I. What is Logic?

Man is the rational animal. Rationality, reason, is his defining faculty. All human acts, in so far as they are directed well, are directed by reason, and reason is itself carried out with skill, according to logic, its governing art. Logic is the art of reasoning well. It is not merely the study of precise arguments or algebra-style statements and inferences, as is sometimes thought. Logic studies and seeks to direct all the ways we use reason to arrive at truth: proof and demonstration, probable arguments, rhetoric, proposing plausible explanations, and keeping out errors in how we reason. Logic also includes how we make definitions, how we classify things, and how we come to know things from experience and by comparison with other things. Logic is not just an art that applies to how we search for truth in itself. It also applies to the decisions we make about how we should act in accord with truth.

The structure of reasoning, and also therefore the structure of logic, is simple. First, based on our experience, we form concepts in our minds which represent things in the world; that is, we come to know what different things are by abstracting from our experience of them. Second, we relate two concepts together in a proposition about the world which we assert or deny. That is, we see that one thing is related to another or not (e.g., A belongs to B). Third, we consider at least two propositions that relate three concepts together (e.g., A belongs to B and B belongs to C), and from these two propositions we come to see the truth of a third proposition (e.g., A belongs to C). Of course, not all reasoning looks or sounds exactly like this, but this is the simplest and defining structure of reasoning. Each of these three, the formation of concepts by abstracting from experience, the assertion (or denial) of propositions, and the deriving conclusions from what we already know, will be treated below.

This understanding of logic developed in the classical tradition of Plato, Aristotle, the Neo-Platonists, and later the scholastics of the medieval universities and their successors in our time (see most recently, e.g., Gerson, The Possibility of Philosophy). This tradition is multiform, and there are many differences in particular points between key authors; but, as with Plato and Aristotle, there is much similarity and harmony across the tradition. The contrasting contemporary view, which holds that logic is primarily the study of valid inference, is predicated on the rejection of the classical tradition’s perspective on logic. Both traditions, as well as those of non-western cultures, will be considered below. In most Catholic theology, the perspective taken is that of the classical tradition. The perspective of the classical tradition is adopted here for that reason, even though various theologians in the twentieth century and before criticized the scholastics for what was thought to be an over-rationalization of theology. The classical tradition is also adopted here because this perspective allows the various aspects of reasoning and the historical developments of logic to be integrated into a coherent picture, in a way that no other tradition has so far achieved.

Logic is primarily an art, the art of reasoning, but it is also a science, the science of how to reason. More precisely, logic is the science of the operations of the mind. Completed operations of the mind produce relations that are purely beings of reason (in Latin, entia rations), also called second intentions (more on this below). It is helpful to compare logic, for example, to medicine. The goal of medicine is health. In that sense, medicine is an art aimed at producing something in the patient. But medicine is also secondarily a science. Logic is a science in that it is a body of certain knowledge about reasoning, specifically about the acts that the mind forms as it proceeds through its reasoning. This is mainly what we study in courses on traditional logic: the doctrine of what logic is and how to reason best in general. Courses in logic also give us practice in key areas of reasoning. These doctrines about how reason works will constitute the third main part of this article. But logic as we actually use it is always related to specific subject matters and situations in which we are seeking the truth. Determining whether a claim is true or false is not proper to logic but to the given discipline that studies that thing.

II. Logic and Thought

The relationship between logic, rationality, and thought needs to be clarified. Rationality is a distinctively human capacity; logic is the art that governs the use of rationality. What, then, is thought? In order to understand what thought is, as we use the terms nowadays, we should ask what the goal of rationality is. The goal of rationality is the discovery of truth, and the guidance of human action according to truths we have discovered. Knowledge is the end or goal toward which our use of rationality aims in a given context. However, as we progress toward knowledge or as we are making our discoveries, we have not yet arrived at knowledge. Thoughts are the acts of the human mind as it deliberates and seeks to discover truth. Seen in this way, it is clear that thought as such is not the fundamental human capacity and logic is not the art of thinking, as it is sometimes said to be. Thought is subordinate to and exists for the sake of our discovering truth. St. Thomas Aquinas puts the point like this: “Thinking, properly speaking, is the movement of the mind as it deliberates, while it has not yet arrived at the full vision of the truth in question” (Summa theologiae II-II, q. 2, a. 1).

Logic governs thought, but this does not mean that thought is somehow unimportant unless thinking relates to something explicitly formulated in formal logic. In fact, the opposite is the case: the true place of thought in human life is as ordered to the discovery of truth through the exercise of rationality. Therefore, we see once again that logic governs the whole process of human reasoning, from principles to conclusions.

But logic clearly does not apply in all situations or inquiries in the same way. To examine how the art of reasoning works differently in different contexts we need to consider the parts of logic.

III. The Parts of Logic

There are different parts of logic, each based on what knowledge or evidence is available to us. In every part we must consider how we arrive at truth through the evidence available to us and how we come to show others that truth. We can also check our reasoning by expressing the truth to ourselves and seeing whether it makes sense based on the evidence we have. That is why logic includes both our inquiry and the statement of the results of our inquiry.

Demonstration

The highest part of logic is demonstration. Demonstrations are what produce intellectual knowledge of what things are, knowledge of the fact that that some one thing is necessarily and per se related to some other thing and knowledge of the causes of those relations. The word “demonstration” (in Greek, apodexis) means “laying out” or “showing.” This is the word artists use when they show students how to paint by painting an example while the students watch; it is also the word that mathematicians use when they prove that something must be true in geometry or calculus—for example, the proofs in Euclid’s Elements are paradigms of demonstration. Aristotle outlines demonstrations in his two books of analytics, the Prior and Posterior Analytics.{1}

A demonstration in its highest form gives us proof of why something is the way it is. This is called a “propter quiddemonstration,” from the Latin propter quid which means “on account of what” or “why,” as in “he showed why all animals must have an opening for taking in food.” We can also demonstrate that something is the case or not; this is called a “quia demonstration,” from the Latin quia which means “that,” as in, “he showed that the moon revolves around the earth.” Knowing the causes of things is the highest form of human knowledge, since we think we know something best and most completely when we both know what it is and why it is the way it is. In the classical tradition of philosophy, this knowledge of the highest kind is called scientia; this is translated in English as “science” or “discipline of learning”, but it does not narrowly refer only to what English speakers think of when they heard the word “science”, namely physics, biology, and the other natural sciences. When we can come to know the causes of things, we come to know them only through demonstrations. When making demonstrations to others, for example as a teacher might do, we must take into account what those others already know, since all learning starts from what we already know. When we study logic, we are coming to know, not primarily about the world, but about how our acts of understanding relate to the world; when we are learning about demonstration, we are coming to know the causes, not of things, but of our knowledge of things. The highest part of logic is formally called “demonstrative science” because through this study we come to have knowledge of demonstration itself, we come to know the causes of coming to know. 

Within demonstrative science, there are two broad parts: formal logic and material logic. When an ordinary person nowadays thinks of logic, he might think of chains of reasoning such as “all Bs are As; and all Cs are Bs; therefore, all Cs are As.” We study this kind of reasoning in formal logic. What we are concerned with in such cases is the arrangement of the terms in the reasoning and whether, from the arrangements whose truth we take to be given or assumed (these are known as “premises”), we can deduce the arrangement of terms that is the conclusion. Such deductions are said to be valid when the truth of the premises necessitates the truth of the conclusion (which is usually marked by words such as “therefore”). That is, for reasoning to be valid, it must be impossible that the premises be true and the conclusion false. The reasoning in the case above is always valid because of the relationships that the terms (i.e., A, B, and C) have to each other. If everything of one kind of thing, B, belongs another kind, A, and if a third kind of thing, C, itself belongs to the first kind, B, then, clearly, everything of the third kind, C, will belong to the second kind, A.

Though formal logic is what is most often studied in logic courses, there is another major (or greater) part of demonstrative science: material logic. This is the study of, first, how demonstration works in general (as studied in Aristotle’s Posterior Analytics), second, how different kinds of premises (like premises that are only probably true) can relate generally to the kinds of true propositions that we can discover through them, and, third, demonstration as it is found within given domains of inquiry (e.g., demonstrations in law or physics). In general, demonstrations are not just valid arguments, where the conclusion follows from the premises; demonstrations reveal what is universal and necessary and what derives from the essential aspects of the subject, and because of that they can reveal why the kind of thing we are studying is the way it is. Each discipline has its own material logic because each discipline has its own terms and principles which relate to the things the discipline studies. Each discipline has, as we say, a subject matter that determines key aspects of how to demonstrate truth in that discipline. Whenever one studies a given discipline, part of what one learns is how reasoning in that discipline operates and how it differs from reasoning in other fields. Consider one contrast: in family counseling, no one considers a family with 1.9 children since no families can have 1.9 children. But in demography, a family with 1.9 children can be reasoned about validly as a statistical average. What the major classes of disciplines have in common across their kinds of reasoning is especially that part of material logic that a logician would study. 

In contemporary analytic philosophy, it is common to use a set of tools called “analysis” to study the terms and principles and forms of reasoning within a given discipline or family of disciplines. A good example of this is the philosophy of science, which, from the classical perspective can be understood as an exploration of the material logic of the modern empirical sciences. Another even more relevant example is contemporary logic. Most modern logic is taught explicitly as “mathematical logic.” From the classical perspective, such logic is specifically claiming itself to be the formal and material logic of mathematics, not formal and material logic in general. Often contemporary logicians will speak as if mathematical logic is somehow paradigmatic and thus able to be applied to many fields. In a sense, this is true. However, from the classical perspective, the reason is not that this style of logic is somehow closer to what reasoning really is than traditional Aristotelian logic. Rather, the reason is that modern mathematical logic is a logical system designed for the modern style of pure and applied mathematics. Since every discipline has quantitative aspects, a logic of this kind will always find application.

But before we can arrive at any kind of demonstration, we must understand the terms we are using, their definitions, and the possible ways to combine terms to form propositions, i.e., sentences that are true or false. Aristotle studies terms and their relation to things in his Categories and he studies how terms can be combined into propositions in De interpretation.{2} These acts, forming terms and combining terms into propositions, are required for all demonstration. It is true that we do come to know terms and their definitions, as well as how two terms can be combined into a true proposition, but such knowledge is often drawn from our sense experience. Demonstration, on the other hand, is the act of coming to know something based on what we already have intellectual knowledge of.

