What is wrong with Boolean style of reasoning? 
Two main languages have been developed to achieve that object: Boole’s
“algebra of logic” and the predicate calculus. Boole’s approach was to
represent classes (e.g., happy creatures, things productive of
pleasure) by symbols and to represent logical statements as equations
to be solved. His formulation proved inadequate, however, to represent
ordinary discourse. A number of nineteenth-century logicians,
including Jevons [80], Peirce [132, 133], Poretsky [139], Schröder
[159], Venn [183], and Whitehead [185, 186], sought an improved
formulation based on extensions or modifications of Boole’s algebra.
These efforts met with only limited success. A different approach was
taken in 1879 by Frege [51], whose system is the ancestor of the
predicate calculus. The latter language has superseded Boolean algebra
as a medium for general symbolic reasoning.
Brown, Frank Markham. Boolean Reasoning: The Logic of Boolean
Equations (Dover Books on Mathematics) (pp. 1-2). Dover Publications.
Kindle Edition.

Can you please explain what does it mean ‘inadequate’ and how did they prove it? What ware the crucial reasons to switch to predicate calculus? What is wrong with ‘’Law of though” as an instrument for reasoning?
 A: One of reasons I can think of is it seems quantification is the first order (step) towards true causality than unquantified Boolean versions, such as the usual material conditional where you cannot tell whether P logically implies Q just by looking at the truth values of P and Q. This is because causality claim means that every logically possible circumstance that makes P true makes Q true. Logical implication implicitly quantifies over all possible circumstances.
For example the sentence "Mary is not at home whenever Jack is singing." is limited in capturing the implicit logical consequence if expressed through the usual Boolean material conditional connective as $Q→P$ where P=Mary is not at home, Q=Jack is singing. But as an English speaker we intuitively know there's some causal connection between P and Q. And for that you have to employ quantification $∀t(Q(t) \rightarrow P(t))$ so that it can capture more logical implication than non-quantified versions even though it still only formally means $∀t(\neg Q(t) \lor P(t))$. In addition to offer truth-functional information, Frege's quantified form also offers more truth-conditional information.
