A World of Interlocking Structures
WE LIVE our lives across space and time, but we can extend our logic across space and time too. One way to see how we can do this is to think for a moment about geometry and arithmetic.
When we look at a geometrical figure, we obviously think about space, but the figure we are looking at, if printed on a page, is essentially timeless—it is static and unchanging. When we carry out an operation in arithmetic, on the other hand, we typically think of a “before” and an “after.” And our use of the operation represents a change. Before the operation, we have, say, a pair of numbers, like five and seven, but after the operation we have a sum: twelve. Thus geometry seems connected in some deep way to our sense of space, but arithmetic seems connected to our sense of time. (This point was made more than two centuries ago by the German philosopher Immanuel Kant.)1
The same can be said for different branches of logic. Aristotle’s logic is closely connected to geometry and to space; when we contemplate a question of whether one class includes another, the classes we contemplate are static. The classes don’t change as we think about them, and, in consequence, we can represent them spatially, with geometrical figures. But there is another sort of logic that considers the changing and dynamic—and concerns our choices in life. When we think about choosing a course of conduct, we think across time to the future, and the reasonableness of our choice then depends on our ability to predict its future consequences. As a result, in addition to a logic of classification as worked out by Aristotle (syllogistic logic), there is also a logic of choice and consequence (propositional logic), which appeared about a century after Aristotle within the ancient philosophical school of the Stoics.
The Stoics’ new approach to logic came from a new imperial world, where sovereign kings had replaced the old democracies of the past and where philosophical introspection had replaced political participation. As large, new autocratic states spread across the region of the eastern Mediterranean, the thinkers of the age tended to dwell less on public and political rights and more on personal and private choices. They focused on the many ways we manage our individual lives. In pursuing these themes, they also discovered a new set of interlocking logical structures. The Stoics saw the whole universe as rationally interconnected, and the structures they investigated turned out to be interconnected, too.
THE STOICS
The Stoics first emerged as a distinct group about a generation after Aristotle, and their chief concern was to find the good life for human beings. The Stoics were primarily moralists. They believed the universe was governed by a law of reason, emanating from God, and that the good life consisted in following this law. In the Stoic view, this rational law was a “law of nature,” meaning it controlled physical events but also applied to morality. People’s moral choices were supposed to conform to it. We might try to evade the moral aspects of the rational law and make excuses for our evasions, but in trying to excuse ourselves we were still conscious, they believed, of the law’s presence. After all, if there were no such law in the first place, why would anyone bother with excuses?
In addition, this law of nature could be demanding, and discovering its implications required that we think logically. According to the Stoics, so long as one strove to do right and maintained a positive attitude, one’s own life was a life worth living. External fate was irrelevant. Pain, defeat, opprobrium, and death made no difference. The Stoics insisted that we do right without excuses and without fear of death. They often admitted that this outlook could be difficult to achieve, and as a result, Stoicism was as much a discipline as a doctrine. Nevertheless, it attracted many people. Like many ancient philosophical schools after the fourth century B.C., the Stoics looked back on Socrates as a patron saint, and they saw an embodiment of their ideal in his acceptance of external fate.
In fact, if we set logic aside for a moment and consider Stoicism as an approach to ethics and morality, Stoicism has probably existed in nearly all cultures and at all times (whether or not it has had a name), and one still sees many expressions of it today. For example, the Stoic emphasis on fearless virtue, combined with a patient acceptance of external fate, appears in the popular “Serenity Prayer,” usually attributed to Reinhold Niebuhr: “God, grant me the serenity to accept the things I cannot change, the courage to change the things I can, and the wisdom to know the difference.”2 A similar emphasis appears in a lecture by the American abolitionist Frederick Douglass, when he puts action and attitude ahead of comforts and acquisitions: “He lives most who thinks the most, feels the noblest, acts the best.”3
But Stoicism didn’t emerge in a vacuum—it became ever more popular because of political changes sweeping the Greek world after the fourth century B.C. After the conquests of Philip of Macedon and his son Alexander the Great, political power shifted steadily away from democratic assemblies and fell into the hands of kings. The subjugation of the old Greek democracies by these new Macedonian warlords put an end to the classical age of Greece and ushered in new styles in literature, the arts, and logic.
A key factor was a change in the nature of war. Philip and his successors managed to outmaneuver and outclass the common citizen-soldiers of the old city-states by confronting them with a new kind of army: paid professionals, who used new weapons and tactics. This new army ultimately overthrew the older political systems of the eastern Mediterranean, and the effect on the world of ideas was to place new demands on artists, writers, and intellectuals.