Dialectic

Despite logic’s power, however, we cannot always arrive at intellectual knowledge; we cannot always find proof for making a demonstration. Sometimes, we cannot come to know what we want to know but we can come to see what is most likely to be true, although we do not know the precise reason why it is likely true. For example, perhaps a doctor treating a sick patient cannot come to know exactly which disease the patient has; but the doctor can come to see that one particular disease is likely to be the cause of the sickness, even though that is only probably true. In these cases, in order to discover as much of the truth as we can, we use logic to consider as carefully as possible whatever things we do know already as well as the opinions of the experts on the matter. When we express truths that we have learned through looking at what is most likely, we use all the probable truths that we have discovered to show what we have found. For instance, we say that an opinion or view is “probable” if it has strong arguments in favor of its truth but those arguments fall short of resolving the central question. In other words, an account of something is probable if that account resolves the relevant question only for the most part. Now, since these views are only probable, the persons to whom we express them may well disagree with what we say; this results in a discussion of both sides of the question to see what really is more probably true. This part of logic is dialectic. Aristotle discusses this in his book Topics. To sum up, dialectic is the use of reason to discover what is probably true in cases where we have not yet found demonstrative proof.

Rhetoric

Sometimes when inquiring into something, the evidence we have as nearly favors one view as the opposing view. In such cases, we seek to find a reason to prefer one view to the other. For example, in inquiring which medicine is best for a disease, a doctor might learn that there is just as much evidence that a certain medicine is good as that it is not good. But, nevertheless, he should study and inquire more in order to be able to determine as best he can whether the medicine is effective or not, although there is no strong evidence either way. In such a case, the doctor may decide that the medicine is good for him to prescribe because the company that makes it is prestigious, and other medicines from this company have proved effective. In cases like these, we consider as carefully as we can whatever will give us a reason to prefer one choice over the other. When aiming to convince others of truth that we have found like this, we try to persuade them of one view over the other by appealing to examples of similar things, the character of the speaker, other circumstances, or even appealing to the emotions of the audience as appropriate. This part of logic is rhetoric. This is mostly what we use in areas of life where many people have strong disagreements, the truth is difficult to find, or when the subject matter under consideration is contingent (e.g., in detailed political debates about what to do in a specific case such as what a given country should do at a certain time to avoid war with an aggressive neighbor). Originally, in his book Rhetoric, Aristotle focused his account of rhetoric mainly on questions related to public life by showing us how to learn the means of persuasion for public oratory in law courts and assemblies. But this part of logic is in fact more general than just public speaking. For example, we can use the art of rhetoric in teaching, in discussions with friends or family, or in business deliberations, to name just a few. Most generally, we use rhetoric when considering questions where we are not inclined by direct evidence to either side of the question and so must appeal to more indirect evidence.

Poetics

Sometimes, one must consider a matter where one does not know the cause of something and does not have a probable explanation of it or even much evidence to incline us toward one or the other view. In this case, the matter under consideration might possibly be true and might possibly be false. The best way to proceed is to propose different possible accounts in an attempt to give an explanation and see which one fits the best. If a patient is found unconscious, for example, a doctor will try to explain how he came to be unconscious by proposing different scenarios (e.g., that the patient fainted). Such possible explanations, which often take the form of stories, can help the doctor decide better how to treat the patient. Once he has proposed a few possible explanations, the doctor can start to see why some of these stories cannot be true; possible explanations therefore can help us inquire when we do not have much evidence. These stories are aimed at helping us try to understand the situation better. We also use stories more generally to convince others who might not be able to reason about the truth, like when we tell a story to a child to teach him to honor his parents. In such a case, we are not just using a story in order to discover something ourselves but also to help someone else discover something.

This part of logic is poetics, and Aristotle discusses one part of this art in his book Poetics. When we use poetics to tell stories that teach truths about the best kind of life to live or about virtue or vice), we call this the art of literature. These are the most important and difficult stories that we tell. Literature is mainly what Aristotle is concerned with in the Poetics, although what he says there can be applied more generally to telling stories of all kinds. Note that this account implies that literature is, properly speaking, part of logic with respect to its goal and its form; but with respect to its content (matters about human life, good stories, stories that we can learn to live well from), literature is subordinate to moral philosophy. (The Poetics speaks of plot and character, not because those are features of every possible story or explanation, but because they specifically relate to Aristotle’s main concern there: stories that inform and direct human action.)

Fallacies

Finally, sometimes our reasoning is flawed, causing us to fail to reach the truth. There are two ways to make a mistake in coming to know: one is to accept something false that is actually true, but the other way is to commit an error in reasoning, such as when someone says, “This person is bad and therefore everything he says must be false,” or “Everyone is doing this and so it must be good to do.” It is the typical errors in reasoning that are considered in this part of logic; we learn these so that we can avoid them. This part is the study of fallacies (errors in reasoning). Aristotle talks about these in his book Sophistical Refutations. This book is aimed especially at refuting those who want to use errors in reasoning to manipulate others. In antiquity, those who especially used fallacies and tricks of reasoning to confuse and manipulate others were sophists.

The Parts of Logic and How They Relate to Each Other

Each of these parts of logic—demonstration, dialectic, rhetoric, poetics, and the study of fallacies—is used in every branch of knowledge and in every area of inquiry. But, more than that, the different parts of logic broadly correspond to the parts of an inquiry in which we begin without knowing the subject matter or question well. We can begin without much knowledge of a subject and then we propose different stories to make sense of what we do know. As we do that and continue to learn more about the subject in other ways, we may then see that we have some reasons to prefer one story over the others and we can keep studying the stories that fit better. Then, through further study, we may find probable explanations that do not prove what the truth is but do make sense of a lot of the evidence. Finally, we may discover a demonstration of what the subject is or what the answer to the question must be. Though all inquiry or reasoning proceeds like this, this is often how it does develop. If nothing else, this shows us a possible method: come up with possible accounts that explain what you want to know about; discover why one is preferable; find out more and see what is most probable, then discover proof for it.

The Place of History in a Discipline

Before considering the history of logic, it is worthwhile to ask what the place of history is in any discipline. Aristotle and St. Thomas Aquinas typically begin their treatments of a subject with a history of how it has been studied so far, and often they refer in detail throughout their studies to the discipline’s history. The goal of the use of history in this way is to elucidate the terms and distinctions and principles of the discipline and to show how and why they have been discovered and applied. In that way, history is a part of every discipline. But there another sense in which history is also a part of nearly every discipline: for any given subject we will also want to consider how the things pertaining to that subject have changed over time (e.g., the history of the oceans in marine biology or the history of the stars in astronomy or the history of man in the various disciplines that study humankind). Most commonly, when we speak of history as a discipline, we mean the history of human affairs related to our community. History in this sense is primarily for the sake of understanding ourselves and how our community came to be the way it is as well as how we can best direct that community toward its ends. In the history of logic below, the goal is partly to reveal how logic has developed formally but also partly to show how logic has evolved as a human art responding to the discoveries and pressures of every age.{3}

IV. History of Logic

The History of Logic I: Logic before and beyond the West

In our time, what is meant by logic is merely formal logic, the study of the arrangement of terms and their relations to each other. Histories of logic are mostly just histories of formal logic. But, as the above outline of the parts of logic shows, there is much more to logic than formal logic. Thus, the history of logic, classically understood, should include much more than just the history of formal logic. Such a history should also consider the history of demonstration, dialectic, rhetoric, and poetics. What that history shows, albeit only in rough outline, is that the history of logic is one of expanding application of central insights to ever wider domains with ever greater nuance. For the purposes of this article, however, we will consider primarily the history of formal and material logic and demonstration.{4}

It is sometimes said that the Ancient Greeks discovered rationality and that Ancient Greece made a distinct shift from myth to reason. A statement like this does not imply that men before or outside Ancient Greece were somehow not rational; it only implies that men before and outside Greece at that time did not discover the human faculty of rationality as such. Men, for example, in the Ancient Near East (Egypt, Assyria, Babylon, among others) did not discover that what distinguishes man from other animals is his rationality, nor did they discover ways to guide or direct the faculty of reason in general. In cultures quite remote from Ancient Greece, however, such as classical India and China, rationality was discovered in many respects, though man’s distinctive rationality was not seen with the clarity and precision that marked ancient Greek thought. No one could read the works of Confucius or Xunzi (his great 13th century commentator) and think that this tradition lacks an awareness of rationality as a central aspect of human nature. Take for example the following passage from Confucius and a related one from his commentator: “To have studied without thinking is of no value. To have thought without having studied is dangerous” (Confucius, Analects 2:15).

If a person lacks the proper model, then he will act recklessly. If he has the proper model but does not fix his intention on its true meaning, then he will act too rigidly. If he relies on the proper model and also deeply understands its categories, only then will he act with comfortable mastery of it. (Xunzi 2) 

This is surely a tradition that is aware of rationality and its fundamental role in human life. Nevertheless, Chinese tradition as well as others are not concerned, as Ancient Greeks were, to delimit precisely the nature of things in general and the nature of man in particular. Chinese reflections on human nature were, as the above illustrate, not primarily aimed at knowing the nature of man in himself, but rather at knowing the best practices for man in carrying out his life in harmony with the ultimate good.

Among Ancient Greek philosophers there is very little like the commentary of Xunzi, nothing so aimed to helping the student practically maintain an awareness of reasoning as an activity of the entire man. Ancient Greeks were more aware of and concerned with understanding those aspects of real things that are universal, that is, the forms or essences or attributes which individual things share with other things. It is more difficult but also more necessary to compare Ancient Greek logic with the logic that was developed by the several Vedic schools of thought in India.{5} Rather than considering Indian logic in detail, since this is an active and growing area of research, the essential point to make here is that Indian philosophers may well have discovered reason as such as Greek thinkers did; but they seem not to have considered reason primarily as a faculty proper to man and as the faculty by which man is directed toward his final end, happiness in God. Indian thought extends consideration of rationality and intellectuality well beyond the domain of the human, to include the entire universe, the divine and the underlying essence of both. Such consideration means, from the perspective of a classical, Aristotelian account of logic, that Indian logic tends to take too wide a scope. That having been said, it must also be recognized that Indian logic was developed formally to a higher degree than anywhere else outside of Greece, Rome, and their inheritors in Islam and the West.

