Logic first order

Of the many variants of logic, one of the most important is first-order predicate logic. Facts in predicate logic are expressed using variables, constants, predicates, and quantifiers. For example, consider the facts shown below ... 

In statements 1 and 2, "For all" is a quantifier, specifically, the universal quantifier Water is a constant A is a variable and Liquid and Flow are predicates. Both statements 1 and 2 are asserted to be tme. First-order predicate logic gets its name because predicates such as Liquid and Flow are allowed, and quantification is only allowed on variables, not predicates. Having represented these facts in the logic system, the system can then be queried ... 

This expression suggests a rate-controlling step in which RM reacts with an intermediate. If so, [Int] °c [RM] /2. To be consistent with this, the initiation step should be first-order in [RM] and the termination step second-order in [Int]. Since O2 is not involved in the one propagation step deduced, it must appear in the other, because it is consumed in the overall stoichiometry. On the other hand, given that one RM is consumed by reaction with the intermediate, another cannot be introduced in the second propagation step, since the stoichiometry [Eq. (8-3)] would disallow that. Further, we know that the initiation and propagation steps are not the reverse of one another, since the system is not well-behaved. From this logic we write this skeleton ... 

It is beyond the scope of this chapter to explain the syntax and semantics of first-order predicate logic, and the reader is referred to Lloyd s text (1987) for a general introduction. However, it is useful to provide some details on the horn clause form. 

Theoretically it has been shown (Thayse, 1988) that the DDP formalism is closely related to a simpler form of horn clause logic, i.e., the propositional calculus. This would suggest that we could use the horn clause form to express some of the types of knowledge we are required to manipulate in combinatorial optimization problems. The explicit inclusion of state information into the representation, necessitates the shift from the simpler propositional form, to the first-order form, since we wish to parsimoniously represent properties that can be true, or take different values, in different states. By limiting the form to horn clauses, we are striving to retain the maximum simplicity of representation, whilst admitting the necessary expressive power. 

First-order predicate calculus admits proof techniques that can be shown to be sound and complete (Lloyd, 1987). The soundness of the proof technique is important because it ensures that our methodology will not deduce results that are invalid. We are less concerned with completeness, because in most cases, although the proof technique will be complete, the theory of dominance or equivalence we have available will be incomplete for most problems. Restricting the first-order logic to be of horn clause form, enables the employment of SLD resolution, a simpler... 

First-order horn clause logic is the representation that has been adopted by workers in the field of explanation-based learning (Minton et al., 1990). Hence, using this representation allows us take advantage of the results and algorithms developed in that field to carry out the machine learning task. 

This is a linear equation if k and x are constants. Now if we follow the logic in Section 2.8.2, we should find that the time constant of a CSTR with a first order reaction is x/(l + kx). 

A second fluorine substituent shields in the ortho- and especially in the para-position, but one in the meta-position deshields, with 1,3-5-trifluorobenzene having the most deshielded fluorines in a polyfluoro-aromatic system (Scheme 3.58). On the other hand, hexafluorobenzene has highly shielded fluorines. The fluorine spectra of these multifluoro-benzenes are second order in nature and their appearance is thus not generally predicable on the basis of first-order logic. 

In this chapter we discuss techniques for program verification and their mathematical justification. The basic idea behind these methods was originally presented by Floyd mathematical formulations and logical justifications were developed by Cooper and Manna, and others, and continued in King s Ph.D. thesis in which he presented the development of a partial implementation for these techniques. A sanewhat different axiomatic approach has been pursued by Hoare et al. The reader who has never made acquaintance with the formalism of the first order predicate calculus should at this point turn to Appendix A for a brief and unrigorous exposition of the material relevant to this chapter. 

Gries95] Gries, D., and F. B. Schneider, Eds. 1995. First-Order Logic and Automated Theorem Proving, 2d ed. New York Springer-Verlag. 

If the aim of an investigator is to determine equilibrium concentrations in samplers, then the residence time (tm) is a logical parameter to compare among samplers. The tm is the mean length of time that a molecule spends in a passive sampling device, where solute exchange follows first-order kinetics. Residence time is given by... 

We chose the first order predicate calculus (PC) as our language for representing synthetic principles. The first order predicate calculus (PC) is a "formal" system of logic.(11)(12)(13) In this context, formal means that it is the form of the arguments that is important, not the actual content. The term "calculus" comes from the meaning "a method of calculation", and does not refer to Newton s differential calculus. 

A. Margaris "First Order Mathematical Logic Blaisdell Publishing Company 1967. 

Fuzzy logic represents a quantification of the first-order predicate logic that is the backbone of KBES. Given that quantification, fuzzy logic improves the KBES capability to solve problems such as car steering [33] and parallel parking. For the latter, the ill-parked car... 

Graphic Method A plot of the data can be used to ascertain the order. If a plot of concentration versus time yields a straight line, the reaction is zero order. A straight line from the plot of logic/ - x) versus time is first order and second order if the plot of 1 /(a - xf versus time is a straight line (where the initial concentrations are equal). 

Remark 1 The mathematical model is an MINLP problem since it has both continuous and binary variables and nonlinear objective function and constraints. The binary variables participate linearly in the objective and logical constraints. Constraints (i), (iv), (vii), and (viii) are linear while the remaining constraints are nonlinear. The nonlinearities in (ii), (iii), and (vi) are of the bilinear type and so are the nonlinearities in (v) due to having first-order reactions. The objective function also features bilinear and trilinear terms. As a result of these nonlinearities, the model is nonconvex and hence its solution will be regarded as a local optimum unless a global optimization algorithm is utilized. 

When several temperature-dependent rate constants have been determined or at least estimated, the adherence of the decay in the system to Arrhenius behavior can be easily determined. If a plot of these rate constants vs. reciprocal temperature (1/7) produces a linear correlation, the system is adhering to the well-studied Arrhenius kinetic model and some prediction of the rate of decay at any temperature can be made. As detailed in Figure 17, Carstensen s adaptation of data, originally described by Tardif (99), demonstrates the pseudo-first-order decay behavior of the decomposition of ascorbic acid in solid dosage forms at temperatures of 50° C, 60°C, and 70°C (100). Further analysis of the data confirmed that the system adhered closely to Arrhenius behavior as the plot of the rate constants with respect to reciprocal temperature (1/7) showed linearity (Fig. 18). Carsten-sen suggests that it is not always necessary to determine the mechanism of decay if some relevant property of the degradation can be explained as a function of time, and therefore logically quantified and rationally predicted. 

If you recall, back in Chapter 5 we discussed half-life in the context of the decay of radioactive nuclei. In that chapter, we defined the half-life as the amount of time it took for one half of the original sample of radioactive nuclei to decay. Because the rate of decay only depends on the amount of the radioactive sample, it is considered a first-order process. Using the same logic, we can apply the concept of half-life to first-order chemical reactions as well. In this new context, the half-life is the amount of time required for the concentration of a reactant to decrease by one-half. The half-life equation from Chapter 5 can be used to determine the half-life of a reactant ... 

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