Modeling propositional logic

With the basic equivalent relations given in Table I (e.g., see Williams, 1988), one can systematically model an arbitrary propositional logic expression that is given in terms of OR, AND, IMPLICATION operators, as a set of linear equality and inequality constraints. One approach is to systematically convert the logical expression into its equivalent conjunctive normal form representation, which involves the application of pure logical operations (Raman and Gross-mann, 1991). The conjunctive normal form is a conjunction of clauses, gi gj A A gj. Hence, for the conjunctive normal form to be true, each clause g, must be true independent of the others. Also, since a clause g, is just a disjunction of literals, Pj V V " V t can be expressed in the linear mathematical form as the inequality. 

Greenwood (1989) stated that the word model should be reserved for well-constrained logical propositions, not necessarily mathematical, that have necessary and testable consequences, and avoid the use of the word if we are merely constructing a scenario of possibilities. A scientific model is a testable idea, hypothesis, theory, or combination of theories that provide new insight or a new interpretation of an existing problem. An additional quality often attributed to a model or theory is its ability to explain a large number of observations while maintaining simplicity (Occam s razor). The simplest model that explains the most observations is the one that will have the most appeal and applicability. 

Baldwin recognises that it is possible to interpret the actual philosophical meaning of the fuzzy truth value restrictions in different ways for different problems. For example he argues that we may wish to think in terms of plausibility, of possibility, of importance, or of dependability as our model or interpretation of truth for a particular problem. Recalling the discussions on dependability in Sections 2.11 and 5.8, it is clear that engineers are not so much interested in the truth of a proposition but in its dependability. Whilst the fuzzy logic notation and fuzzy truth values are retained in the rest of this chapter and in Chapter 10, the interpretation should be that fuzzy truth restrictions are fuzzy restrictions on the dependability of a proposition. 

CPNs model the system structure and its dynamic behavior in the same model. The dynamic behavior is modelled thanks to token evolution. After each transition firing, some tokens are consumed and some other are produced. This notion of production/consumption cannot he expressed in classical logic, that is why the mill was preferred. On the other hand, unlike Ordinary Petri Nets, token in CPN isofacertain type (color)and belongs to a set of this type (color set) and is transformed by arc expressions. So the translation from CPN to mill must respect these properties. That is why, the FirstOrder mill (MILL I) is used for the translation. CPN Places are expressed in mill i by imary relation symbols (Propositional variables) which allow... 

Figure 4 presents the syntax of the propositional temporal logic we use in Backus-Naur form, and also hints at the intended semantics [13]. The truth of a temporal formula is defined with respect to a so-called model, which is an... 

This is a mathematical law (law of logic), i.e. a proposition that can be rigorously proved without further assumptions. Interestingly, however, it is found that the effective shear and bulk moduli of multiphase materials always lie between the arithmetic and the harmonic mean. This is a physical law (law of nature), i.e. a finding which can be (and has been in micromechanics) rigorously proved for model materials with well defined microstructures. Of course, since its proof is based on model assumptions, its applicability to real materials is and remains, strictly speaking, a question of experience. 

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