1) ?P(c = b?P(c)) 2) ?x( f (b) = x)) (note: ? means backwards E)

Question

Suppose we have an FOL with names/constants a,b,c,d, variables x,y, z, function symbols
f (x),g(x),h(x,y), predicate symbols P(x),Q(x),R(x,y),x = y, the usual connectives and the quantifiers
∀,∃. For each of the following, indicate whether or not it’s a simple term, complex term,
well-formed formula [wff.], non-atomic sentence or an atomic sentence, or none of these things,
and, if they are, explain why, and if not, also explain why, pointing out all the reasons why they fail
to be so. (State also whether there are the right number of brackets. If a bracket is missing, the expression
is not properly formed.) In the case of wffs. (which, of course, include sentences), then you
should say which of the variables, if any, is bound and which free.

Answer 1

Gino, the statement as you have written it is not, to my knowledge, a wff in predicate/first-order logic.

 

∃P(c = b∃P(c)) 2) ∃x( f (b) = x))

 

It has many problems, to the point that it isn't really clear precisely what it is trying to say.