When Philip was still a teenager, his own country of Macedonia had been dominated by the city-state of Thebes to the south, and the Thebans held him as a royal hostage. But his years as hostage also gave him a chance to study the new military tactics of the Theban “Sacred Band,” a special infantry corps of highly-trained youths. In war, the Sacred Band of Thebes, unlike other Greek troops, formed an especially dense phalanx (a massed infantry formation) and attacked at oblique angles. When Philip was later released and rose to power in Macedonia, he copied these tactics and introduced other changes too. He armed each of his infantrymen with a new, long pike in place of a conventional spear. The result was to make his new Macedonian phalanx nearly unbeatable in frontal attacks. And though the new pikes also made the Macedonian phalanx crucially vulnerable to assault on its flanks (since the pike was hard to turn in a new direction without colliding with one’s colleagues), Philip countered this difficulty by covering the flanks with masses of conventional spearmen and by developing more sophisticated mobile forces—cavalry, archers, and other specialized units. The ultimate effect was to emphasize a greater variety of arms and a level of specialized training that only professional soldiers could hope to master. Philip hired large numbers of foreign mercenaries, drilled them incessantly, and paid them with captured gold. (One of his early moves as king was to seize the gold mines of his eastern neighbors in Thrace.)
Philip then exploited these innovations to overwhelm the old city-states of Greece. He ensured the loyalty of his mercenary troops by forming many of his army’s most elite units from Macedonian nobles, who had a stake in preserving royal authority. The Greek cities tried to keep pace with him by hiring foreign mercenaries of their own, but their democratic majorities generally resisted anything like domination of their military apparatus by an elite corps of hereditary aristocrats. Philip was essentially building an international force of military professionals held in place by his local nobility. Meanwhile, the city-states were still trying to preserve the sovereignty of large democratic majorities whose soldiering could never be more than part-time.
This shift from citizen armies to professional ones undermined the power of the old Greek democracies and tipped the balance toward new, royal governments like those of Philip, his son Alexander, and Alexander’s successors. The new military-political system Philip and his son were forging eventually came to dominate the eastern Mediterranean, and with it came a new kind of logic.
After the death of Alexander in 323 B.C., his empire broke into pieces as his former generals carved out large, new dominions for themselves; these expansive new monarchies then persisted for centuries. One of Alexander’s generals, a former friend and bodyguard, quickly seized Egypt and became Ptolemy I, whose Macedonian descendants ruled in the style of pharaohs down to the death of Cleopatra in 30 B.C. Another general, the former commander of Alexander’s footguards, became Seleucus I, whose family grabbed much of the old Persian Empire. A third element of Alexander’s domains broke off to become the kingdom of Macedonia, which changed hands many times but nevertheless dominated the old Greek cities.
In arts, letters, and logic, the impact of this new political order would shift attention away from public and political undertakings because so few people could have political undertakings in the first place. Instead, the new focus was on the personal and private. In sculpture, there was greater emphasis on the emotional, the extreme, and the idiosyncratic, culminating in the artistic style we now call “Hellenistic.”4 In earlier times, the city-states had commissioned idealized representations of gods, heroes, and even ordinary citizens to express the dignity of the state, and the serene countenances of these figures constituted the “severe style” of the fifth century B.C. But in the new order of things, the chief patrons of the arts were now wealthy individuals who valued novelty and expressiveness, and the sculpture they favored celebrated the domestic, the intimate, the agonizing, the comic, the tragic, and the grotesque.
There were also changes in drama; playwrights could no longer risk ridiculing the leading political figures of the day in the manner of the ancient Athenian Aristophanes, and they rarely discussed supreme political power in the style of Sophocles. Rather, theater focused increasingly on the domestic problems of ordinary people, and a new comedy of manners emerged that revolved around the intrigues of yearning lovers, churlish slaves, ignorant masters, and long-lost relatives. (This “new comedy,” Latinized for Roman audiences by Terence and Plautus, supplied Shakespeare with the plot for his Comedy of Errors and became the historical basis of modern romantic comedy.) On the whole, the general tendency was away from public and political themes and toward introspective ones, and Stoicism mirrored this trend.
As political power became more autocratic, philosophical writers dwelled less on political problems, more on spiritual ones. Personal duty rather than political duty became the primary focus. When a ruler was Stoic, so that personal and political duties coincided, the emphasis of Stoicism was never on changing the political system but rather on administering it justly. The Roman emperor Marcus Aurelius, whose Meditations from the second century A.D. has come down to us, inherited Stoicism in this form, and the Macedonian ruler Antigonus II studied it in much the same way in the third century B.C. (In answer to the suggestion that he was a god, Antigonus replied, “The man who carries my chamber pot knows better.” And he chastened his son, who had apparently mistreated someone else, with the words, “Do you not understand, boy, that our kingship is a noble servitude?”)5
In logic, the effect of these tendencies was a change in emphasis—from the old logic of static classification in the manner of Aristotle to a new logic that examined choice and consequence. Ethics, for the Stoics, was always a matter of choosing well, and the Stoics always insisted that argumentation shapes the morality of our choices. Indeed, for the Stoics, ethics was impossible without logic.6 Of course, one might easily suppose that the logic of choice discovered by the Stoics could have been explored just as well in any historical period; nevertheless, the real impetus for their efforts, which required much energy and concentration, came from their stress on personal morality—the same stress that continually filled their school with students. In Stoicism’s early years, both teachers and students alike were convinced that a life of personal morality required attention to logic. They also believed that reason ought to rule the passions just as reason ruled the cosmos. So far as the spirit of the time was concerned, this focus on the personal, in contrast to the political, was part and parcel of an autocratic age.