For the student of scripture and the Catholic tradition, a word about rationality and modes of thought in the Ancient Near East is important as a contrast with later Greek and properly Western modes of reasoning.{6} The Ancient Near Eastern traditions had a strong and beautiful literature of proverbs and of advice from elders to the young. But the student will not find in this literature anything like general guidance for reasoning or the search for wisdom. The proverb and counsel literature reveals a search for wisdom that is fundamentally personal, that is, a search for wisdom that is related to, drawn from, and ending in other persons, human or divine. The sapiential literature of the Ancient Near East does not represent a search for causes or essences or general forms. Even the vision of wisdom itself and the advice given to those who seek her is laid out as a kind of story, a story of a young man searching the streets of a great city for something and being met by two women: one alluring and loud (Folly), the other noble and reserved (Wisdom). Since the discovery of the literatures of the Ancient Near East, scholars who study these traditions in detail have established key results for interpreting how men in these cultures thought of knowledge and the search for it and, therefore, implicitly, how they understood rationality and its guiding art, logic. Men in the Ancient Near East did not search for universal causes and would not have understood them; their world was a world of persons, all the way down. Every experience of life and every mode of living, even to the state bureaucracies that managed city life, was thought of as being governed by other persons, whether kings and merchants or gods and demons, whether spirits of ancestors or kings who became divine. Nothing in their experience was impersonal; everything was from or related to a person—very often two persons—one natural and one supernatural. The daily and yearly cycles of the natural world were attributed to spiritual and cosmic powers, not to abstract entities of nature like causes or the laws of physics. Despite this, however, there is one abstract way by which men of the Ancient Near East believed that they could find truth: the use of words.

In the Old Testament there is special care for the divine name and for the names of persons in general, especially kings. The name of a thing was taken to have power over that thing. Egyptian kings, for example, would make war on their enemies by writing the names of the nations and kings they opposed on clay pots and ritually smashing the words; this was not just a declaration of war but a way of seeking to strike their enemies, in a way analogous to a Voodoo doll. The scribes of various of these regimes and periods seem to have reflected on this understanding of the nature of names and then to have supposed that underneath any given thing and its name was an impersonal connection, one between that name and the thing on the one hand and other things and their names to which the original thing was itself related. For example, a scribal pen is related to a reed and so writing must be related to reeds; a cow is related to the deity to whom cows are offered in sacrifice. A common practice among ancient scribes reveals to us that they saw connections like this: across several cultures, these ancient scribes began experimenting with exploring the names of things by changing the words in order to discover linguistic connections with other names and through those other names to discover what they thought were real connections with the things that these names signified.{7} This idea of the connection between name and essence shows up as part of the background of Ancient Greek thought on language and in a common practice of Greek thinkers: the reference to etymology. When Socrates in Plato’s Cratylus is exploring the possibility that names and things are related by nature, he is considering something quite close to the view that scribes in several cultures in the Ancient Near East thought themselves to have discovered: the name of a thing reveals the truth about that thing. These historical strands show both that ancient peoples before Greece inhabited a world that they understood quite differently from how we understand our world and also that ancient peoples before Greece were exploring some of the central issues that lie behind all human rationality and the art that governs its use. We will consider the intimate connection between words and concepts below.

Greek philosophers were not the first to inquire into the causes of things. But they were the first to propose causes and explanations based on the natures or essences of things. And if the classical view of logic is right, this is precisely what enabled the defining part of logic to be discovered in Greece: demonstration as the coming to know by means of causes; it is only when causes are being sought that demonstration as such can be discovered. Comparative studies of India and China have revealed that demonstration as understood in Greece is lacking in every other known culture.{8} Other cultures have discovered and reflected upon inference and argument and developed elaborate theories of how these work and how they should work. Hence, other cultures have discovered something analogous to formal logic and dialectic. But they did not develop demonstration as proof through causes. And this is, in one way, not surprising, since even now the dominant modern idea of what logic is itself also does not center around demonstration. Logic in our time is more a matter of argument, definition, and inference than of real demonstration of the truth of things by means of their causes. This is why, from the modern formal perspective, it can be said that Indian logic is more general than and superior to logic as developed in Greece. To this, the classical tradition would respond that Indian logic, like modern mathematical logic, only considers part of what logic is and leaves out the principal goal of logic and the part that makes it what it is: demonstration.

This prehistory of logic is useful here, especially for the student of theology, because it can help us see more clearly what logic is and what it is not and how it relates to the more general pursuit of wisdom, by way of comparison.

The History of Logic II: Logic in Ancient Greece and Rome

The first demonstration in history, according to tradition, was Thales’s proof that any triangle inscribed in a semicircle is a right triangle. Two centuries later, Aristotle produced the first texts outlining the characteristics of demonstration and how to achieve it and what the principles and foundations of demonstration are. As Aristotle outlines it, a demonstration is reasoning that produces knowledge. He starts from the obvious fact that we do know some things and that we can come to know other things. He then asks in detail what the process of arriving by reasoning at certain knowledge entails. For instance, Aristotle considers where any process of arriving at certain knowledge must begin, namely with things we already know—and this implies that some things can be known without proof and that we cannot prove everything. Additionally, Aristotle shows that we cannot prove some conclusion (C1) and then use that conclusion (C1) to prove another conclusion (C2) and then use that second conclusion (C2) to prove the first (C1), i.e., demonstration cannot go in a circle. He also shows that demonstration cannot go on forever, that we must at some point arrive at our conclusion, since otherwise there would be no demonstration. These seem so obvious to modern students that they may not seem to warrant mentioning. However, note that what Aristotle is doing in articulating these is setting out the boundaries of what demonstration is and where it can or cannot go.

By tradition, Aristotle’s logical writings are called the Organon (the “instrument”). These are the Categories (on words and what we can say); the De interpretatione (on sentences and their relations to each other); the Prior Analytics (on the form of reasoning); the Posterior Analytics (concerning the subject matter of a given chain of reasoning and how it relates to demonstration); the Topics (on ways to give probable arguments on a given topic); and the Sophistical Refutations (on errors and tricks in reasoning). Avicenna and St. Thomas Aquinas also include the Rhetoric (on public speaking and persuasion) and the Poetics (on storytelling, plat, and literature) as part of Aristotle’s logical works.

Aristotle drew on Plato, and Plato especially learned from Socrates. But Plato was not Socrates’ only student. Taking the heritage of Socrates in a different and more practical direction, the school of the Stoics developed a rival logic to that of Aristotle. In particular, the Stoics were not primarily concerned with demonstration but rather with argument and inference in general. Aristotle believed that we have a demonstration when we know the cause of why one thing relates to another thing; since both things in question are represented in logic by terms, Aristotelian logic is considered a logic of terms. The Stoics, by contrast, were concerned primarily with truth of any kinds and how to arrive at any kind of truth from any beginning (and not only truth in a highest form as knowledge of the cause). The Stoics developed a logic of propositions. A logic of propositions outlines how, given a certain set of propositions, we can infer from that set some other proposition that must be true if the propositions in that set are true. For example, if God exists, then he would seek to help mankind. If it is in fact true that God exists, then we can infer based on these two that God does in fact seek to help mankind. Note that we are inferring based on the following pattern: If A is true, then B is true; A is in fact true; therefore, B must also be true. This may seem simplistic compared to the Aristotelian concern for demonstrating the cause, but bear in mind that propositional logic can apply more generally to any kind of reasoning about any kind of subject. The key difference is that Stoic logic is concerned with what it is possible to infer, not with how we can come to know. Of course, it will be clear that, from an Aristotelian perspective like the one adopted in this article, propositional logic and inference are considered by logic and there is nothing inappropriate in studying them. But they are confined primarily to dialectic, the examinations of truths and arguments when we have some evidence one way or the other but do not have proof. The Stoic approach to logic is closer to the kind of approach developed in the Indian traditions and also closer to the approach of contemporary formal logic.

Logic did not develop further in fundamental ways in the ancient world. Nevertheless, two strands of thought about logic and coming to know are important to articulate: that of the Neoplatonists and that of St. Augustine. The Neoplatonists did not advance the study of logic by adding new tools, techniques, or theses about logic. They advanced the application of Aristotelian logic throughout the Platonic way of doing philosophy, in particular by writing commentaries on Plato’s dialogues and on Aristotle’s logical works and treatises on central Platonic themes. They also sought to show that Aristotelian logic and Aristotelian philosophy in general were in harmony with a broadly Platonic way of doing philosophy; this meant that the Neoplatonists used Aristotelian logic in their studies of Plato and Plato in their studies of Aristotelian logic, which mostly took the form of commentaries on his logical works.{9}

By St. Augustine’s day (late 300s and early 400s AD), one of the most fashionable schools of philosophy was that of the skeptics. Aristotelian logic continued to be known but had been displaced from its central role centuries earlier by rhetoric. Skeptics sought to refute anyone who held firmly to a truth and to show that we should suspend judgment on all questions. For example, skeptics believed that we cannot really perceive anything and that we should not assent to any truth, no matter how insignificant. Part of the reason that these claims were so successful in St. Augustine’s time is just that philosophical arguments were already very old and disagreements already deep and for every argument on one side it seemed that there was an argument on the other side too—a feeling quite common in our time. Part of logic, dialectic (the art of disputing about probable truths), was therefore dominant at this time, and it was used to refute or at least call into doubt many philosophical truths. St. Augustine, having escaped from skepticism himself, sought to use these same tools to find truths that would be absolutely certain and at the same time also meaningful. An Academic skeptic, for example, might give arguments that the world does not exist and then also give arguments that the world does exist, the goal being to cast doubt on what we experience and take ourselves to know. St. Augustine responded to this by pointing out that this very example shows that I know something with absolute certainty: I know that the world either exists or does not exist. To take another example, I know that there is either one world or many worlds and that, if there is one world or many, any such world is either finite or infinite (see Against the Academics 3.10.22–23). By this, St. Augustine proposes certainties with which we can begin our own inquiry.

The History of Logic III: Logic in the Medieval and Modern World (1050–1900)

Aristotelian logic coupled with Augustinian insights was dominant in the West from the 1100s to the mid-1800s. It only came to be replaced in schools and curricula in the early 20th century by formal mathematical logic, for reasons we will see shortly. In 1900, undergraduate students taking logic would have studied Aristotelian logic, principally terms, definitions, classification, the relation of terms in propositions, syllogisms, and fallacies.{10} This means that Aristotelian logic served as the logic of the western world from around 300 BC to 1900 AD, a span of two thousand two hundred years, during which most of the foundational advances in every modern discipline, from law to classical physics, were made. 

