-value combinations for the (...) formulas whose independence is at issue, and that the consequence relation ‘says’ nothing special about how those formulas are related that it does not say about arbitrary formulas. Each of these proposals returns counterintuitive verdicts in certain cases—the truth-value inspired approach classifying certain cases one would like to describe as involving failures of independence as being cases of independence, and the de Jongh approach counting some intuitively independent pairs of formulas as not being independent after all. In final section, a modification of the latter approach is tentatively sketched to correct for these misclassifications. The attention is on conceptual clarification throughout, rather than the provision of technical results. Proofs, as well as further elaborations, are lodged in the ‘longer notes’ in a final Appendix. (shrink)

We prove two results intended to streamline proofs about cellularity that pass through mutual algebraicity. First, we show that a countable structure M is cellular if and only if M is \-categorical and mutually algebraic. Second, if a countable structure M in a finite relational language is mutually algebraic non-cellular, we show it admits an elementary extension adding infinitely many infinite MA-connected components. Towards these results, we introduce MA-presentations of a mutually algebraic structure, in which every atomic (...) formula is mutually algebraic. This allows for an improved quantifier elimination and a decomposition of the structure into independent pieces. We also show this decomposition is largely independent of the MA-presentation chosen. (shrink)

We introduce multifunction algebras B i τ where τ is a set of 0 or 1-ary terms used to bound recursion lengths. We show that if for all ℓ ∈ τ we have ℓ ∈ O then B i τ = FP Σ i−1 p , those multifunctions computable in polynomial time with at most O )) queries to a Σ i−1 p witness oracle for ℓ ∈ τ and p a polynomial. We use our algebras to obtain independence (...) results in bounded arithmetic. In particular, we show if S 2 i proves Σ j b = PH for some j⩾i then S 2 i ≼ B S 2 . This implies if P NP ≠ P NP then S 2 1 does not prove the polynomial hierarchy collapses. We then consider a subtheory, Z , of the well-studied bounded arithmetic theory S 2 = ∪ i S 2 i . Using our algebras we establish the following properties of this theory: Z cannot prove the polynomial hierarchy collapses. In fact, even Z+ Π ̂ 0 b -consequences of S 2 cannot prove the hierarchy collapses. If Z ⊆ S 2 i for any i then the polynomial hierarchy collapses. If Z proves the polynomial hierarchy is infinite then for all i , S 2 i ⊢ Σ i p ≠ Π i p. (shrink)

Two algebraic structures, the contrapositionally complemented Heyting algebra (ccHa) and the contrapositionally |$\vee $| complemented Heyting algebra (c|$\vee $|cHa), are studied. The salient feature of these algebras is that there are two negations, one intuitionistic and another minimal in nature, along with a condition connecting the two operators. Properties of these algebras are discussed, examples are given and comparisons are made with relevant algebras. Intuitionistic Logic with Minimal Negation (ILM) corresponding to ccHas and its extension |${\textrm {ILM}}$|-|${\vee }$| for (...) c|$\vee $|cHas are then investigated. Besides its relations with intuitionistic and minimal logics, ILM is observed to be related to Peirce’s logic and Vakarelov’s logic MIN. With a focus on properties of the two negations, relational semantics for ILM and |${\textrm {ILM}}$|-|${\vee }$| are obtained with respect to four classes of frames, and inter-translations between the classes preserving truth and validity are provided. ILM and |${\textrm {ILM}}$|-|${\vee }$| are shown to have the finite model property with respect to these classes of frames and proved to be decidable. Extracting features of the two negations in the algebras, a further investigation is made, following logical studies of negations that define the operators independently of the binary operator of implication. Using Dunn’s logical framework for the purpose, two logics |$K_{im}$| and |$K_{im-{\vee }}$| are discussed, where the language does not include implication. The |$K_{im}$|-algebras are reducts of ccHas and are different from relevant algebraic structures having two negations. The negations in the |$K_{im}$|-algebras and |$K_{im-{\vee }}$|-algebras are shown to occupy distinct positions in an enhanced form of Dunn’s kite of negations. Relational semantics for |$K_{im}$| and |$K_{im-{\vee }}$| is provided by a class of frames that are based on Dunn’s compatibility frames. It is observed that this class coincides with one of the four classes giving the relational semantics for ILM and |${\textrm {ILM}}$|-|${\vee }$|⁠. (shrink)

