Aristotle, Definitions, and Math, Pt. 1

I’ve been thinking of rebooting my Aristotelian metaphysics series, and I thought I might put this as the preamble or something, so you might also consider this a draft of that.

This stuff is mostly just observations of analogies; I’m not sure if I would consider any of it strictly “proven.”

EDIT: Realized I forgot to add tags.


Aristotelian thought distinguishes two ways in which a given attribute can exist in multiple things: formally and analogically. An attribute exists in two things formally if it is in both of them in exactly the same way; so two red objects both have redness formally, since there are no two ways that something can be red (if we’re specific about what hue we mean by “red”). On the other hand, an attribute exists in two things analogically if it exists in both of them in different ways. For example, both humans and octopuses can be considered to have “hands” in a sense, but obviously a human’s hands are very different from an octopus’s tentacles.

Aristotelian thought also gives us an archetypal form of definition. This form works by considering a genus of things that are assumed to be known to the listener, and delimiting a species from within that genus by means of a specific difference that is common to everything in that species. So for example, a mammal can be defined as a species of animal whose females bear live young and feed their newborns with milk. (Of course, this isn’t technically accurate because platypuses and echidnas lay eggs.) This definition takes a category, animals, that is assumed to be known to the listener, and then delimits the category of mammals by means of their common characteristic of bearing live young and nursing their newborns. Here, the genus is animals, the species is mammals, and the specific difference is bearing live young and nursing their newborns.

Notice that “genus” and “species” are relative terms, since a category can be a genus relative to another, narrower category and a species relative to another, broader category. Further, genera are recursive, since a species within a genus is itself (potentially) a genus; the genus is made up of genera.

Now, math makes a lot of use of things called “sets,” which are rather vaguely defined collections of objects. As you work with them, you gain an intuitive grasp of them, but they’re never really rigorously defined. All you can really say about a set is, as the Wikipedia page says, that it’s a collection of well-defined objects (ironic, considering that the set itself is vaguely defined). Thus a set can contain anything. You can define a set consisting of the numbers 7, 12, 13, 19, and 20, just because you like those numbers. Or you can define a set consisting of red, green, and blue. Or you can refer to the set of all even numbers, or the set of all rational numbers, etc. Or the set of all people who wear their hair in a topknot.

A set can also be broken down into subsets, where every member of the subset is also in the original set (referred to as a superset). So the set containing 2, 4, and 6 is a subset of the even numbers, while the even numbers are a superset of the set containing 2, 4, and 6.

Incidentally, it’s also perfectly acceptable to have a set of sets. In fact, the set of all subsets of a given set is called the power set of that set.

This brings up an interesting question: Is it possible to form a set of all sets? As it turns out, the answer is no, because it results in Russell’s paradox. Every set is either a member of itself or not; for convenience, we can refer to these as self-inclusive sets and self-exclusive sets. The set of all self-exclusive sets would then have to be a subset of the set of all sets. But is the set of all self-exclusive sets a self-exclusive set, or a self-inclusive set? If it’s self-exclusive, then it would have to be a member of itself—which then implies that it must be self-inclusive. By the same token, if it’s self-inclusive, then that means that it’s not a member of itself, which means that it must be self-exclusive. Either way, we get a contradiction. Therefore, the set of all self-exclusive sets can’t possibly exist, and therefore the set of all sets, which must be a superset of the former, also can’t exist.

This leads us to the concept of classes, which is even more vaguely defined than sets. Basically, a class is a group of objects that all have something in common somehow, but that we can’t necessarily represent as a set. “All sets” would then be a class, but not a set.

Now, one interesting point that’s often glossed over in math textbooks is that there’s a very obvious difference between sets like “7, 12, 13, 19, and 20” and sets like “the even numbers.” Formal math doesn’t have a term for distinguishing these two types of sets as far as I know, but for convenience, let’s call the former type of set a scoop (from the action of arbitrarily scooping random things out of a jar) and the latter a proper set. We can then say that a scoop only exists because somebody decided it does, while a proper set actually has a kind of inner coherence. Why is this?

Well, thinking back to Aristotle gives us a clue. The members of a scoop don’t necessarily have anything in common. But the members of a proper set have some common characteristic that they all share formally. And we can take this as a kind of “definition” of proper sets: A proper set is a grouping of objects that all share some common characteristic formally (but see below—I don’t think it’s actually possible to give a rigorous definition of proper sets).

