Changes

This note continues a previous discussion on the problem of dealing with change and diversity in logic-based intelligent systems.  For ease of reference, I begin by summarizing essential material from previous reports.

This note continues a previous discussion on the problem of dealing with change and diversity in logic-based intelligent systems.  For ease of reference, I begin by summarizing essential material from previous reports.

Table 1 outlines the notation that I use for propositional calculus.  Explained as briefly as possible, I am using only two basic kinds of truth-functional connectives, both of variable ''k''-ary scope.

Table 1 outlines a notation for propositional calculus based on two types of logical connectives, both of variable <math>k\!</math>-ary scope.

* A bracketed list of propositional expressions in the form <math>(e_1, e_2, \ldots, e_{k-1}, e_k)</math> indicates that exactly one of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> is false.

# For the other, I use a bracket of the form (&nbsp;,&nbsp;,&nbsp;,&nbsp;) as a connective which says that exactly one of its ''k'' arguments is false.

All other truth-functional connectives can be obtained in a very efficient style of representation through combinations of these two forms.

 

All other propositional connectives can be obtained in a very efficient style of representation through combinations of these two forms.  Strictly speaking, the concatenation form is dispensable in light of the bracketed form, but it is convenient to maintain it as an abbreviation of more complicated bracket expressions.

This treatment of propositional logic is derived from the work of [[C.S. Peirce]] [P1, P2], who gave this approach an extensive development in his graphical systems of predicate, relational, and modal logic [Rob].  More recently, these ideas were revived and supplemented in an alternative interpretation by G. Spencer-Brown [SpB].  Both of these authors used other forms of enclosure where I use parentheses, but the structural topologies of expression and the functional varieties of interpretation are fundamentally the same.

This treatment of propositional logic is derived from the work of [[C.S. Peirce]] [P1, P2], who gave this approach an extensive development in his graphical systems of predicate, relational, and modal logic [Rob].  More recently, these ideas were revived and supplemented in an alternative interpretation by G. Spencer-Brown [SpB].  Both of these authors used other forms of enclosure where I use parentheses, but the structural topologies of expression and the functional varieties of interpretation are fundamentally the same.