Meaning of the logical connectives

The meaning of the implication/conditional operator $A\rightarrow B$ is simply that of a relation of truth values (as Kant described the hypothetical judgment in the CPR). It has nothing to do with causality, inference or relevance.  If we take $0$ as false and $1$ as true then $A\rightarrow B$ is simply the proposition which states that the truth value of $A\leq $  truth value of $B$. What is paradoxical about the fact that for any proposition $A$ we have that 0 $\leq$ truth value of $A$ ? What is paradoxical about the fact that given any two propositions $A$ and $B$ we have that either truth value of $A\leq$ truth value of $B$ or truth value of $B\leq $ truth value of $A$ ? Relevancy is irrelevant in the face of propositions regarding the relationship of the truth-values of propositions - which are purely  mathematical. Logical connectives are in a way a reflection-into-self of logic, they are propositions - having truth values - about the truth values of propositions. This is clear even in the semantics of linear logic, interpreted as a many-valued logic.  And the many-valued truth value of $A\& \sim A$ can be seen for instance the the result of a voting process. There can be a draw between $A$ and $\sim A$ and this itself be a value.

In general implication means that there is some computable function that takes terms inhabiting in $A$ into terms inhabiting $B$.  That is, we can compute $B$s in terms of $A$s.  Connectives are semantically truth-value based or in general witness based. Their legitimacy and value is untouched.  We can however think of an additional, alternative theory of intensional connectives, relevance, inference and causality. Notice that if an effect is unique to its cause then classical logical connectives cannot capture causality.

Category theory has since decades developed a useful tool for dealing with contextualism and pragmatics: fibered category theory.

Even Girard's linear logic can be understood in terms of phase semantics; as an algebraic many-valued logic.  $\multimap$ is interpreted much like in realizability or dependent type theory.