Changes

Working under the functional perspective, the formal system known as ''propositional calculus'' is introduced as a general system of notations for referring to boolean functions.  Typically, one takes a space <math>X\!</math> and a coordinate representation <math>T : X \to \mathbb{B}^n\!</math> as parameters of a particular system and speaks of the propositional calculus on a finite set of variables <math>\{ \underline{\underline{x_i}} \}.\!</math>  In objective terms, this constitutes the ''domain of propositions'' on the basis <math>\{ \underline{\underline{x_i}} \},\!</math> notated as <math>\operatorname{DOP}\{ \underline{\underline{x_i}} \}.\!</math>  Ideally, one does not want to become too fixed on a particular set of logical features or to let the momentary dimensions of the space be cast in stone.  In practice, this means that the formalism and its computational implementation should allow for the automatic embedding of <math>\operatorname{DOP}(\underline{\underline{X}})\!</math> into <math>\operatorname{DOP}(\underline{\underline{Y}})\!</math> whenever <math>\underline{\underline{X}} \subseteq \underline{\underline{Y}}.\!</math>

Working under the functional perspective, the formal system known as ''propositional calculus'' is introduced as a general system of notations for referring to boolean functions.  Typically, one takes a space <math>X\!</math> and a coordinate representation <math>T : X \to \mathbb{B}^n\!</math> as parameters of a particular system and speaks of the propositional calculus on a finite set of variables <math>\{ \underline{\underline{x_i}} \}.\!</math>  In objective terms, this constitutes the ''domain of propositions'' on the basis <math>\{ \underline{\underline{x_i}} \},\!</math> notated as <math>\operatorname{DOP}\{ \underline{\underline{x_i}} \}.\!</math>  Ideally, one does not want to become too fixed on a particular set of logical features or to let the momentary dimensions of the space be cast in stone.  In practice, this means that the formalism and its computational implementation should allow for the automatic embedding of <math>\operatorname{DOP}(\underline{\underline{X}})\!</math> into <math>\operatorname{DOP}(\underline{\underline{Y}})\!</math> whenever <math>\underline{\underline{X}} \subseteq \underline{\underline{Y}}.\!</math>

The rest of this section presents the elements of a particular calculus for propositional logic.  First, I establish the basic notations and summarize the axiomatic presentation of the calculus, and then I give special attention to its functional and geometric interpretations.

The rest of this section presents the elements of a particular calculus for propositional logic.  First, I establish the basic notations and summarize the axiomatic presentation of the calculus, and then I give special attention to its functional and geometric interpretations.

This section reviews the elements of a calculus for propositional logic that I initially presented in two earlier papers (Awbrey, 1989 & 1994).  This calculus belongs to a family of formal systems that hark back to C.S. Peirce's "existential graphs" (PEG) 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.

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.

 

One of the benefits of this notation is that it facilitates the development of a "differential extension" (DEX) for propositional logic that can be used to reason about changing universes of discourse.

A "propositional language" (PL) is a syntactic system that mediates the reasonings of a "propositional logic" (PL).  The objects of a PL, 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 Bn >B and remaining subject only to suitable choices of the parameter n.

A "propositional language" (PL) is a syntactic system that mediates the reasonings of a "propositional logic" (PL).  The objects of a PL, 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 Bn >B and remaining subject only to suitable choices of the parameter n.