And recalling the Aristotelian contrast of formal vs. analogical and the mathematical contrast of set vs. class immediately brings another connection to mind: A class would just be a grouping of objects that all share some common characteristic analogically.

And now that we’ve gotten ourselves into a math-and-Aristotle-y sort of mood, we might as well go a bit further. Recall how genera and species behave recursively—any species within a genus can potentially be a genus itself, and any genus can potentially be a species of another genus. Well, notice that the relation of supersets and subsets behaves in exactly the same way—any subset can potentially be a superset of another set, and any superset can potentially be a subset of another set. And further notice that a species is delimited by some characteristic that all its members share formally. In other words, a species is a proper set. So a definition is nothing other than a delimitation of one proper set from within another.

And this shows why it’s not possible to define proper sets—we would have to delimit the proper set of all proper sets from some other proper set, which is impossible because, as shown above, there is no proper set of proper sets. But the collection of all sets is a class, which tells us that proper sets are an analogical concept.

Against the Ontological Argument

I’ve always thought the ontological argument was invalid, and I’ve known what the problem is, but I’ve never really been able to express it. Today, I found a book at the library that took the problem right out of my head and expressed it perfectly:

“St. Anselm says: If the most perfect being that can be conceived did not exist, it would be possible to conceive of a being which has all the qualifications of the former, plus existence, so that this latter being would then be more perfect than the most perfect being that can be conceived. I admit that if this being did not exist, and was not conceived as self-existing, it would be possible to conceive one more perfect. But I deny the assertion that if it did not exist, though it was at the same time conceived as self-existing, then it would be possible to conceive of a more perfect being. Hence it is not logical to conclude: ‘Therefore God exists’; all that can be logically concluded is: Therefore, God must be conceived as self-existing, and in truth does so exist, and is entirely independent of any other being, if He exists.” (Reginald Garrigou-Lagrange, God: His Existence and His Nature, vol. 1, ch. 2)

The problem is that the argument fails to distinguish between whether God is conceptualized as existing and whether God actually exists. The argument shows that God conceptualized as actually existing is greater than God conceptualized as not existing. But just because one must conceptualize God as existing if one is to conceptualize Him at all, doesn’t mean God has to exist independently of a person’s concept of Him.

One More Thing: Boolean Conditionals

Uncle1

I don’t mean to beat a dead horse, but I just thought of another thing to say about Boolean logic. When I wrote my first post on Boolean logic, I just dismissed the idea that conditionals are true if the condition never comes true by saying, “I don’t see in what sense it’s true that if Mars is green, then I don’t need glasses.” But recently I actually thought of some reasoned counter-arguments that aren’t just appeals to common sense.

Under the Boolean interpretation, complex propositions are analyzed in terms of truth tables. In a truth table, the possible combinations of truth values (truth or falsity) of the simple propositions that make up the complex propositions are listed, along with the truth value of the resulting complex proposition in each case. The truth table for a conditional proposition looks like this:

p  |  q  |  If p, then q
T     T     T
T     F     F
F     T     T
F     F     T

So the statement “If I am 18 years old, then I am legally an adult” is true because the first proposition, “I am 18 years old,” is true, and the second, “I am legally an adult,” is as well. But the statement “If Kobe Bryant is with the LA Lakers, then Ronald Reagan was not a Republican” is false because the first proposition, “Kobe Bryant is with the LA Lakers,” is true while the second proposition, “Ronald Reagan was not a Republican,” is false. For convenience, the first proposition is called the antecedent, and the second is called the consequent.

So here’s my first problem with this interpretation of conditional statements. Suppose I said, “If I have eaten sushi, then I am Japanese.” It just so happens that I have never eaten sushi in my life, so the antecedent is false. Therefore, this conditional statement is true in the Boolean interpretation.

But then suppose I were to pay a visit to the sushi station in my college’s cafeteria and have some sushi for the first time in my life. Now the antecedent has become true. But obviously even if I were to eat sushi, I wouldn’t become Japanese, so the consequent would remain false. So now the antecedent is true and the consequent is false. If you look at the truth table, this means that the conditional as a whole is false.

