Quantifier in Integer Programming/Logic

Question

It is common to write constraints with something like $x_s \leq y \quad \forall s \in X$ In Integer Programming, where x_s and y is a variable. However my tutor said that this not so absolutly correct since the quantifier is coming from logic and doesn't make sense in this form. He suggested to write for all as word instead. Is this correct?

Answer 1

It is arguably a matter of style.

My -- quite strong -- personal opinion agrees with your tutor, though. The $\forall$ symbol is not a works-everywhere drop-in replacement for the words "for all". It's a sign that specifically belongs to formal, symbolic logic, so if you use it at all, you should follow the grammar of logical formulas. That means that "$\forall s$" must go in front of the property that you state holds for every value of $s$ -- not afterwards, and not with any weird spacing.

If, in your thinking, "for all $s$" comes after the property that holds for all $s$, then you're thinking with English grammar, and you should then notate your thoughts with English words and spelling too.