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'''''NOTE.'''  This section is currently under construction.  In the meantime, see [[Logical Graph]].''

'''''NOTE.'''  This section is currently under construction.  In the meantime, see [[Logical Graph]].''

The development of differential logic is greatly facilitated by having a conceptually efficient calculus in place at the level of [[boolean-valued functions]] and elementary logical propositions.   

The development of differential logic is greatly facilitated by having a conceptually efficient calculus in place at the level of [[boolean-valued functions]] and elementary logical propositions.  A calculus that is very efficient from both conceptual and computational standpoints is based on just 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 parenthesized sequence of propositional expressions in the form <math>\texttt{(} e_1 \texttt{,} e_2 \texttt{,} \ldots \texttt{,} e_{k-1} \texttt{,} 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.

 

| [[Image:Cactus Graph Lobe Connective.jpg|500px]]

{| align="center" cellpadding="6" width="90%"

{| align="center" cellpadding="6" width="90%"

| A concatenated sequence 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 concatenated sequence 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 through combinations of these two forms.  Strictly speaking, the parenthesized form is sufficient to define the concatenated form, making the latter formally dispensable, but it is convenient to maintain it as a concise way of expressing more complicated combinations of parenthesized forms.  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.

All other propositional connectives can be obtained through combinations of these two forms.  Strictly speaking, the parenthesized form is sufficient to define the concatenated form, making the latter formally dispensable, but it is convenient to maintain it as a concise way of expressing more complicated combinations of parenthesized forms.  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.

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