Islamic polities of the classical period recognized the importance of seaweeds in their daily life. Their men of science, craftsmen, and navigators used them for medicinal purposes, manufacturing, and navigation. The agar components were used in treating pathological conditions such jaundice, spleen, kidney and skin ailments, and malignancies. As food, we stress that our conclusions derive from Qur'ān-based commentaries and Muslim religious law that encouraged seafaring and exploiting the resources of the sea. Concerning navigation, sailors could identify coastal trunk routes, (...) shallows, and various marine phenomena; shipwrights used agar compounds as a protective coating against the Greek Fire. Like their Greco-Roman counterparts, Muslim physicians, chemists, botanists, and professional sailors of this period were acquainted with numerous species of seaweeds and could appreciate the actual scientific importance of each type as well as the aquatic environment where these species lived and developed. Their scholarly literature consists of several generic Arabic and Arabicized terms to denote seaweeds and the terms variations appeared to be physical rather than linguistic. (shrink)

The capability approach is gaining momentum as a theory of corporate responsibility and business ethics at a time when the UN Guiding Principles have become a most important framework. A novel approach is now emerging that seeks to understand and specify human rights obligations of businesses within the framework provided by the capability approach. This article partially examines the triad corporate responsibility–human rights–capability approach by exploring the relationship between human rights and capabilities. Thus, it offers conceptual and practical implications for (...) a human rights perspective on corporate responsibility. The bulk of our thesis focuses on studying Nussbaum’s and Sen’s claim that human rights are entitlements to capabilities by means of a discussion of the notion of human dignity. In particular, we show the capability approach’s ability to dissipate theoretical and practical challenges posed by the notion of dignity. This article does not offer a detailed explanation of the implications of our investigation for human rights responsibilities of business. However, an outline of its potential contribution toward unfolding those obligations, conceptually and practically, is provided. (shrink)

We prove a local normal form theorem of the Gaifman type for the infinitary logic L∞ωω whose formulas involve arbitrary unary quantifiers but finite quantifier rank. We use a local Ehrenfeucht-Fraïssé type game similar to the one in [9]. A consequence is that every sentence of L∞ωω of quantifier rank n is equivalent to an infinite Boolean combination of sentences of the form ψ, where ψ has counting quantifiers restricted to the -neighborhood of y.

This paper introduces a new Ehrenfeucht-Fraïssé type game that is played on two classes of models rather than just two models. This game extends and generalizes the known Ajtai-Fagin game to the case when there are several alternating moves played in different models. The game allows Duplicator to delay her choices of the models till the very end of the game, making it easier for her to win. This adds on the toolkit of winning strategies for Duplicator in Ehrenfeucht-Fraïssé type (...) games and opens up new methods for tackling some open problems in descriptive complexity theory. As an application of the class game, it is shown that, if m is a power of a prime, then first order logic augmented with function quantifiers F1n of arity 1 and height n < m can not express that the size of the model is divisible by m. This, together with some new expressibility results for Henkin quantifiers H1n gives some new separation results on the class of finite models among various F1n on one hand and between F1n and H1n on the other. Since function quantifiers involve a bounded type of second order existential quantifiers, the class game solves an open problem raised by Fagin, which asks for some inexpressibility result using a winning strategy by Duplicator in a game that involves more than one coloring round. (shrink)

For each truth-functionally complete set of connectives, we construct a sound and complete natural deduction system containing no axioms and the smallest possible number of inference rules, namely one.

We give an axiomatization of first‐order logic enriched with the almost‐everywhere quantifier over finitely additive measures. Using an adapted version of the consistency property adequate for dealing with this generalized quantifier, we show that such a logic is both strongly complete and enjoys Craig interpolation, relying on a (countable) model existence theorem. We also discuss possible extensions of these results to the almost‐everywhere quantifier over countably additive measures.

