Theory logical probability

One can see that the truth values in fuzzy logic strongly resemble the stochastic values from the theory of probabilities. However, methods based on the use of statistics are not considered fuzzy by the orthodox fuzzy theory protagonists. Instead of using probability values, fuzzy theory works with possibility values. It is argued that both values are substantially different and that the latter have to be evaluated by methods other than statistical. Our understanding, however, is that at a very fundamental level, both values have essentially the same nature. 

Secondly we can use it to measure the degree of belief in the truth of a statement or hypothesis. The first is the probability of events, the second is logical probability. If we wish to discuss the dependability of a theory in terms of probability we must therefore use logical probability and not the probability of events. 

The objective interpretation of probability calculus (Popper, 1976 48, and Appendix IX, Third Comment [1958]) is necessary because no result of statistical sampling is ever inconsistent with a statistical theory unless we make them with the help of. .. rejection rules (Lakatos, 1974 179 see also Nagel, 1971 366). It is under these rejection rules that probability calculus and logical probability approach each other these are also the conditions under which Popper explored the relationship of Fisher s likelihood function to his degree of corroboration, and the conditions arise only if the random sample is large and (e) is a statistical report asserting a good fit (Farris et ah, 2001). In addition to the above, in order to maintain an objective interpretation of probability calculus, Popper also required that once the specified conditions are obtained, we must proceed to submit (e) itself to a critical test, that is, try to find observable states of affairs that falsify (e). 

The claim conld be made that systematics can proceed without underlying universal laws, for what is reqnired is nothing bnt a method that allows us to choose a preferred hypothesis from a set of competing hypotheses of relationships relative to some theory such as evolutionary theory. Indeed, we do have a method at our disposal that allows us to do just this, but is it Popperian in its logic The conformity of parsimony analysis with Popper s falsificationism has been asserted in terms of Popper s concepts of logical probability, explanatory power, degrees of corroboration and severity of test. Let us look at these concepts in more detail. 

Kolmogorov, A. N. (1968a). Logical basis for information theory and probability theory. IEEE Transactions on Information Theory, 14(5), 662-664. Available from http //ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm arnumber= 1054210, doi 10.1109/TIT.1968.1054210... 

ET Jaynes. Probability Theory The Logic of Science, http //bayes.wustl.edu/etj/prob.html. 1999. 

A successful method to obtain dynamical information from computer simulations of quantum systems has recently been proposed by Gubernatis and coworkers [167-169]. It uses concepts from probability theory and Bayesian logic to solve the analytic continuation problem in order to obtain real-time dynamical information from imaginary-time computer simulation data. The method has become known under the name maximum entropy (MaxEnt), and has a wide range of applications in other fields apart from physics. Here we review some of the main ideas of this method and an application [175] to the model fluid described in the previous section. 

Event trees are used to perform postrelease frequency analysis. Event trees are pictorial representations of logic models or truth tables. Their foundation is based on logic theory. The frequency of n outcomes is defined as the product of the initiating event frequency and all succeeding conditional event probabilities leading to that outcome. The process is similar to fault tree analysis, but in reverse. 

Finite-additive invariant measures on non-compact groups were studied by Birkhoff (1936) (see also the book of Hewitt and Ross, 1963, Chapter 4). The frequency-based Mises approach to probability theory foundations (von Mises, 1964), as well as logical foundations of probability by Carnap (1950) do not need cr-additivity. Non-Kolmogorov probability theories are discussed now in the context of quantum physics (Khrennikov, 2002), nonstandard analysis (Loeb, 1975) and many other problems (and we do not pretend provide here is a full review of related works). 

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