Online research in philosophy

In this paper we consider properties, related to model-completeness, of the theory of integrally closed commutative regular rings. We obtain the main theorem claiming that in a Boolean algebra B, the truth of a prenex Σn-formula whose parameters ai partition B, can be determined by finitely many conditions built from the first entry of Tarski invariant T(ai)'s, n-characteristic D(n, ai)'s and the quantities S(ai, l) and S'(ai, l) for $l < n$. Then we derive two important theorems. One (...) claims that for any Boolean algebras A and B, an embedding of A into B preserving D(n, a) for all a ∈ A is a Σn-extension. The other claims that the theory of n-separable Boolean algebras admits elimination of quantifiers in a simple definitional extension of the language of Boolean algebras. Finally we translate these results into the language of commutative regular rings. (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)

Summary In especially the Sneed-Stegmüller structuralist theory a so-called problem of theoretical terms emerges. But this problem bases on a questionable presupposition (Introductory remark). And the structuralist solution of this problem, the so-called Ramsey-Sneed-solution, is also problematic ((i), (ii), (iii)). Beyond this the structuralist assertion is problematic, that the problem of theoretical terms and his Ramsey-Sneed-solution is empirically relevant (iv). On the basis of the discussed systematic and empirical defects of the problem of theoretical terms and its solution, the so-called (...) non-statement view2, i. e. this solution, will be refused (Concluding remark). (shrink)

Introduction: Middle-Earth, The lord of the rings, and international relations -- Order, justice, and Middle-Earth -- Thinking about international relations and Middle-Earth -- Middle-Earth and three great debates in international relations -- Middle-Earth, levels of analysis, and war -- Middle-Earth and feminist theory -- Middle-Earth and feminist analysis of conflict -- Middle-Earth as a source of inspiration and enrichment -- Conclusion: international relations and our many worlds.

The equivalence connective in ukasiewicz logic has its algebraic counterpart which is the distance function d(x,y) =|x–y| of a positive cone of a commutative -group. We make some observations on logically motivated algebraic structures involving the distance function.

A first-order dynamic non-commutative logic (DN), which has no structural rules and has some program operators, is introduced as a Gentzen-type sequent calculus. Decidability, cut-elimination and completeness theorems are shown for DN or its fragments. DN is intended to represent not only program-based, resource-sensitive, ordered, sequence-based, but also hierarchical (tree-based) reasoning.

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)

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 commutative BCK-algebra with the relative cancellation property is a commutative BCK-algebra (X;*,0) which satisfies the condition: if a ≤ x, a ≤ y and x * a = y * a, then x = y. Such BCK-algebras form a variety, and the category of these BCK-algebras is categorically equivalent to the category of Abelian ℓ-groups whose objects are pairs (G, G 0), where G is an Abelian ℓ-group, G 0 is a subset of the positive cone generating (...) G + such that if u, v ∈ G 0, then 0 ∨ (u - v) ∈ G 0, and morphisms are ℓ-group homomorphisms h: (G, G 0) → (G′,G′0) with f(G 0) ⫅ G′0. Our methods in particular cases give known categorical equivalences of Cornish for conical BCK-algebras and of Mundici for bounded commutative BCK-algebras (= MV-algebras). (shrink)

We recall the notions of weak and strong Euler characteristics on a first order structure and make explicit the notion of a Grothendieck ring of a structure. We define partially ordered Euler characteristic and Grothendieck ring and give a characterization of structures that have non-trivial partially ordered Grothendieck ring. We give a generalization of counting functions to locally finite structures, and use the construction to show that the Grothendieck ring of the complex numbers contains as a subring the ring of (...) integer polynomials in continuum many variables. We prove the existence of a universal strong Euler characteristic on a structure. We investigate the dependence of the Grothendieck ring on the theory of the structure and give a few counter-examples. Finally, we relate some open problems and independence results in bounded arithmetic to properties of particular Grothendieck rings. (shrink)

This paper presents a new correctness criterion for marked Danos-Reginer graphs (D-R graphs, for short) of Multiplicative Cyclic Linear Logic MCLL and Abrusci's non-commutative Linear Logic MNLL. As a corollary we obtain an affirmative answer to the open question whether a known quadratic-time algorithm for the correctness checking of proof nets for MCLL and MNLL can be improved to linear-time.

The paper describes the isomorphic lattices of quasivarieties of commutative quasigroup modes and of cancellative commutative binary modes. Each quasivariety is characterised by providing a quasi-equational basis. A structural description is also given. Both lattices are uncountable and distributive.

