What Goedel said of PM (Principia Mathematica, Whitehead/Russell, 1919):
"What is missing, above all, is a precise statement of the syntax of the formalism. Syntactical considerations are omitted even in cases where they are necessary for the cogency of the proofs"[5]."
-- cited by wiki.
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Yes, but this is a reference to a concise statement of the syntax of a formal (artificial) language, not of a natural language.
ReplyDeleteToo true! I must say I had occasion to discuss, elsewhere, what Whitehead/Russell explicitly say about the 'tilde' or negation-sign. What P. Smith, in his "Logic" calls the 'squiggly':
ReplyDelete~
---.
This of course originates from work in mathematics. "~" is like the descendant of the MINUS sign in mathematics.
So, while indeed Whitehead/Russell care just for the 'formal' language, they are setting the bases for this idea, taken up by Grice in "Logic and Conversation" that there IS an easy correlation between this 'formal' counterpart and its natural correlate or analogue. Notably
~ "not"
--- This is the FIRST pair identified by Grice in the opening passage of "Logic and Conversation". So I would think that those formalists who were reading "PM" were just thinking that, with minor caveats, Whitehead/Russell were providing for the formal schemata (as per logical FORM) of what something like "English" is all about.
As discussed by Atlas, in his book on Implicature (Clarendon), then, Whitehead/Russell commit themselves to ~, qua 'contradictory negation' being at the base of our 'not' expressions.
And allowing for
~~p iff p
as 'negation' elimination, as it were. Reductio ad absurdum being negation introduction, rather. These introductions and eliminations, for the formal counterparts, are the ones Goedel is criticising as not being too obvious. And indeed, as Grice is writing in 1967, he can of course make avail of works like Quine's and -- his favourite -- Benson Mates -- as they were working on the actual construction of explicit formal systems with explicit formation-rules and 'interpretation' rules.
Goedel was indeed a genius IF he was able to detect that "PM" was lacking in syntactic clarifications, yet spend no time in proving the indecidability issues for the quantification fragment of the language "PM" was providing. Or something!