Paradoxically, the shift from citizen assemblies to sovereign kings reinforced the idea of the equality of peoples. Both citizens and noncitizens were now equally powerless in the political arena; only kings and their appointees could exercise real authority. Thus the Stoics were far less prone to view some social groups as born to rule and others as born to obey. Instead, the citizens who now concerned them were the “citizens of the world” (or, as the Stoics put it, the “cosmopolites”), and these citizens consisted of all men and women of goodwill, whatever their nationality.
This insistence on equality, combined with an emphasis on introspection, focused the attention of these thinkers on what was introspectively common to all peoples—citizens or noncitizens—and on what all people might discover if only they reflected quietly on their own mental habits. Writers of the period became more interested in a common human nature under the “rational law,” and they believed that part of this nature was to participate in an eternal realm of rational, moral choice. The key that unlocked the door to this realm was logic.
THE LOGIC OF CHOICE
Consider for a moment how our choices appear to us as a matter of logical form. For example, we often ask ourselves whether we ought to do one of two different things, A or B. If we do A, then we might cause C; yet, if we do B, we might cause D. And perhaps we can’t do both A and B. As a result, whereas for Aristotle the key logical expressions were quantifying words like “all” and “some,” which we apply to classes, the key logical expressions for the Stoics were different: they were the connectives “either/or,” “if-then,” and “not both.” The result was a new group of logical relations. In fact, the Stoic connectives had always been common in ordinary reasoning; they just hadn’t been systematically investigated. We see these connectives at work, for example, in the old Socratic dialogues of Plato.
The Socrates of Plato’s dialogues often asks how we ought to live, and his further analysis then turns out, often, to be an assessment of consequences. (To avoid a damaging consequence, his reasoning suggests, we must sometimes choose a different way of life.) The key notions of choice and consequence are thus central to Socrates’ reasoning, and they even appear in the form of the dialogues.
Socrates typically asks some fellow Athenian if he believes a thing, and when the Athenian says he does, Socrates often shows that the belief in question leads logically to something false or absurd—from which he then infers that the belief in question must also be false. When a belief is refuted by showing that it leads to a falsehood, the rhetorical technique is often called reductio ad absurdum.7 But the underlying connective is “if-then.” The classical Greeks had a long tradition of arguing in this manner, beginning with the view of one’s opponent and deriving an absurd consequence—thereby showing that the opponent’s view must be false.8 The technique had probably long flourished in Greek law courts, where witnesses could be cross-examined, and it also played an early role in Greek mathematics.9
The Stoics seem to have relied on the technique a great deal, and their great contribution was to investigate it as a matter of logical form. Of course, any argument can be conceived as an if-then proposition if we think of it as a relation between its premises and its conclusion (if the premises are true, then the conclusion should also be true). But the if-then investigated by the Stoics was typically more specific. The Stoics were concerned, above all, with the moral demands of rational, eternal law and in examining the consequences of our actions by the light of this law. As a result, their arguments not only had consequences but were about consequences. The emphasis on choice and consequence led them to investigate a new set of logical forms.
The forms they explored (or, as some logicians would say, the “schemata” they explored)10 all depended on “either/or,” “if-then,” or “not both.” But the Stoics also made a further, crucial discovery: all the new forms were in some way interconnected. Interconnection was one of their grand themes, since, according to their view, the whole world was rationally interconnected. The law of nature ruled a connected universe, and logic revealed the implications of this law to its citizens. In consequence, when the new logical forms turned out to be connected among themselves, the Stoics pursued them with great zeal.
Stoicism first appeared as a philosophical school through the teachings of Zeno of Citium, a Cypriot merchant who had come to Athens in the late 300s B.C. and had taken up philosophy. When Zeno first offered public discourses, he was too poor to hire a lecture hall, and so he began lecturing in a public arcade or stoa (from which we get the word “Stoic”).11 At his death in or around 265, the philosophical movement that had grown up around him passed to a former professional boxer, Cleanthes (originally from Assos in Asia Minor), and then to Cleanthes’ student Chrysippus (originally from Soli, also in Asia Minor). And it was under Chrysippus (pronounced CRY-sip-us) that Stoic logic became fully developed.
