I started to reply to some of the issues in recent notes. And I
realized that some examples would help clarify the many issues about
continuity and semiosis.
For the first example, see the attached diagram frere2.gif. At
the top is traditional notation for one bar of music. That diagram
states a proposition that is true of any music that any musician
would play by reading that diagram.
To emphasize the fact that music notation can express propositions,
frere2.gif also contains a conceptual graph (CG) that expresses the
same proposition as that bar of music. That CG could be mapped to
logic as expressed by an existential graph or the algebraic notation.
For further explanation, see http://jfsowa.com/pubs/eg2cg.pdf
All four notations -- traditional music notation and its translation
to CGs, EGs, or predicate calculus -- consist of a finite number of
symbols and icons. Some of the symbols are names, which have an
indexical effect of relating the notation to a physical instrument
and the music it generates.
The vertical dimension of the lines of music is an icon that represents
tones ranging from lower to higher. The horizontal dimension is an
icon of the temporal sequence of the tones. The duration of each tone
is shown by symbols for quarter notes, half notes...
In the CGs, EGs, or predicate calculus, the iconic conventions are
mapped to dyadic predicates named Dur (duration) or Next. For both
the EGs and CGs, the nodes can be freely moved around. But in
frere2.gif, the boxes and circles are placed in an iconic pattern
that mimics the icons in music notation. But the iconic placement
of CG or EG nodes, which is an aid to the reader, is irrelevant to
the logic -- any nodes of the CG or EG could be moved in any direction
without affecting the semantics.
We can relate this to what Peirce wrote in 1902 (CP 2.320). He
started with two visual examples, a portrait and a photograph:
A man's portrait with a man's name written under it is strictly a
proposition, although its syntax is not that of speech, and although
the portrait itself not only represents, but is, a Hypoicon. But the
proper name so nearly approximates to the nature of an Index, that
this might suffice to give an idea of an informational Index.
A portrait plus an index asserts a proposition (dicisign). In effect,
the image is used as a kind of predicate. In that same paragraph,
he introduced the word 'quasi-predicate' for images used in this way:
A better example is a photograph. The mere print does not, in itself,
convey any information. But the fact, that it is virtually a section
of rays projected from an object otherwise known, renders it a Dicisign.
Every Dicisign, as the system of Existential Graphs fully recognizes,
is a further determination of an already known sign of the same object.
It is not, perhaps, sufficiently brought out in the present analysis.
It will be remarked that this connection of the print, which is the
quasi-predicate of the photograph, with the section of the rays,
which is the quasi-subject, is the Syntax of the Dicisign;
This paragraph explains the nature of continuity in semiosis:
1. Every predicate defined in logic or language is specified by
an expression with a finite number of symbols. There is only
a discrete (countable or denumerable) infinity of definitions
for logical predicates or logical subjects.
2. But visual images (such as photographs or visual percepts) and
sound images and percepts (such as the musical passages that
correspond to frere2.gif) vary along a continuum -- an uncountable
infinity of possibilities for which the logical predicate is true.
3. When an image is "connected" with an index, it serves as a
quasi-predicate to state a proposition or dicisign. Any instance
of that continuum of pseudo-predicates may be connected with an
index to state a proposition.
4. But note Peirce's term 'quasi-subject' for the intersection of
a continuum of light rays with the plane of the photograph. There
is an uncountable infinity of potential intersections. A slight
movement of the camera might have caused any of them to become
the image recorded by photograph.
In summary, Peirce based his semeiotic on a solid foundation of
mathematics and logic. A seme is a predicate or a quasi-predicate.
There is no overlap between a seme and a subject or quasi-subject.
Peirce would reject any attempt to blur those distinctions as
"loose talk" -- as he did with loose definitions of pragmatism.
John
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