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)

The class of all Artinian local rings of length at most l is ∀ 2 -elementary, axiomatised by a finite set of axioms Art l . We show that its existentially closed models are Gorenstein, of length exactly l and their residue fields are algebraically closed, and, conversely, every existentially closed model is of this form. The theory Got l of all Artinian local Gorenstein rings of length l with algebraically closed residue field is model complete and the (...) theory Art l is companionable, with model-companion Got l. (shrink)

We develop an elimination theory for addition and the Frobenius map over rings of polynomials. As a consequence we show that if F is a countable. recursive and perfect field of positive characteristic p, with decidable theory, then the structure of addition, the Frobenius map x $\rightarrow$ $x^{p}$ and the property 'x $\epsilon$ F', over the ring of polynomials F[T]. has a decidable theory.

We show that all QE rings of prime power characteristic are constructed in a straightforward way out of three components: a filtered Boolean power of a finite field, a nilpotent Jacobson radical, and the ring Z p n or the Witt ring W 2 (F 4 ) (which is the characteristic four analogue of the Galois field with four elements).

Let ${\mathbb{BRL}}$ denote the variety of commutative integral bounded residuated lattices (bounded residuated lattices for short). A Boolean retraction term for a subvariety ${\mathbb{V}}$ of ${\mathbb{BRL}}$ is a unary term t in the language of bounded residuated lattices such that for every ${{\bf A} \in \mathbb{V}, t^{A}}$ , the interpretation of the term on A, defines a retraction from A onto its Boolean skeleton B(A). It is shown that Boolean retraction terms are equationally definable, in the sense that there (...) is a variety ${\mathbb{V}_{t} \subsetneq \mathbb{BRL}}$ such that a variety ${\mathbb{V} \subsetneq \mathbb{BRL}}$ admits the unary term t as a Boolean retraction term if and only if ${\mathbb{V} \subseteq \mathbb{V}_{t}}$ . Moreover, the equation s(x) = t(x) holds in ${\mathbb{V}_{s} \cap \mathbb{V}_{t}}$ . The radical of ${{\bf A} \in \mathbb{BRL}}$ , with the structure of an unbounded residuated lattice with the operations inherited from A expanded with a unary operation corresponding to double negation and a a binary operation defined in terms of the monoid product and the negation, is called the radical algebra of A. To each involutive variety ${\mathbb{V} \subseteq \mathbb{V}_{t}}$ is associated a variety ${\mathbb{V}^{r}}$ formed by the isomorphic copies of the radical algebras of the directly indecomposable algebras in ${\mathbb{V}}$ . Each free algebra in such ${\mathbb{V}}$ is representable as a weak Boolean product of directly indecomposable algebras over the Stone space of the free Boolean algebra with the same number of free generators, and the radical algebra of each directly indecomposable factor is a free algebra in the associated variety ${\mathbb{V}^{r}}$ , also with the same number of free generators.A hierarchy of subvarieties of ${\mathbb{BRL}}$ admitting Boolean retraction terms is exhibited. (shrink)

We calculate the Cantor-Bendixson rank of the Ziegler spectrum over a commutative valuation domain R proving that it is equal to the double Krull dimension of R.

Let n ≥ 3. The following theorems are proved. Theorem. The theory of the class of strictly upper triangular n × n matrix rings over fields is finitely axiomatizable. Theorem. If R is a strictly upper triangular n × n matrix ring over a field K, then there is a recursive map σ from sentences in the language of rings with constants for K into sentences in the language of rings with constants for R such that $K (...) \vDash \varphi$ if and only if $R \vDash \sigma(\varphi)$ . Theorem. The theory of a strictly upper triangular n × n matrix ring over an algebraically closed field is ℵ 1 -categorical. (shrink)

We show that Diophantine equivalence of two suitably presented countable rings implies that the existential polynomial languages of the two rings have the same "expressive power" and that their Diophantine sets are in some sense the same. We also show that a Diophantine class of countable rings is contained completely within a relative enumeration class and demonstrate that one consequence of this fact is the existence of infinitely many Diophantine classes containing holomophy rings of Q.

Axiomatics which do not employ rules of inference other than the cut rule are given for commutative product-free Lambek calculus in two variants: with and without the empty string. Unlike the former variant, the latter one turns out not to be finitely axiomatizable in that way.

This volume is the product of the Proceedings of the 9th International Congress of Logic, Methodology and Philosophy of Science and contains the text of most of ...

The volume contains 37 invited papers presented at the Congress, covering the areas of Logic, Mathematics, Physical Sciences, Biological Sciences and the ...