Chrysippus, leader of the Stoics at Athens from 232 onward, was the one to arrive at the key insight that all the new forms were logically interconnected. The simplest of Chrysippus’s forms are highly intuitive, and you will probably recognize many of their features as already occurring in your thinking. But Chrysippus sought to connect and elaborate these forms so as to develop what logicians now call “propositional logic” (a term whose meaning we shall explain later on), and it also turns out that this new, propositional logic runs our digital computers. (Without the sort of logic pioneered by Chrysippus, modern computer processing would be impossible.) For a start, then, let’s look at six of these new forms and then consider how they work in a modern computer. The names we give them now are the ones traditionally assigned by logicians.
1. Modus Ponens (“The Method of Affirming”)
The form is as follows:12
If A, then B.
A.
Therefore, B.
If the Great Oz has spoken, then you must come back tomorrow.
The Great Oz has spoken.
Therefore, you must come back tomorrow.
2. Modus Tollens ( “The Method of Denying”)
The form is as follows:
If A, then B.
But not B.
Therefore, not A.
Here is an example:
If hearts will ever be practical, then they must first be made unbreakable.
But hearts will never be made unbreakable.
Therefore, they will never be practical.
(Notice that the word “not” can be replaced with other negative expressions, such as “never,” “no,” “nobody,” or “no one.” Here is another example: “If you are to get help, then you must certainly see the Wizard; but no one sees the Wizard—not nobody, not no how; therefore, you are not to get help.”)
3. The Hypothetical Syllogism
The form is as follows:
If A, then B.
If B, then C.
Therefore, if A, then C.
Here is an example:
If I were king of the forest, then I’d rule with a royal growl.
If I ruled with a royal growl, then chipmunks would genuflect to me.
Therefore, if I were king of the forest, then chipmunks would genuflect to me.
(The Cowardly Lion’s thinking)
4. The Conjunctive Syllogism
The form, in two versions, is as follows:
| Not both A and B. | Not both A and B. |
| A. | B. |
| Therefore, not B. | Therefore, not A. |
Here is an example:
5. The Dilemma
The form, in two versions, is as follows:
| Simple dilemma | Complex dilemma |
| If A, then B. | If A, then B. |
| If C, then B. | If C, then D. |
| But A or C. | But A or C. |
| Therefore, B. | Therefore, B or D. |
Here is an example:
If you say what is just, then men will hate you; and if you say what is unjust, then the gods will hate you. But you must say one or the other. Therefore, you will be hated.
(An ancient priestess discussing her child’s prospects for a political career)14
The form, in two versions, is as follows:
| A or B. | A or B. |
| But not A. | But not B. |
| Therefore, B. | Therefore, A. |
Here is an example:
Lincoln tells you the States will be either all free or all slave. He is against making them all slave. It follows that he is for making them all free.
(Stephen A. Douglas)15
Now all these forms are interconnected in a special way: they can be interpreted so that none are valid unless all are. This is so because they all contain compound propositions. But to understand their interconnections—as vital in a modern computer as they were to the ancient Chrysippus—we first need to consider just what a “compound proposition” is.
THE NATURE OF COMPOUND PROPOSITIONS
Consider, for example, the last form on the list: the disjunctive syllogism. This form is called a syllogism because it has two premises working jointly, but it is disjunctive because the first line has the form,
A or B.
A and B are both simple propositions, but when joined by a connective like “or,” the resulting whole is a compound.
Logic traditionally uses three types of compounds. The first type, using the connective “or,” is called a disjunction, and the elements joined are its disjuncts. (Thus A and B mentioned in the previous paragraph are disjuncts.) Here is another disjunction, again with two disjuncts:
(In ordinary speech, the connective “or” is sometimes ambiguous since it can be used inclusively to mean “A or B or both” or exclusively to mean “A or B, but not both.” Chrysippus preferred the exclusive use, but in their most rigorous work today logicians use the connective inclusively. If a modern logician then wants to convey a sense of exclusion, the logician uses the inclusive form but adds an additional, qualifying formulation: “A or B, but not both A and B.”) The basic idea of disjunction is choice.
The second traditional compound is the conjunction; a conjunction joins simple statements with the connective “and,” and its individual elements are conjuncts. The following conjunction consists of three conjuncts:
A and B and C.
Julius Caesar’s famous description of his campaign in Asia Minor is a conjunction of three conjuncts, with the ands omitted:
I came, I saw, I conquered.
The idea of conjunction is the idea that there can be combinations; anyone who sees that more than one thing can be true at the same time has embraced a conjunction.
The third type of compound is the hypothetical or conditional (or, in the language of classical symbolic logic, “material implication”). The hypothetical uses the connective “if-then,” though, in ordinary English, the word “then” is often omitted. In The Call of the Wild by Jack London, John Thornton expresses his sentiments toward the brutal prospector Hal in the form of a hypothetical:
If you hit that dog again, I’ll kill you.
The idea behind the hypothetical is that many things have consequences.