Now, the thing is, if the conditional was true before I ate sushi and then became false after I ate sushi, then how was it true to begin with? The whole point of conditionals is to predict what will happen when the condition becomes true. A conditional that starts out true but then turns false when its antecedent is fulfilled is completely useless. If the conditional statement will turn false when its antecedent is fulfilled, then it should be false to begin with. If I say, “If it rains tomorrow, I will die,” and then it rains tomorrow but I don’t die, you wouldn’t say that the conditional started true and then turned false. You would just say that I was wrong to say in the first place that I would die if it rained. But this is not what the truth table analysis says.

Second, notice that in the sushi example we could tell, even though the antecedent hasn’t come true, that if it were to come true the conditional would be false. If humans really thought in terms of truth tables, then you’d expect that we wouldn’t be able to come to that conclusion. We should still think that “If I have eaten sushi, then I am Japanese” is a true statement, since the antecedent is still false.

This shows that humans do not in fact think of conditionals in terms of truth tables. So how do we think of them? I would contend that we think of them in terms of Aristotle’s four causes.

For example, take “If Socrates is a (healthy) man, then he can walk.” We know that this is true because part of what it means to be a (healthy) man is to be able to walk; simply by being a man Socrates has the potential to walk. In Aristotelian terms, Socrates’s being a man is the formal cause of his being able to walk.

The next cause in the list is the material cause… but I’m actually not entirely sure about this one. In the Physics, Aristotle defines the material cause as “that out of which something is made and which exists in [the thing even after it is actually made].” I think an illustration of material causality in a conditional would be something like, “If Socrates is a man, then Socrates is subject to gravity.” Here, the consequent doesn’t follow from Socrates’s being a man insofar as he is a man, but from Socrates’s being a man insofar as a man is something made up of particles that carry mass. Here, the consequent doesn’t follow from what Socrates is (viz. a man) so much as from what he is made of.

For antecedents that describe change, two causes come into play. Take “If a match is struck, it will catch fire,” for example. Here, the match’s being struck is the cause (in the colloquial modern English sense) of its catching fire. Thus the match’s being struck is what in Aristotelian terminology is called the efficient cause of its catching fire. But besides the efficient cause, the final cause also comes into play here. A lot of people think that the final cause just refers to the purpose of something. That is one meaning of final causes. But the more basic meaning is simply what a thing is directed toward, regardless of whether it’s directed consciously or not. In this case, catching fire is the final cause of striking a match. Whenever you strike a match, unless something else gets in the way, the result is always fire, never a duck, or music, or anything else. If the act of striking a match were not inherently such that it always produced that outcome all else being equal, then you’d expect that it might just as well produce fire as anything else. The fact that it does inherently produce fire is all Aristotle means by final causality. It’s because of this inherent ordering that we can confidently say that if I strike a match, it will catch fire; whereas when there’s no connection of final causality between one thing and another, we can’t say for sure whether the presence of the one entails that of the other.

Under this analysis, it’s easy to see why it’s false that if Mars is green, then I don’t need glasses. The antecedent here has no connection to the consequent at all. Here, since we aren’t talking about change, the causes in question are the formal and material causes. But the formal cause of my needing glasses is my eyeball being too long for its lens to focus light onto my retina, and the formal cause of that would be my eye’s stiffness which causes it to retain the shape it happened to take when I was growing up. The efficient cause of my eye taking this shape probably has something to do with genetics and/or my tendency to hold up handheld video games and books too close to my face. Mars being green has no causal connection, formal, material, efficient, or final, either to my needing glasses or to the length of my eyeball, since those are already sufficiently explained by other things. This, not anything to do with truth tables, is the reason we can tell easily that “if Mars is green, I don’t need glasses” is false.

Ok, now I think I’m done with Boolean logic.

Existential Import, Pt. 2

Okay, I finally got time to write the rest of my thoughts on the Boolean interpretation. If you haven’t already, you might want to see this post so you can understand what the heck I’m talking about; in particular, you’ll want to at least read up to “but there’s one problem with the Boolean interpretation: it uses two different senses of ‘true’ and ‘false,'” and then read the edit at the end. Speaking of which, if you’ve already seen the original post but haven’t seen the edit, you should probably take a look at that to make sense of this.

