Lemma 1.9 in Cori and Lascar's Mathematical Logic: is part 2 of the proof incomplete?

[omoplata]

"Mathematical Logic" by Cori and Lascar: Incomplete proof of Lemma 1.9?

I have a question on the book "Mathematical Logic: Propositional calculus, Boolean Algebras, predicate calculus" by Rene Cori and Daniel Lascar.

Proof of Lemma 1.9 given on http://books.google.com/books?id=JB...tical logic cori&pg=PA15#v=onepage&q&f=false" is in three parts (bulleted list). Part 2 is where they prove that $o[\neg F] \geq c[\neg F]$ for any propositional formula $F$. $o[\neg F]$ is the number of opening parentheses in $\neg F$ and $c[\neg F]$ is the number of closing parentheses in $\neg F$.

My argument is that this cannot be proven YET for ANY formula $F$, because it hasn't been proven yet for formulas containing parentheses or the symbols $\wedge , \vee , \Rightarrow , \Leftrightarrow$. That is done in part 3. Part 2 proof is only correct for formulas containing propositional variables (since part 1 proves $o[\neg P] \geq c[\neg P]$ for any propositional variable $P$ ) and the symbol $\neg$.

Propositional formulas and propositional variables are defined in http://books.google.com/books?id=JB...atical logic cori&pg=PA9#v=onepage&q&f=false".

Am I correct or am I missing something?

 

[micromass]

Hi omoplata!

You basically apply lemma 1.6 here (sadly I cannot see the book past that point).

 

[omoplata]

Hello micromass

Only page 13 is missing. I uploaded it to http://i1105.photobucket.com/albums/h359/jacare_omoplata/page13.jpg" .

 

[omoplata]

They haven't mentioned lemma 1.6 in the proof :/

 

[micromass]

They haven't mentioned lemma 1.6 in the proof :/

No, they haven't, but that's what they're using. They said that the prove it through induction, and lemma 1.6 basically describes how you need to prove something through induction.

In your example, we have Y(F) to be the statement o(F)=c(F)...

 

[omoplata]

Oh, OK. I get it now. Thanks.