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Logical Counterparts (Complementary Concepts)

In the course of a treatment of the so-called problem of evil John Mackie introduces some important arguments that apply beyond the restricted subject in which they are introduced. I shall have more than one occasion to refer to Mackie’s distinctions, so I want to isolate the core of his argument here unembellished with any theodicious trappings.

Background: The setting is the familiar problem of evil. If we assume that god is all good, all powerful and all knowing (and assuming we understand what these dubious terms might mean), then how could the universe, which is presumably god’s creation, contain evil? Mackie comes down on the side that says that it couldn’t for, if it did, the qualities of being all good and being all powerful would contradict each other. But, since the universe does contain evil, there is no such thing as an all good, all powerful and all knowing entity, and so there is no such thing as god. One response to this position is that good presupposes evil (more precisely lack of good; the fact that the concept of moral neutrality constitutes a sufficient counterpart to the concept of good undoes this response from the git go). The idea of a good thing or action has no meaning unless we also understand what an evil thing or action might be. Thus it is logically necessary that there be evil in the universe. One limitation on god’s power (a limitation accepted obviously by those who subscribe to this response, but not by all goddists) is that god cannot do what is logically impossible. Since good without evil is logically impossible, god cannot create or maintain a universe without evil in it.

Let me add a cautionary note before getting started. The traditional discussion of good and evil is confused because unfortunate physical events and a certain type of disapproved behavior are lumped together as evil.

Mackie’s Response: Mackie (in Martin & Monnier pp, 64-66 and 74-75) distinguishes between three different assertions. The first is that if everything in the universe were in fact good, then “there would be no need” in any language for the predicate “good.” This might be made a little bit stronger to the effect that in practice we probably would not be able to understand what the predicate “good” meant. The second is that, if the universe were only made up of good things and actions, “no one would notice it.” This is “less persuasive,” to use Mackie’s phrase, since the universe does not contain unicorns that are simultaneously horned and not horned, and yet we can “notice” (not observe, since you can’t observe a simultaneously horned and hornless unicorn) that fact by simply trying to qualify a horned unicorn as “not horned.” The third assertion is that, if the universe did not contain evil, then it would not contain good either. This last assertion is obviously absurd. It is like saying that Newton’s First Law of Motion is invalid because there is nothing in the physical universe that does not obey it. These three assertions might be loosely qualified as the logical, epistemological and ontological consequences of our having a concept such as the concept of “good.”

What is Logical? More properly stated, what is the meaning of “logical” and to what sort of things does it apply. The predicate “logical” applies to assertions and the inferences we draw using assertions. It is a qualification of language and not of physical objects or acts (with the exception of certain sorts of speech acts). While objects and acts can be possible, only assertions and inferences can be logically permissible. To say that an event is logically possible is to say that the assertion that that event may occur is logically permissible. Those who call acts such as creating or maintaining a universe “logically impossible,” speak incorrectly. The correct way of stating this is that it is a logical contradiction to assert that god can create a universe containing good but not containing evil. Sloppiness in matters like this leads to a great deal of speculative foolishness.

A different point is that many philosophers, most often English philosophers, use “logical” as a qualification of many different sorts of linguistic issues. They do not use the term in the strict sense of having to do only with mathematical logic and formal deductive systems. Thus the meaning of an individual non-syncategorematically defined word is in their usage a logical issue. There is nothing wrong with this usage especially since the meaning of a term often contributes to inferences that can be drawn from assertions of which that word is a part. But we should be aware that this sort of logical argument does not gain one bit of the purported rigor of formal systems. A logical argument in this broader sense is supported entirely and exclusively by the evidence that is presented in its behalf.

Mackie’s Errors: Mackie’s basic points are perfectly valid. But there are a couple of occasions where he commits minor errors. It would be a good idea to clear these up since goddists are prone to seize on red herrings (a messy business) as a supposed refutation of a perfectly valid argument.

On pp. 76-77 Mackie concedes that it is “formally possible” for god to create a universe with “first order evils” such as physical pain for the sake of more valuable “second order goods” such as triumphing over pain. Formally possible? Not for an all good god. An all good god could easily create a universe without physical pain but where we could triumph over one good for a greater good. Little Johnny could catch three fish in the river on Tuesday - which is good – and then catch four fish on Wednesday and so triumph over Tuesday’s good. (If anyone who has already abandoned the goddist mindset feels we are descending into the purest jabberwocky, I happen to agree. We just have to provisionally accept some goddist assumptions about good, gooder and goodest to show that the goodest god can create gooder things without the help of bad things.)