Now notice that the hypothetical, like the disjunction, commits the speaker to neither element of the compound but only to the whole. Thus John Thornton hasn’t promised to kill Hal, nor has he said that Hal will hit the dog again. Instead, he only says that if Hall hits the dog again, then he’ll kill Hal. Hal replies with a disjunction:
Get out of my way, or I’ll fix you.
Again, Hal doesn’t promise to fix Thornton, he merely promises to do so if Thornton fails to get out of the way. These examples illustrate the logical equivalence between hypotheticals and disjunctions. Hal’s disjunction can be rewritten as a hypothetical:
If you don’t get out of my way, then I’ll fix you.
And Thornton’s hypothetical can be rewritten as a disjunction:
Don’t hit that dog again, or I’ll kill you.
In each case, the two formulations are equivalent in that the paired sentences are necessarily both true or both false—they are true or false together.
Schematically, the compound
A or B
can be construed as logically equivalent to the compound
If not A, then B,
or again, to the compound
If not B, then A.
And the compound
If A, then B
can be construed as logically equivalent to the compound
Either not A or B.16
Once we see the equivalence of these compounds, we can see the deep, fundamental connection between the disjunctive syllogism and a world of related logical necessities, the world first explored systematically by Chrysippus.
In essence, what Chrysippus discovered was that many of our most common methods of argument are logically tied; suitably interpreted, each is valid only if the others are. But let’s look at Chrysippus’s insight in greater detail.
INTERLOCKING FORMS OF ARGUMENT
The explorations of Chrysippus and his followers took place about a century after Aristotle, but their consequences were never fully exploited from a mathematical perspective until the nineteenth century. (More precisely, Chrysippus saw a great many logical connections among the simple forms we have already noted, but the specific equivalence of these forms was mostly developed in modern times—and only when logicians interpreted conditionals in a way that made them systematically convertible into disjunctions.)17 Still, Chrysippus plainly realized that he was once more entering a world of connected implications and symmetries—a world of related logical necessities.
For example, if we look carefully at modus ponens and the disjunctive syllogism, we can see intuitively that both are logically valid. We sense their validity instinctively by contemplating only a few examples. But it also turns out that, with the right interpretation, the validity of each of these forms turns on the validity of the other. And this must be so, because the premises of one can be interpreted as logically equivalent to the premises of the other. We can see this point if we write out the two forms schematically.
Here is modus ponens:
If A, then B.
A.
Therefore, B.
But remember that its first line, “If A, then B,” can be construed as logically equivalent to “Either not A or B” (just as “If you hit that dog again, I’ll kill you” is logically equivalent to “Don’t hit that dog again, or I’ll kill you”). In consequence, we can pair modus ponens with a new form that nevertheless has an initial premise that is logically equivalent. On the left is modus ponens, and on the right is the new form:
| Modus ponens | New form |
| If A, then B. | Either not A or B. |
| A. | A. |
| Therefore, B. | Therefore, B. |
In this comparison, one can see that neither of these forms is valid unless the other is (since all lines on the right are either identical or logically equivalent to lines on the left), but the new form is actually a disjunctive syllogism.
Of course, it doesn’t quite look like a disjunctive syllogism—not exactly. But this is because it is disguised, so to speak, and we can lift the disguise if we write it out with a few modifications. In the forthcoming comparison, therefore, our initial version of the new form has been placed on the left, and a modified version is on the right:
| Initial version | Modified version |
| Either not A or B. | Either [not A] or B. |
| A. | Not [not A]. |
| Therefore, B. | Therefore, B. |
Notice what we’ve done. In the modified version on the right, in the second line, we’ve replaced “A” with “Not [not A].” This is the principle of double negation, and we shall need to come back to it in a moment. We have also put brackets around all the instances of “not A” so as to highlight the hidden structure. If we then compare this modified version on the right with a standard disjunctive syllogism in one of its traditional incarnations, we can see the sameness of form. For clarity, then, let’s put the modified version on the left and a traditional version of the disjunctive syllogism on the right:
| Modified version | Traditional version |
| Either [not A] or B. | Either A or B. |
| Not [not A]. | Not A. |
| Therefore, B. | Therefore, B. |
This is the same thing twice. The only difference between these last two schemas is that the modified version has “not A” wherever the traditional version has “A.” But we can still see the same basic pattern; the logical form of the argument doesn’t depend on which simple propositions it contains but only on whether these propositions appear in the same configuration. And here the configurations are the same. This is part of Chrysippus’s insight.
But what about our use of double negation? We must still ask whether using double negation here is logical because the manipulations just carried out depend on it. It is logical only if “Not [not A]” really is logically equivalent to “A.” (That is, the manipulations are logical only if “Today is Tuesday” really is equivalent to “It is not the case that today is not Tuesday.” But how do we know this?)18
THE LAWS OF CONTRADICTION AND EXCLUDED MIDDLE
To establish this further point, we shall need two more assumptions crucial to logical analysis. The first new assumption will guarantee a viable sense of negation, and the second will make it the sort of negation we can double. The first assumption is that none of these propositions of argument can be true and false at the same time. This is the law of contradiction. The second assumption is that each of these propositions must be at least true or false. This is the principle traditionally called the law of excluded middle.19 The special role of these laws in all logic, no matter how complicated, is something we shall come back to in the next chapter, but if you do indeed grant the two assumptions—the two laws—then the principle of double negation follows.
