The
treatment and recuperation of hydric resources portray recurrent
challenges and demand alternative processes and technologies. The
removal of hardness and metal ions from water are necessary in certain
sources of human consumption and for industrial purposes. In this
work, sodium-modified vermiculite was investigated toward the removal
of calcium ions in aqueous medium. The effects of adsorbent dosage,
contact time, initial concentration of calcium ions, initial pH, and
temperature were systematically studied for both raw and modified
vermiculite. pH 10 was shown to be the most appropriate to conduct
the experiments. Raw, sodium-modified, and postadsorption vermiculite
were characterized by microstructure analysis (scanning electron microscopy,
Fourier transform infrared spectroscopy, X-ray fluorescence, and X-ray
diffraction). The maximum adsorption capacity was found in sodium-modified
vermiculite, resulting in about 80 mg g–1. Kinetic
studies were carried out to relate the experimental data to pseudo-first-order,
pseudo-second-order, and Elovich models. Isotherm models (Freundlich,
Langmuir, and Redlich–Peterson) were employed to describe the
softening process, and thermodynamic parameters ΔG°, ΔH°, and ΔS° were determined. Results revealed a spontaneous endothermic
adsorption process with pseudo-second-order kinetics and corroborated
the sodium-modified vermiculite adequacy for water softening.
The characterizing properties of a proof-theoretical presentation of a given logic may hang on the choice of proof formalism, on the shape of the logical rules and of the sequents manipulated by a given proof system, on the underlying notion of consequence, and even on the expressiveness of its linguistic resources and on the logical framework into which it is embedded. Standard (one-dimensional) logics determined by (non-deterministic) logical matrices are known to be axiomatizable by analytic and possibly finite proof systems as soon as they turn out to satisfy a certain constraint of sufficient expressiveness. In this paper we introduce a recipe for cooking up a two-dimensional logical matrix (or -matrix) by the combination of two (possibly partial) non-deterministic logical matrices. We will show that such a combination may result in -matrices satisfying the property of sufficient expressiveness, even when the input matrices are not sufficiently expressive in isolation, and we will use this result to show that one-dimensional logics that are not finitely axiomatizable may inhabit finitely axiomatizable two-dimensional logics, becoming, thus, finitely axiomatizable by the addition of an extra dimension. We will illustrate the said construction using a well-known logic of formal inconsistency called mCi. We will first prove that this logic is not finitely axiomatizable by a one-dimensional (generalized) Hilbert-style system. Then, taking advantage of a known 5-valued non-deterministic logical matrix for this logic, we will combine it with another one, conveniently chosen so as to give rise to a -matrix that is axiomatized by a two-dimensional Hilbert-style system that is both finite and analytic.
The bilateralist approach to logical consequence maintains that judgments of different qualities should be taken into account in determining what-follows-from-what. We argue that such an approach may be actualized by a two-dimensional notion of entailment induced by semantic structures that also accommodate non-deterministic and partial interpretations, and propose a prooftheoretical apparatus to reason over bilateralist judgments using symmetrical two-dimensional analytical Hilbert-style calculi. We also provide a proof-search algorithm for finite analytic calculi that runs in at most exponential time, in general, and in polynomial time when only rules having at most one formula in the succedent are present in the concerned calculus.
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