Note: Barnes (pp. 455-456) mentions the problem of evil in one of its earliest formulations by the Presocratic philosopher Diagoras. Barnes divides the philosophical formulation into two types. The first, presumably Diagoras' version of the problem, is disposed of by the example of a "benevolent ruler" who must opt for some evils in favor of a higher good. The reason is that constant intervention by the ruler in matters of justice will do more harm to the state than the occasional occurrence of injustice. The second type includes natural evils (as well as human free choice although the admission of freedom of choice should also be accounted for in the benevolent ruler example). The second type does not admit of the concept of a higher good and does not succumb to the example of the benevolent ruler. Clearly the response to the first type only holds for a god who is not omnipotent. The omnipotent Xtian god succumbs to the problem of evil, because there is no logical inconsistency in describing a universe where humans have free choice but still always do the right thing. The state description of that universe would include propositions such as (1) Agent a is free to do either φ or not-φ at t, where φ denotes a good action and not-φ a corresponding evil action, and (2) a does φ at t_i  where t_i  follows t. Presumably the Xtian or Mohammedan god could create such a universe and must do so if it is also all good.

On pp. 64-65 Mackie clouds the important concept of logical counterparts (a better term would be “conceptual complements”) with an unnecessary and fallacious concession that there are such things as absolute concepts, i.e. concepts that do not have a complement. This concession, which he does not need, is meant to support his argument that good can exist without evil despite the fact that the concept of evil (or rather lack of good) is a conceptual complement to the concept of good. However, it is important to understand that every concept has a complement (just as every set (for the purposes of this argument) has a complementary set). Mackie states that the absolute concept “great,” for example, could just mean “having at least a certain size.” But this concept does indeed have a complement. If “great” means having a measurement larger than n for some real number n, then “small” means having a measurement less than or equal to n. It is important to understand the pervasiveness of conceptual complements or logical counterparts because Mackie’s argument with respect to good and evil can be turned on its head in other contexts. For example, one can argue with perfect justification that “undesigned” is a conceptual complement to “designed,” but, using Mackie’s argument, assert that everything in the universe could in fact be designed (by god. Get it?) However, design arguments rely on pointing to something that is designed or appears to be designed and inferring that some other thing (or rather every other thing) is in fact designed even though it does not appear to be designed. But then we lose the basis for recognizing design, since there is nothing special about appearing to be designed that would induce us to believe that it is designed. The epistemological problem is not overcome by simply adding a negation sign in front of the term “designed” just as we could add a negation sign in front of the term “horned unicorn” since in this case the proof depends our being able to recognize, to actually see undesigned things. Logically it is possible that everything in the universe is designed, but the proof that everything in the universe is in fact designed – which depends on observation and not mere logical possibility – fails because it depends on our being able to perceive things as distinct from each other (Some are designed and some are not designed) and not distinct (All are designed) at the same time. The idea that every concept has a complement does not help in this case since what we are deprived of is not the complementary concept but rather criteria for the recognition of instances of the complementary concept. A similar fallacy is Berkeley’s assertion that everything is “mental” even though some things appear to be non-mental. Although the assertion that everything is in the mind is logically permissible (assuming that assertion has meaning), nevertheless Berkeley deprives himself of the basis for proving that everything is mental (if indeed that is his intention) because he eliminates a criterion for distinguishing between the mental and the non-mental. You haven’t really proved that everything in the universe is green just by saying that things that don’t look green really are green. All you are doing is redefining “green” as “all.”

Conceptual Complementarity – Generalizations: As I mentioned above, conceptual complementarity and the fallacies that could be committed from a misuse of complementary concepts in an argument has a broader scope than the limited and goofball debate of how an all good entity could create, tolerate or maintain a universe that contains evil. So it is worthwhile to make a few general points here as a point of reference for future use.

The structure of conceptual complements is obviously related to the structure of classes (or sets; the differences between classes and sets (Cf. Church pp. 28-29) don’t bear on the present issues) and their complements. Clearly concepts quite often specify the predicate that defines class membership. The fit is not perfect since the members of classes can be specified by enumeration while concepts are often vague and their meaning is a matter of progressive discovery – inductive, as Mill would say. However, assuming a more or less adequate fit between a concept and its corresponding set, the structure of class inclusion parallels that of conceptual definition.