Here’s how the principle can be derived (it’s a bit tricky):
The law of contradiction tells us that “A” can’t be both true and false at the same time. Neither can “not A.” But if by “not A” we mean that “A” is indeed false, then the truth of “not A” always implies that “A” can’t be true, since “A” can’t be true and false simultaneously. Therefore, by the law of contradiction, “A” and “not A” can’t both be true at the same time; it is one or the other but not both. Neither, however, can they both be false. Why not? Because if “A” is indeed false, then this will already fulfill the condition we just laid down, by definition, for “not A” to be true.
We can go further, all the way to the principle of double negation, so long as we can also invoke the law of excluded middle.
The law of excluded middle tells us that each of these propositions, “A” and “not A,” must be at least true or false. Thus, given that they can’t both be true and can’t both be false, each must be false whenever the other is true. So “A” and “not A” are logical contradictories; exactly one is true, and exactly one is false.
To reach the principle of double negation, we need one more step. We need to run through once more the same reasoning we just sketched, but this time we must apply it to a different pair of propositions: not to the pair “A” and “not A,” but to the pair “not A” and “not not A.” That is, if we substitute the expression “not A” for “A” in the reasoning we just sketched and also substitute “not not A” for “not A,” then what we discover is that the same relation must hold between “not not A” and “not A”—they will be logical contradictories. Exactly one will be true, and exactly one will be false; and this will imply that “not not A” is indeed logically equivalent to “A” because each of these last two propositions is a logical contradictory of “not A.” Thus the last two propositions, “A” and its double negation, will each be true whenever the other is.
These manipulations are delicate, to be sure, because they are all manipulations of form, and none of them would have been possible without Aristotle’s insight that logical force turns on logical form. What we really see here, though, is something more basic. We see just how tightly the primary structures of logic can be tied together. So long as each proposition of the analysis is true or false but not both at the same time, all the main structures studied by Chrysippus can be interpreted so that each is valid only if the others are. They hang together and must hang together. We need negation, to be sure, but once we have it, we can express the logical equivalent of conjunction in terms of negation and disjunction, and we can express the logical equivalent of disjunction in terms of negation and conjunction. We can also express the logical equivalents of both conjunction and disjunction in terms of negation and the hypothetical.20
Just to drive this point home, here is another of these curious equivalences:
Not both A and B
is logically equivalent to
[Not A] or [Not B].
(For example, “Today is not both Tuesday and Thursday” is equivalent to “Today is not Tuesday, or today is not Thursday.”)
This is one of De Morgan’s laws (after the nineteenth-century English logician Augustus De Morgan), and it follows from our earlier assumptions: If A and B can’t both be true, then at least one of them must be false. This implies that either “not A” is true or “not B” is true. Once we make this move, however, we can then convert the conjunctive syllogism into a disjunctive one, like this:
| Conjunctive syllogism | Disjunctive syllogism |
| Not both A and B. | [Not A] or [not B]. |
| A. | Not [not A]. |
| Therefore, not B. | Therefore, [not B]. |
Again, it is the patterns that matter, and the pattern on the right is the same as that of a standard disjunctive syllogism. Thus the conjunctive syllogism and the disjunctive syllogism are logically equivalent.
We can even throw modus tollens into the mix. In the following comparison, you will see that each premise on the left is equivalent to the one on the right:
| Conjunctive syllogism | Modus tollens |
| Not both A and B. | If A, then [not B]. |
| B. | Not [not B]. |
| Therefore, not A. | Therefore, not A. |
(Remember that the first line of the conjunctive syllogism can be converted into a disjunction via De Morgan’s law, and that this disjunction can then be construed as equivalent to a hypothetical, just as it was for Hal and Jim Thornton. Thus “Not both A and B” becomes “Not A or not B,” which then becomes “If A, then not B.”) Of course, the new modus tollens on the right looks complicated, but it corresponds to the traditional one in terms of form. Compare the new modus tollens to the traditional one, and you will see the same hidden structure:
| Traditional modus tollens | New modus tollens |
| If A, then B. | If A, then [not B]. |
| Not B. | Not [not B]. |
| Therefore, not A. | Therefore, not A. |
This is just part of the peculiar world of interlocking compounds Chrysippus entered into, and it suited his Stoic philosophy perfectly. To his mind, what he was exploring was the universe’s rational order, and to further the investigation, he and his followers picked out a few logical forms to stand as axioms and then systematically inferred others.21
MORE INTERLOCKING FORMS
There are still two more among the six basic forms we started with earlier in this chapter: the dilemma and the hypothetical syllogism.
| Simple dilemma | Hypothetical syllogism |
| If A, then B. | If A, then B. |
| If C, then B. | If B, then C. |
| But A or C. | Therefore, if A, then C. |
| Therefore, B. |
We can pull these last two forms into our interlocking system too, if we first point out that the dilemma is actually two instances of modus ponens glued together. If we take the simple dilemma in its standard form,
If A, then B.
