In any case, you should warn them about the different notation in the textbook.
This question is related to Should students be asked to use more than one notation for the derivative in an introductory calculus class? - In my answer, I wrote "I think, it is important that students should be flexible and open-minded to notation" and went into detail there (It will not be the first time that they will be confused with notation, once they read a textbook on their own or they have a different teacher, a similar problem arises).
However, in your situation, the problem is that the textbook is older and the notation there is out of date. But, also this will not be the last time that students will have a textbook or an article with outdated notation!
I think, in general it depends on the "type" students you have. If this a class of engineers who will only have a few courses in mathematics, what was said above might not be true - and in this case, you should probably change your textbook; but if you have a class of math majors who will eventually do their master or even more in mathematics (which I think you have since you mentioned logic), at some point you have to talk with about notation. Most textbook have some pages where they explain the notation needed to read the book and the rest of the notation is explained when intoducing. Explain your students how to use that and explain them the (dis)advantages of your notation and the textbook's notation. At least, they will learn to be more flexible (which is good for them in future as mentioned above) and they will also learn that mathematics is a historical progress and notation is part of this progress as well. If you have the feeling that your message does not arrive, you can give them examples from fields and notations they already know, e.g., Bourbaki introduced some important notation like $\varnothing$.