There are different flavors of complementarity to a given concept. Inclusion in a complementary class is usually specified only as non-membership in the class the complementary class complements. However, non-membership can be specified in different ways. Let’s call a concept C. The complement of C can be divided into the following groups: (1) the contrapositive concept to C, (2) concepts falling under a higher more inclusive concept than C, and (3) concepts categorically unrelated to C. Contrapositives are concepts that are utterly opposed to some other concept; they are inimical to some other concept. Anything that falls under a contrapositive concept is somehow directly contrasted to another concept. Although I use, and perhaps abuse, a logical term here, the concept of a contrapositive is not a concept that figures in the theory of formal deductive logic (Obviously conceptual contrapositivity as I am using it in this essay is not the contrapositivity of the traditional non-formal Square of Oppositions). The reason seems to be that any given contrapositive is not formally derived from another concept as a complement is. Not all concepts have contrapositives. The concept of having a certain dimension, for example, is not directly opposed to anything. Neither is the concept of living in Seattle. The case of color concepts is not clear. (The use of colors as supposedly clear examples from Aristotle through Locke and the sense-data theorists is unfortunate not only because of the secondary quality nature of colors named by common color names but also because colors may indeed lack true contrapositives. Whether magenta is a contrapositive to green, for example, is partly an empirical issue and partly a matter of definition.) The theodicious example of good has a clear contrapositive, viz. evil. Another example might be the conceptual pair saving the princess/abducting the princess. In the end, however, there remains something vague about contrapositives. These concepts characterize things that are not merely different from the things characterized by the original concept, but are somehow distinctively different and opposed in a way that has to do with the meaning of the concept and its contrapositive. (We make such a distinction, for example, when we say, “Pope Pacelli was not just a helpless bystander. He was positively evil.”) The idea of extensionally specifying a contrapositive class is no help because you can put anything in a class without specifying a concept for that class. Defining a class by its contrapositive concept begs the question, since we want to know what it is about the contrapositive concept that makes it a contrapositive (However, once defined, we could turn a contrapositive into a complement by specifying a class that contains only a concept and its contrapositive, such as a class that contains only good and evil things, a sort of Either You're With Us Or You're Against Us class). So at this stage I don’t think we will find much help in set theory. Even the examples are not entirely clear. The concept of the NFC could be regarded as a contrapositive to the concept of the AFC as long as we limit the universe to NFL conferences. But no one conference in college football is conceptually contrapositive to another. Similarly the Bloods and the Crips are contrapositives. They might be complements in a class of contenders for the LA drug trade. In a wider class that includes the Latin Kings the Crips are part of the complement to the Bloods. Contrapositives seem to crop up with alarming frequency in areas where we make value judgments. There is not only good/evil, but also beautiful/ugly, just/prejudiced etc. No one term of any such pairing logically requires the other term, since we can get along with the not-good, the not-beautiful, the not-just and so on.

Other concepts are complementary to a given concept not because they are contrapositive to that concept but because, together with the original concept, they fall under a higher concept, much like species fall under a genus. The concept of morally neutral actions is partially complementary to the concept of good actions because both of them fall under a higher concept that includes both. Eating dinner, for example, is, absent qualifying circumstances, a morally neutral action. So is the act of an innocent bystander who walks past the tower where the princess is imprisoned. Depending on how broadly we want to take the group of concepts falling under a higher concept more inclusive than C, categorically unrelated concepts may just be a special case of the former. A concept is categorically unrelated to C because it is not meaningful to characterize the unrelated concept in any way that is relevant to C. The concept of the beating of my heart is an example because it does not involve volition. The concept of a rock is another example because it is not the sort of thing to which we can attach moral predicates (For one thing, it is not an event, like an avalanche or abducting the princess and it is not a state of affairs like famine.) One important difference between contrapositive concepts and the other two flavors of complementary concepts (We might call them strict complements) is that, given a concept C, an item might be neither a C nor an instance of C’s contrapositive, but for any item, that item must be either a C or an instance of C’s complete complement. In (2) the item must be an instance of the higher concept that C and the complement of C fall under. In (3) it must not be an instance of some (not any) higher concept that C falls under. The class of all items that are instances of (1) or of every complementary concept of flavor (2) or every complementary concept of flavor (3) but not instances of C might constitute the complete complement of the class C (Plato provides a strong argument that at least one contrapositive, non-being, does not belong to the class complementary to being, i.e. things that don’t exist, since non-being may very well exist). If there is a name for this class, it is the name of the complete complementary concept of C. It is often best expressed by simple negation.

