The quantifier phrases not every and not only can be taken to mark negations of generalizations stated with every and only; they therefore cite the existence of counterexamples. Similarly, though less naturally, words like some and a can be taken to mark the negations of generalizations stated with no (although a may sometimes be used to the same effect as every).
Although some sentences containing both quantifier phrases and words marking connectives cannot be analyzed as truth-functional compounds, many can. It is clear how to do this when the sign for a connective is used to combine separate generalizations, but the analysis may be more problematic in other cases. For example, every X and Y can be understood to indicate a conjunction of generalizations but so does no X or Y. A claim of either sort can be analyzed as a single generalization, but its restricting predicate must use disjunction (i.e., it amounts to the quantifier phrase everything that is X or Y). This recalls, and can be traced to, the properties of conjoined conditionals with a common consequent. Something similar happens when or appears in the quantified predicate of a negative generalization.
In sentences where any and every are alternatives that convey different meanings, the use of any can be understood to indicate a generalization whose scope is wider than some other operation, and the use of every will indicate a generalization whose scope is narrower than that same operation.