Some ideas in computer science are so familiar that it is hard to remember that they were once new. The idea of logic is one. Calculating methods from the ancient world had limited influence on the design of today’s computers, but the principles of two-valued logic underpin digital computing.
Computers are general-purpose. Most chess-playing computers or inventory-keeping computers are just generic computers that can be used for games or for business. A bit that one day indicates whether the knight on square f3 is white or black might the next day indicate whether this book is in stock. The logical rules embedded in the computer’s hardware can be used for manipulating both kinds of information. These abstract ideas of things, properties, and logical rules for reasoning about them did not always exist. These were Aristotle’s ideas, and they were necessary first steps toward computations about things and their properties.
Aristotle (384–322 BCE) was the great systematizer. In many works, mostly lost, he analyzed and categorized everything imaginable. His Prior Analytics presented the world’s first system of logic. Its purpose is to infer conclusions from premises in a way that depends only on the form of the argument, not on the persuasiveness of the speaker or on anything not mentioned in the premises. Aristotle’s explanation of a logical deduction is the root of all modern logic.
Aristotle’s technical vocabulary makes for difficult reading. Fortunately, the archaic details are not important to us. He works with the idea of a predicate, a property that a thing may or may not have. This gave rise to the modern idea of a thing being a member of a set, the set of all things having that property. Aristotle’s notion of “belonging” can be understood as the subset relation. For example, to say that property A belongs to none of the Bs is to say that no member of the set B is a member of A, that is, that B is disjoint from A. So the example “if A is predicated of every B and B of every C, it is necessary for A to be predicated of every C” is a statement of the transitivity of the superset relation: If A ⊇ B and B ⊇ C, then A ⊇ C. Aristotle had no such notation at his disposal, but those who formalized logic and set theory stood on his shoulders.
There had been convincing mathematical arguments before Aristotle, but Aristotle was the first to abstract the form of such arguments from their content. In doing so, he showed how to reason mechanically, by matching propositions to general templates and inferring the conclusions that necessarily followed. Aristotle neither designed nor built logical calculating machines, but his vocabulary hints that he was describing a computational process. The word translated here as “deduction” is συλλογισμός, which means a “reckoning up” or “computation.” “Syllogism,” the English rendition of Aristotle’s term, now means something narrower, the particular forms of deduction that Aristotle explains in this work.
Aristotle also exhibits a method for showing that certain purported forms of inference are not universally valid, by the use of counterexamples. He invites the reader to draw a general inference from the premises A ⊇ B and B ∩ C = ∅. In fact, no necessary conclusion about the relation of A and C can be inferred from these two premises. For if we take A = animals, B = horses, and C = men, then the premises are satisfied (horses are animals, but no horse is a man), and A ⊇ C (men are animals); but if A = animals, B = men, and C = stones, then again the premises are satisfied (men are animals, but no man is a stone), but A is disjoint from C (no stone is an animal). (In a modern argument, we would have to finish by noting that C cannot simultaneously be a subset of and disjoint from A, as long as C is nonempty.) This is the method used to this day to refute conjectures and to prove the independence of hypotheses.
WE must first state what our inquiry is about and what its object is, saying that it is about demonstration and that its object is demonstrative science. Next, we must determine what a premise is, what a term is, and what a deduction is, and what sort of deduction is complete and what sort incomplete; and after these things, what it is for something to be or not be in something as a whole, and what we mean by “to be predicated of every” or “predicated of none.”
A premise, then, is a sentence affirming or denying something about something. This sentence may be universal, particular, or indeterminate. I call belonging “to every” or “to none” universal; I call belonging “to some,” “not to some,” or “not to every,” particular, and I call belonging or not belonging (without a universal or particular) indeterminate (as, for example, “the science of contraries is the same” or “pleasure is not a good”).
A demonstrative premise is different from a dialectical one in that a demonstrative premise is the taking of one or the other part of a contradiction (for someone who is demonstrating does not ask for premises but takes them), whereas a dialectical premise is the asking of a contradiction. However, this will make no difference as to whether a deduction comes about for either man, for both the one who demonstrates and the one who asks deduce by taking something either to belong or not to belong with respect to something. Consequently, a deductive premise without qualification will be either the affirmation or the denial of one thing about another, in the way that this has been explained. It will be demonstrative if it is true and has been obtained by means of the initial assumptions; a dialectical premise, on the other hand, is the posing of a contradiction as a question (when one is getting answers) and the taking of something apparent and accepted (when one is deducing), as was explained in the Topics.
What a premise is, then, and how deductive, demonstrative, and dialectical premises differ, will be explained more precisely in what follows; let the distinctions just made be sufficient for our present needs.