Here is the story of this period in brief. Most of Aristotle’s logical works were lost to Europe after the fall of the Western Roman Empire. They were rediscovered in the eleventh and twelfth centuries and increasingly came to importance in the curriculum of the burgeoning medieval universities. These universities, begun first in Bologna in the late eleventh century and then in France and England and throughout Europe, taught students the artes liberales (liberal arts) of grammar, logic, and rhetoric (the trivium) and arithmetic, music, geometry, and astronomy (the quadrivium); later, the universities added other central works of Aristotle and so also taught, among other things, natural philosophy, ethics, and metaphysics. Students passed through these lower studies (the arts) to the higher studies of law, theology, and medicine. Aristotelian logic played a central role in all of these disciplines. James Franklin, discussing the history of science, shows that medieval scholars laid the foundations for most modern disciplines, from accounting and economics to classical physics and linguistics, and they did so by applying logic to the analysis of concepts and experience, especially discovering first principles and defining what would become key terms for later discoveries (see Franklin, “Science by Conceptual Analysis: The Genius of the Late Scholastics”). By the 1700s, however, Aristotelian logic had begun to be challenged in various ways and was often considered no longer adequate for the advancement of learning.

Aristotelian logic achieved extraordinary elaboration, sophistication, technicality, and insight in the period from around 1150 to 1500. Though once referred to as the part of the “Dark Ages,” there is nothing “dark” about logic during these centuries. The brilliance of medieval logic can easily be compared with the finest and most sophisticated mathematical logic, and any fair judgment of how the two traditions compare may well have to favor medieval logic for the simple reason that it went on by far longer than mathematical logic has and that it also anticipated many, though not all, developments in modern logic as well as having some developments which have no parallel in contemporary logic (Parsons, Articulating Medieval Logic, 259–276). For Thomists, the lectures and commentaries of St. Albert the Great on Aristotle’s logical works are both a great witness to the depth of this tradition and also an often-overlooked starting point for the in-depth study of logic.{11} To get a sense for the sophistication of logic in the Middle Ages, the interested reader should consult even just the table of contents of the Summa de Dialectica of the fourteenth century nominalist Jean Buridan. Buridan was certainly one of the greatest lights in this tradition, but there were many like him. In the medieval style of scholarly writing, the concern was almost exclusively the exposition of truth and arguments for and against different views, and there was little place for exploring history, giving detailed examples, or citing one’s opponents in a judicious way. A work like Buridan’s Summa de Dialectica, which in the English translation numbers over four hundred pages of block text, contains a massive fund of arguments and direct reasoning carried out almost without stint on every aspect of logic, from inference patterns to defining terms and from paradoxes of truth to the meanings of words. Modern papers and even many modern books have two or three central arguments; Buridan considers thousands—and so did his fellows at the medieval universities of Europe.{12}

The structure of logical doctrine and how it was taught in the Middle Ages stayed more or less constant. It began with terms and concepts (as well as signs), then considered propositions in various forms, and then syllogisms or chains of reasoning, and finally fallacies, as well as, frequently, various topics related to technical problems concerning what terms or propositions mean. The logical work of John of St. Thomas, especially his Ars Logica, written in the early seventeenth century, can serve as a mature witness of the tradition, and one that also draws especially on St. Thomas Aquinas, which makes his work especially useful for theology. John of St. Thomas divides his work into formal and material logic. He outlines the subject matter of formal logic by means of the three acts of the intellect: simple apprehension (grasping an essence or something universal), judgement (saying that one thing is related to something else or not), and reasoning (drawing a new proposition from ones we already know). Material logic he treats, not schematically, but in detail through extended disputed questions (e.g., on the subject and nature of logic, on the universal, on signs, on demonstration, and on unity and distinction in the philosophical sciences). The essentials of this doctrine are expounded later in this article.

In later centuries, it might be thought, this use of logic was opposed or somehow overturned by early natural scientists such as Galileo. But Galileo himself was a careful student of Aristotelian logic, as was William Harvey, who discovered the circulation of the blood (see Wallace, Galileo and His Sources and Cunningham, “I Follow Aristotle”: How William Harvey Discovered the Circulation of the Blood). Likewise, many early modern natural scientists were themselves deeply engaged with Aristotelian logic, even with the theory of demonstration, as they articulated and developed their methods of experiment and of the mathematical modeling of nature. Nevertheless, there was active and deep criticism of Aristotelian logic by authors such as Francis Bacon and Rene Descartes. Aristotelian logic was too formal and unconnected with experience, they said, and it did not have the power to help us discover nature on its own terms. Syllogisms and definitions of essences and demonstrations were seen by some as trivial and more of a hinderance to the development of knowledge than a help. Aristotelians were criticized also from another direction: that of the renaissance humanists. The humanists studied history and literature and sought to contextualize arguments and even to consider the history relevant to understanding the great figures of philosophy. The detail that they began to discover also suggested to many that the Aristotelian-style of inquiry by general terms and principles was obscuring aspects of reality, especially in the domain of human affairs. The question of the role of history in our understanding and articulation of truth found no ready answer in these centuries.

These criticisms of Aristotelian logic were related to a central problem that Aristotelians at this time faced (Sgarbi, The Age of Epistemology: Aristotelian Logic in Early Modern Philosophy, 1500–1700).{13} The problem was this: if a demonstration is a proof of why something is true by means of revealing the cause of the thing, how do we come to know what the cause of the thing really is? The Aristotelians were not asking whether we can come to know anything or whether demonstration is possible at all; they took it as obvious that we can come to know things and that demonstration can reveal the cause. Their question concerned the process whereby we come to discover the cause. We might observe the phases of the moon, for example, and come to see that the moon is a sphere that is rotating around the earth. But, as anyone who remembers their first experience of physics in high school will recall, these kinds of claims are not obviously true; they are not something that is apparent to everyone. So, how does the mind begin from disparate observations and infer to the general cause behind them all? This was not just a theoretical question but a practical one. Faced with so many new discoveries and advances, and so many corresponding questions and problems, Renaissance thinkers were concerned with finding a method, the right method for proceeding from principles to substantial discoveries, answering questions like the origins of the rainbow, the cause of thunder and lightning, and so on. The simplest explanation of how the mind moves from all the experiences and facts it has collected to knowing the cause of the thing is that it does so by passing through a period of concerted inquiry over every related truth. This period was called a meditatio or negotiatio, a “meditation” or “negotiation of the intellect”—negotio in Latin means the state of being busy and was one of the words for work in general. But what happens during this period of mental effort? And what is the best way to structure our efforts to come to draw a knowledge of causes out of our experience and other knowledge? And how is the best method connected with the insights that we can truly achieve as we discover the causes of things?

But, at this time, the more traditional Aristotelian approach to demonstration had already been called into question by nominalism. Nominalism is “the doctrine according to which universals (and, therefore, essences or natures) are merely names and have no existence in reality or in the mind. In its radical form, nominalism holds that everything is wholly particular in nature. Thus, there would be no natures that transcend the individual and enjoy commonality (i.e., universality) with other things” (ECT Lexicon, “Nominalism”). On the nominalist view, which was popularized in the via moderna and became the dominant approach to philosophical questions in the later Middle Ages, the problems with Aristotelian-style demonstrations go much deeper. There are no essences existing in things, nominalists say; this means that the terms of our demonstrations cannot have the universality that Aristotle and Aquinas believed they had. But it also means that we cannot abstract those universals from the things we experience—and this calls into question the reliability of our senses themselves.

These questions, how the intellect discovers causes and whether the senses (and intellect) are reliable, set up much debate in modern philosophy down even to our own day. Both relate to logic, as will be clear below. Modern philosophy sets up one series of problems, and modern mathematics sets up a parallel but related series of problems. The answers come together in the next stage of the history of logic. To understand the later development of logic, a sketch of these debates is necessary, one touching briefly on Descartes, Hume, and Kant.{14} Descartes, the seventeenth-century French philosopher and mathematician, posed the problem of the reliability of our senses in a particularly acute way but then tried to solve the problem by saying that we can know what God is like just from our ideas of him and that God would not allow our senses to lead us astray systematically. Descartes further answered the question of how the intellect can know causes by saying that the best method to achieve insights and know causes is through the use of mathematics in our study of everything in nature. Here Descartes is the key figure in these two streams that dominate much of modern thought: the use of mathematics to study the world through the modern empirical sciences and the search for how to ground the reliability of our senses in modern philosophy.

David Hume, the eighteenth-century Scottish Enlightenment philosopher, rejected Descartes’s idea of God and thus opened the way for direct skepticism about everything we experience. But Hume went even further and extended skepticism to the acts of the intellect itself: assuming that our senses do report things to us accurately, they can never reveal the causes of things. He used the example of something simple and obvious to our experience, something we think we understand easily: billiard balls, i.e., the game of pool. When we strike a ball with the cue stick and it rolls across the surface of the table toward another ball, we may well expect that the ball we hit will strike the other ball when it reaches it; but we do not in fact have proof of this; it is entirely possible that something happen to stop the ball rolling or even that, for one strange time, the ball that is struck simply does not move. It is true that our experience tells us that this has never happened in the past, but that, Hume said, is all that experience itself actually teaches. At the time of Hume’s writing, his critique of our ability to know was felt to be a deep challenge to human life and to the progress of science. But Hume’s radicalism did not go unanswered.

Immanuel Kant responded to the skepticism of Descartes and Hume by arguing that they were right to cast doubt on our reason’s ability to prove things about reality beyond itself; Kant motivated this in his readers by a presentation of “antinomies,” arguments on both sides of important issues such as the existence of God and the truth of key moral principles, good arguments that Kant thought proved that we can have equally good arguments on both sides of most important questions. But, Kant argued, this does not mean that we cannot know truth or discover certainties. Kant believed that he had found truth and certainty in the mind itself, not in the things we experience. He believed that by turning inward to ourselves as subjects of experience, we could discover that our experiences and the categories we form based on those experiences have a form or character that is given to them by the mind itself because that is somehow the prerequisite for having any experiences at all. So, for Kant, it is the mind that gives the categories and principles; and Kant thinks that the forms of what Aristotelians think of as demonstrations are as certain and beautiful as they are because they are simply the result of the mind contriving its own acts according to the ways that its thinking must in fact be. The mind encounters the world of things but can never know things in themselves; it can only interpret them wholly in terms of its own acts. This philosophical idealism in the style of Kant dominated the nineteenth century, and in its rejection in England arose the next great chapter in the history of logic.

