An Algebra for Pattern Logic

All patterns within the first consideration of ADEPT LION may be interpreted as a ratio between two variables.  The Thing channel is the numerator of the ration and the Use channel is the denominator.

 The pattern above is labeled in ADEPT LION as OEH\\. This stands for the "Obscure Entity of aetHer" and is named the "Inclusion trickle of the Notion rivulet".  It is the primary building block of many more complex relationships in pattern logic. 

The lowercase Greek alphabet is used in pattern logic to identify a participating entity in the logical relation that is captured by the gray circle labeled "alpha" at the center that receives a thing input (with square arrowhead) from "beta"  and a use input (round arrowhead) from "gamma".  Dashed lines indicate that these entities are indeterminate.

The statements derived from the patterns may then be manipulated using common mathematical operations into specific forms of expression.  This produces a fascinating diversity of logical relation.  Any arrangement of ADEPT entities that form a valid arrangement in pattern language can be interpreted through pattern logic using this method.  A common relational pattern is the "braid" of inter-connected arrows but other patterns include "nested", "redundant" and "cyclical".

In our most simple of examples, there is no need to manipulate the ratio of beta to gamma in order to get it into a form that can be interpreted as a relation of logical inclusion.  For more on the rules of transforming algebraic expressions into logical statements click the button below.

Once a pattern has undergone any necessary algebraic manipulation its expression is transformed into a collection of one or more logical statements.  These statements are infinitely expressive, but rather quickly fill out the formal components of first order logic and other logical models such as mereology, description logics, modal logic and Aristotelian syllogisms.

The logical statements that corresponds to our example can be given various spoken expressions:

• "alpha is 'beta included in gamma'"
• or in mereological form "alpha is 'the beta part of gamma'" 
• or in Aristotelian form " alpha is 'gamma belongs to beta'" 
• or "alpha is 'gamma's inclusion of beta'"

The Logical statements can then be used to construct Frame-Region Diagrams which provide an interpretation that aids in visualizing the meaning of the patterns.  Because the possible patterns extend beyond known systems of symbolic logic while remaining computable these diagrams provide a useful grounding for interpretation.  For instance, the quantifiers of syllogistic logic find early expression in pattern logic in what are called the "dyadic braids" and are also present an alternative method of reasoning over the conditional and unconditional valid syllogistic forms while expanding the scope of syllogistic expressiveness.