In this paper we extend to non-classical set theories the standard strategy of proving independence using Boolean-valued models. This extension is provided by means of a new technique that, combining algebras (by taking their product), is able to provide product-algebra-valued models of set theories. In this paper we also provide applications of this new technique by showing that: (1) we can import the classical independence results to non-classical set theory (as an example we prove the independence of (...) $\mathsf {CH}$ ); and (2) we can provide new independence results. We end by discussing the role of non-classical algebra-valued models for the debate between universists and multiversists and by arguing that non-classical models should be included as legitimate members of the multiverse. (shrink)

For (finitary) deductive systems, we formulate a signature‐independent abstraction of the weak excluded middle law (WEML), which strengthens the existing general notion of an inconsistency lemma (IL). Of special interest is the case where a quasivariety algebraizes a deductive system ⊢. We prove that, in this case, if ⊢ has a WEML (in the general sense) then every relatively subdirectly irreducible member of has a greatest proper ‐congruence; the converse holds if ⊢ has an inconsistency lemma. The result extends, in (...) a suitable form, to all protoalgebraic logics. A super‐intuitionistic logic possesses a WEML iff it extends. We characterize the IL and the WEML for normal modal logics and for relevance logics. A normal extension of has a global consequence relation with a WEML iff it extends, while every axiomatic extension of with an IL has a WEML. (shrink)

The notion of shape invariance in supersymmetric quantum mechanics is examined in relation with the generalized oscillator algebra. Shape invariance is reformulated as fermion-number independence of a parameter function and seen as a symmetry under a shape-related parameter transformation. It is also shown how shape invariance is implied in the dynamical group approach.

We deepen the study of conjoined and disjoined conditional events in the setting of coherence. These objects, differently from other approaches, are defined in the framework of conditional random quantities. We show that some well known properties, valid in the case of unconditional events, still hold in our approach to logical operations among conditional events. In particular we prove a decomposition formula and a related additive property. Then, we introduce the set of conditional constituents generated by $n$ conditional events and (...) we show that they satisfy the basic properties valid in the case of unconditional events. We obtain a generalized inclusion-exclusion formula, which can be interpreted by introducing a suitable distributive property. Moreover, under logical independence of basic unconditional events, we give two necessary and sufficient coherence conditions. The first condition gives a geometrical characterization for the coherence of prevision assessments on a family F constituted by n conditional events and all possible conjunctions among them. The second condition characterizes the coherence of prevision assessments defined on $F\cup K$, where $K$ is the set of conditional constituents associated with the conditional events in $F$. Then, we give some further theoretical results and we examine some examples and counterexamples. Finally, we make a comparison with other approaches and we illustrate some theoretical aspects and applications. (shrink)

It is proposed in this paper to develop a method by which the most general problem of the algebra of propositions is solved. This problem is to construct all propositions whose truth is independent of the form of the variables. As might be expected this method will enable us to determine without the use of matrices the consistency and independence of propositions, except in the case of those fundamental properties, which taken together define consistency itself. In the discussion which (...) follows no limitations are placed on the meaning of the terms and the word proposition is used simply in the sense of a variable. If we were to say, for example, that our variables must be restricted to zero or one values, we would be introducing a limitation which would reduce the generality of our science. A logic may be false in the sense that its postulates lead to contradictory theorems, or it may be false in the sense that it lacks generality, and the situation is not saved by calling it a two-valued, or a three-valued logic, or a logic of “propositions,” or by designating it by some other euphuism. (shrink)

We study expansions of an algebraically closed field K or a real closed field R with a linearly independent subgroup G of the multiplicative group of the field or the unit circle group \\), satisfying a density/codensity condition. Since the set G is neither algebraically closed nor algebraically independent, the expansion can be viewed as “intermediate” between the two other types of dense/codense expansions of geometric theories: lovely pairs and H-structures. We show that in both the algebraically closed field and (...) real closed field cases, the resulting theory is near model complete and the expansion preserves many nice model theoretic conditions related to the complexity of definable sets such as stability and NIP. (shrink)

The notion of absolute independence, considered in this paper has a clear algebraic meaning and is a strengthening of the usual notion of logical independence. We prove that any consistent and countable set in classical prepositional logic has an absolutely independent axiornatization.

