Rick, (01)
To relate Peirce's terminology to Tarski's, it's useful to treat
Peirce's triangle of sign-interpretant-object as a generalization
of Frege's triangle Zeichen-Sinn-Bedeutung, which has been
translated in various ways. One popular triad of terms is
sign-intension-extension. (02)
Tarski's version of model theory relates the sign directly to the
object or extension while ignoring the interpretant or intension.
Tarski's student Richard Montague developed a model-theoretic
semantics for a "fragment" of English in which he represented both
the intension and the extension: (03)
1. Montague represented the intension of an English word, phrase,
or sentence s in English as a function from a possible world w
to the extension of s in the world w. (04)
2. For a word like 'dog' the intension f=int('dog') would map
a world w to f(w), which would be the set of all dogs in w. (05)
3. For a sentence like 'The sky is blue', the intension
g=int('The sky is blue') would map w to g(w), which would be
the truth value of 'The sky is blue' in the world w. (06)
4. Montague defined a grammar for his fragment that would start
with primitive intension functions for each word and derive
lambda expressions that would combine the intension functions
for the words in each phrase or sentence. The result would be
the intension functions for the phrase or sentence as a whole. (07)
Montague's definition was brilliant, but his fragment of English was
limited to 37 words, and it omitted many serious issues. Hans Kamp,
who was one of Montague's students, defined a more extensive system
called Discourse Representation Theory, in which the intension was
represented by a diagram called a Discourse Representation Structure.
For first-order logic, Kamp's DRS diagrams are isomorphic to Peirce's
existential graphs. I mentioned that similarity to him, and Kamp
acknowledged the point in his later publications. (08)
More recently, Kamp has expressed misgivings about the adequacy of
any of these logics for representing full natural language semantics: (09)
http://www.illc.uva.nl/lia/farewell_kamp.html
A Farewell to Logic in Action: Hans Kamp (010)
John (011)
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