For those who may be interested, Edmund Burke offers up a charming example in his discussion of the contrapositives, pain and pleasure. He points out that pain is not simply the absence of pleasure nor pleasure the absence of pain. For the purposes of his discussion he chooses to call the absence of pain "delight" (although his use of this term is not consistent). In distinguishing positive pain and positive pleasure Burke makes largely the same point as I am trying to make here.

Aristotle’s discussion of ἐναντίοι in Anal. Pr. I xliv may exemplify the error of confusing contrapositives with complements. The passage in question is confusing because it is unclear as to how he understands being healthy. If health is the absence of disease, i.e. if it is the complement of disease, then Aristotle’s illustration is valid at least as far as the impossibility of being healthy and diseased at the same time is concerned. If, on the other hand, health is some distinct state not defined in this passage but not as broad as the complement of disease, i.e. if health is a contrapositive of disease, then his illustration is not valid. A less ambiguous example of this fallacy can be found at De Gen et Corr. II 6 333b 28-30 where Aristotle argues that if compelled motion occurs in the world then natural motion must also occur.

In Categories XI 14a 7 ff.Aristotle makes a point about particulars characterized by opposite qualities to the effect that the existence of one such particular does not imply the existence of its opposite. His examples are actually complements but his point is if anything more valid for contrapositives like good and evil. More on Aristotle below.

The old saw that atheism is just another religion is an example of confusing contrapositives and complements. The phrase is, of course, little more than a goddist slogan and bears not a jot of resemblance to a credible argument. However, it is a good example of a logical fallacy. It consists in confusing membership in a complementary class with membership in some perhaps undefined contrapositive class. To say that atheism is the contrapositive religion to some sort of goddist sect is equivalent to saying that, if someone is not an Episcopalian, he must be a Methodist, or, if he is not a Democrat he must be a Republican, if someone rejects string theory, he must be advocating some alternative, if the liquid isn’t blue it must be green. Part of the complete complement is to be colorless, unconvinced, politically neutral or disbelieving. Complementarity with respect to beliefs involves a modality where the negation can move from inside to outside the scope of the epistemic operator without entailing further change in the embedded content. Extensionally speaking, the propositional attitudes expressed when the negation is inside the epistemic operator constitute a proper subset of the propositional attitudes expressed when the negation is outside the epistemic operator. Belief that ¬p and not believing that p are indeed different. But neither entails belief that q for some q unless q is defined as ¬p. Electrical synapse communication theory is a good example. Not believing that nerves communicate across synapses electrically is not exactly the same as believing that nerves do not communicate across synapses electrically. Someone who never thought about the matter can be characterized by the first but not by the second. The class defined by the second is a proper subset of the class defined by the first. But neither propositional attitude entails belief in some alternative theory. Someone may simply think there is not enough evidence to justify the electrical theory without ever having entertained the pharmacological theory. Likewise, the set of those who believe that there is no god is a proper subset of the set of those who do not believe that there is a god. Included in the full complement are mountains to which it is inappropriate to ascribe beliefs, animals to whom it may be inappropriate to ascribe a belief of this sort, humans who never thought about the matter and agnostics who thought about it but didn’t reach a conclusion. Nobody who actually professes a religion belongs to this complement, since professing a theistic religion would seem to require believing in the existence of a god. So the set of atheists belongs to the full complement of those who don’t believe and to its proper subset consisting of those who believe that there is no god. It does not belong to the set of believers in a god, a set that could include beliefs in all sorts of batty gods.

In fact it is not at all clear that all concepts require a contrapositive. Mackie has shown that the existence or even the concept of good does not require the existence of evil. It is equally true that an understanding of the concept of good does not require an understanding of a concept of evil. It simply requires an understanding of the concept of not good. For why should the concept of good require or logically necessitate a concept of evil and not just a concept of not-good, the true complement of the concept of good, any more than (setting aside the physics of the color spectrum – an empirical qualification) the concept of blue logically necessitates a concept of green over and above a simple concept of not-blue? Obviously there is no reason why. So, in the final insult to the god-can’t-do-the-logically-impossible argument, it turns out that it is not even logically necessary that good and evil come in a pair. (By way of aside, Veitch (not Descartes) made the same mistake when he translated a passage in the Méditations to read that there can’t be mountains without valleys.) Note this is not the case with the designed/undesigned pairing since the concept undesigned is a true conceptual complement to the concept designed. That is why creationists are obliged to show us an example of something undesigned if they wish to define any concept of being designed.