In [9] we introduced a new framework for asymptotic probabilities, in which a $\sigma-additive$ measure is defined on the sample space of all sequences $A = $ of finite models, where the universe of An is {1, 2, .., n}. In this framework we investigated the strong 0-1 law for sentences, which states that each sentence either holds in An eventually almost surely or fails in An eventually almost surely. In this paper we define the strong convergence law for formulas, (...) which carries over the ideas of the strong 0-1 law to formulas with free variables, and roughly states that for each formula φ(x), the fraction of tuples a in An, which satisfy the formula φ(x), almost surely has a limit as n tends to infinity. We show that the infinitary logic with finitely many variables has the strong convergence law for formulas for the uniform measure, and further characterize the measures on random graphs for which the strong convergence law holds. (shrink)

This article focuses on the idea of ‘human rights city’ and explores its practice. It starts from the concepts of human rights cities and subsidiarity to explain what a human rights city is and delves into the existing literature identifying the challenges to guarantee human rights in local contexts, such as the legal framework, education and training, the institutional structure, and the resources. Our article is based on an empirical-based study of Madrid Human Rights Plan (2017–2019). We carried out semi-structured (...) interviews, focus group, and participant observation to grasp the obstacles that civil servants encounter in the implementation of the Plan. We identify five types of obstacles that we classify as conceptual, ideological, legal, organisational, and budgetary. Our study questions the idea that the local context, just because of its proximity to the citizenry, is the best equipped level of government to guarantee human rights. Without proper training and resources human rights can be lost in translation. Through the identification of such obstacles in a specific case study, we contribute to the academic debate on human rights in practice with the aim of fostering its guarantee in local contexts. (shrink)

We introduce a new framework for asymptotic probabilities of sentences, in which we have a σ-additive measure on the sample space of all sequences A = {A n } of finite models, where the universe of A n is {1,2... n}, and use this framework to strengthen 0-1 laws for logics.

We introduce a new framework for asymptotic probabilities of sentences, in which we have a σ-additive measure on the sample space of all sequencesA= {} of finite models, where the universe ofis {1,2, …, n}. and use this framework to strengthen 0-1 laws for logics.

This paper studies the expressive power that an extra first order quantifier adds to a fragment of monadic second order logic, extending the toolkit of Janin and Marcinkowski [JM01].We introduce an operation existsn on properties S that says "there are n components having S". We use this operation to show that under natural strictness conditions, adding a first order quantifier word u to the beginning of a prefix class V increases the expressive power monotonically in u. As a corollary, if (...) the first order quantifiers are not already absorbed in V, then both the quantifier alternation hierarchy and the existential quantifier hierarchy in the positive first order closure of V are strict.We generalize and simplify methods from Marcinkowski [Mar99] to uncover limitations of the expressive power of an additional first order quantifier, and show that for a wide class of properties S, S cannot belong to the positive first order closure of a monadic prefix class W unless it already belongs to W.We introduce another operation alt on properties which has the same relationship with the Circuit Value Problem as reach has with the Directed Reachability Problem. We use alt to show that Πn⊈ FO, Σn⊈ FO, and Δn+1⊈ FOB, solving some open problems raised in [Mat98]. (shrink)

We obtain an almost everywhere quantifier elimination for (the noncritical fragment of) the logic with probability quantifiers, introduced by the first author in [10]. This logic has quantifiers like $\exists ^{ \ge 3/4} y$ which says that "for at least 3/4 of all y". These results improve upon the 0-1 law for a fragment of this logic obtained by Knyazev [11]. Our improvements are: 1. We deal with the quantifier $\exists ^{ \ge r} y$ , where y is a tuple (...) of variables. 2. We remove the closedness restriction, which requires that the variables in y occur in all atomic subformulas of the quantifier scope. 3. Instead of the unbiased measure where each model with universe n has the same probability, we work with any measure generated by independent atomic probabilities pR for each predicate symbol R. 4. We extend the results to parametric classes of finite models (for example, the classes of bipartite graphs, undirected graphs, and oriented graphs). 5. We extend the results to a natural (noncritical) fragment of the infinitary logic with probability quantifiers. 6. We allow each pR , as well as each r in the probability quantifier $(\exists ^{ \ge r} y)$ , to depend on the size of the universe. (shrink)

Gaifman's normal form theorem showed that every first-order sentence of quantifier rank n is equivalent to a Boolean combination of “scattered local sentences”, where the local neighborhoods have radius at most 7n−1. This bound was improved by Lifsches and Shelah to 3×4n−1. We use Ehrenfeucht–Fraïssé type games with a “shrinking horizon” to get a spectrum of normal form theorems of the Gaifman type, depending on the rate of shrinking. This spectrum includes the result of Lifsches and Shelah, with a more (...) easily understood proof and with the bound on the radius improved to 4n−1. We also obtain bounds for a normal form theorem of Schwentick and Barthelmann. (shrink)