Restricted Closure Research Articles

Questions, topics and restricted closure

Single-premise epistemic closure is the principle that: if one is in an evidential position to know that \(\varphi\) where \(\varphi\) entails \(\psi\), then one is in an evidential position to know that \(\psi\). In this paper, I defend the viability of opposition to closure. A key task for such an opponent is to precisely formulate a restricted closure principle that remains true to the motivations for abandoning unrestricted closure but does not endorse particularly egregious instances of closure violation. I focus on two brands of epistemic theory (each the object of sustained recent interest in the literature) that naturally incorporate closure restrictions. The first type holds that the truth value of a knowledge ascription is relative to a relevant question. The second holds that the truth value of a knowledge ascription is relative to a relevant topic. For each approach, I offer a formalization of a leading theory from the literature (respectively, that of Jonathan Schaffer and that of Stephen Yablo) and use this formalization to evaluate the theory’s adequacy in terms of a precise set of desiderata. I conclude that neither theory succeeds in meeting these desiderata, casting doubt on the viability of the underlying approaches. Finally, I offer a novel variant of the topic-sensitive approach that fares better.

Paradox, Closure and Indirect Speech Reports

Bradwardine’s solution to the the logical paradoxes depends on the idea that every sentence signifies many things, and its truth depends on things’ being wholly as it signifies. This idea is underpinned by his claim that a sentence signifies everything that follows from what it signifies. But the idea that signification is closed under entailment appears too strong, just as logical omniscience is unacceptable in the logic of knowledge. What is needed is a more restricted closure principle. A clue can be found in speech act pluralism, whereby indirect speech reports are closed under intersubstitutivity of co-referential terms. The conclusion is that solving the semantic paradoxes does not require revision of logic, thus saving logic from paradox.

Closure of A Priori Knowability Under A Priori Knowable Material Implication

The topic of this article is the closure of a priori knowability under a priori knowable material implication: if a material conditional is a priori knowable and if the antecedent is a priori knowable, then the consequent is a priori knowable as well. This principle is arguably correct under certain conditions, but there is at least one counterexample when completely unrestricted. To deal with this, Anderson proposes to restrict the closure principle to necessary truths and Horsten suggests to restrict it to formulas that belong to less expressive languages. In this article it is argued that Horsten’s restriction strategy fails, because one can deduce that knowable ignorance entails necessary ignorance from the closure principle and some modest background assumptions, even if the expressive resources do not go beyond those needed to formulate the closure principle itself. It is also argued that it is hard to find a justification for Anderson’s restricted closure principle, because one cannot deduce it even if one assumes very strong modal and epistemic background principles. In addition, there is an independently plausible alternative closure principle that avoids all the problems without the need for restriction.

Abstract 3382: Three-Dimensional Color Doppler Echocardiography for Direct Measurement of Mitral Regurgitation Vena Contracta Area: A Clinical Experience

Introduction: 3D color Doppler allows a direct measurement of Vena Contracta (VC) area in patients with mitral regurgitation (MR), without the need for geometric assumptions. Hypothesis: 3D color Doppler VC area provides a more accurate assessment of MR severity than the conventional 2D color Doppler VC diameter. Methods: Patients with ≥ mild MR underwent 2D and 3D color Doppler and a quantitative echo-Doppler study. 2D VC diameter was measured from a zoomed parasternal long-axis view. 3D VC area was acquired using real-time 3D (Philips) and measured off-line (TomTec). MR severity was assessed both quantitatively, as effective regurgitant orifice area from pulsed Doppler [EROA = (mitral stroke volume - aortic stroke volume)/MR time velocity integral], and semi-quantitatively as recommended by the American Society of Echocardiography. Results: 61 patients (65 ± 15 years) were studied. MR severity was mild in 25%, moderate in 36%, and moderate to severe or severe in 39%. Mitral valve pathology was prolapse in 33%, restricted closure in 44% and annular calcification in 7%. Regurgitant MR jet was eccentric in 49% of patients. For all patients, 3D VC area correlated significantly with EROA, better than 2D VC diameter (figure ). The advantage of 3D VC area over 2D VC diameter was more pronounced in eccentric jets (r=0.76, p< 0.001 vs. r=0.5, p =0.006, respectively) and in moderate to severe or severe MR (r=0.65, p<0.001 vs. r =0.13, p=0.54, respectively). Conclusions : Measurement of VC area is feasible with real-time 3D and provides a simple parameter that accurately reflects MR severity, particularly in eccentric and clinically significant MR where geometric assumptions may be challenging.

