Classical logic is divided into two components, propositional calculus and predicate calculus. The second is an extension of the first and is more powerful, as it allows us to define statements such as Gx to be "x is a god" (the uppercase letter is the predicate), and introduces the existential and universal quantifiers.
However, belief statements are not of the form allowed in classical logic, because they are not "ordinary" predicates, but mental state predicates. If treated as such, they lead to contradictions. Let Bxyz= "x believes y is z", s= "superman", c= "Clark Kent", l= "Lois Lane", Wxy= "x works with y"
(1) Wlc | Given (Translation: "Lois Lane works with Clark Kent."
(2) ~Blcs | Given (translation: "Lois Lane doesn't believe Clark Kent is Superman")
(3) ∀x(c=x→c=s) |Given (translation: for all x, if x is Clark Kent, then x is Superman)
(4)Wls | from 1 & 3, substitution (translation: "Lois Lane works with Superman")
(5) ∀x(Wlx &∀y{y=s →~Blyx})| from 1 & 2 (translation: "for all x, if Lois Lane works with x AND for all y, if y is Superman then Lois Lane doesn't believe x is superman")
(6) ∃x(Wlx & ∀y{y=c →Blyc}) | from 1, 3, & 5 and substitution (translation: there exists an x such Lois Lane works with x and for all y, if y is Clark Kent then Lois Lane believes y is Clark Kent")
(7) ∃x(Wlx & ∀y{y=s →Blys}) | from 1, 3, 5 & 6 and substitution (translation: there exists an x such Lois Lane works with x and for all y, if y is Superman then Lois Lane believes y is Clark Kent")
(8) ∀x({Wlx & x=c} →Blxs)| from 7 (translation: "for all x, if Lois Lane works with x AND x is Clark Kent, then Lois Lane Believes x is superman")
***(9) Blcs | from 8
Here we have a straightforward contradiction. Because we treated belief like any other predicate, we proved a contradiction. Many like this could be offered (and in fact the above could be simplified but I used a somewhat extended version as a teaching tool), but the point is that predicate calculus ("classical logic") requires the identity operator to actually identify something as being itself (Superman IS Clark Kent, they're just two different names for the same person), while beliefs do not. It is possible for Lois Lane to work with Clark Kent and not believe she works with Superman.
Hence, other logics exist which allow epistemic predicates.