a fundamental clarification about predicate expression (formula)

Question

I have few foundation questions to be clear about expression involving predicates.

$\forall n\in \Bbb N.p(n) \tag {1.2}$

Here the symbol $\forall$ is read “for all.” The symbol $\Bbb N$ stands for the set of nonnegative integers, namely, $0, 1, 2, 3, \ldots$ (ask your instructor for the complete list). The symbol $\in$ is read as “is a member of,” or “belongs to,” or simply as “is in.” The period after the $\Bbb N$ is just a separator between phrases.

Q1 : what is the correct mathematical term for expression like above ? is it Predicate formula?

Q2: When to use "." to separate phrases? because in the same doc I have seen that there are more than one term such as $\forall n\in \Bbb N$ without separated using ".", is there different meaning in such declarations? http://courses.csail.mit.edu/6.042/spring12/mcs.pdf

Answer 1

There is not one single correct term for expressions like $\forall n \in \Bbb N. p(n)$. "Predicate formula" seems as adequate as the more common "(well-formed) formula of predicate logic". A distinction worth pointing out is:

• (Well-formed) Formula (of predicate calculus): Any expression that you would call a "predicate formula", i.e. that can be formed using the rules of formation for predicate logic; a listing of those is e.g. here (there are as many variations in notation for predicate calculus as there are books on it, so don't mind the small differences with what you may have encountered before).

• Sentence (of predicate calculus): A formula where all variables are "quantified over". For example, $\forall n \in \Bbb N. p(n)$ is a sentence, while $\forall n \in \Bbb N. p(n,m)$ is not (the $m$ is not quantified over).

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As for the use of the dot, it is simply a syntactical/convenience device to help you extract the intended reading of the formula. Almost everyone is very sloppy in the use of these whenever possible, to keep formulae as readable as possible without being ambiguous. Thus, the following are all expressions of the same thing (I've encountered them all):

1. $\forall n \in \Bbb N.p(n)$;
2. $\forall n \in \Bbb N: p(n)$;
3. $(\forall n \in \Bbb N)\, p(n)$;
4. $\forall n \in \Bbb N\, (p(n))$;
5. $\forall n \in \Bbb N\, p(n)$;
6. $\forall n \in \Bbb N\,pn$.

Outside of the context of formal logic, many authors take the liberty to write down things that are not strictly speaking admissible as formulae, but nonetheless their reading is (or at least is intended to be) completely clear. So if you are asking yourself questions like:

Is this expression just a sloppy version of that other, formally correct expression?

then I can assure you that it is safe to assume that the answer is "Yes".