It is ironic that, at the same time as modern philosophers like Hume and Kant were questioning the possibility of knowledge, mathematicians and empirical scientists were carrying out the greatest large-scale investigation of nature the world had ever known. This is not the place to discuss this well-known historical movement, but suffice it to say that this was one of the greatest periods of discovery in human history and has continued down to our time. And that fact may well prompt a question: Where was logic in the scientific revolution? The answer is that, from the classical perspective, the logic of the classical tradition (essences, genus, species, propositions, and proof) lay behind these developments: it was the logical work and system developed in the later medieval period that allowed for the clear articulation of definitions, the validation of proofs in every discipline, and the methods of analysis. It is true that this period did not see fundamentally new discoveries in formal logic; rather, from the classical perspective, this period saw major developments in the material logic of the modern natural sciences.

The History of Logic IV: The Origins and Limits of Mathematical Logic

From thirteenth to the nineteenth centuries, the foundations of what would become a new logic, which for a time would largely replace Aristotelian logic, were laid. It began only as one discipline, mathematics, and then through the work of natural philosophers of the fourteenth century came to be applied more and more to the study of nature itself. In the late thirteenth century, Roger Bacon said that the book of nature is written in the language of mathematics, following Aristotle himself who had said that quantity is the first property of material substance. This assertion was increasingly recognized and exploited for studying the world.{15} Over time, but especially during and after the seventeenth century, mathematics became less a single discipline and more a host of related disciplines that all followed something like a similar method and employed similar kinds of reasoning to gain fresh insight into old problems and to pose and solve entirely new problems. One thing this new use of mathematics seemed to bring to questions in the study of nature was something like absolute certainty. Natural philosophers began to think that, through mathematics, we could advance beyond the realm of natural things in their nearly infinite variety and endless variation and derive from their study conclusions with perfect certainty. The older, Aristotelian understanding of certainty as having degrees in different domains came to be ignored or forgotten. Mathematics and mathematically informed science grew in power and scope throughout these centuries. Descartes’s emphasis on mathematics as the core of a method to know the world seemed to be vindicated. Mathematics began to displace logic, not as a formal study of proposition, inference, and argument, but as a method and a spirit of inquiry which was carried forward by certain means, in particular the kind of abstraction we find in algebra and the kind of symbolization of quantities that has come to be widespread.

But problems were just below the surface. The mathematics in use by the end of the nineteenth century was becoming increasingly abstract and ever more remote from anything like a simple, realistic interpretation. Mathematicians were studying strange kinds of geometry and comparing the size of different kinds of infinity. And these very abstractions began to raise doubts about the meaning of the underlying mathematics and about how to interpret mathematics, ultimately about what mathematical objects are and what grounds the truth of mathematical claims. Similarly, as mathematics began through inquiries like abstract algebra to be even more remote from experience, the certainty of mathematics itself began to be called into question. 

The problem was simple, and it began with basic questions that even beginning mathematics students have: What is algebra? What do algebraic statements really mean? What are negative numbers and polynomials? How can we understand non-Euclidean geometry? How can an infinite number of infinitely small points make up a finite line length? Questions like these raised fundamental doubts about the subject matter of mathematics and what ultimately grounds mathematical truths. Mathematicians and philosophers, especially influenced by reemerging skepticism, did not want to locate the truths of mathematics in the observable world, in part because Kant’s philosophy had turned inquiry toward the mind and the world that the mind constructs. Instead, therefore, of turning to the world to ground mathematics, they sought to find a way to prove that mathematics could simply be reduced to mental operations, that is, to logic; they sought to show that all mathematical truths were simply complex logical forms, and that as such their foundation would be as secure as the same foundation that grounds even our most basic acts of reason. In particular, Gottlob Frege, who developed the kind of logical system that would come to dominate the field, sought to prove that arithmetic is entirely derived from and based on the formal principles of the mind. But the program failed. Despite the elaborate construction of modern mathematical logic systems, logic, it turned out, could not be reduced to mathematics. In the 1930s, Kurt Gödel proved, first, for any given consistent system of mathematical logic, there are truths that cannot be proved within that system. Thus, such truths can scarcely be said to be grounded in any given system of logic. Second, Gödel proved that it is impossible to use any given system of mathematical logic of this kind to prove that system’s own internal consistency. An inconsistent system is one in which we can prove that some proposition is true and also that it is not true. Thus, if Gödel is right, then we cannot prove all the truths of mathematics, and no logical system of the relevant kind can prove itself own consistent. Mathematical logic was constructed in order to be the foundation of mathematical truth, a task for which it proved inadequate. But what had been developed was a powerful and expressive symbolic tool, the use of which came to dominate Anglophone philosophy down to the third decade of the twenty-first century.

The History of Logic V: Logic in the Twentieth and Twenty-First Centuries (1950–2020s)

There are two periods in which a form of logic has seen massive application and research: the high Middle Ages (1150–1500) and the twentieth and early twenty-first centuries. The difference between them is the result of a difference in first principles, and we can see this difference by asking a question: What is the most foundational and also the most useful truth-bearing tool of the human mind? For the Middle Ages, following the classical tradition, it was language. For the twentieth and twenty-first centuries, following the modern approach epitomized by Descartes, it was mathematics. Both periods share a high degree of technical sophistication, which makes both periods harder to summarize. The following indicates in brief the two main developments of contemporary logic relevant to a comparison with Aristotelian logic.

Many of the founding figures of modern logic, such as Bertrand Russell, were specifically concerned with rejecting the Kantian idea that the world is unknowable through our senses. These logicians believed that the world consisted of individual facts that we can come to know and organize together in logical arguments and systems of conclusions in order to generate a science based wholly in observation of the world, but a science that they explicitly believed would be without essences and natures. They sought to avoid the uses of essences and natures because they are not observable in a direct way by the senses in the way that, for example, color or sound is. They sought to eliminate natures and essences by means of logical descriptions of individuals and their directly observable properties. (The view of twentieth century logicians is a form of nominalism.)

To do this, they employed the system Frege developed first to state the simplest kind of facts. These consisted of individual things, their names, sets that we can group them in, and names of those sets. A simple construction of a system can serve as an illustration. We posit individual things named a, b, c, etc. We posit that individuals belong to sets named F, G, H, etc. If an individual belongs to a set, we can form a simple atomic sentence like this: we can symbolize “a is F” as Fa. We write the set name first, F, because these sets and their relations to each other are the primary concern of this logic. These relations, elaborately composed, are what, in this view, constitute science. If some set has at least one member in it, we can say “something is F” and symbolize this as (∃x)Fx, where “∃x” is read “there is an x such that,” and x stands for any member of F that we like. (Just as in linear equations, e.g., y = x + 1, x stands for any real number; so too here x stands for any member of the set.) If every individual that we are considering is a member of the set F, we can say “everything is F” and symbolize this as (∀x)Fx, where “∀x” is read “for all x.” So far, this is not a particularly rich system. But when we add connectives that we can use to join statements together, like “and” symbolized as &, “or” as ∨, “not” as ~, and “if-then” as →, then we get a system of great expressive power. It might not seem so right away, but the skill that lies behind modern mathematical logic is the ability to take ordinary classes of sentences of interest to philosophers and logicians and put them in symbolic form. One can surely see the appeal of a logical system in which everything could be formulated just based on what we can all observe. All reasoning could be put explicitly in terms of observable things and classes of observations. Once again, however, problems were not long in coming.{16}

To see both the expressive power of the system and also the central problem with it, consider a statement like “all men are mortal.” From the above, it will not be apparent how to symbolize this, but a little exploration will reveal that it can be modeled like this: (∀x)(Fx → Gx), read “for all x, if x is F, then x is G,” where F is the set of men and G is the set of mortal things. Of course, this does not mean the same thing as “all men are mortal;” instead, it is better understood as a way of representing “all men are mortal” with just sets, individuals, and connectives like “and” and “if-then”: if this system was all one had to express oneself with, one could get in some ways close to “all men are mortal” with (∀x)(Fx → Gx) interpreted in this way. The advantage of that representation of “all men are mortal” is that it is perfectly straightforward to understand and uses only very simple notions and symbols whose meaning can be perfectly fixed by stipulation. So, if one is concerned to make only statements and inferences that are perfectly clear and unambiguous, then this system will be attractive. Similarly, if one is philosophically committed to the existence only of individual things and their relations with other things by means of sets, likewise this system will be attractive. Another advantage, although less apparent, is that this system allows us to produce simple algorithms for evaluating arguments in mathematics and even in ordinary language, and such algorithms admit of being realized mechanically in digital computer systems. To see how, just imagine that we made a truth-table with all possible values of true and false assigned to each part of the statement and to the whole. Because we can make a truth table like this, such a system of symbols permits us to realize and evaluate these logical forms mechanically with only one device, an array of on-off switches (on for “true” and off for “false”) on an electronic circuit.

The key problem here, however, is just that “all men are mortal” means that everything that is man is mortal and imputes mortality to man in a universal way. It does not mean that, for everything we might consider, if one of those things is a man, then that thing is mortal. “All men are mortal” means more than this. But to early developers of modern logical systems, it was impossible for “all men are mortal” to mean more than “for all x if x is a man, then x is mortal.” The reason this was impossible is that the only things that exist, in their view, are atomic facts or individual things and sets of individuals. Thus, mortality is just a set of mortal things, and a set of mortal things cannot be imputed to humankind in any sense other than to say that the set of men is a subset of the set of mortal things. Again, their view of reality was that the world consisted of individuals that we come to know as members of sets, and we organize statements about these sets together to generate a science based wholly on observation of the world. This early view was known as “logical positivism.” The logical positivists made their central claim explicit: any statement, they said, that is not able to be verified directly by the senses is a meaningless statement. This meant the cutting away of traditional metaphysics and most other aspects of philosophy. But logical positivism, as the logical positivists themselves soon saw, is self-refuting: their criterion of truth and meaning, that only statements that can be verified with the senses have any real meaning, was itself not able to be verified with the senses and thus by their own doctrine their central principle was meaningless. The effort to start from atomic simples and construct a complete science through mathematical logic had failed, this time in a deeper way.

Lest this all seem to tell against modern symbolic logic so strongly that the reader begins to wonder why one should bother with it at all, let us note two things: first, modern mathematics is the most elaborate and technical discipline ever devised and has almost universal reach and widespread application in nearly every discipline of learning; and the most abstract mathematics has, time and again, turned out to be useful for understanding something direct and concrete about the world. Second, it is mathematical logic, built into electronic circuits, that makes the digital age possible. It is perhaps strange that a system with key problems at the heart of it could come to be the most powerful systems for the direction and encoding of reasoning. As a metaphysical system, modern mathematical logic is a failure; but as a practical tool for the precise representation of a great many statements and the evaluation of complex argument strings there has never been a symbolic system more powerful.

