Changes

The ''kleene star'' <math>\mathfrak{A}^*</math> of alphabet <math>\mathfrak{A}</math> is the set of all strings over <math>\mathfrak{A}.</math>  In particular, <math>\mathfrak{A}^*</math> includes among its elements the empty string <math>\varepsilon.</math>

The ''kleene star'' <math>\mathfrak{A}^*</math> of alphabet <math>\mathfrak{A}</math> is the set of all strings over <math>\mathfrak{A}.</math>  In particular, <math>\mathfrak{A}^*</math> includes among its elements the empty string <math>\varepsilon.</math>

The "surplus" !A!^+ of an alphabet !A! is the set of all positive length strings over !A!, in other words, everything in !A!* but the empty string.

The ''kleene plus'' <math>\mathfrak{A}^+</math> of an alphabet <math>\mathfrak{A}</math> is the set of all positive length strings over <math>\mathfrak{A},</math> in other words, everything in <math>\mathfrak{A}^*</math> but the empty string.

A "formal language" !L! over an alphabet !A! is a subset !L! c !A!*.  If z is a string over !A! and if z is an element of !L!, then it is customary to call z a "sentence" of !L!.  Thus, a formal language !L! is defined by specifying its elements, which amounts to saying what it means to be a sentence of !L!.

A ''formal language'' <math>\mathcal{L}</math> over an alphabet <math>\mathfrak{A}</math> is a subset of <math>\mathfrak{A}^*.</math>  In brief, <math>\mathcal{L} \subseteq \mathfrak{A}^*.</math> If <math>s\!</math> is a string over <math>\mathfrak{A}</math> and if <math>s\!</math> is an element of <math>\mathcal{L},</math> then it is customary to call <math>s\!</math> a ''sentence'' of <math>\mathcal{L}.</math> Thus, a formal language <math>\mathcal{L}</math> is defined by specifying its elements, which amounts to saying what it means to be a sentence of <math>\mathcal{L}.</math>

One last device turns out to be useful in this connection.  If z is a string that ends with a sign t, then z · t^-1 is the string that results by "deleting" from z the terminal t.

One last device turns out to be useful in this connection.  If <math>s\!</math> is a string that ends with a sign <math>t,\!</math> then <math>s \cdot t^{-1}</math> is the string that results by ''deleting'' from <math>s\!</math> the terminal <math>t.\!</math>

In this context, I make the following distinction:

In this context, I make the following distinction:

The informal mechanisms that have been illustrated in the immediately preceding discussion are enough to equip the rest of this discussion with a moderately exact description of the so-called ''cactus language'' that I intend to use in both my conceptual and my computational representations of the minimal formal logical system that is variously known to sundry communities of interpretation as ''propositional logic'', ''sentential calculus'', or more inclusively, ''zeroth order logic'' (ZOL).

The informal mechanisms that have been illustrated in the immediately preceding discussion are enough to equip the rest of this discussion with a moderately exact description of the so-called ''cactus language'' that I intend to use in both my conceptual and my computational representations of the minimal formal logical system that is variously known to sundry communities of interpretation as ''propositional logic'', ''sentential calculus'', or more inclusively, ''zeroth order logic'' (ZOL).

The ''painted cactus language'' !C! is actually a parameterized family of languages, consisting of one language !C!(!P!) for each set !P! of ''paints''.

The ''painted cactus language'' !C! is actually a parameterized family of languages, consisting of one language !C!(!P!) for each set !P! of ''paints''.