Mathematical Logic by Cori and Lascar : Possible typo?

[omoplata]

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

This is not about an exercise but about the conceptual content of the book. So I did not post this in the "Coursework and Homework questions" forum. I hope I'm not breaking the rules.

On Lemma 1.6 on http://books.google.com/books?id=JB...tical logic cori&pg=PA12#v=onepage&q&f=false" there is a part that says,
"...if $\mathcal{Y}(W)$ and $\mathcal{Y}(V)$ are true, then $\mathcal{Y}(\neg F)$, $\mathcal{Y}(F \wedge G)$, $\mathcal{Y}(F \vee G)$, $\mathcal{Y}(F \Rightarrow G)$, $\mathcal{Y}(F \Leftrightarrow G)$ are also true.".

I think there is a typo there and it should be,
"...if $\mathcal{Y}(W)$ and $\mathcal{Y}(V)$ are true, then $\mathcal{Y}(\neg W)$, $\mathcal{Y}((W \wedge V))$, $\mathcal{Y}((W \vee V))$, $\mathcal{Y}((W \Rightarrow V))$, $\mathcal{Y}((W \Leftrightarrow V))$ are also true.",
WITH THE ADDITION OF THE EXTRA PARENTHESES.

I've attached a picture from the next page of the rest of the proof, because that might help, and the google book omits that page.

Propositional Formulas are defined in Definition 1.2 on http://books.google.com/books?id=JB...atical logic cori&pg=PA9#v=onepage&q&f=false".

Is this a typo is there something I don't understand?

 

[Stephen Tashi]

I think the use of 'F' and 'G' instead of "W" and "V" is a typo. I don't see the need for extra parentheses. Does the book distinguish between Y(X) and Y((X)) ?

 

[omoplata]

Well, I didn't see it at first, but the book says on http://books.google.com/books?id=JB...tical logic cori&pg=PA10#v=onepage&q&f=false" that extra parentheses can be omitted.

The extra parentheses are applied only when there is a binary connective between two symbols. Like $\mathcal{Y}((W \wedge V))$. The problem is according to the definition of a formula in http://books.google.com/books?id=JB...atical logic cori&pg=PA9#v=onepage&q&f=false", $F \wedge G$ is not a formula, but $(F \wedge G)$ is ($F$ and $G$ are also formulas here). So the property $\mathcal{Y}$ applied to that formula is $\mathcal{Y}((F \wedge G))$.

But, since they say that extra parentheses can be omitted, I guess the extra parentheses are not needed.