Mention “logic” around the typical university today and you are likely to educe a variety of thoughts in your audience: perhaps something having to do with computer programming: loops, if, else, then statements, and so on; perhaps something to do with “postgenderist” ideology railing against the “illogic” of the political “right-wing”; maybe a very dry textbook; or the name Gottlob Frege; or perhaps, if someone has taken a course called “Logic”—or stumbled upon the whiteboard leavings of such a class—something that looks a lot like this: ̚ ⱻx(Rx&Px)→ ⱻxQx, ̚ ⱻx(Rx&Px) ⸠ ⱻxQx. This lattermost sense of logic has much more to do with computer programming than postgenderist ideology—such being an abuse of the term—and it has quite a lot to do with dry textbooks and Gottlob Frege, whose rather dry 1879 paper, Begriffschrift, was a watershed moment in the conception of a new approach to logic for the ensuing 20th century. Frege’s work as a whole was a great advance in the field of formal logic, for he introduced a system more agile in dealing with both predicates and propositions than the traditional formal logic which had preceded it.
But, unfortunately, Frege conceived of his logic on the model of mathematics. I say this is unfortunate not because of the many and often-confusing (or confused) variations in abstract symbolization which have arisen since Frege (including those of Frege himself), but because of the consequent disconnection of logic from the reality of the λόγος. This disconnect likely follows the general divisiveness characteristic of modern philosophy, which separates mind and body, subject and object, nature and culture; why not logic from λόγος?
Of course, to see what I mean, it is perhaps helpful to think about what logic really is. The λόγος, of course, is the intelligibility of the real; not as residing in intellects, but as transcending both the intellect which grasps it and the reality in which it is grasped. Logic, on the other hand, is or ought to be the study of this process by which the mind strives in grasping this reality.
I believe it an unfortunate reaction that—given that ultramodernizing analytic philosophers donned the mantle of logicians—many Thomists of the 20th century rebelled against the advances that followed Frege (and even more unfortunate that Charles Peirce’s contributions went largely unnoticed or at least underappreciated by all quarters). Thus while there was an emphasis on the Aristotelian logic, and even some-albeit-insufficient interest shown in the logical advances of later Scholastics, such as the Conimbricenses and John Poinsot, this interest came late and only to the sides, despite the importance we find Thomas Aquinas granting logic in q.6 of the tertia pars of his Super Boethium de Trinitate:
|Ad tertium dicendum quod in addiscendo incipimus ab eo quod est magis facile, nisi necessitas aliud requirat. Quandoque enim necessarium est in addiscendo incipere non ab eo quod est facilius, sed ab eo, a cuius cognitione sequentium cognitio dependet. Et hac ratione oportet in addiscendo a logica incipere, non quia ipsa sit facilior ceteris scientiis, habet enim maximam difficultatem, cum sit de secundo intellectis, sed quia aliae scientiae ab ipsa dependent, in quantum ipsa docet modum procedendi in omnibus scientiis. Oportet autem primo scire modum scientiae quam scientiam ipsam, ut dicitur in II metaphysicae.||To the third it must be said that we begin in learning from that which is most easily done, unless something is required of necessity. For sometimes it is necessary in learning to begin not with that which is easier, but with that the cognition of which cognition of subsequent matters depends. And for this reason it is necessary in learning to begin from logic, not because it is easier than other sciences, for it has the greatest difficulty, since it is concerned with second intentions, but because the other sciences depend upon it, insofar as it teaches the mode of proceeding in every science; and it is necessary first to know the mode of a science before that science itself, as said in Metaphysics II.|
To put this otherwise: we cannot have any science—philosophical or empirical, cenoscopic or idioscopic—unless we have logic, and ignorance of logic will lead to error in the conduct of those sciences. Thus, even though logic is the most difficult of sciences, “since it is concerned with second intentions”, it is necessary for all sciences, since science operates through the mind. In the proemium of his commentary to Aristotle’s Posterior Analytics, Aquinas asserts: “ars quaedam necessaria est, quae sit directiva ipsius actus rationis, per quam scilicet homo in ipso actu rationis ordinate, faciliter et sine errore procedat” – “a certain art is necessary, which is directive of the very act of reason, namely, that through which humans may proceed orderly, more easily, and without error in the act of reasoning itself”; that is, logic.
What this does not mean is that someone needs to be able to interpret ̚ ⱻx(Rx&Px)→ ⱻxQx, ̚ ⱻx(Rx&Px) ⸠ ⱻxQx or any other symbolically formulated logical proof. That is but one way to characterize the operations of the mind—an at times helpful way, but not a necessary one, and not always the best, or even a helpful way. For one thing, it strictly limits the range of logical investigation to what can be conveyed by empty symbols. When testing for deductive validity, this is extremely valuable. When considering the relation of the logical to abductive inference or “retroductive” testing, an overly-rigid symbolic mentality may be a hindrance. In other words, formal logic as the study of varied forms of deductive validity is only as good as it enables us in reasoning as a whole. As Charles Peirce put it (1893: “The grammatical theory of judgment and inference”, Collected Papers 2.444fn), “the illative relation”—that is, the relation in which the mind moves from one term or proposition to another, discerning the truth between the two of them—“is the primary and paramount [logical] relation.” Deduction is but one means for illation.
What the above need for an art of logic, an art that governs our understanding of human reasoning and allows us to have a genuine philosophical science, does mean is that we require critically reflective habits. To be critically reflective is to think about how you are thinking; that is, to deal with what the Scholastics called “second intentions”. A first intention is the relation following from a concept to the object it makes known. A second intention is the relation of the mind to that first relation whereby the object is made known, and therefore, indirectly, to the concept whereby it is known. Among the common notions which fall within second intentions are genus, species, specific difference, property, and accident, often called the predicables because they distinguish how (rather than what—called the predicaments) we speak of things. The ideas of “genus” and “species”, etc., do not exist outside the human mind, but rather identify how we conceptualize of things that do so exist. If I say, “Human beings exist within the genus ‘animal’”, “the genus ‘animal’” does not identify a reality existent outside the mind—even though animality really does exist in each and every animal. Rather, I am identifying the way in which I conceptualize this reality; namely, as a genus.
Through understanding how we are thinking of things—which goes into far more nuance than genus and the other predicables, embracing all manner of socially-constituted realities (which are not non-natural, but an extension of nature either fitting to or deviant from)—we can improve our understanding of them; rooting out false ideas we have unconsciously adopted. Thus, Charles Peirce considered logic (said broadly, synonymous with semiotics) to be the “normative science of truth”: i.e., the science which discerns and exposits the norms by which the truth is grasped and made known.
This sort of logic is what we need, today, more than ever.
 Different editions, commentators, and translators divide the text in different ways. The Busa edition (at http://www.corpusthomisticum.org/cbt.html, places the relevant text as pars 3, q.6, a.1, ad.13. Maurer’s translation places it as q.6, a.1, reply to the second question (b), ad.3 (p.69).
 Distinctions of logic as either an art or a science have received a great deal of attention; for a science is certain knowledge while an art is knowledge concerning something to be done, something which can be otherwise. Put as succinctly as possible, the art of logic is the conduct of reasoning itself, while the science is the knowledge of how that reasoning occurs. To practice logic as an art requires investigating it as a science, though not all great scientific logicians are great logical artists; and vice versa.