Changes

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This usage allows us to refer to the blank as a type of character, and also to refer any blank we choose as a token of this type, referring to either of them in a marked way, but without the use of quotation marks, as I just did.  Now, since a blank is just what the name "blank" names, it is possible to represent the denotation of the sign " " by the name "blank" in the form of an identity between the named objects, thus:

This usage allows us to refer to the blank as a type of character, and also to refer any blank we choose as a token of this type, referring to either of them in a marked way, but without the use of quotation marks, as I just did.  Now, since a blank is just what the name "blank" names, it is possible to represent the denotation of the sign " " by the name "blank" in the form of an identity between the named objects, thus:

{| align="center" cellpadding="8" width="90%"

{| align="center" cellpadding="8" width="90%"

A sequence of length <math>k > 0\!</math> is typically presented in the concatenated forms:

A sequence of length <math>k > 0\!</math> is typically presented in the concatenated forms:

s_1 s_2 ... s_(k-1) s_k,

<math>s_1 s_2 \ldots s_{k-1} s_k\!</math>

or

or

s_1 · s_2 · ... · s_(k-1) · s_k,

<math>s_1 \cdot s_2 \cdot \ldots \cdot s_{k-1} \cdot s_k</math>

with s_j in !A!, for all j = 1 to k.

with <math>s_j \in \mathfrak{A}</math> for all <math>j = 1 \ldots k.</math>

Two alternative notations are often useful:

Two alternative notations are often useful:

In this context, I make the following distinction:

In this context, I make the following distinction:

# By "deleting" an appearance of a sign, I mean replacing it with an appearance of the empty string "".

# To ''delete'' an appearance of a sign is to replace it with an appearance of the empty string "".

# By "erasing" an appearance of a sign, I mean replacing it with an appearance of the blank symbol " ".

# To ''erase'' an appearance of a sign is to replace it with an appearance of the blank symbol "&nbsp".

A "token" is a particular appearance of a sign.

A ''token'' is a particular appearance of a sign.

 

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

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

family of languages, consisting of one language !C!(!P!) for

each set !P! of "paints".

The alphabet !A!  =  !M! |_| !P! is the disjoint union of two sets of symbols:

The alphabet !A!  =  !M! |_| !P! is the disjoint union of two sets of symbols: