Changes

This note continues a previous discussion on the problem of dealing with change and diversity in logic-based intelligent systems.  It is useful to 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.  It is useful to begin by summarizing essential material from previous reports.

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

Table&nbsp;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.

 

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

* A concatenation of propositional expressions in the form <math>e_1~e_2~\ldots~e_{k-1}~e_k</math> indicates that all of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> are true, in other words, that their [[logical conjunction]] is true.

| A concatenation of propositional expressions in the form <math>e_1\ e_2\ \ldots\ e_{k-1}\ e_k</math> indicates that all of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> are true, in other words, that their [[logical conjunction]] is true.

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.

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.

It is important to note that the last expressions are not equivalent to the triple bracket <math>(x, y, z).\!</math>

It is important to note that the last expressions are not equivalent to the triple bracket <math>(x, y, z).\!</math>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:96%"

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"

|+ '''Table 1. Syntax and Semantics of a Calculus for Propositional Logic'''

|+ <math>\text{Table 1.}~~\text{Syntax and Semantics of a Calculus for Propositional Logic}</math>

|- style="background:ghostwhite"

|- style="background:#f0f0ff"

! Expression

! Expression

! Interpretation

! Interpretation