Thinking systems of this kind typically fall into at least one of the following three motivational categories (and most of the grand logic programs of the 20th century aimed for all three):
The aim here is Descriptive and not say Justificatory - e.g. the basic axiom schemata here describe a pattern of thinking that does need necessarily track truth but rather describes the ways that thoughts, true or false, follow sequentially (rather than as a result of logical consequence).
I will depict through several axioms a few kinds of thinking (techniques) that when combined with other formal systems, imbue them with richer power:
i. @, >, ^ - these symbols represent an item of thought (which is quite approximate but will suffice).
ii. | - denotes a temporal transition in thinking.
iii. Thus, > | > represents the same item of thought over a single transition.
The same symbol on both sides of the | specify the same item of thought.
iv. Other marks will be introduced as well, specific to the exact operation axiomatized.
i. "The house on the street" - "the dining room is next to the front entrance in that house on the street".
ii. Here the - represents a relational symbol.
i. Every stream starts with a single symbol of thought: >.
ii. Each of the axioms above have a dual opposite (except for Identity and Replacement).
iii. Thinking proceeds through distinction (difference) or symmetry breaking (as described in Spencer-Brown's Laws of Form).
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