TITLE: Second order logic question.
QUESTION [6 upvotes]: I'm reading Michael Potter's book "Set Theory and its Philosophy" and where he's explaining why he chose to use first-order predicate calculus with identity instead of second order logic to reason about sets, there is this paragraph on page 13 where I need some clarification about what exactly he's saying about second order logic.  The paragraph is as follows:

It is undoubtedly significant, however, that a formalization of first-order logic is available at all. This marks a striking contrast between the levels of logic, since in the second-order case only the formation rules are completely formalizable, not the inference rules: it is a consequence of Godel’s first incompleteness theorem that for each system of formal rules we might propose there is a second-order logical inference we can recognize as valid which is not justified by that system of rules.

My questions:
1) Whats do we mean by the formation rules and the inference rules of a logic?  I.e. whats the difference between the two types of rules?
2) Whats meant by "...for each system of formal rules we might propose
there is a second-order logical inference we can recognize as valid which is
not justified by that system of rules."  I.e. how do we recognize something as valid if its not justified by the logics system of rules?

REPLY [2 votes]: 1)  By "formation rules", he just means rules which tell you when a string of symbols counts as a well-formed formula or a sentence. 
By "inference rules" he means like in a proof system. There are a number of different proofs systems available for first-order logic, e.g. Hilbert axiomatic systems, Fitch-style natural deduction systems, sequent calculi, etc. (I'm assuming you're familiar with some of these?) All of these systems are sound and (more importantly) complete for first-order semantics, i.e. a set of sentences $\Gamma$ has a model if and only if $\Gamma$ is consistent (i.e. there is no proof from some sentences of $\Gamma$ to a contradiction). However...
2)  Second-order logic, by contrast, has no such proof system. Any sound proof system you propose for second-order logic will inherently be incomplete, i.e. there will be a set of sentences $\Gamma$ from which you cannot deduce a contradiction, yet which have no model. (Put another way: there are some $\Gamma$ and $\varphi$ such that $\Gamma \vDash \varphi$ but $\Gamma \not\vdash \varphi$.)
Potter's point here is just that there is some such set of sentences in second-order logic, which is both unsatisfiable yet consistent; it's not really a comment about what we can or can't recognize to be valid.