One recurring pattern in the history of philosophy seems to be the substitution of a canonic for logic by certain groups of empiricists. 'Canonic' is a word we get from the Epicureans. Epicurus had insisted on three criteria by which we measure whether a claim is true or false: sensations, prolepses (preconceptions or anticipations), and passions. Sensations were the fundamental standard; prolepses arise when we have a great many sensations and therefore begin to anticipate on the basis of our experience; and passions were pleasures and pains. Each of these had immediate evidence (enargeia). On the basis of this we make judgments which may be true or false depending on how they conform to or diverge from these basic starting-points. The Epicureans, of course, developed these ideas at some length: each of the criteria, for instance, was explained in terms of our being affected by the simulacra of things. This "test-science of truth," to use Zeller's apt name for it, is really just a set of guidelines for judging claims on the basis of empirical experience; it wholly occupies the place that logic held for (e.g.) the Stoics.
But there are other cases; Hume is a shining one. As a result of his study of causation, and particularly necessary connection, Hume develops a list of "general rules" by which we may know when things are causes and effects (I paraphrase from Treatise 1.3.15):
(1) Cause and effect must be contiguous.
(2) Cause must be prior to effect.
(3) Cause and effect must be constantly united.
(4) The same type of cause always produces the same type of effect, and the same type effect always arises from the same type of cause. (As he usually puts it, like effects imply like causes, like causes imply like effects.)
(5) Where several different objects produce the same type of effect, it must be by means some quality common to them all.
(6) A difference in the effects of resembling objects must proceed from some point in which they differ.
(7) When any object increases or diminishes in correspondence with the increase or diminishment of its cause, it is an effect compounded of several different effects that are derived from different components of the cause.
(8) An object that fully exists for a time without its effect is not the sole cause of that effect, but only when combined with some other principle.
"Here," says Hume, "is all the LOGIC that I think proper to employ in my reasoning; and perhaps even this was not very necessary, but might have been supply'd by the natural principles of our understanding" (18.104.22.168). It seems clear enough that this is essentially a canonic, in the same family as the Epicurean canonic, although it has the advantage of providing some direct treatment of what is usually regarded as a weakness in the Epicurean canonic, namely, its lack of a well-developed account of how things may be explained.
One of the questions that arises in considering cases like these is what would push an empiricist in this direction. A very tempting and plausible suggestion is that this is caused by polarization: empiricists will go this far when there is an opposing position that is able to stake its claim using logic, which is not taken as a set of guidelines for making judgments about sensory experience but as something exhibiting necessity, universality, etc., that cannot be found in sensory experience -- when there are "scholastic head-pieces and logicians," as Hume calls them. This the Epicureans certainly had, in the Stoics, and the early modern empiricists, in the rationalists. What was key was not that Stoics and rationalists appealed to logic but that they appealed to features of it as constitutive of rationality and (with at least some plausibility) as confirming their own position. This forces the empiricist to provide an alternative route for deciding what is true and false, one that does not give succor to the enemy. Explaining the effect in terms of such an opposition seems to be right; but it is difficult to say if such a conflict is both necessary and sufficient.