Heinrich Behmann (1891-1970) obtained his Habilitation under David Hilbert in Göttingen in 1921 with a thesis on the decision problem. In his thesis, he solved - independently of Löwenheim and Skolem's earlier work - the decision problem for monadic second-order logic in a framework that combined elements of the algebra of logic and the newer axiomatic approach to logic then being developed in Göttingen. In a talk given in 1921, he outlined this solution, but also presented important programmatic remarks on (...) the significance of the decision problem and of decision procedures more generally. The text of this talk as well as a partial English translation are included. (shrink)

The projection latticesP(ℳ1),P(ℳ2) of two von Neumann subalgebras ℳ1, ℳ2 of the von Neumann algebra ℳ are defined to be logically independent if A ∧ B≠0 for any 0≠AεP(ℳ1), 0≠BP(ℳ2). After motivating this notion in independence, it is shown thatP(ℳ1),P(ℳ2) are logically independent if ℳ1 is a subfactor in a finite factor ℳ andP(ℳ1),P(ℳ2 commute. Also, logical independence is related to the statistical independence conditions called C*-independence W*-independence, and strict locality. Logical independence ofP(ℳ1,P(ℳ2 turns (...) out to be equivalent to the C*-independence of (ℳ1,ℳ2) for mutually commuting ℳ1,ℳ2 and it is shown that if (ℳ1,ℳ2) is a pair of (not necessarily commuting) von Neumann subalgebras, thenP(ℳ1,P(ℳ2 are logically independent in the following cases: ℳ is a finite-dimensional full-matrix algebra and ℳ1,ℳ2 are C*-independent; (ℳ1,ℳ2) is a W*-independent pair; ℳ1,ℳ2 have the property of strict locality. (shrink)

What makes our utterances mean what they do? In this work I formulate and justify a structural constraint on possible answers to this key question in the philosophy of language, and I show that accepting this constraint leads naturally to the adoption of an algebraic formalization of truth-theoretic semantics. I develop such a formalization, and show that applying algebraic methodology to the theory of meaning yields important insights into the nature of language. ;The constraint I propose is, roughly, (...) this: the meaning of an utterance of a natural language sentence is derived from the place of that sentence within a system that includes other sentences, and is structured in virtue of primitive semantic facts about complete sentences. This constraint calls for a formal semantics program where the meaning of natural language sentences is represented by an appropriate assignment of the elements of an algebraic structure to sentences, an assignment that represents facts concerning complete sentences. I call the above constraint 'The Algebraic Thesis', and the corresponding formal semantics program 'Algebraic Semantics.' ;I introduce AT by carving it out of the holistic views of language of Quine and Davidson, and I show that it is independent of other important features of these philosophers' views. I then argue for AT, taking into account recent work by Brandom, Dummett, Fodor and Lepore, Gaifman and Perry. In developing Algebraic Semantics I present structures of two basic types: Boolean algebras and cylindric algebras; these structures are construed in the context of AT in a novel way: they are treated as applying to classes of sentences as numbers apply to physical objects in measurement. Applying algebraic methodology to formal semantics and to the theory of meaning, I develop these results: ; An account of the semantics of necessity and possibility that avoids both Quine's dismissal of these notions and their reduction by David Lewis to quantification over possible worlds; ; The application of the model-theoretic notion of expansion to capture the process of our learning progressively more of the meaning of expressions in our first languages. (shrink)

In this paper, we study the field of algebraic numbers with a set of elements of small height treated as a predicate. We prove that such structures are not simple and have the independence property. A real algebraic integer is called a Salem number if α and are Galois conjugate and all other Galois conjugates of α lie on the unit circle. It is not known whether 1 is a limit point of Salem numbers. We relate the (...) simplicity of a certain pair with Lehmer's conjecture and obtain a model‐theoretic characterization of Lehmer's conjecture for Salem numbers. (shrink)

It was proved recently that Telgársky’s conjecture, which concerns partial information strategies in the Banach–Mazur game, fails in models of \. The proof introduces a combinatorial principle that is shown to follow from \, namely: \::Every separative poset \ with the \-cc contains a dense sub-poset \ such that \ for every \. We prove this principle is independent of \ and \, in the sense that \ does not imply \, and \ does not imply \ assuming the consistency (...) of a huge cardinal. We also consider the more specific question of whether \ holds with \ equal to the weight-\ measure algebra. We prove, again assuming the consistency of a huge cardinal, that the answer to this question is independent of \. (shrink)