Partly due to the failure of the larger philosophical project, starting in the 1960s and down to our own time, systems of symbolic logic began to be stated in abstract terms as a completely independent formal language; these systems are stated using a metalanguage, usually English, and then the elements of the formal language—its semantics and its syntax—are precisely specified. Formal languages themselves become the object of study. For example, logicians will prove that classes of these formal languages are complete(meaning that every proposition that is true in every possible interpretation can be proved using the language), and that these formal languages are consistent (meaning that it is impossible to use the axioms and rules of the system to prove that a given proposition is true and then use the same axioms and rules to prove that the given proposition is false). This study of the formal language is metatheory. Its goal is to make formal languages and their systems of inference as well-understood as possible. 

Second only to the study of formal languages more generally, the most important development in formal logic in the twentieth century was the construction of modal logic and a semantic theory (i.e., theory of meaning) that allows for consideration of what is possible, necessary, or impossible rather than just what is actual. Predicate logic does not have the expressive power to make claims about the possible, actual, necessary, or impossible; however, these are fundamental features of human reasoning and of reality, and modal logics extend the expressive power of predicate logic to include expressions about possibility and necessity. Modal logic came to be applied widely throughout philosophy to study commands (deontic logic), beliefs (doxastic logic), and probabilities (probability logic). The technicalities are not important for our purposes. What is important is just this: precisely at this moment of the most technical advancement of modern mathematical logic, some important logicians began to object that these new tools introduced into logic matters that we cannot be certain about because we cannot experience them (e.g. possibilities, necessities, etc.). As W. V. Quine argued, in introducing these very tools, we have reintroduced Aristotelian essences back into modern philosophy, something analogous to those essences that Bacon and Descartes sought to purge from our use in favor of only empirically verified claims (Quine, “Reference and Modality”). Quine was right: this logic made possible a return to Aristotelian-style essences in whole areas of philosophy, trends which were taken up across many areas starting in the 1960s, from metaphysics and philosophy of mind to ethics and even logic itself, in the final set of developments relevant here.

Most recently, Aristotelians themselves have proposed formal-style systems for the regimentation of natural language in a way that is both faithful to traditional logic and to philosophical realism and also able to extend the expressive power of modern formal logic. Systems developed by Gyula Klima for the sake of formalizing differences between the views of medieval logicians have been applied to larger philosophical problems, even modeling the supposition and signification relations in medieval logic within a modern formal system (see Klima, Nova Ars Artium). The most notable consideration of Aristotelian logic by a modern logician is the development of the system Linguish by Terrence Parsons to capture the medieval Aristotelian reasoning systems in a framework that is linguistic, as the medieval system was, but also accessible to modern logicians (see Parsons, Articulating Medieval Logic). There are critics of formal style symbolic systems like Klima has developed, and Parsons’s more narrowly language-based approach may be more amenable to them.

Finally, a word on this from an Aristotelian perspective like that adopted here. Modern formal logic primarily studies and formalizes arguments of every kind and uses its formalizations to study inference in general. In that way, it is like the logic of the Stoics or the Vedic schools of India, not aimed at demonstration but rather at turning all possible inference into a precise set of rules, and thus its primary role is in dialectic, since we are interested in the varieties of possible inference about a thing mainly when we cannot know its cause. It is also worth noting that one of the most active areas of the study of modern symbolic logic is in the study of formal languages, their syntax and semantics. For an Aristotelian, such study is not proper to logic but to the general study of language, what to Aristotle and the scholastics would have been the science of grammar.

V. Logic and the Operations of the Mind

In the remaining sections of this article, we turn to a more detailed look at the fundamental acts of the mind and discuss how logic directs them toward their common goal of knowing truth. Each act can be stated simply like this: the first operation of the mind is defining what something is (or understanding its essence); the second operation of the mind is composing and dividing (affirming or deny something of something else); and the third operation of the mind is reasoning from what is known to what is not yet known (inference).

A simple model of these three can help us establish the right general picture. Imagine that we have experiences from which we abstract concepts A, B, and C. This occurs by the first operation of the intellect: we can know what things are by experiencing them. But we may also see that A, B, and C are related to each other; and we can then form propositions like "some B is A” when some of what belongs to B also belong to A or “no C is A” when nothing that belongs to C belongs also to A. This is the second operation of the intellect by which we affirm or deny ways that our terms, A, B, and C, can combine to form true propositions. Finally, we can also see that if we know how some of our terms relate, most simply in two propositions relating three terms, e.g., no C is A and some B is A, then, just because of what these propositions mean, we also know another proposition, e.g., here some B is not C. This inference to what we do not know is the third operation of the intellect. 

What follows here is not a comprehensive introduction to the Aristotelian-Thomistic account of the three acts of the mind. Rather, it is a general presentation specifically aimed at motivating and explaining the principles and key conclusions of this approach for contemporary readers. Again, note the justification for following the classical tradition given above: it is the classical tradition that so deeply influences Catholic theology and nothing else, and it is the classical tradition that can integrate together the various other traditions and developments of logic. (For example, modern mathematical logicians have nothing to say about the larger understanding of logic as the art of reason, but a perspective based on the classical tradition can state how mathematical logic fits into logic in general as to the history of logic and as to its relation to the parts of logic, both as outlined above.) The following account emphasizes the first operation of the mind since this is where most of the philosophically important distinctions lie, and this is also where the classical tradition has differed most significantly from other traditions. The reader should consult a standard textbook or manual for a more detailed treatment of the second and third operations.{17}

The First Operation of the Intellect: Signs, Words, Categories, Predicables, and Simple Apprehension

The first operation of the mind is apprehending what something is, that is, understanding the essence of a thing. But this raises a question: What do we mean by “thing” (Latin: res)? Informally, a thing is anything that we can speak about or that can be an answer to the question, “What are you talking about?” This could be a color, an attitude, a thought, an animal, a species of animal, a rock, a cloud, an association of lawyers, and so on. The difference between a thing and a being is that when we consider something as a thing we are considering it as having an essence; whereas when we consider something as a being we are considering it, not primarily with respect to what it is, its essence, but primarily with respect to its existence (whether or not and in what way it exists). To take the example of an association of lawyers, when we consider it as a thing we are considering it as to its essence (what it is, a professional association in law); when we consider an association of lawyers as a being, we are considering whether it exists (since we could be talking about a potential association that we are founding or one that has disbanded) and how it exists (since a human association only exists in the habits and agreements of men; it does not exist independently of human agreements). A thing, then, is anything that can be spoken about.

The first operation of the mind, therefore, is the apprehension of the essence of anything that can be spoken about. This reveals something surprising: the centrality of words and language in logic. So, what are words? Informally, we most often use words to speak about things. Words, therefore, are signs of things. But what are signs? We can distinguish at once two kinds of signs: natural signs (e.g., smoke as a sign of fire or tracks in the sand as a sign of an animal having passed by) and conventional signs (e.g., the sounds we use to express our words or the arrows we paint on posts to tell drivers that the road is bending). These latter signs are conventional because they have their meaning due to agreement of a social group. But there is another, third kind of sign. When something is brought to our attention, we employ a concept of that thing, and concepts are a kind of representation. To represent something is to make that thing present to a cognitive power, a power capable of attending to it. But concepts are more than mere representations; concepts are representations in themselves that always represent something other than themselves. They are not representations that employ something else in order to represent, in the way that spoken words use sound or photographs use ink on paper; what they are is pure representation, whose very form is the representation. Concepts are formal signs. We can compare these to the natural signs that, for example, even animals can understand, such as smoke as a sign of fire or the howling of wolves as a sign of danger. Natural signs represent something to humans and animals through what humans and animals can perceive; natural signs signify because the sign is caused by what it signifies (e.g., smoke is caused by fire). We come to know the sign first and then what it signifies. Instrumental signs represent to us also only through what we can perceive; but instrumental signs can be used to signify anything we like; they do not need to be causally related to what they signify. Formal signs, that is, concepts, represent to the human mind only through themselves, without a medium. Concepts are caused by our attending to the things that our concepts are formed to represent, and our act of coming to know the given things themselves is the act of forming a concept of those things. Since we come to know signs through our senses as animals do, but through our intellectual power to represent without a medium, we can understand the nature of signs and we can form signs for any purpose. This all permits us to give a general definition of sign: “A sign is that which represents something other than itself to a cognitive power” (John of St. Thomas, Summulae, bk. I, ch. 2).

More precisely, the human mind abstracts from a given thing the intelligible species and through this intelligible species the mind forms its concept, representing the essence of the thing. The intelligible species is found in how the mind captures what is experienced, and the concept is the end or term of the act of understanding the thing.{18} This abstraction of the essence of a thing and the corresponding formation of our concept of it is the first act of the mind. A concept is called a “term” because it can be one of the terminations of a simple proposition by being either the subject or the predicate: for example, in “A is B”, A and B are both terminations or ends or extremes of the proposition that are joined together by the copula “is.”

Words as sounds that refer to the same thing differ from language to language (e.g., “cat” in English and “billee” in Hindi both referring to cats). But concepts of the same thing are the same for everyone. This is a counterintuitive claim in our time, but it is also a foundational truth of the whole classical tradition and necessary for any account of logic. Given how counterintuitive it is for us, this truth deserves attention here. It should first be noted that the greatest linguist of the twentieth century and the founder of modern linguistics, Noam Chomsky, holds essentially this view, albeit for different reasons than Aristotle and St. Thomas (Chomsky, “New Horizons in the Study of Language and the Mind”). The reason that concepts are the same for everyone is easy to see: if concepts are likenesses of things, then two concepts that are likenesses of the same thing will each represent that thing; and, in so far as they are representations of that thing, they will be the same, and, in so far as they do not capture what is of the essence of that thing, then they are different, and those concepts represent different things. If, for example, someone believes that faith is only a feeling and someone else believes that faith is a virtue, then a discussion between the two about faith will not be about the same subject, even though it might seem to be at first. If such a discussion turns to the question of the nature or essence of faith itself, then that implies that there is an essence and definition of faith. It is this thing that is now the subject of the discussion. In such a discussion, the goal is to put truths that the interlocutors know into an order for the sake of revealing the unified nature or essence of the thing that lies behind them. Most simply, think of two photographs of the same thing: is it possible for two photographs of the same thing to be photographs of different things? Clearly not, although in the case of different photographs how they depict the same object may differ. This comparison reveals a key point. Concepts are different from photographs and other material representations in precisely this: concepts are formal signs that represent through themselves and without a medium; since there is no medium, there is nothing that can differentiate two concepts of the same thing; and, therefore, what differentiates one concept from another are the things that each concept represents.

