Axiomatic theories definition

Definition of Axiomatic Theories. An axiomatic theory is an attempt to formalize an intuitive theory. Geometry was intuitive before Euclid wrote "The Elements . An intuitive theory is defined as a body of knowledge which attempts to express relationships and causality between objects, but is not formal. Most modern science is still intuitive, even though it may represent many of it s findings in exact mathematical formulae. As long as the entire corpus of knowledge is not expressed in a single formal system, it will remain intuitive. 

In modern axiomatic theory, postulates and axioms are defined simply as given statements. By the definition of an axiomatic theory the concept of truth is not considered relevant to its construction. If we can derive a theory which seems to mirror reality as reported by our current experimental knowledge, then we consider the postulates to be successful in some sense of the word. If the theory derived from the postulates clash drastically with our observations, the postulates can be thrown away as non-relevant . If the differences are slight, or if the theory predicts new experiments which should show differences from what the intuitive theory would predict, we can even call the axiomatic theory interesting . 

The use of formal metliods mandates a scientific approach. One possible approach is to formalise specification knowledge as an axiomatic system. Tlie CREATIV toolset uses a new model and a new definition of the specification process, together with an axiomatic theory to support specification knowledge. All operations in tlie system are provable and traceable we have built the reasoning component of tlie CREATIV toolset on the basis of this theory. 

Let s formulate the description of the modem catastrophe theory in the form of interpretation of the dynamic system of emergency situations control. Plotting the theoretical basis of the system within the frame of this theory is defined by original axioms. All the rest statements of the theory are logical conclusions of axioms. In the basis of the axiomatic synthesis of models of catastrophes lies the method of interpretation. Each definition and relation of the system s axiomatic theory is correspondent to some formalized description. 

Finally, reductionism is closely tied to the so-called syntactic approach to theories, an approach which treats theories as axiomatic systems expressed in natural or artificial languages. Indeed, closely tied may be an understatement, since deduction is a syntactic affair, and is a necessary component of reduction. Once philosophers of science began to take the semantic approach to theories seriously, the very possibility of reduction became moot. For the semantic approach treats theories as families of models, and models as implicit definitions, about which the only empirical question is whether they are applicable to phenomena. For reduction to be obtained among models semantically characterized requires an entirely different conception of reduction, and whether such a conception would capture anything of interest about inter-theoretical relations is questionable. 

Why More Systems Haven t Been Axiom it i zed. Geometry is unique in that it can be expressed in a simple logic, the results are either true or false, and that the actual experiments were capable of being done with thought alone. In chemistry there was not sufficient knowledge to enumerate the basic definitions and postulates. The recent explosion of knowledge in chemistry has made it feasible to begin the process of axiomatization of chemical theories. 

We refrain from entering here upon a thoroughgoing discussion of the preceding definition of temperature from the thermodynamic and axiomatic point of view (a complete treatment for tlie generalized statistics introduced, by the quantum theory is given in Appendix XXIX, p. 336), and merely add a brief remark on the units in which temperature is measured. 

The axiomatic definition of probability uses the set theory. A certain event X is the event that occurs in every trial. The union A + B of two events A and B is the event that occnrs when A or B both occur. The intersection AB of the events A and B is the event that occurs when both events A and B occur. The events A and B are mutually exclusive if the occurrence of one of them excludes the occurrence of the other. Three postulates are given. The probability P(A) of an event A is... 

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