Jeff, Mike, and Jon,
Mathematics is diagrammatic reasoning, and EGs are a version of
logic that uses a more flexible and versatile system of diagrams
than Peirce's algebra of 1885 or any algebra since then.
But the diagrams are fundamental. Any words used to describe
the diagrams are useful *only* in teaching and explaining the
diagrams. Mathematicians, especially Peirce, always think in
terms of the diagrams, not the words that describe the diagrams.
I agree with Jeff's comments and Mike's emphasis of what Jeff said:
MS quoting JD:
I take the EGs to be topological in character. As a formal system,
they are based on the notion of relations of composition and
transformation that hold between areas on a sheet of assertion that
is, itself, continuous. Various discontinuities are introduced onto
the sheet to represent what is existing and discrete as individuals,
but the continuity of this type of logical system is central and not
peripheral.
As an example, I drew the attached diagram EGgiving.png. It shows
four different EGs for the sentence "Sue gives a child a book."
1. The EG on the upper left is the simplest. The relation named
Gives is a triad with three subjects: Sue, Child, and Book.
2. The EG on the upper right is transformed from #1 in two ways:
(a) The verb 'gives' may be nominalized to the gerund Giving
by hypostatic abstraction. That creates a tetra-identity.
(b) Three dyads, named Agent, Recipient, and Theme, are linked to
the three subjects of Give or the gerund Giving. In linguistics,
those dyads are called case relations or thematic roles.
3. The EG on the lower left shows that the tetra-identity in #2
may be replaced by two teridentities. It has no effect on
the semantics of the EG or its translation to the algebra.
4. The EG on the lower right replaces the triad Gives in #1
with the tetrad Covenant. It shows that Covenant is linked
to Child by two different dyads: Co-agent and Recipient.
These examples show how a single EG can replace a vague cloud
of verbiage with a clear and precise diagram. Since diagrams are
Peirce's natural way of thinking, especially about math & logic,
the diagrams must always take precedence over the words.
During the past century, there have been many new developments in
linguistics, logic, and the many branches of cognitive science.
For example, the labels for thematic roles are very convenient for
highlighting the relationships. Compare EG #4 in EGgiving.png to
Peirce's explanation:
CSP
Thus, A gives B to C becomes A makes the covenant D with C
and the covenant D gives B to C. (CP 1.474)
EG #4 shows that Covenant is a tetrad, and the labels Agent,
Co-agent, Recipient, and Theme show how the covenant is related
to each of the three participants. The EG also shows that the
covenant is related to the child in two distinct ways.
JAS
My purpose (as usual) is to interpret Peirce by attempting to
harmonize each passage that I encounter with his corpus taken
as a whole, in accordance with my systematizing and regularizing
tendencies.
But the diagrams are fundamental and *permanent*. For first-order
logic, the linear diagrams of 1885 and the 2D diagrams of EGs
have *identical semantics*. That semantics is also identical to
Frege's version of 1879 and to *every* version of classical FOL
for the past 140 years.
That must be the foundation for any kind of harmonizing,
systematizing, and regularizing. It is also a solid foundation
for teaching Peirce's logic, semeiotic, and philosophy to any
students and professors who began their education with any other
kind of notation or terminology.
Remember that Peirce admitted that his "left-handed brain" made
it difficult for him to translate his thoughts into words. That
means that the diagrams are the primary evidence for what he was
thinking, and the words are secondary.
In EG #4, for example, the word 'covenant' has several senses.
Google, for example, lists the following synonyms: contract,
compact, treaty, pact, accord, deal, bargain, settlement, concordat,
protocol, entente, agreement, arrangement, understanding, pledge,
promise, bond, indenture, guarantee, warrant; undertaking, commitment.
But EG #4 shows that the covenant is an agreement or commitment
by the agent with or to the co-agent that the co-agent would be
the recipient of the book. That pattern of relationships can
be seen at a glance from EG #4. There is no need for Googling.
When in doubt, draw a diagram. That's what Peirce would do.
John
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