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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
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This section reviews the elements of a calculus for propositional logic that I initially presented in two earlier papers (Awbrey, 1989 and 1994). This calculus belongs to a family of formal systems that hark back to C.S. Peirce's ''existential graphs'' (ExG) and it draws on ideas from Spencer Brown's ''Laws of Form'' (LOF). A feature that distinguishes the use of these formalisms can be summed up by saying that they treat logical expressions primarily as elements of a ''language'' and only secondarily as elements of an ''algebra''. In other words, the most important thing about a logical expression is the logical object it denotes. To the extent that the object can be represented in syntax, this attitude puts the focus on the ''logical equivalence class'' (LEC) to which the expression belongs, relegating to the background the whole variety of ways that the expression can be generated from algebraically conceived operations. One of the benefits of this notation is that it facilitates the development of a ''differential extension'' for propositional logic that can be used to reason about changing universes of discourse.
This section reviews the elements of a calculus for propositional logic that I initially presented in two earlier papers (Awbrey, 1989 and 1994). This calculus belongs to a family of formal systems that hark back to C.S. Peirce's ''existential graphs'' (ExG) and it draws on ideas from Spencer Brown's ''Laws of Form'' (LOF). A feature that distinguishes the use of these formalisms can be summed up by saying that they treat logical expressions primarily as elements of a ''language'' and only secondarily as elements of an ''algebra''. In other words, the most important thing about a logical expression is the logical object it denotes. To the extent that the object can be represented in syntax, this attitude puts the focus on the ''logical equivalence class'' (LEC) to which the expression belongs, relegating to the background the whole variety of ways that the expression can be generated from algebraically conceived operations. One of the benefits of this notation is that it facilitates the development of a ''differential extension'' for propositional logic that can be used to reason about changing universes of discourse.
A ''propositional language'' is a syntactic system that mediates the reasonings of a ''propositional logic''. The objects of the language and the logic, that is, the logical entities denoted by the language and invoked by the operations of the logic, can be conceived to rest at various levels of abstraction, residing in spaces of functions that are basically of the types <math>\mathbb{B}^n \to \mathbb{B}\!</math> and remaining subject only to suitable choices of the parameter <math>n.\!</math>
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Persistently reflective engagement in logical reasoning about any domain of objects leads to the identification of generic patterns of inference that appear to be universally valid, never disappointing the trust that is placed in them. After a time, a formal system naturally arises that commemorates one's continuing commitment to these patterns of logical conduct, and acknowledges one's conviction that further inquiry into their utility can be safely put beyond the reach of everyday concerns. At this juncture each descriptive pattern becomes a normative template, regulating all future ventures in reasoning until such time as a clearly overwhelming mass of doubtful outcomes cause one to question it anew.
Persistently reflective engagement in logical reasoning about any domain of objects leads to the identification of generic patterns of inference that appear to be universally valid, never disappointing the trust that is placed in them. After a time, a formal system naturally arises that commemorates one's continuing commitment to these patterns of logical conduct, and acknowledges one's conviction that further inquiry into their utility can be safely put beyond the reach of everyday concerns. At this juncture each descriptive pattern becomes a normative template, regulating all future ventures in reasoning until such time as a clearly overwhelming mass of doubtful outcomes cause one to question it anew.