Changes

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 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 ordinary parentheses are needed for other purposes an alternate typeface <math>\texttt{(} \ldots \texttt{)}</math> may be used for logical operators.

The briefest expression for logical truth is the empty word, usually denoted by <math>\varepsilon\!</math> or <math>\lambda\!</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>((~))\!</math>", or, especially if operating in an algebraic context, by a simple "<math>1\!</math>". Also when working in an algebraic mode, the plus sign "<math>+\!</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>\varepsilon\!</math> or <math>\lambda\!</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:

<center>

<p><math>\begin{matrix}

x + y & = & (x, y)

<math>\begin{matrix}

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

\end{matrix}</math>

<math>\begin{matrix}

x + y + z

& = &

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

\end{matrix}</math>

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>(x, y, z).\!</math>

<br>

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