"You're Too Formal" (Putnam to Grice)

Grice was somewhat offended. He had gone all the way from the Oxford of Austin to live peacefully among the formal Americans (a manner of speaking) and in a corridor at Harvard, Hilary Putnam stops Grice and ejaculates:

"You are too formal"

-- "I didn't know he meant my truth-assignments in a model. It couldn't be my attire".

Anyway, this from

Sakharov, Alex. "Interpretation." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/Interpretation.html

-- Alex doesn'care to quote Grice, but then that's vice versa, so what gives?

Alex writes:

"An interpretation of first-order logic consists of a non-empty domain and mappings for function and predicate symbols. Every -place function symbol is mapped to a function from to , and every -place predicate symbol is mapped to a function from to the set comprised of two values true and false.

(x)Px ("for all x,Px") is true if it is true for any element of P as value of P at free occurrences of x in P. Otherwise, it is false.

(Ex)Px ("there exists an x such that P") is true if it is true for at least one element of P as value of P at free occurrences of x in P. Otherwise, it is false.

"A formula is called satisfiable if it takes at least one true value in some interpretation. A formula whose truth table contains only false in any interpretation is called unsatisfiable. The Löwenheim-Skolem theorem establishes that any satisfiable formula of first-order logic is satisfiable in an (aleph-0) domain of interpretation. Hence, aleph-0 domains are sufficient for interpretation of first-order logic.

REFERENCES:
Chang, C.-L. and Lee, R. C.-T. Symbolic Logic and Mechanical Theorem Proving. New York: Academic Press, 1997.
Kleene, S. C. Mathematical Logic. New York: Dover, 2002.
Mendelson, E. Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall, pp. 12 and 57, 1997.

----

Corollary:

-- Grice's example:

-- We are having a party for Marmaduke Bloggs.
-- Who is he?
-- The man who climbed Mt. Everest on hands and knees.
-- Oh, he doesn't exist. He was totally invented by Merseyside journalists.
-- But we are having a party in his honour.
-- You are having a party in your dishonour, you mean.
-- Are you saying that someone isn't coming to the party.
-- You heard me distinctly: someone isn't coming, and he is Marmaduke Bloggs.




CITE THIS AS:

Sakharov, Alex. "Interpretation." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/Interpretation.html