Another important distinction to keep in mind is that between (1) the logical distinction between complementary concepts, (2) the issue of whether anything actually exists that can be characterized by a concept or its complement, and (3) the circumstances under which we can meaningfully recognize that a thing is characterized by a concept or its complement. I referred to these above as logical, ontological and epistemological issues. We can pretty much form any concept we want. Whether our concept is meaningful, whether anything actually falls under our concept and whether we can communicate to anyone else what our concept means is another story. The fact that we form a concept (such as “bodacious”) does not mean that any existing thing (There is an important problem with the predicability of existence which can be safely set aside in the present discussion) actually falls under the concept or its complement. And - This is Mackie’s point – the fact that some existing things may fall under a given concept (such as good things) does not mean that there necessarily have to be existing things (such as evil things) that fall under its complement. Or that it isn’t entirely possible that someone somewhere could give it another try and reproduce the universe of existing things such that no existing things fall under its complement (The inhabitants of the latter universe would probably have no understanding of the contrapositive conceptual pair good/evil). On the other hand, unless examples, existing or non-existing, can be produced of objects that fall a concept’s complement, it is impossible to make the original concept comprehensible or meaningful. This is the fallacy of those who argue that everything falls under a given concept. They are simply redefining their original concept as identical to the concept “all” (“All” has its own significant definitional problems that need not be addressed here). It is the fallacy committed by design theorists when they say that everything is designed. They take away the distinctive quality “looking strangely designed” that characterized some initial example (such as a watch in a field or propeller driven spermatozoa).

Let’s take a closer look at the distinction between arguments about logic or, more properly speaking, language and arguments about what actually exists. Two of the most famous arguments from the history of philosophy erroneously draw an ontological conclusion from a verbal or semantic distinction. Socrates’ argument for the immortality of the soul (Phaedo, pp. 104B ff.) is based on a “logical,” i.e. lexical distinction between a body and what gives that body life. Nothing in his argument proves that what gives the body life actually exists or, if the soul does exist, that it is the soul that gives the body life. Both assumptions are conceded by Cebes and his concession gives away most of the game. There are other errors in Socrates’ argument. He fails to prove that whatever engenders (ἐγγένεται) a feature such as heat will not admit (δέχεσθαι) its opposite such as cold. He takes that as a given and Cherubino, for one, provides at least a plausible introspective counterexample in favor of a roughly simultaneous presence of heat and cold. Socrates also mistakenly assumes that a cause of a feature must itself be characterized by that feature (In The Sophist Plato would come to see the error of this assumption). In concluding that, because the soul causes life, the soul must be immortal, he assumes that the soul must have life since immortality is obviously a feature of life. But that assumption is unfounded. A very Platonic sentiment would have it that virtue engenders virtuous behavior, but that does not mean that virtue itself (an abstract entity) is either virtuous or non-virtuous. More concretely, Abel Magwitch may have caused Pip to be a gentleman by way of inheritance although he himself was not a gentleman. Nevertheless, my intent for present purposes is to highlight Socrates’ error in proceeding from a definitional distinction between what lives and what engenders life to an unfounded conclusion that the distinct definitions refer to actually existing entities (ὄντα).

Descartes' two substance theory is based on the same error. His contention that matter can’t doubt is based on a definition, not on an observation. It’s based on how he wants to define certain words and his decision to use two words, “mind” and “body,” with different meanings that may in fact designate just one thing. That decision by itself does not have ontological import. What's doing the doubting could be doubting matter, or, to put it less prejudicially, both terms “mind” and “matter” could refer to the same object. They simply highlight different features of that object in accordance with Descartes’ stipulative definitions. A linguistic or semantic distinction between mind and matter is perfectly compatible with ontological monism (Think of the notion of emergent properties popular with physicists today). Indeed whatever is proved by the Cogito remains proved if we conclude that what was doing the doubting was matter all along and there is no need to propose a new entity, namely the mind, to perform the act of doubting. We could regard Spinoza's dual aspect monism as a correction of the lapse in Descartes’ reasoning. Indeed Spinoza’s version of the ontological proof concludes no more than that something exists.