Boole’s entire system rests on the claim that logical propositions are primarily concerned with asserting or denying existence. This is the basis of his claim that A’s and E’s are not contrary and his claim that all statements about subjects that don’t exist are true.

But this presupposition is not sound, which is evident from the simple fact that logical propositions are syntactically different from existential statements. In English, existential statements begin with “there is/are.” Logical propositions do not begin with “there is/are.” Therefore, logical propositions are not existential statements. Nor would appeals to different languages help. In Ancient Greek and Latin, the languages in which logic was first extensively studied in the West, existential statements took the form of the 3rd person of “to be” (ἔστι or εἰσί in Greek, est or sunt in Latin) and the nominative form of the thing that is said to exist. But this is not the way logical propositions were phrased in Greek or Latin either. In those languages, logical propositions were phrased in sentences of the form “P belongs to S” (“P ὑπάρχει τῷ S” in Greek). Thus logical propositions have not historically been thought of as existential statements, nor would they be thought of in this way now if this interpretation were not imposed on them by modern logicians.

And then there’s the obvious fact that A statements are positive statements. Yes, “all unicorns have horns” does imply that there are no unicorns that have no horns. But it seems odd that that should be interpreted as the primary meaning of that statement, when anyone would tell you that it is a statement about what unicorns do have, not about what kinds of unicorns don’t exist.

This is not to say that logical propositions say nothing about existence, but they do so only incidentally. Suppose, for example, that I told you that a triangle is a figure consisting of three lines joined together at the endpoints. Now, it just so happens that using that information, you can deduce that a triangle is also a figure whose interior angles add up to 180 degrees. But this doesn’t mean that I told you that a triangle is a figure whose interior angles add up to 180 degrees; or, if I did, I did so only incidentally, as a side effect of what I was really trying to tell you. Similarly, though “all unicorns have horns” also incidentally means that there are no unicorns that lack horns, that is not what it means per se. This is why the Boolean interpretation carries some plausibility; an A proposition does necessarily carry some incidental existential information. But this, as the Scholastics* would say, is a property, not its essence; it follows from what the statement is, but it does not define the statement.

(*Have I ever talked about the Scholastics? In case I haven’t, they were the proponents of the medieval tradition of thought derived from Aristotle.)

So if logical propositions are not existential statements, the question of what kind of statements they really are remains to be answered. I would argue that logical propositions are statements about the natures of things, in other words what it means to be a given thing. This seems to be supported by the fact that anyone would interpret a statement in the form “S is P” or “S does P” as a statement about the subject, and specifically about what kind of thing the subject is, not about what actually exists or does not. So, an A proposition states that the nature of a thing necessarily entails that it have some attribute, an I proposition states that the nature of a thing at least allows for it to have some attribute, an E proposition states that the nature of a thing absolutely excludes its having some attribute, and an O proposition states that the nature of a thing at least allows that it should sometimes lack some attribute. From this, all the relationships on the Square of Opposition can easily be seen to follow. Corresponding A and E propositions are contrary because the nature of a thing cannot both include and exclude an attribute. An I proposition is the subaltern of its corresponding A because if the nature of something necessarily includes an attribute, then of course it must have that attribute all the time, let alone some of the time. Corresponding A and O propositions are contradictory because stating that the nature of something always includes some attribute necessarily denies that its nature ever allows for it to lack that attribute, and conversely, saying that a thing’s nature allows for it to lack an attribute necessarily denies that it must always have that attribute. An O proposition is the subaltern of its corresponding E because saying that the nature of something absolutely excludes some property entails that it must lack that attribute all the time, let alone some of the time. E and I statements are contradictory because stating that the nature of something never allows it to have some attribute necessarily entails a denial that its nature allows for it to have that attribute even some of the time. I and O propositions are subcontraries because if a thing’s nature does not allow for it to have some attribute even some of the time, then it must always lack that attribute, which by subalternation means that it must lack that attribute at least some of the time. And lastly, by the same token, if a thing’s nature does not allow for it to lack some attribute even some of the time, then it must always have that attribute, which by subalternation means that it must have that attribute at least some of the time.

