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In this Subsection, I describe the syntax of a family of formal languages that I intend to use as a sentential calculus, and thus to interpret for the purpose of reasoning about propositions and their logical relations. In order to carry out the discussion, I need a way of referring to signs as if they were objects like any others, in other words, as the sorts of things that are subject to being named, indicated, described, discussed, and renamed if necessary, that can be placed, arranged, and rearranged within a suitable medium of expression, or else manipulated in the mind, that can be articulated and decomposed into their elementary signs, and that can be strung together in sequences to form complex signs.  Signs that have signs as their objects are called ''higher order signs'', and this is a topic that demands an apt formalization, but in due time. The present discussion requires a quicker way to get into this subject, even if it takes informal means that cannot be made absolutely precise.

In this Subsection, I describe the syntax of a family of formal languages

that I intend to use as a sentential calculus, and thus to interpret for

the purpose of reasoning about propositions and their logical relations.

In order to carry out the discussion, I need a way of referring to signs

as if they were objects like any others, in other words, as the sorts of

things that are subject to being named, indicated, described, discussed,

and renamed if necessary, that can be placed, arranged, and rearranged

within a suitable medium of expression, or else manipulated in the mind,

that can be articulated and decomposed into their elementary signs, and

that can be strung together in sequences to form complex signs.  Signs

that have signs as their objects are called "higher order" (HO) signs,

and this is a topic that demands an apt formalization, but in due time.

The present discussion requires a quicker way to get into this subject,

even if it takes informal means that cannot be made absolutely precise.

As a temporary notation, let the relationship between a particular sign z

As a temporary notation, let the relationship between a particular sign <math>s\!</math> and a particular object <math>o\!</math>, namely, the fact that <math>s\!</math> denotes <math>o\!</math> or the fact that <math>o\!</math> is denoted by <math>s\!</math>, be symbolized in one of the following two ways:

and a particular object o, namely, the fact that z denotes o or the fact

that o is denoted by z, be symbolized in one of the following two ways:

1.  z  >-> o,

# <math>s \rightarrow o.</math>

 

# <math>o \leftarrow s.</math>

    z  den o.

 

2.  o  <-<  z,

 

    o ned  z.

Now consider the following paradigm:

Now consider the following paradigm: