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The syllogisms
When a model
is built under the principles of reasoning called “ultra-optimization,” a
result may be the discovery of ultra-optimized conditions
on the model’s space of independent variables; each such condition is called a
“pattern.” Discovery of patterns reduces, from the maximum, the missing information in the inference that is made by the model to the outcome of a statistical event, given the condition.
In the event
that the missing information is reduced to nil, then the patterns are paired
with the outcomes. Each pattern-outcome pair is then associated with a certain
kind of model; each inference made by this model is of the form.
If the pattern P
is true
then the outcome O is true
P is true
Therefore, O is true
This
inference is a type of “syllogism.”
The set of
events is empty for which P
is true and O is false. From this fact follows the
existence of a second kind of model; each inference made by this model is of
the form:
If
the pattern P
is true
and then the outcome O
is true
O
is false
Therefore,
P
is false.
This
inference is a second type of syllogism. The first type of syllogism is called modus ponens while the second is called modus tollens.
According to Jaynes*, deductive reasoning
amounts to the repeated application of the two types of syllogism.
The logic
underlying construction of the syllogisms is the probabilistic
logic. In this construction, reduction of the missing information to nil
constrains the values of the probabilities of states to 0 and 1, thus reducing
the probabilistic to the so-called “deductive” logic. In the domain of validity
of the deductive logic, ultra-optimization is equivalent to the principle of
reasoning which states that an inference is correct if it conforms to a
syllogism. From the perspective of the probabilistic logic, this inference is
correct because it is ultra-optimized.
By itself,
the deductive logic contains no explanation for the origins of the languages on
which the two syllogisms operate. Under Christensen’s theory of knowledge, the
words, phrases and sentences of a language are the names which the speakers of
this language ascribe to patterns discovered by an approximation to
ultra-optimization. Under Christensen’s theory, languages evolve from the
continuing discovery of new patterns. It is in the discovery of these patterns
that knowledge is created. Communication is the mechanism by which we convey
knowledge. Pattern discovery is the mechanism by which we create knowledge.
*Jaynes, E.T., Probability Theory: The Logic of Science, West Nyack, NY, USA: Cambridge University Press, 2003. p. 4.