The semantics of propositional logic involves so-called connectives, including the negation "$\neg$", the conjunction "$\wedge$", the disjunction "$\vee$", the inclusion "$\Rightarrow$", and the equivalence "$\Leftrightarrow$". These connectives are important concepts in mathematics on their own. Therefore, they deserve separate definitions. We will introduce them now step by step, starting with a formal definition of connectives.
Let $L$ be a formal language with a syntax of propositional logic $PL0$. Let $L'\subset L$ be the subset of all strings of $L$ which are propositions, i.e. which have a semantics of $PL0$. A $k$-nary connective is an operation "$\circ$" used to connect $k$ ($k=1,2,\ldots$) propositions
Propositions being Boolean constants or a Boolean variables are called prime propositions.
Any proposition involing a connective $p:=p_1\circ\ldots\circ p_k$ is called a compound proposition.
Definitions: 3 4
Examples: 5 6
Lemmas: 7 8 9
Proofs: 10 11 12 13 14