Jon AS,
In modern predicate logic, is the variable a predicate, a proposition,
or an argument (in Peirce's sense)? Clearly none of these, so either
a fourth division is required or the first one must be widened.
Every sentence in every version of logic, formal or informal,
consists of symbols -- ordinary words for informal logic or
special symbols for formal (mathematical) logic.
Those symbols (ordinary words or mathematical symbols) are
combined in patterns (icons) in one or more dimensions.
The resulting combinations are bigger symbols.
In standard notation for modern predicate logic, an upside-down
A or E represents the quantifier, a lowercase letter represents
each variable, and an uppercase letter represents each predicate.
Is each of these symbols a predicate, a proposition, or an argument?
Each of them, by itself, is a symbol. When combined in patterns
(icons) of one or more dimensions, they form larger symbols. There
are no new kinds of signs -- just more kinds of symbols.
But we have to distinguish the notation (symbols and icons) from what
those symbols and icons represent in any universe of discourse (UoD).
Def: An *identifier* is a symbol that represents something in the UoD.
In the algebraic notation, an identifier is called a variable.
In EGs, an identifier is called a line of identity.
Def: A *predicate* is a symbol that represents a relation in the UoD.
In the algebra, a predicate is a character string that may be
followed by one or more variables. In EGs, a predicate is a
character string that has one or more pegs that may be attached
to lines of identity.
Def: A *sentence* is a symbol that represents a proposition in the
UoD. In the algebra, a sentence is a linear combination of symbols
called quantifiers, variables, predicates, Boolean operators, and
parentheses. In EGs, a sentence is a graph (an icon) constructed
from subgraphs of two kinds: predicates whose pegs are attached
to lines of identity; or oval icons, called negations, that contain
other subgraphs.
Def: An *argument* in either the algebraic notation or the EG notation
is a symbol that consists of a linear sequence (icon) of sentences.
For brevity, these definitions avoid details of the syntax, but I'm
assuming the syntax that Peirce specified in 1885 for the algebra
or later for the EGs.
Summary: Logic notation (either algebraic or EG form) consists
of symbols and combinations of symbols. The base notation has
four kinds of symbols:
1. identifiers refer to entities in some universe of discourse (UoD).
2. predicates refer to relations (patterns) of entities in the UoD.
3. sentences refer to propositions that are true or false about
combinations of entities and relations in the UoD.
4. A few other symbols are used to combine the identifiers and
predicates to form sentences.
John
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