It seems the logic/ontology confusion extends far beyond the bounds of relatively trivial disputes from popular theology. The Plato and Descartes examples show it reaches into the very basics of what we have come to consider philosophy. One way of defining fuckosophy would be as a type of argument that does not succumb to this error.

Aristotelian Excursus:  The distinction between complements and contrapositives is related to Plato's distinction between ἕτερος and ἐναντίος (The Sophist 257B ff.). Plato's purpose was to argue that τὸ μὴ ὄν could be (i.e. exist) since it does not belong to the class complementary (ἕτερος) to τὸ ὄν (while things that truly do not exist do belong to the complementary class) even though it is "opposite" (ἐναντίος) to τὸ ὄν. His distinction is meant to defuse the sophists' use of Parmenidean arguments regarding being and non-being as a way of justifying a kind of complete relativism. This is not the context to explore the problems with Plato's solution. But it is worth pointing out that what exactly characterizes a contrapositive concept is not entirely clear just as Plato's understanding of what it means to be ἐναντίος is not clearly specified.

Aristotle’s similar distinction between τὸ μὴ εἶναι τοδὶ and ἐῖναι μὴ τοῦτο (Anal. Pr, I, xlvi  51b ff.), while correct in spirit, is somewhat deficient in detail (The deficiency lies in the fact that there is nothing in the grammar of the phrases τὸ μὴ εἶναι τοδὶ and εἶναι μὴ τοῦτο to indicate which one means what. We need to look at complete sentences and Aristotle’s arguments and then somewhat arbitrarily assign a meaning to one or the other). The essence of the distinction is that something that is not-F is different from something that is not F, where “F” stands for a quality of some sort.  This sounds like the distinction between simply not being good, for example, and being positively not-good or evil. In other words, it sounds like the distinction between Plato’s ἕτερος and ἐναντίος or my distinction between complementaries and contrapositives. But while τὸ μὴ εἶναι τοδὶ does indeed correspond to my complementary, εἶναι μὴ τοῦτο is a different concept altogether. Εἶναι μὴ τοῦτο is part of a concept's complete complement but it is also a contrapositive of sorts. The clue that this is the case appears in Aristotle’s proof. Aristotle argues that τὸ μὴ εἶναι τοδὶ and εἶναι μὴ τοῦτο behave differently in certain modal, quantificational and oblique contexts. If, Aristotle argues, “not being able to walk” meant exactly the same thing as “being able to not-walk,” then they should both be true or false of the same subject (at the same time). Similarly “being able to walk” should not be concurrently applicable (ὑπάρχειν) to a subject that is able to not-walk, just as it cannot apply to a subject that is not able to walk. Yet presumably a subject can be simultaneously able to walk to the opera and to not-walk to the opera since he can take the bus. The other example is that a person can at the same time understand the not-good and understand the good but no one can simultaneously understand the good and not understand the good (Aristotle’s habit of using controversial concepts like the good as “obvious” examples is glaring here as it is throughout his writings. In what seems to be an insert Aristotle says later that the concept of being equal (ἴσος) behaves in the same way as that of being good). Aristotle sums up his point in the Sybillic utterance,τῶν γὰρ ἀνὰ λόγον ἐάν θάτερα ᾖ ἕτερα, καὶ θάτερα. Roughly this says that if two distinct verbal expressions mean the same thing, then they always behave in the same way, or, stated in more modern terms, synonymous terms can be substituted in modal and oblique contexts without change of truth value (not exactly true, but true enough in the context of the issue concerning Aristotle).

In ll. 29 ff. Aristotle clarifies the issue by observing that genuine complementaries (ἀπόφασεως) apply to the same subject. Anything that is wood must either be white wood or wood that is not white. But something that is not wood at all cannot be wood that is not white even though it can be something that is not white wood. Perhaps the latter could be a genuine complementary to white wood in some unspecified genus. The same holds true of being good. This is similar to the point I made above that it is a logical error to assume that good and evil exhaustively characterize moral qualities. It is not the same point, however, since things that are not white wood do not have the same relation to white wood that evil has to good. The difference lies in the fact that something like evil is designated by a term with its own meaning while things that are not white wood are defined purely in terms of what they are not plus membership in some genus. A leaf is something that is not white wood. A leaf is also not a good man. But when we say that the pope is evil we mean something more specific than that he is just not a good man. Aristotle calls our complementary the ἀπόφασις of a concept or its negation. Further down (VI 52b l. 15) he calls the complementary the ἀντικείμενον or something that cannot apply at the same time to the same subject. But in this later passage he does not give a separate name to τὸ μὴ τοῦτο. Clearly τὸ μὴ τοῦτο is not a contrapositive in our sense in the same way that evil is a contrapositive of good. Taking the bus is not related to walking in that singular sort of antagonistic manner.

