Jerry, Clark, list,
In my response to Jeff B.D., I was defending the claim that board
games are versions of mathematics. But I definitely do *not* restrict
math to board games or to set-theoretic models.
Jerry
Many mathematicians reject set theory as a foundation for mathematics
Yes. Peirce did and so do I. The four board games I cited illustrate
diagrammatic reasoning. But those diagrams use only discrete set
theory. Peirce also considered continuous diagrams, and so do I.
I would also allow diagrams for any mathematical structures anyone
might propose or discover -- including quantum-mechanical diagrams.
JFS
Thanks for the reference. On page 134, Béziau makes the
following point, and Peirce would agree:
JYB
Universal logic is not a logic but a general theory of different
logics.
Jerry
Analyze this quote. Is [JYB] saying anything more beyond
a contradiction of terms?
Peirce's semiotic is a general theory of all kinds of sign systems.
Those systems include, as special cases, all natural languages and
all versions of formal logic. I agree with Montague that the
underlying semantics of NLs and formal logics are essentially the
same, but I would add that formal logics are weaker than NLs.
I interpreted JYB as saying that universal logic is a theory about
logics in the same sense that CSP's semiotic is a theory about logics.
But JYB's notion of universal logic is weaker than CSP's semiotic.
JYB
This general theory is no more a logic itself than is
meteorology a cloud.
Jerry
What the hell is this supposed to mean? Merely an ill-chosen metaphor?
My interpretation of JYB: Universal logic is to any particular logic
as meteorology is to clouds.
Jerry
Chemical isomers are not mathematical homomorphisms because of the
intrinsic nature of chemical identities. Thus, this reasoning is
not relevant to the composition of Boscovichian points.
I would not impose any restrictions on the kinds of diagrams or the
mappings that define similarity. If you can define a Boscovichian
diagram for chemistry, I believe that Peirce's notion of diagrammatic
reasoning could accommodate that diagram.
Implication: Instead of defining a special kind of logic for every
kind of subject matter, I would just change the kinds of diagrams
-- quantum mechanical diagrams, Boscovichian diagrams, or whatever
mathematical structures anyone might discover or imagine.
JLRC
Semiotics is not, in my view, a foundation for logic which is
grounded on antecedent and consequences.
That is a Fregean view of logic, not a Peircean view. For his
Begriffsschrift, Frege chose implication, negation, and the
universal quantifier as his primitives.
For his algebraic logic, Peirce started with Boolean algebra and
added quantifiers. But he later switched to existential graphs.
The early version distinguished Alpha (Boolean) from Beta (which
added the line of identity). But he later started with relational
graphs (existence and conjunction) and added ovals for negation.
For beginning students, Boolean algebra is too abstract. It just
represents an NL sentence with a single letter like 'p'. Peirce's
relational graphs are a better starting point because they can be
translated to and from actual NL sentences. As a pedagogically
sounder approach, I follow Peirce's later tutorials (circa 1909).
See the first 25 slides of http://www.jfsowa.com/talks/egintro.pdf
Note slides 3 and 4 which come from Peirce's own intro in MS 145.
In slide 8, I discuss one of CSP's examples that has a direct
mapping to and from RDF -- the basic notation for the Semantic Web.
Many people believe RDF is a good starting point for logic. I hate
the RDF notation, but I use the comparison to show semantic webbers
how a real logic can be defined on top of something like RDF.
Also note CSP's rules of inference (slide 25). They are grounded
in the need to preserve truth (as determined by endoporeutic). And
they apply equally well to Kamp's Discourse Representation Structures,
which Kamp designed for NL semantics.
Note slide 31, which presents two *derived rules of inference*
that are implied by the rules in slide 25. These derived rules
emphasize generalization and specialization. I believe that it is
more appropriate to say that logic is a theory of generalization
and specialization. That includes implication as a special case
(p implies q iff p is more specialized than q).
There is much more to say, some of which I say in the slides
http://www.jfsowa.com/talks/ppe.pdf . See slides 39 to 60.
In particular, note slide 59 about Turing oracles.
Clark
The problem with the game theoretical view of mathematics is
the question of realism.
I'm not sure what you mean by "game theoretical view".
There are three options, with some similarities among them:
1. The idea that games like chess are mathematical systems.
2. The point that Peirce's endoporeutic may be characterized
as an example of Hintikka's game theoretical semantics.
3. Wittgenstein's debate with Turing. (I prefer LW's side.)
Clark
there’s a difference between how we use the language of
mathematics and what the objects of mathematics are. That
is what are the relationship between the game and reality.
The issues of nominalism vs. realism are orthogonal to all three
of these kinds of games.
Clark
I think we have to think through carefully what sort of game we
are playing if we’re going to use that as our metaphor.
Yes. But I believe that both nominalists and realists could adapt
any of the three "game theoretical views" to metaphors that are
compatible with their ways of thinking and talking.
Summary: What I'm trying to emphasize is the fundamental
importance of diagrammatic reasoning for logic, mathematics,
language, science, and everyday life. The model-theoretic
semantics used to define truth in formal logics is a special
case of diagrammatic reasoning.
John
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