If C, then B.
But A or C.
Therefore, B.,
then it is possible to rewrite it as a contemplation of two different situations, each subjected to modus ponens:
| If A, then B. | If C, then B. | |
| A. | or | C. |
| Therefore, B. | Therefore, B. |
The first two lines of the dilemma have been relegated to left- and right-hand columns, and the alternatives of the original dilemma’s third line come next, separated by the word “or.” Then we draw an inference about the situation on the left and the situation on the right. But since either situation might prevail—we don’t know which—the strategy of the dilemma is to use the connective “or” to glue the situations together. Put another way, the situation on the left is what you get if you land on one side of the fence, and the situation on the right is what you get if you land on the other. (The complex dilemma is then a small variation on this theme.)22
As for the hypothetical syllogism, we can pull it into the system by pointing out that it is equivalent to a dilemma, though a rather unusual one, this time built from a combination of modus ponens and modus tollens. The premises of the hypothetical syllogism are in this form:
If A, then B.
If B, then C.
But notice what happens if B is true. If B is true, then we can infer C by modus ponens:
Modus ponens
If B, then C.
B.
Therefore, C.
On the other hand, suppose B isn’t true; in that case, we can infer “not A” by modus tollens:
Modus tollens
If A, then B.
Not B.
Therefore, not A.
But the original premises don’t tell us which of these two situations obtains; we don’t know whether B is true or untrue. All the same, since we know that at least one of them must obtain (because, by the law of excluded middle and the definition of negation, either B or not B must be true),23 it follows that we can combine all the information that we do know by treating this analysis as a contemplation of two different situations. And this is just the way we built our first dilemma:
| Modus ponens | Modus tollens | |
| If B, then C. | If A, then B. | |
| B. | or | Not B. |
| Therefore, C. | Therefore, not A. |
If we now combine this information into a single argument, we get a peculiar dilemma:
If A, then B.
If B, then C.
B or not B.
Therefore, not A or C.
This is actually a hypothetical syllogism—which just happens to be wearing, once more, a sort of disguise—and we can lift the disguise in two steps: first, by replacing the conclusion with the logically equivalent formulation, “Therefore, if A, then C.” (Remember that “Not A or C” can be construed as equivalent to “If A, then C.”)
If A, then B.
If B, then C.
B or not B.
Therefore, if A, then C.
Then, to arrive at standard form, we need merely omit the third line of the argument on the grounds that it goes without saying; after all, the proposition “B or not B” is a logical truth following directly from the law of excluded middle and the meaning of “not.” It thus remains true even if unuttered. In consequence, we have this:
Hypothetical syllogism
If A, then B.
If B, then C.
Therefore, if A, then C.
Until the design of digital computers, any interest in a system like this was largely theoretical; nobody needed all the manipulations of Chrysippus to handle the ordinary situations of life. But the discovery of the system still showed the existence of these unexpected symmetries, rather like those of the old square of opposition. And, once again, the discovery raised the deeper question of why it all works.
We might say it works because of the way we have defined our key logical operations: disjunction, conjunction, the hypothetical, and negation. As long as we define these things with an eye to the law of contradiction and the law of excluded middle so that everything must be true or false but not both simultaneously, all the relations hold good; it all fits. But our sense of logical necessity, of course, still lies behind all of it; we still need a sense of what follows from what before we can put such a system together.
Specifically, whatever definitions we choose to lay down, we still need to draw valid conclusions from the definitions, and this means we must still assume in advance that some inferences are valid. We can invent logical expressions and give definitions to them, but we can only discover their resulting interrelations. As a result, we can explore and describe the interlocking world of Chrysippus, but we can never fully justify or explain it without taking some of it for granted.24 Such, then, was the abstract world that the ancient Chrysippus discovered, but how does all this play out in a modern computer?
THE BASIS OF COMPUTER LOGIC
It all depends on the compounds we looked at a moment ago: the disjunction, the conjunction, and the hypothetical. The first thing to notice about these compounds is how different they are from the logic developed by Aristotle. Aristotle’s logic is largely devoid of compounds. Aristotle was surely aware of compounds, but he didn’t pick them out for extensive study. In effect, then, what we really have here (in the hands of Aristotle and Chrysippus) are two different kinds of logic. The two kinds are by no means incompatible; they complement each other well, but they concern different things. We see this if we look once more at modus ponens. In the case of modus ponens,
If A, then B.
A.