One other problem with the Boolean interpretation: If all universal statements about things that don’t exist are true, what do we do about statements like “all unicorns exist”? I asked my logic teacher about this, and he said that supporters of the Boolean interpretation solve this problem by saying that existence is not a predicate. Now, I assume that if asserting existence of something is not a predication, then it would have to be classified as an existential statement. But if we’re going to oppose predicates and existential statements, and argue that “all unicorns have horns” is a valid statement while “all unicorns exist” is not because the former is a predicate as opposed to an existential statement while the latter is an existential statement as opposed to a predicate, then Boolean logic falls apart because, as was said earlier, Boolean logic depends on the presupposition that all logical propositions are kinds of existential statements. If “all unicorns have horns” is not denying the existence of a certain type of unicorn but predicating a certain attribute of unicorns, then A and E propositions become contrary again because we can’t both predicate an attribute of something and deny it. Taking this route is trying to have the cake and eat it too.

Contrast this with the way this problem can be dealt with if we go with the traditional interpretation. In the traditional interpretation, a statement like “all unicorns exist” would be stating that the nature of unicorns necessarily entails existence. Because we do not hold that A statements are essentially negative existential statements, this does not lead us to a contradiction. Now, as it turns out, the only thing that exists by nature is Pure Act unmixed with any potency, a.k.a. God, so the statement “all unicorns exist” couldn’t possibly be true. (I’ll probably talk more about that in a later post, after I finally get around to talking about Aristotelian metaphysics.) But we are not forced to deny the validity of the statement a priori; we are still allowed to at least ask the question of whether “all unicorns exist” is a true statement.

Lastly, one other thought on why the Boolean interpretation might seem to carry some plausibility. I can think of four different meanings of the present tense in English. One of them, which doesn’t really come into play in logic, is to indicate an imminent action: “Hand over the money, or the kid gets it!” Other times, the present tense signifies an action that a person does habitually; for example, “I eat pizza every Tuesday.” For some verbs that indicate a kind of state rather than an action, it indicates that that state is presently occurring: “I know that you killed Jethro,” or “I have a toothache.” And sometimes, the present tense indicates a truth that always holds regardless of time; “Triangles have three sides” and “Nice guys finish last,” for example, are not statements about any particular time, but statements that apply to all times. The Ancient Greek and Latin present tenses also carried all these meanings except for the one about imminent action. In addition, they also included what English expresses as the present progressive. Now, I think this is one of the reasons Boole came to such a different conclusion from Aristotle and the Scholastics: the latter interpreted the present tense in the timeless sense, while Boole interpreted it in the presently-occurring-state sense. So, I think that Aristotle would have interpreted “All dogs are living things” as “Dogs are always living things,” whereas Boole seems to interpret it as “Currently, all dogs are living things.” The former interpretation necessarily implies that we are talking about natures rather than about any particular instances of the thing in question, because the only sense in which a thing exists timelessly is in that its nature or form exists timelessly (not as a separate thing (substance in Aristotelian terms), but as an idea conceived of by some mind). The latter interpretation, on the other hand, makes more sense if we are talking about particular, existing things rather than about natures. After all, natures themselves do not exist as particular, changeable things that might have an attribute at one time but not another. So it seems to me that Boole arrived at the conclusions he did at least partly because of how he interpreted the present tense.

Ok, I’m pretty sure that’s all I have to say on this subject. Now I can finally move on to other topics without feeling bad.

Things I Think Other People Are Wrong About: Existential Import

Today in logic class, we went over the Boolean interpretation of logic. Which I don’t like. So now I’m going to write about it before I forget.

Traditional logic divides all propositions into four categories: A (universal affirmative; “all dogs are mammals”), I (particular affirmative; “some* dogs are mammals”), E (universal negative; “no dogs are mammals”), and O (particular negative; “some* dogs are not mammals”).

*In logic, “some” does not exclude “all,” nor does it imply more than one. In normal speech we would assume that if someone chose to say “some dogs (rather than all dogs or a dog) are mammals,” then he must mean that there are also some dogs that are not mammals, and that there is more than one dog that is a mammal. Formal logic would make neither of these assumptions.