A kleene-like characterization of languages accepted by systolic tree automata

We present a Kleen-like characterization of the class of languages accepted by systolic binary tree automata, L(SBTA). This characterization uses union, intersection, restricted concatenation, restricted concatenation closure, and finite substitution closure. The restrictions we impose on the operator of concatenation are purely in terms of the length of the words to be concatenated.

A characterization of constant-time cellular automata computation

This paper investigates three different classes of one-dimensional CA languages that can be accepted in constant time. For each class of the languages, two characterizations are given, one in terms of finite sets and the other in terms of a subclass of deterministic finite automata. It is also shown that the constant-time CA languages have very restricted closure properties.

Toral rank one Lie algebras

Let L be a finite-dimensional Lie algebra over an algebraically closed field F of characteristic p > 0. Assume that H is a Cartan subalgebra of L. We say that L has toral rank m relative to H if dim,PA = m, where A is the set of roots with respect to H, P is the prime field, and PA is the P-vector space spanned by A. In this work we study Lie algebras of toral rank one. The motivation to investigate this class of algebras comes from the problem of classifying simple Lie algebras. If A is a finite dimensional simple Lie algebra with Cartan subalgebra H, and if CI is a root of H, then the subalgebra A(” = CieP A, has toral rank one relative to H. In [12, 133 Wilson has shown that if a simple algebra A has characteristic p > 7, then the restricted closure w of H in Der A has the form R= T+ Z, where I is the nil radical of B and T is the maximal torus, which is central in j?. It follows that ad R on each root space A, can be simultaneously upper triangularized. Indeed, I is a Lie algebra of nilpotent transformations on A,; and thus by [8, Theorem 3.31, it annihilates a nonzero vector. The space of such vectors is T-invariant and so T + I has a common eigenvector u on A,. One can proceed inductively on AJFu to argue that each root space A, has a basis relative to which ad R is upper-triangular. With that in mind, we say in what follows that a Cartan subalgebra His triangularizable if there is a basis for each root space relative to which ad H is upper-triangular. We show in Section 1 that if L is a Lie algebra having a toral rank one Cartan subalgebra H, and if H is triangularizable, then L modulo its maximal solvable ideal either is (0) or consists of a simple toral rank one Lie algebra S together with derivations of S in H. In the first case we say that the core of L is (0) and in the second case that the core of L is S. Wilson [ 141 has classified the simple toral rank one Lie algebras. Using

Structure theory of faithful rings. II. Restricted rings

The first paper of this series(2) concerned itself mainly with closure operations on a lattice. This paper applies these results to the global structure theory of a faithful ring and its modules. A ring R is called faithful if aR = 0 implies a= 0. Let ? (?2) be the lattice of right (two-sided) ideals of R, and c={A; AC?', AnA =0, A=All}. The set i: becomes a Boolean algebra with the obvious definition of the union operation. In case 3Y is complete, R is called a restricted ring. Being complete, 9Y induces a closure operation f on ? and ?V. It is shown that if f is homogeneous, then there exist irreducible rings Ai such that i AiCR C As, where j ( *) designates the discrete (full) direct sum and 47 is a universal extension ring of Ai. A reducible ring is shown to be restricted, where R is reducible if and only if for every pair A, B of ideals of R with zero intersection, there exist ideals A'DA and B'DB also with zero intersection such that A'nB'I-0. Modules of a faithful ring are studied in the fourth section. It is shown that every suitably restricted closure operation on ? induces a closure operation on Y, the lattice of submodules of a R-module M. If 'A, A CR, designates the annihilator of A in M, then it is shown in the fifth section that 3c 'A; A EDY } is a Boolean algebra. In the final two sections, it is assumed that the ring R and the R-module M have the property that for every nonzero element x in R or M there exists a nonzero A G2 such that xa0 for every nonzero aEA. It is shown that R is restricted, aind that every A in ? (M1Z) has a unique maximal essential extension A, (At). For the closure operation s on ? so defined, 5 is proved to be the center of the lattice 28, and similarly for xC in St. Imbedding ll in its unique minimal injective extension M, it is proved that the lattice o't is isomorphic to the lattice of principal right ideals of the centralizer C of R over M. If s is atomic, e is a full direct sum of primitive rings with minimal right ideals. 1. Faithful rings. If R is a ring, then ?(R)(2'(R)) will designate the lattice of all right (two-sided) ideals of R. We shall upon occasion write S or ?' if the ring in question is obvious. For each subset A of R, AI (Ar) will