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 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>\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 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.

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 George 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 George 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.

While working with expressions solely in propositional calculus, it is easiest to use plain parentheses for logical connectives.  In contexts where parentheses are needed for other purposes &ldquo;teletype&rdquo; parentheses <math>\texttt{(} \ldots \texttt{)}\!</math> or barred parentheses <math>(\!| \ldots |\!)</math> may be used for logical operators.

While working with expressions solely in propositional calculus, it is easiest to use plain parentheses for logical connectives.  In contexts where parentheses are needed for other purposes &ldquo;teletype&rdquo; parentheses <math>\texttt{(} \ldots \texttt{)}</math> or barred parentheses <math>(\!| \ldots |\!)</math> may be used for logical operators.

The briefest expression for logical truth is the empty word, usually denoted by <math>{}^{\backprime\backprime} \boldsymbol\varepsilon {}^{\prime\prime}\!</math> or <math>{}^{\backprime\backprime} \boldsymbol\lambda {}^{\prime\prime}\!</math> in formal languages, where it forms the identity element for concatenation.  To make it visible in this text, it may be denoted by the equivalent expression <math>{}^{\backprime\backprime} \texttt{((~))} {}^{\prime\prime},\!</math> or, especially if operating in an algebraic context, by a simple <math>{}^{\backprime\backprime} 1 {}^{\prime\prime}.\!</math>  Also when working in an algebraic mode, the plus sign <math>{}^{\backprime\backprime} + {}^{\prime\prime}\!</math> may be used for [[exclusive disjunction]].  For example, we have the following paraphrases of algebraic expressions by bracket expressions:

The briefest expression for logical truth is the empty word, usually denoted by <math>{}^{\backprime\backprime} \boldsymbol\varepsilon {}^{\prime\prime}</math> or <math>{}^{\backprime\backprime} \boldsymbol\lambda {}^{\prime\prime}</math> in formal languages, where it forms the identity element for concatenation.  To make it visible in this text, it may be denoted by the equivalent expression <math>{}^{\backprime\backprime} \texttt{((} ~ \texttt{))} {}^{\prime\prime},</math> or, especially if operating in an algebraic context, by a simple <math>{}^{\backprime\backprime} 1 {}^{\prime\prime}.</math>  Also when working in an algebraic mode, the plus sign <math>{}^{\backprime\backprime} + {}^{\prime\prime}</math> may be used for [[exclusive disjunction]].  For example, we have the following paraphrases of algebraic expressions by bracket expressions:

{| align="center" cellpadding="6" style="text-align:center"

{| align="center" cellpadding="6" style="text-align:center"

|}

|}

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

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

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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"

|+ style="height:30px" | <math>\text{Table 1.} ~~ \text{Syntax and Semantics of a Calculus for Propositional Logic}\!</math>

|+ style="height:30px" | <math>\text{Table 1.} ~~ \text{Syntax and Semantics of a Calculus for Propositional Logic}</math>

|- style="height:40px; background:ghostwhite"

|- style="height:40px; background:ghostwhite"

| <math>\text{Expression}~\!</math>

| <math>\text{Expression}</math>

| <math>\text{Interpretation}\!</math>

| <math>\text{Interpretation}</math>

| <math>\text{Other Notations}\!</math>

| <math>\text{Other Notations}</math>

|-

|-

| &nbsp;

| &nbsp;