Aristotle makes another related but ultimately quite different distinction in Bk II of the Prior Analytics (Par. xv). The difference is not immediately clear, however, for two reasons. The first is that Aristotle is considering a logical (κατὰ λέξιν) issue while Plato was speaking ontologically (i.e. about things, not words). The second is that Aristotle uses his terms somewhat differently. As far as the first source of unclarity is concerned, there is a broad isomorphism between the ontological and the logical, as Aristotle effectively acknowledges when he distinguishes the κατὰ τὴν λέξιν μόνον (even though μόνον here does not mean that the two propositions in question do not really say something about the world; it simply means that they are not true “opposites”).

The second source of unclarity is more tangled. Both Plato and Aristotle use ἀπόφασις to mean “complementary,” at least in the passages in question (The Loeb translators sometimes translate ἀπόφασις as “negation” or “negative”). However, for “complementary” Plato also uses ἕτερος and Aristotle ἀντικείμενος. Plato uses ἐναντίος to mean a concept of something that cannot be characterized by its opposite concept. Thus non-being is ἕτερος but not ἐναντίος since both can be characterized as existing (“Ὁπόταν τὸ μὴ ὄν λέγωμενον…οὐκ ἐναντίον τι λέγομεν τοϋ ὄντος, ἀλλ’ ἕτερον μόνον.” Ibid). Presumably good is ἐναντίος to evil since we cannot characterize evil as good. Using the terminology I introduced above, being and non-being are not necessarily contrapositives. I say “not necessarily” because there is a great deal of vagueness as to what the words “being” and “non-being” actually mean. I would be inclined to say that things that exist and things that don’t exist are complementary classes and that it is somewhat fruitless to talk of non-being at all much less try to figure to which of those classes it belongs. Nevertheless, if we stick with the somewhat loose characterization of a contrapositive as something completely inimical to something else, then there is a sense in which we could view non-being as the contrapositive of being even if the two are not strictly speaking ἐναντίοι. Aristotle uses ἐναντίος in a related way, but while Plato’s ἐναντίος is something that cannot be characterized by a concept that applies to a certain something else, Aristotle’s use of the term is purely logical, that is what is ἐναντίος is a universal negative proposition, for example, in contrast with the appropriate universal positive proposition. One problem in this case is that Aristotle uses another term, ἀντικείμενος, in two different ways: (1) to mean propositions about complementary classes (roughly equivalent to a proposition and its negation); and (2) to mean the grouping that includes both propositions about complementary classes (which he also calls ἀντικείμεναι) and the grouping of universal positive propositions and universal negative propositions (which he calls ἐναντίαι XV ll. 22 ff.) (Tredennick translates (1) as "contradictory" in conformity with current English usage and corrects Aristotle's equivocation, so to speak, by translating (2) as "opposite.")

So in Plato’s The Sophist we have a distinction between ἕτερος and ἐναντίος. In the Prior Analytics we have a distinction between ἐναντίος and ἀντικείμενος, but Aristotle also calls the two together ἀντικείμεναι and doesn’t use the term ἕτερος at all in this context. Is your head spinning yet? Well, there is another problem. Remember Aristotle’s other distinction between τὸ μὴ εῖναι τοδὶ and εῖναι μὴ τοῦτο? Τὸ μὴ εῖναι τοδὶ appears to be a de re version of ἀντικείμενος in the sense of “complementary” or “contradictory.” But εῖναι μὴ τοῦτο is clearly not analogous to universal negative propositions or to Plato’s ἐναντίος. It just means something different but related in the sense that walking and taking the bus are different from each other and yet related. Plato’s terminology is somewhat clear, but Aristotle’s is confused to say the least. The easiest way to see what he means by the ἐναντίος/ ἀντικείμενος distinction (obviously taking ἀντικείμενος in the specific and not generic sense) is to state it in modern standard formal notation. The following assertions are ἐναντίαι: (x)(Fx) and (x) ¬(Fx). The following assertions are ἀντικείμεναι: (x)(Fx) and ¬ (x)(Fx). As are: ∃x(Fx) and ¬∃x(Fx). Since ¬∃x(Fx) is equivalent to (x) ¬(Fx), that proposition is ἐναντίος to one sort of affirmative proposition and ἀντικείμενος to another as Aristotle states in XV ll. 35 ff.

