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