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Axiomas is the set of basic proposition crucial to a theory. There is no theory without axioms even if not explicitly stated and taken for granted.

The rules of a theory and the predictions of a model are always true only within the limits set by the axioms. Not properly stating axioms is often a source of utter confusion or even bitter dispute. The reason of not properly stating the underlying axioms often is not sheer neglect by rather unawareness of its existence. Now I will give some examples.

Lets start simple with example A. Given a box that contains red balls only. In the realm of that box it is always true that any ball taken from that box will be red.

As a  matter of fact from this box can only be taken rad balls.

Now we extend our model including other boxes that contain other colors. Possibly, we may come up with an other theory as depicted in the next picture. The rule of which is the color of the ball taken from a box matches the color of the box.

From each box can  only balls  with matching colors can be drawn.

This theory, we call it example B, is more general as it includes theory A. Mathematically we may say that theory A is a subset of theory B.

A \in B

The probability of a ball's color depends upon the proportion.

If we study more boxes we may come to the conclusion that box and ball color do not necessarily match in all circumstances. An even more generalized theory would say. The color of the balls that eventually can be taken from a box depends on the color of the ball that were put in it in the first place. This theory, we may want to call it C, is still quite simple but much more abstract and versatile than A and B. Theory C embraces B and A.

B \in C and A \in C

Causality is the issue in this example. Only balls that were put into tha box can be drawn from it.

But we may push further. Let consider boxes that contain a mixture of balls of different colors. Then our next theory D will go as such: The probability of a ball bearing a certain color depends on the ratio of ball of that color in the box.

\textrm{Probability}(\textrm{blue}) = \frac {\textrm{Number of blue balls}}{\textrm{Number of all balls}}

or using simplified mathematical symbols

P(x) = \frac {N_x}{\sum_{i=1}^T N_i} \; \textrm{with} \; x \in T = \{blue, green, red, \cdots \}

The sample drawn from the box is quite representative.

If we do not know the exact composition of a box' content, we may take a sample and induce the probability of a certain color from that sample. Such a prediction is even more general as it works with boxes (collections) that are uncountable. For instance we may count the collection of possibilities of rolling a die (6) or tossing a coin (2), but we will never be able to count all the possible offspring of a cat. Thus in order to make predictions about her kittens we need other methods for instance analyzing the sample of her last litter. We call this finally theory E.

p(x) \approx \frac {N_x} {N_T} \; \textrm{with} \; N_x = \textrm{number of x,} \: \textrm{and} \: N_T =\textrm{sample size}

A relationship exists among all these theories.

A \subseteq B \subseteq C \subseteq D \subseteq E

Two conclusion can be made from this little thought experiment.

  1. By considering a more general proposition we were able to develop am more general theory.
  2. A more general theory is not necessarily more exact. More about that is discussed on the abstract proviso page.

Tags: Theory


(c) Mato Nagel, Weißwasser 2004-2013, Disclaimer