Words, we began by saying, are signs of things. But concepts are also signs of things. What is the order among words, concepts, and things? According to St. Thomas Aquinas, the order is this (based on his commentary on Aristotle’s De interpretatione I, lectio 2): we form concepts to represent things; words signify concepts immediately; but words do not immediately signify things; this is because what really exists are singulars; so, words signify ultimately the things that concepts represent but they do so only by means of our concepts. It is not enough for us to have concepts in order to reason; we must utter those concepts and their relations to ourselves as we reason through what we seek to know. Much of modern linguistics, again following Chomsky, agrees with this basic doctrine: language is for a deeper purpose than only communication in social groups; it is for representing the world to ourselves, a purpose on which any communication in social groups depends (Chomsky and Berwick, Why only Us: Language and Evolution).

To make this order as explicit and clear as possible, the human mind employs words as the material instruments that signify concepts (which are not material). Since logic is the art of reasoning and the science of the acts of the mind (beings of reason, entia rations), logic is concerned with words but not as to the correctness of their use. Rather, logic is concerned with words only in so far as reasoning is carried out by means of words signifying and expressing those acts of the mind which conform to reality.

The first operation of the mind is the apprehension of the essence of anything that can be spoken about, and words are the instruments whereby we express these essences. What then are general kinds of things that we can say about something with our words? A few preliminary points must be addressed. First, we need to note that two different words can sometimes have the same meaning (e.g., “car” and “automobile”), and sometimes related meanings (e.g., like “courage” and “courageous”), and sometimes what looks like the same word can have different meanings (e.g., “bat” meaning the flying creature that lives in caves and “bat” meaning what you hit a baseball with). Second, we can say things that are simple, like “red” or “bear,” and we can also say things that are complex, like “that old blue soccer ball by the door.” The complex things we can say are built of the simple things we can say.

So, what are the primary kinds of simple things we can say? (In the following, the category names are in bold for clarity.) First, substances are those things that are not said of other things nor do they exist in other things. These are primary substances. Substances in this sense do, however, have essences; and these essences exist instantiated in individual substances. These essences are also considered substances but only in a secondary way, i.e., as secondary substances. We can say that something is an individual thing (e.g., “this dog here”) or that it is a kind of thing (e.g., “cat”). We do not say “this dog here” as the predicate of any sentence but only as the subject of a sentence. We can, however, say “cat” as the predicate, as when we say that “Missy is a cat.” Substances are what we say things about, either because we are speaking about an individual real thing or because we are speaking about some kind or essence of real things or about the essence of something related to real things. Substance is the category on which all others depend, because everything else inheres in or exists in substances.

Secondary substances, essences, or natures and universal or general terms more broadly have relations to one another, as outlined by Porphyry in his Introduction to the Categories. These are the principal kinds of predicates that can be said of a subject, called “the five predicables.” First, one essence or general term can be a species of a more general essence or term, which is the genus of the thing (the use of “species” and “genus” here applies to relations between two general terms of any kind). What differentiates one species from another species within the same genus is the differentia (e.g., what differentiates man from other animals is rationality). For any essence or universal term that is a species, it is possible for other universal terms either to flow from the essence of the thing or not. If the universal flows from the essence of its species, it is a property of that species (e.g., the ability to laugh is a property of man because it flows from man’s nature, which entails that the ability to laugh belongs to only our species, all together, at the same time, and always). If a universal term does not flow from the nature of the species, then that universal is, like tall in man, only an accident of the species, i.e., it is something that only happens to be (Latin: accidens) in the species but is not essentially.

St. Thomas Aquinas explains how the other categories relate to each other and to substance (see Commentary on Aristotle’s Metaphysics V, lect. 9, nos. 889–91). Substances have things said about them either as inhering in the substance in some way (e.g., the way the size or color of a thing inheres in it) or as affecting the substance (e. g. the way doing something changes a substance or the way being somewhere affects a substance). What inheres in a substance can either inhere absolutely or relatively. What inheres absolutely in a substance either inheres in the substance’s form as a quality (e.g., “white” or “hot” or “healthy”); or it inheres in the substance’s matter as quantity (e.g., “four parts” or “ten feet long”). What inheres in a substance relatively is a relation, like (e.g., “being twice the size of” or “being the son of”). What affects a substance either affects it by being in the substance in some way or it affects it in a way extrinsic to the subject. What affects a subject by being in it is action, like “running” or “studying” or “growing,” and passion (e.g., “carrying” or “attacking” or “being sprayed”). What affects a substance in a way extrinsic to the substance can affect it extrinsically in a way that somehow measures the subject, either in time (e.g., “now” or “two hours ago”), or in position; and there are two aspects of being in position, being in place (e.g., “in Paris” or “at your kitchen table”) or having a certain posture or orientation (e.g., “lying down” or “crumpled up” or “bending over”). Finally, what affects a subject can do so in a way that does not measure the subject but somehow the subject has possession of what affects it (e.g., when we say someone is “wearing shoes” or “has knowledge of animals” or “is courageous”). These nine, together with substance, are the ten categories of Aristotle.

It is useful for students to note that categories other than substance do not typically predicate what is of the essence of a substance. These other nine categories are categories of accidents. Since the principal goal of logic is demonstration, it should be noted that we come to demonstrations by abstracting from the other nine categories and considering primarily the category of substance, in particular what pertains to substance secondarily, namely, genus, species, differentia, and property. (One exception to this is any consideration of material bodies as such, which is common especially to physics, since the other nine categories are properties of material bodies as such, and thus detailed study of each reveals further the nature of material body.)

There is a subtle difference between the ten categories (primary substance and the other nine) and the five predicables (which treat of secondary substance). The ten categories are directly about real things outside the mind, but the five predicables are, in the view of Aristotle and St. Thomas Aquinas, not directly about real things outside the mind; rather, they are produced by the mind as a consequence of its way of knowing reality. But, to be clear, this does not mean that, as Kant believed, they are products of the mind whose reality is wholly given by the mind, since real things themselves are unknowable. Rather, they are products of the mind precisely as having a foundation in reality. Concepts are often called “intentions” (Latin: intentiones) because the directing of the mind to a thing is an intention (a tending toward the thing). Among concepts some are the result of the first intention of the mind to a thing, as when we discover what white is; these are first intentions. But some concepts are the result of the mind turning toward the thing a second time, as when we discover that white is a species of color; these are second intentions. Logic can be said to be the science of second intentions in that what logic studies are the ordered relations between concepts that the mind has produced as its way of knowing reality. Logic does not properly study concepts that are directly about reality, since these are studied by the various disciplines; what logic studies is the concepts and relations between acts of the mind and whether these are well-formed and adequate to what they represent.{19}

The Second Operation of the Intellect: Judgment

The connections between the first operation of the intellect and the second and third are made clearest by John of St. Thomas:

When something obscure or unknown needs an explanation through a statement that explains it and takes away that obscurity, such a statement is called “the means of demonstrative knowledge,” because every act of clarifying obscurity or making something known is either said to be demonstrative knowledge or is aimed at demonstrative knowledge. Now, there are two things that a statement can manifest to our intellect, a simple thing or a complex truth. Something simple, like man, sky, earth, etc., is explained by giving a definition if there is obscurity about the essence of the thing; and something simple is explained by a making division if there is confusion about the parts or the units that are contained in the thing. But a complex truth, if it is obscure or doubtful, is made manifest by proof. And proof is discourse and inference and so is argumentation. Thus, the modes of demonstrative knowledge are adequately divided into definition, division, and argumentation. (John of St. Thomas, Summulae, bk. II, ch. 2)

The mind abstracts the essence of a thing from our experience of that thing. Such an essence can be expressed as a definition: “a definition is a statement that sets forth the essence of a thing or the meaning of a term” (John of St. Thomas, Summulae, bk. II, ch. 3). We can also clarify a thing or a term by setting out the order of its parts or of what is contained in the thing; we do this by division: “a division is a statement that distributes a thing into its members or a term into its meanings” (ch. 4). We have seen several divisions above; for example, the division of the parts of logic, the division of the five predicables (genus, species, etc.), and the division of the ten categories as they relate to material substances. Divisions are useful and deserve careful study, as are the principles behind making good definitions.{20}

The second operation of the intellect, in its simplest form, is the judgment that affirms or denies that a predicate is in a subject. Affirmation and denial (or negation) are the two kinds of proposition. Properly speaking, the second operation of the intellect does not concern the formation of just any kind of sentence, with subject and predicate. Rather, it concerns the formation of affirmations and denials that are the means of demonstrative knowledge, that is, affirmations and denials that reveal the essences and causes of things. This rules out sentences like commands, questions, etc. But, crucially, it also largely excludes sentences that express truths which are not the means of demonstration. Propositions that can be the means of demonstrations are mostly general or categorical, since science is the knowledge of general truths and thus does not consider individual or singular things directly; nevertheless, scientific knowledge of something can certainly be applied to individual things to help us understand them better in a particular context or to predict what they will do. (Of course, uniquely, sacred doctrine can extend to individual and singular things insofar as such individual and singular things are divinely revealed.)

Given its importance for theology, a technical word is in order about what the terms in a proposition can refer to (and what they refer to is said to be what they “supposit for” or their “supposition”). There are different ways that a term inside a proposition can refer to or stand for what it signifies. First, and most commonly, a concrete common term (like “squirrel” in a proposition such as “squirrels gather nuts”) refers to whatever the term is typically used to refer to; this is called the term’s “personal supposition” (from Latin persona, which means “role”, since the concern here is for the term in its most common role). For example, the personal supposition of “squirrels” is individual squirrels and the personal supposition of “white” is white things, since the main role of the word “squirrels” is to refer to squirrels and “white” to refer to white things. Second, a term can refer to the nature of the thing that the term signifies; this is called “simple supposition.” For example, “man” in “man is a rational animal” does not stand directly for individual men but for the universal nature of humankind. Third, a term can be used to refer to its own linguistic aspects; this is called “material supposition.” For example, “angel” has material supposition in “‘angel’ has five letters” and “‘angel’ is a noun”, since in both cases we are talking about the word as a linguistic item, not as to its meaning. In contemporary writing, it is common to denote material supposition by the use of quotation marks. In general, the consideration of supposition allows us to be precise about what terms refer to in a proposition.{21}

When we affirm that a predicate is in a subject, we can most simply either assert that the predicate is in everything of the subject (if S is the subject and P the predicate, we can symbolize this as “all S is P”) or that the predicate is only in some of the subject (“some S is P”). When we deny that a predicate is in a subject, similarly, we can most simply either assert that the predicate is in nothing of the subject (“no S is P”) or that the predicate is not in some of the subject (“some S is not P”). These four propositions have two features: whether they are affirmations or denials, called their “quality”; and whether they are asserted of everything in the subject (universal) or only of some things in the subject (particular), called their “quantity.”