In the end Aristotle does not even refer to Plato. Apparently he either thought his logical distinctions were not relevant to Plato’s issues or else that the Prior Analytics was not the appropriate locus to discuss them. Nevertheless, Aristotle’s logical analysis provides valuable support for Plato’s undoing of Parmenides’ conundra and their sophistical misuse. In addition, we have discovered, apophantically so to speak, four phenomena. The first is what I called a contrapositive or something absolutely inimical to something else. The second is an opposite that is nevertheless characterized by what it is opposite to. The third is the contrary relation in the Square of Oppositions. The fourth consists of concepts that are related in that in some contexts they cannot apply to the same subject (Jones cannot walk and ride a bicycle at the same time) and in other contexts they can (Jones can be able to walk and to ride a bicycle at the same time.) Interestingly enough all three are contrasted somewhere in the history of philosophy with true complementarity. But whether anything actually corresponds to the first two apparent phenomena is open to doubt. Non-being may just be generated by the concepts of being and negation in our perfervid imaginations in the same way that horned-unhorned unicorns are. Likewise contrapositives may be no more than convenient labels, such as can be attached to sports “rivalries.” Evil, where it is not simple lack of good or misfortune or malevolence is very likely just another Xtian myth invented to keep the peasantry under control. (Clearly the problem of evil is better named the problem of misfortune or the problem of antagonism in order to clean up the argument and avoid the implication of Satanism whenever the car won’t start.) We have also been reminded of Aristotle’s important distinction between complemenatrity within a genus and thoroughgoing complementarity.

Aristotle’s terminology is a little muddled because he uses the same term ἀντικειμένας and its conjugational forms as both a generic term to cover all the types of opposition in the Square of Oppositions and to refer specifically to complementary opposition. This is particularly evident in Anal. Pr. II xv esp. 63b 23-81.  However, the context makes Aristotle’s intentions and meanings in this instance somewhat clearer. Note that ἐναντίας in this passage refers to a logical opposition between “Everything is…” and “Nothing is …” and not to contrapositivity as I have been using the concept. On the other hand,  Anal. Pr. II xxvi is vague about whether ἀντικειμένας means the same thing as ἐναντίας or whether it is a more generic term including ἐναντίας and other types of opposition. Indeed ἐναντίας seems to mean several different things in this passage, and, to add to the confusion, it is used in his example as well as being part of the subject he is trying to clarify. Either Aristotle was just sloppy in his terminology or the passage we have is the result of bad note taking by some student or both.

One final thought. It is interesting how Plato could make a valid distinction in order to argue one philosophical point (How non-being does not have to be a member of the class of things that don’t exist) and I (in all humility) could make a similar distinction to argue another philosophical point (That a concept of evil is not a necessary logical consequence of the concept of good in the same way that a concept of non-good is). Plato’s point could perhaps be better stated in the language of category mistakes, i.e. the predicate “exists” does not result in a well-formed proposition if it is predicated of a concept such as being. I don’t see a better way to formulate my point, but I’m sure there is one.

Stirring Conclusion: Logical counterparts or conceptual complements are linguistic phenomena. They are part and parcel of how our predicates and the words we use in general have meaning. They have no greater ontological significance than the original concept. Even though some concept ranges over things that exist, we cannot thereby conclude that that concept’s counterpart must also range over existing things. Concepts are like fences. They separate our thoughts. You can’t really have a fence without assuming that there are two areas that have been fenced off from one another. But the conceptual fence does not divide the real world, or at least it doesn’t do so directly. What it divides is how we conceive the world. So while “the good” cannot meaningfully be conceived without “the not-good,” it is entirely possible for the real world to in fact contain only good things.

Much of Maxju theology is a kind of conceptual alchemy. It is the attempt to create existing things from the fool’s gold of pure and often empty concepts. Like alchemy it shall, one fondly hopes, be pushed aside one day and consigned to the basement laboratories of cranks and pseudo mystics.

Scholium: Interestingly Baudelaire noticed this distinction which so many academic philosophers miss. Cf. his admittedly lapidary comments on contraire and contradiction in his Salon de 1846 (p. 456).