Therefore, B.,
it doesn’t matter which propositions we substitute for the variables A and B as long as we substitute them consistently. The argument is still valid as a matter of form. But the thing to notice is that the variables do indeed stand for whole propositions.
With Aristotelian syllogisms, on the other hand, the situation is different. To be sure, a syllogism’s validity is still the same, whatever you substitute for its variables:
All As are Bs.
All Bs are Cs.
Therefore, all As are Cs.
But this time, the variables don’t stand for whole propositions. Instead, they stand for classes. The variables are terms within propositions (a term being the name for a class). Thus, whereas modus ponens is valid merely in virtue of how the propositions fit together (hence, the expression “propositional logic”), Aristotle’s logic relies on a more complicated “logic of terms.” (The validity of a syllogism turns on things inside each proposition, on the terms themselves. By analogy, we might say that Chrysippus’s logic combines atoms into whole molecules and then traces out the implications, but Aristotle’s logic goes about dissecting each atom in turn and then relates the parts of these atoms to the parts of other atoms.) It is only Chrysippus’s logic, propositional logic, that translates easily into computer circuitry.
The reason for this last point is a crucial feature of the compounds we have been examining. The feature is this: with the compounds, the truth or falsity of the parts determines the truth or falsity of the whole. Take, for example, the disjunction. A disjunction is true or false depending entirely on the truth or falsity of its elements. Specifically, a disjunction is true if at least one of its disjuncts is true; otherwise, it is false. (In the language of logic, the disjunction is thus “truth-functional.”) But notice also that, in saying this, we are dealing with only two values, true and false, and the simplicity of this choice will make the transition to electronic circuitry comparatively easy. If the only possibilities for a proposition are true or false, then we can represent this choice quite mechanically.
Now before we consider how this is done, we need to dispel a common confusion. The confusion is that, in confining ourselves to only two values—true and false—we are somehow asserting that everything in life must be either true or false. On the contrary, we assert no such thing, and we don’t deny for a moment that much of life has nothing to do with being true or false. Still less do we offer any theory of what truth and falsity, in themselves, really are. The analysis of truth and falsity is an interesting problem, one we shall come back to in the next chapter, but the thing to see at present is that our minds do indeed manipulate relationships of this type, relationships where parts determine wholes. And for the moment, it is the manipulations that matter. So long as the manipulations are possible—and plainly they are—it will be a straightforward matter to build a machine that mimics this behavior.
In particular, the two values of true and false can be represented electronically as a circuit switched on or off.25 Such circuits can then be combined in complex arrays so that, if various elements within the array are switched on, their compounds are switched on, and the compounds can then represent the logical relations of disjunction, conjunction, and the implications represented by hypotheticals. This is the basis of modern digital computing. As the electronic elements switch on and off, the machine’s behavior mimics an exceedingly complex inference in which sets of conclusions lead to other conclusions.
With Aristotle’s logic, this effect would be impossible. Why? Because the parts of a categorical syllogism represented by the variables A, B, and C aren’t “true” or “false” in the first place. Instead, A, B, and C are merely names of classes (classes like “dogs,” “mortals,” “Athenians,” or “Thebans”); they are not statements. But since the variables merely stand for classes, we no longer have the same intimate connection between the truth of the variables and the truth of the whole.
What we see as a result is a close similarity between some of our methods of reasoning and the behavior of digital computers. And though this fact is sometimes masked by a computer scientist’s tendency to speak in terms of a binary code, a series of ones and zeros, the scientist’s code is simply another way of expressing the same logical truths. The code happens to come from the nineteenth-century English logician and mathematician George Boole, whose aim was to codify the relations studied much earlier by Chrysippus (albeit with greater abstraction and sophistication). Later generations built on Boole’s insights, and we shall consider in chapter 9 how these efforts eventually gave rise to programmable computers. But the logic that made it all possible was the interconnected logic of an interconnected universe, discovered by the ancient Chrysippus, who labored long ago under an old Athenian stoa.
We don’t know how Chrysippus met his end. One story is that he died from drinking wine; another says he died from a fit of violent laughter. Such odd tales were common in the ancient world. (The first of the Stoics, Zeno of Citium, was supposed to have ended his life at the age of ninety-eight by holding his breath. All these stories come down to us from the ancient biographer Diogenes Laertius.)26 All we know of Chrysippus for sure is that his teachings and logic were much admired.
Living in a time of vast, autocratic regimes, Chrysippus dwelled, like all Stoics, on the rational choices of individuals, and he believed human reason at its best was a form of participation in the eternal. Assessing truth and falsity and cleaving to the laws of contradiction and excluded middle, he sought to make the life of human beings as orderly as the cosmos itself. For Chrysippus, the eternal law of nature was the secret of the starry heavens above, and it was the moral law within. According to an old Athenian saying, “If the gods have any use for logic, it is the logic of Chrysippus.”27 However that may be, the logic of Chrysippus is certainly the logic of our machines.