In general, this gives us the four simplest categorical propositions:

All S is P
No S is P
Some S is P
Some S is not P

These four proposition forms are related to one another in the following ways, among others: 

Contraries: “all S is P” affirms P of every S and “no S is P” denies P of every S; so, they are opposites in that what one affirms the other denies; but the same is true of “some S is P” and “some S is not P;” propositions opposed in this way are contraries.

Contradictories: “all S is P” affirms P of every S and “some S is not P” denies P of some S; these two are opposites in a different way, in that what one affirms the other contradicts (meaning that the second affirms exactly that which entails that the first is false); propositions opposed in this way are contradictories. 

There are other relations too and these relations of opposition allow us to construct the square of opposition, which has been a mainstay of logical study for centuries. Any standard textbook or handbook can provide comprehensive details, as well as details about topics like conversion, ampliation, and modal propositions.

It is important to note that these four are general forms of proportions, formed by the simple act of affirmation or denial of either universal or particular subjects. These propositions, therefore, are not ones that Aristotle just happened to choose to study when he could have focused on another set of propositions. From the classical perspective, these four relate general terms (essences) together in the simplest ways, and therefore the study of them reveals the basics of the structure of the assertions and denials we can make in demonstration.

Finally, it is in the proposition that truth is most properly found. "To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, and of what is not that it is not, is true” (Aristotle, Metaphysics IV.7). Truth properly pertains to what is asserted or expressed (i.e., the proposition); and that is why truth does not properly pertain to the first operation of the intellect. Nevertheless, “Truth has its foundation in reality but truth is completed through the action of the intellect, when the thing is understood to be how it is,” as St. Thomas Aquinas says (I Sent. 19. 5. 1 c). Thus, truth can be said to be in the first operation of the intellect, i.e., in a concept, insofar as the given act of the intellect conforms to the way the thing it represents really is. Truth is primarily in the mind, but it is secondarily in things, A thing can be said to be more truly what it is insofar as the given thing conforms more closely to the idea or exemplar of it in the mind of the Creator.{22}

The Third Operation of the Intellect: Inference and Demonstration

Reasoning is a locution in which, after certain things have been granted (the premises), certain other things must also of necessity be granted (the conclusion). In Greek and other languages following it, this has the name “syllogism.” In ordinary usage, this word has come to mean a set form of argument, but its original meaning was the equivalent of “reasoning” in English. Inference refers to the movement of the mind from the granted premises to the conclusion that must also be granted on the basis of the truth of the premises (in any way this is possible). Inference is part of every demonstration. Demonstration is inference in which the premises are true and in which they reveal either that the conclusion is true or, most properly, why the conclusion must also be true. Demonstration, coming to know, is what makes inference possible and is that for the sake of which inference exists, since we infer in order to know. Inference that is not demonstrative in the strict sense, like dialectical or rhetorical inference, is for the sake of demonstration in the fullest sense; these modes of inferring are part of how we come to know truths in the fullest way. Demonstration is the cause of inference in the same way that sight as accurate perception of color is the cause of all other varieties of visual perception: even poor vision or night-blindness or color-blindness are themselves made possible by sight as accurate perception of color, since accurate perception is the goal of eyes, even if they cannot always reach that goal.

If we ask what is the simplest and most perspicuous form of argument, the answer will be the one with the fewest terms and the most direct ways that terms relate to one another. In the second operation of the intellect, we join two terms in a judgment. Two terms allow us to form only a single proposition, not a chain of reasoning. But what about relations among three terms? This is the simplest form of the third operation of the intellect. The third operation of the intellect takes together three terms, related across two propositions, and from these alone draws out a third proposition that must also be true on the basis of the first two. These two-premise and one conclusion arguments came to be called “syllogisms” narrowly construed (and “syllogism” now usually does not mean reasoning in general but specifically these forms).

To use the most general possible forms, we can state a generalization of the syllogism like this: A and B are related in some way (by inclusion or exclusion of all or some of one term with respect to the other); B and C are related in some way; and so a syllogism is formed in all and only those cases where the two above relations imply of necessity that A and C are related in some way. There are a limited number of cases of this, and traditional logic has considered them all.

A careful look at the generalization of the syllogism above, relating A, B, and C in general, reveals that one of the three terms is the term that allows us to state the relation between the other two, in this general case, B; this is the middle term, so-called because it mediates the relation between the other two. In a demonstration, this mediating (middle) term is essential because it is this term, B, that can reveal the cause of the relation between the two terms of the conclusion, A and C. Aristotelian logicians therefore arrange the valid cases of the above general form of reasoning with respect to where the middle term is in the two premises: the reason for this is that how the mediating term relates to the subject and predicate of the conclusion is what determines what kind of truth can be demonstrated in that arrangement of terms. For example, if the middle term is the predicate of both propositions, we can only validly infer a negative conclusion (e.g., in the case of “all A is B; all C is B”); we cannot validly draw any conclusions, and in the case of “no A is B; all C is B,” we can only draw the negative conclusion “no A is C.” But since demonstration is chiefly concerned with the cause, negative conclusions will have limited use since a negative conclusion can at best reveal what is not the cause. The different arrangements of the middle term across the two premises are called “figures of the syllogism.” The premise that states the relation between the middle term and the predicate of the conclusion is the major premise because the predicate of a proposition is taken to have a wider scope than the subject. The premise that states the relation between the middle term and the subject of the conclusion is the minor premise because the subject of a proposition is taken to have a narrower scope than the predicate.

It has traditionally been held by Aristotle and logicians like St. Albert the Great that all forms of categorical reasoning can be expressed as syllogisms, narrowly construed. For a long time, this claim was not called into question. Beginning in the Renaissance, modern authors began to wonder whether it is true and then later to argue that it is false. This might seem like an unimportant point, but it is not: if categorical reasoning in general cannot be reduced to syllogisms narrowly construed, then that possibility seems to cast doubt on the whole enterprise of basing traditional logic of terms representing the essences of things. This is an active area of controversy, in particular with respect to mathematics. To see this, consider an argument like this: “Line AB is greater than line BC; line BC is greater than line CD; therefore, line AB is greater than line CD.” What is the middle term? Is it “greater than line BC” or is it “line BC”? If we say “greater than line BC”, then that middle term is absent from the second premise, where we only have “line BC”, not “greater than BC”; and if we say “line BC”, that middle term is absent from the first premise, where we have “greater than line BC.” The problem is that we seem to have three concepts in each premise, not two: one line, the relation of greater than, and another line. The simplest answer to this challenge is that we must distinguish between formal and material logic: formal logic is the study of the arrangement of terms and how the arrangements of terms alone in two propositions that have been granted can necessitate that we grant the truth of a third proposition, the conclusion. But in the case considered here, we are considering the form of the syllogisms when they have certain complex terms from mathematics as their subjects and predicates; this suggests that the question is not whether formal syllogisms alone are adequate for all the purposes of mathematics, but whether formal syllogisms employing the necessary mathematical terms, postulates, axioms, and principles (i.e., the material logic of mathematics) are adequate for all the purposes of mathematics. This only suggests, however, that there is or may be a solution to the more general problem above when studied in light of the material logic of mathematics. Research continues on the possible limits of syllogistic logic.{23}

Properly speaking, however, a demonstration is not just a syllogism. In fact, John of St. Thomas distinguishes two modes of demonstration: induction and scientific syllogism (Summulae, bk. III., ch. 2). Induction is the movement of the mind to the proof of a universal by means of a sufficient consideration of the things that contain the given universal. For example, if we seek to prove that a certain insect can swim, we might proceed with a consideration one by one of a sufficient number of members of the species and test whether each can swim or not. Once we have examined a sufficient number, we can induce that this species of insect can swim. The goal of induction is to discover and prove universal truths by means of examining the individual things that have the given essence. But we can also use the reverse of induction, going from a universal truth to the individual things that it is true of in order to prove that something is not true of a given essence. For example, if we mistakenly believe that some species of insects can swim, then we can study individuals of the species to show that this is not true. In ordinary arguments, we often use incomplete inductions, in which we present an argument based on an example. This is an incomplete induction because one example does not prove something true of everything of the given kind. A proof by induction is a demonstration of the fact (quia), not a demonstration of the cause (propter quid). (See above, “Demonstration” in “Parts of Logic.”)

Demonstrations, in the strictest sense, is a syllogism in which the truth of the premises shows why the conclusion is necessary. Demonstrations are valid syllogisms (see “validity” above) that have true premises that express the necessary causes of the conclusion. (Demonstrations also must meet other conditions, e.g., that the premises be better known than the conclusion; for these details, see ECT lexicon entry for “Posterior Analytics”.) Ultimately, a demonstration is an account of why (or that) a proposition must be true given the fact that two other related propositions are also true. Science and demonstrative knowledge are principally and primarily about general and universal things, such as essences of natural things or their properties.

The paradigmatic example of demonstrative knowledge is always mathematics, especially the mathematics of Euclid’s Elements. This also reveals the importance of the problem raised above about whether syllogisms are sufficient for mathematics. Mathematical demonstrations are paradigmatic because their content is formal, general, and necessary and because we can know the definitions of mathematical terms with certainty and clarity. But we can also perform demonstrations in other areas, e.g., from the study of nature or even the study of history. Aristotle himself considers a demonstration of the cause of why the Athenians became involved in the Persian War (Posterior Analytics II.11). When the question of the exact nature of demonstrating the cause became more and more pressing, the later medieval tradition proposed that mathematics and metaphysics have the highest kinds of demonstrations (with absolute certainty) but that there can be demonstrations with lower levels of certainty in disciplines such as the study of nature (with physical certainty) and in the study of human life and affairs (with moral certainty). The classical tradition, especially in the case of Aristotle, is clear that we should only seek for that level of precision that is appropriate to a given domain (i.e., we should not expect the same level of certainty and precision in politics as in mathematics).

Note that the above does not consider in any detail how demonstration relates to and is used by sacred theology. That is an immense and difficult topic. Many debates remain, and they cannot be treated of here. Suffice it to say that the question of the role of demonstration in theology is itself a theological question with profound implications.