We start from the assumption that the real valued observables of a quantum system form a Jordan algebra which is equipped with a compatible Lie product characterizing infinitesimal symmetries, and ask whether two such systems can be considered as independent subsystems of a larger system. We show that this is possible if and only if the associator of the Jordan product is a fixed multiple of the associator of the Lie product. In this case it is known that the two (...) products can be combined to an associative product in the Jordan algebra or its complexification, depending on the sign of the multiple. (shrink)

This is an expository paper on the problem of independent axiomatization of any set of sentences. This subject was investigated in 50's and 60's, and was abandoned later on, though not all fundamental questions were settled then. Besides, some papers written at that time are hardly available today and there are mistakes and misunderstandings there. We would like to get back to that unfinished business to clarify the subject matter, correct mistakes and answer questions left open by others. We shall (...) deal with results of many authors. However, they will be exposed in a different manner, with complete proofs and, often, with refinements and supplements. Some questions will be brought up to date and related to other questions in logic. New results and questions will also be added. Stress will be laid on constructive aspects; that is, we will examine the problem of the possibility of independent axiomatization as well as algebraic means by use of which independent sets of axioms can be given. (shrink)

In an earlier paper the present author proved that de Finetti coherence is preserved under taking products of coherent books on two finite sets of independent events. Conversely, in this note it is proved that product is the only coherence preserving operation on coherent books. Our proof shows that the traditional definition of stochastically independent classes of events actually follows from the combination of two more basic notions: boolean algebraic independence and de Finetti coherent betting system.

Let ${\mathbb{B}}$ and ${\mathbb{C}}$ be Boolean algebras and ${e: \mathbb{B}\rightarrow \mathbb{C}}$ an embedding. We examine the hierarchy of ideals on ${\mathbb{C}}$ for which ${ \bar{e}: \mathbb{B}\rightarrow \mathbb{C} / \fancyscript{I}}$ is a regular (i.e. complete) embedding. As an application we deal with the interrelationship between ${\fancyscript{P}(\omega)/{{\rm fin}}}$ in the ground model and in its extension. If M is an extension of V containing a new subset of ω, then in M there is an almost disjoint refinement of the family ([ω]ω) V (...) . Moreover, there is, in M, exactly one ideal ${\fancyscript{I}}$ on ω such that ${(\fancyscript{P}(\omega)/{{\rm fin}})^V}$ is a dense subalgebra of ${(\fancyscript{P}(\omega)/\fancyscript{I})^M}$ if and only if M does not contain an independent (splitting) real. We show that for a generic extension V[G], the canonical embedding $$\fancyscript{P}^V(\omega){/}{\rm fin}\hookrightarrow \fancyscript{P}(\omega){/}(U(Os)(\mathbb{B}))^G$$ is a regular one, where ${U(Os)(\mathbb{B})}$ is the Urysohn closure of the zero-convergence structure on ${\mathbb{B}}$. (shrink)

Abelian Logic is a paraconsistent logic discovered independently by Meyer and Slaney [10] and Casari [2]. This logic is also referred to as Abelian Group Logic (AGL) [12] since its set of theorems is sound and complete with respect to the class of Abelian groups. In this paper we investigate the pure implication fragment A→ of Abelian logic. This is an extension of the implication fragment of linear logic, BCI. A Hilbert style axiomatic system for A→ can obtained by adding (...) the axiom A (dubbed the ‘axiom of relativity’ by Meyer and Slaney) to BCI, as follows: B (α → β) → ((γ → α) → (γ → β)) C (α → (β → γ)) → ((β → (α → γ)) I α → α A ((α → β) → β) → α MP α, α → β ⇒ β. (shrink)

We give an exposition of results of Baldwin–Shelah [2] on saturated free algebras, at the level of generality of complete first order theoriesTwith a saturated modelMwhich is in the algebraic closure of an indiscernible set. We then make some new observations whenM isa saturated free algebra, analogous to (more difficult) results for the free group, such as a description of forking.