We prove the triviality of the Grothendieck ring of a Z-valued field K under slight conditions on the logical language and on K. We construct a definable bijection from the plane K 2 to itself minus a point. When we specialized to local fields with finite residue field, we construct a definable bijection from the valuation ring to itself minus a point.

The field K((G)) of generalized power series with coefficients in the field K of characteristic 0 and exponents in the ordered additive abelian group G plays an important role in the study of real closed fields. Conway and Gonshor (see [2, 4]) considered the problem of existence of non-standard irreducible (respectively prime) elements in the huge "ring" of omnific integers, which is indeed equivalent to the existence of irreducible (respectively prime) elements in the ring K((G ≤ 0 )) of series (...) with non-positive exponents. Berarducci (see [1]) proved that K((G ≤ 0 )) does have irreducible elements, but it remained open whether the irreducibles are prime i.e.; generate a prime ideal. In this paper we prove that K((G ≤ 0 )) does have prime elements if G = (R, +) is the additive group of the reals, or more generally if G contains a maximal proper convex subgroup. (shrink)

One of the main theorems of the paper states the following. Let R-K-M be finite extensions of a rational one variable function field R over a finite field of constants. Let S be a finite set of valuations of K. Then the ring of elements of K having no poles outside S has a Diophantine definition over its integral closure in M.

The aim of this paper is to provide an addendum to a paper by Rose with the same title which has appeared in an earlier issue of this Journal [2]. Our new result is: Theorem. A ring of characteristic zero which admits elimination of quantifiers in the language {0, 1, +, ·} is an algebraically closed field.

Some works of fiction are widely held by critics to have little value, yet these works are not only popular but also widely admired in ways that are not always appreciated. In this paper I make use of Kendall Walton’s account of fictional worlds to argue that fictional worlds can and often do have value, including aesthetic value, that is independent of the works that create them. In the process, I critique Walton’s notion of fictional worlds and offer a defense (...) of the study and appreciation of fictional worlds, as distinguished from the works of fiction with which they are associated. (shrink)

In a number of publications, John Earman has advocated a tertium quid to the usual dichotomy between substantivalism and relationism concerning the nature of spacetime. The idea is that the structure common to the members of an equivalence class of substantival models is captured by a Leibniz algebra which can then be taken to directly characterize the intrinsic reality only indirectly represented by the substantival models. An alleged virtue of this is that, while a substantival interpretation of spacetime theories falls (...) prey to radical local indeterminism, the Leibniz algebras do not. I argue that the program of Leibniz algebras is subject to radical local indeterminism to the same extent as substantivalism. In fact, for the category of topological spaces of interest in spacetime physics, the program is equivalent to the original spacetime approach. Moreover, the motivation for the program--that isomorphic substantival models should be regarded as representing the same physical situation--is misguided. (shrink)

Royce’s sustained interest in technical logic is beyond doubt. One of his first publications, which appeared while he was still teaching at the University of California at Berkeley, was a logic primer, and many of the productions of his later career were articles on logic. Indeed, it can well seem that Royce spent at least ten or eleven years working almost exclusively on logic following his attendance at Peirce’s 1898 Cambridge Conference Lectures, entitled Reasoning and the Logic of Things. During (...) this period he filled dozens of notebooks with minute explorations of Boolean functions and relations, investigating them mostly by using fourcircle Venn diagrams. Less obvious than Royce’s devotion to logic .. (shrink)

Logic, Methodology and Philosophy of Science VII.

This volume draws together contributions from researchers whose work builds on the theory developed by Martin-Lof over the last twenty-five years.

v. 1-3. Section papers, plenary sessions papers -- [v. 4] Abstracts.

Let K be a function field of one variable over a constant field C of finite transcendence degree over C. Let M/K be a finite extension and let W be a set of primes of K such that all but finitely many primes of W do not split in the extension M/K. Then there exists a set W' of K-primes such that Hilbert's Tenth Problem is not decidable over $O_{K,W'} = \{x \in K\mid ord_\mathfrak{p} x \geq 0, \forall\mathfrak{p} \notin W'\}$ (...) , and the set (W' $\backslash$ W) ∪ (W $\backslash$ W') is finite. Let K be a function field of one variable over a constant field C finitely generated over Q. Let M/K be a finite extension and let W be a set of primes of K such that all but finitely many primes of W do not split in the extension M/K and the degree of all the primes in W is bounded by b ∈ N. Then there exists a set W' of K-primes such that Z has a Diophantine definition over O K ,W', and the set (W' $\backslash$ W) ∪ (W $\backslash$ W') is finite. (shrink)