I call that a term into which a premise may be broken up, i.e., both that which is predicated and that of which it is predicated (whether or not “is” or “is not” is added or divides them).
A deduction is a discourse in which, certain things having been supposed, something different from the things supposed results of necessity because these things are so. By “because these things are so,” I mean “resulting through them,” and by “resulting through them” I mean “needing no further term from outside in order for the necessity to come about.”
I call a deduction complete if it stands in need of nothing else besides the things taken in order for the necessity to be evident; I call it incomplete if it still needs either one or several additional things which are necessary because of the terms assumed, but yet were not taken by means of premises.
For one thing to be in another as a whole is the same as for one thing to be predicated of every one of another. We use the expression “predicated of every” when none of the subject can be taken of which the other term cannot be said, and we use “predicated of none” likewise.
Now, every premise expresses either belonging, or belonging of necessity, or being possible to belong; and some of these, for each prefix respectively, are affirmative and others negative; and of the affirmative and negative premises, in turn, some are universal, some are in part, and some indeterminate.
It is necessary for a universal privative premise of belonging to convert with respect to its terms. For instance, if no pleasure is a good, neither will any good be a pleasure. And the positive premise necessarily converts, though not universally but in part. For instance, if every pleasure is a good, then some good will be a pleasure. Among the particular premises, the affirmative must convert partially (for if some pleasure is a good, then some good will be a pleasure), but the privative premise need not (for it is not the case that if man does not belong to some animal, then animal will not belong to some man).
First, then, let premise AB be universally privative. Now, if A belongs to none of the Bs, then neither will B belong to any of the As. For if it does belong to some (for instance to C), it will not be true that A belongs to none of the Bs, since C is one of the Bs. And if A belongs to every B, then B will belong to some A. For if it belongs to none, neither will A belong to any B; but it was assumed to belong to every one. And similarly if the premise is particular: if A belongs to some of the Bs, then necessarily B belongs to some of the As. (For if it belongs to none, then neither will A belong to any of the Bs.) But if A does not belong to some B, it is not necessary for B also not to belong to some A (for example if B is animal and A man: for man does not belong to every animal, but animal belongs to every man). …
Having made these determinations, let us now say through what premises, when, and how every deduction comes about. (We will need to discuss demonstration later. Deduction should be discussed before demonstration because deduction is more universal: a demonstration is a kind of deduction, but not every deduction is a demonstration.)
Whenever, then, three terms are so related to each other that the last is in the middle as a whole and the middle is either in or not in the first as a whole, it is necessary for there to be a complete deduction of the extremes. (I call that the middle which both is itself in another and has another in it—this is also middle in position—and call both that which is itself in another and that which has another in it extremes.) For if A is predicated of every B and B of every C, it is necessary for A to be predicated of every C (for it was stated earlier what we mean by “of every”). Similarly, if A is predicated of no B and B of every C, it is necessary that A will belong to no C. However, if the first extreme follows all the middle and the middle belongs to none of the last, there will not be a deduction of the extremes, for nothing necessary results in virtue of these things being so. For it is possible for the first extreme to belong to all as well as to none of the last. Consequently, neither a particular nor a universal conclusion becomes necessary; and, since nothing is necessary because of these, there will not be a deduction. Terms for belonging to every are animal, man, horse; for belonging to none, animal, man, stone. Nor when neither the first belongs to any of the middle nor the middle to any of the last: there will not be a deduction in this way either. Terms for belonging are science, line, medicine; for not belonging, science, line, unit.
Thus, it is clear when there will and when there will not be a deduction in this figure if the terms are universal; and it is also clear both that if there is a deduction, then the terms must necessarily be related as we have said, and that if they are related in this way, then there will be a deduction.
If one of the terms is universal and the other is particular in relation to the remaining term, then when the universal is put in relation to the major extreme (whether this is positive or privative) and the particular is put in relation to the minor extreme (which is positive), then there will necessarily be a complete deduction; when, however, the universal is put in relation to the minor extreme, or when the terms are related in any other way, this is impossible. (I call that extreme the “major” which the middle is in and that extreme the “minor” which is under the middle.) For let A belong to every B and B to some C. Then, if to be predicated of every is what was said in the beginning, it is necessary for A to belong to some C. And if A belongs to no B and B to some C, then it is necessary for A not to belong to some C. (For it has also been defined what we mean by “predicated of no” so that there will be a complete deduction.) Similarly also if BC should be indeterminate, provided it is positive (for it will be the same deduction whether an indeterminate premise or a particular one is taken). …
Reprinted from Aristotle (1989), with permission from Hackett Publishing Company, Inc.