| <math>\text{True}\!</math>

| <math>\text{True}</math>

| <math>1\!</math>

| <math>1</math>

|-

|-

| <math>\texttt{(~)}\!</math>

| <math>\texttt{(} ~ \texttt{)}</math>

| <math>\text{False}\!</math>

| <math>\text{False}</math>

| <math>0\!</math>

| <math>0</math>

|-

|-

| <math>x\!</math>

| <math>x</math>

| <math>x\!</math>

| <math>x</math>

| <math>x\!</math>

| <math>x</math>

|-

|-

| <math>\texttt{(} x \texttt{)}\!</math>

| <math>\texttt{(} x \texttt{)}</math>

| <math>\text{Not}~ x\!</math>

| <math>\text{Not}~ x</math>

|

|

<math>\begin{matrix}

<math>\begin{matrix}

\\

\\

\lnot x

\lnot x

\end{matrix}\!</math>

\end{matrix}</math>

|-

|-

| <math>x~y~z\!</math>

| <math>x~y~z</math>

| <math>x ~\text{and}~ y ~\text{and}~ z\!</math>

| <math>x ~\text{and}~ y ~\text{and}~ z</math>

| <math>x \land y \land z\!</math>

| <math>x \land y \land z</math>

|-

|-

| <math>\texttt{((} x \texttt{)(} y \texttt{)(} z \texttt{))}\!</math>

| <math>\texttt{((} x \texttt{)(} y \texttt{)(} z \texttt{))}</math>

| <math>x ~\text{or}~ y ~\text{or}~ z\!</math>

| <math>x ~\text{or}~ y ~\text{or}~ z</math>

| <math>x \lor y \lor z\!</math>

| <math>x \lor y \lor z</math>

|-

|-

| <math>\texttt{(} x ~ \texttt{(} y \texttt{))}\!</math>

| <math>\texttt{(} x ~ \texttt{(} y \texttt{))}</math>

|

|

<math>\begin{matrix}

<math>\begin{matrix}

\mathrm{If}~ x ~\text{then}~ y

\mathrm{If}~ x ~\text{then}~ y

\end{matrix}</math>

\end{matrix}</math>

| <math>x \Rightarrow y\!</math>

| <math>x \Rightarrow y</math>

|-

|-

| <math>\texttt{(} x \texttt{,} y \texttt{)}\!</math>

| <math>\texttt{(} x \texttt{,} y \texttt{)}</math>

|

|

<math>\begin{matrix}

<math>\begin{matrix}

\end{matrix}</math>

\end{matrix}</math>

|-

|-

| <math>\texttt{((} x \texttt{,} y \texttt{))}\!</math>

| <math>\texttt{((} x \texttt{,} y \texttt{))}</math>

|

|

<math>\begin{matrix}

<math>\begin{matrix}

\end{matrix}</math>

\end{matrix}</math>

|-

|-

| <math>\texttt{(} x \texttt{,} y \texttt{,} z \texttt{)}\!</math>

| <math>\texttt{(} x \texttt{,} y \texttt{,} z \texttt{)}</math>

|

|

<math>\begin{matrix}

<math>\begin{matrix}

\end{matrix}</math>

\end{matrix}</math>

|-

|-

| <math>\texttt{((} x \texttt{),(} y \texttt{),(} z \texttt{))}\!</math>

| <math>\texttt{((} x \texttt{),(} y \texttt{),(} z \texttt{))}</math>

|

|

<math>\begin{matrix}

<math>\begin{matrix}

\\

\\

\texttt{(} x \texttt{,(} y \texttt{,} z \texttt{))}

\texttt{(} x \texttt{,(} y \texttt{,} z \texttt{))}

\end{matrix}\!</math>

\end{matrix}</math>

|

|

<math>\begin{matrix}

<math>\begin{matrix}

\\

\\

\text{are true}.

\text{are true}.

\end{matrix}\!</math>

\end{matrix}</math>

|

|

<p><math>x + y + z\!</math></p>

<p><math>x + y + z</math></p>

<br>

<br>

<p><math>\begin{matrix}

<p><math>\begin{matrix}

\\

\\

x'y'z~ &

x'y'z~ &

\end{matrix}\!</math></p>

\end{matrix}</math></p>

|-

|-

| <math>\texttt{(} w \texttt{,(} x \texttt{),(} y \texttt{),(} z \texttt{))}\!</math>

| <math>\texttt{(} w \texttt{,(} x \texttt{),(} y \texttt{),(} z \texttt{))}</math>

|

|

<math>\begin{matrix}

<math>\begin{matrix}

<br>

<br>

'''Note.''' The usage that one often sees, of a plus sign "<math>+\!</math>" to represent inclusive disjunction, and the reference to this operation as ''boolean addition'', is a misnomer on at least two counts.  Boole used the plus sign to represent exclusive disjunction (at any rate, an operation of aggregation restricted in its logical interpretation to cases where the represented sets are disjoint (Boole, 32)), as any mathematician with a sensitivity to the ring and field properties of algebra would do:

'''Note.''' The usage that one often sees, of a plus sign "<math>+</math>" to represent inclusive disjunction, and the reference to this operation as ''boolean addition'', is a misnomer on at least two counts.  Boole used the plus sign to represent exclusive disjunction (at any rate, an operation of aggregation restricted in its logical interpretation to cases where the represented sets are disjoint (Boole, 32)), as any mathematician with a sensitivity to the ring and field properties of algebra would do:

<blockquote>

<blockquote>

The expression <math>x + y\!</math> seems indeed uninterpretable, unless it be assumed that the things represented by <math>x\!</math> and the things represented by <math>y\!</math> are entirely separate;  that they embrace no individuals in common.  (Boole, 66).

The expression <math>x + y</math> seems indeed uninterpretable, unless it be assumed that the things represented by <math>x</math> and the things represented by <math>y</math> are entirely separate;  that they embrace no individuals in common.  (Boole, 66).

</blockquote>

</blockquote>