Traditionally, A and E propositions are said to be contraries; this means that they can both be false, but they can’t both be true. “All dogs are mammals” and “no dogs are mammals” can both be false if some dogs are mammals and some aren’t. But if “all dogs are mammals” is true, then “no dogs are mammals” is false, and if “no dogs are mammals” is true, then “all dogs are mammals” is false.

An I proposition is said to be the subaltern of its corresponding A; this means that the I proposition is contained in the A. So if it is true that all dogs are mammals, then it is also true that some dogs are mammals. However, the converse is not true, so if all you know is that some dogs are mammals, you can’t say whether or not all dogs are mammals. If the I is false, then the corresponding A is false; if not even some dogs are mammals, then it can’t be true that all dogs are mammals.

Similarly, an O proposition is the subaltern of its corresponding E; if “no dogs are mammals” is true, then “some dogs are not mammals” is also true, and if “some dogs are not mammals” is false, then “no dogs are mammals” is also false.

I and O propositions are said to be subcontraries. This means that corresponding I and O propositions can both be true, but they can’t both be false. So if it is true that some dogs are mammals, it might also be true that some dogs are not mammals. However, if “some dogs are mammals” is false, then it cannot also be false that “some dogs are not mammals.” In fact, if “some dogs are mammals” is false, then that means that no dogs are mammals, let alone some dogs.

Corresponding A and O propositions are said to be contradictory. This means that they cannot both be true, and they cannot both be false; to affirm one simply is to deny the other, and to deny one simply is to affirm the other. If “all dogs are mammals” is false, then that means that some dogs are not mammals, and if “some dogs are not mammals” is false, then that means that all dogs are mammals.

Similarly, corresponding E and I propositions are contradictory. For “no dogs are mammals” to be false simply is for “some dogs are mammals” to be true, and for “some dogs are mammals” to be false simply is for “no dogs are mammals” to be true.

Diagramming all these relationships results in what is known as the Square of Opposition.

square-of-opposition
(image from plato.stanford.edu)

Now, some time in the 1800’s, George Boole came along and argued that the traditional Square of Opposition is inaccurate. His reason was that particular propositions carry existential import, whereas universal propositions do not. In other words, according to Boole, “some unicorns have horns” implies that at least one unicorn exists, since the quantifier “some” means “at least one,” while “all unicorns have horns” does not imply one way or the other whether or not unicorns exist.

Under this interpretation, subalternation becomes invalid. “All unicorns have one horn” might be true, but “some unicorns have one horn” is still false because there is not at least one unicorn in existence that actually has one horn. Similarly, “no unicorns have horns” might be true, but “some unicorns do not have horns” is still false because there are no existing unicorns that actually lack horns.

Furthermore, I and O propositions are not actually subcontraries; “some unicorns have horns” and “some unicorns do not have horns” can both be false. If there are no unicorns, then “some unicorns have horns” is false because there are no individual unicorns that actually have horns. But then, if there are no unicorns, “some unicorns do not have horns” is also false, because there are no individual unicorns that actually lack horns either.

And yet furthermore, what I find most mystifying about the Boolean interpretation, A and E propositions are not contrary. Unfortunately, I wasn’t paying attention when my teacher went over this part. But according to this website, the reasoning is as follows. If someone said to you, “All swans are white,” you would falsify that claim by pointing out at least one non-white swan. But suppose someone said, “All unicorns have one horn.” You can’t falsify that because there’s no hornless unicorn to point out. Again, if someone were to say, “No unicorn has one horn,” well, that claim isn’t falsifiable either. So, if neither claim is falsifiable, then apparently that means that they’re… both… true…? =/

(The website cited in the last paragraph says that subcontrariety is maintained in the Boolean interpretation, but I remember my instructor saying that it isn’t. So I’m going with that. Of course, if the website contains one mistake, it might contain more, so maybe the logic in that last part is also a misrepresentation of the Boolean interpretation. But I definitely remember my teacher saying that there is no contrariety in the Boolean square of opposition, and I can’t imagine how you can possibly deny the contrariety of A and E propositions, so any other argument would probably be just as unbelievable to me. In fact, I just looked into my textbook just now, and it simply asserts without explanation that “all unicorns have wings” and “no unicorns have wings” can both be true if there are no unicorns. The chapter on conditionals did not seem to me to clarify their reasoning. So, yeah. College logic courses, dude.)

NOTE: But see edit below; I just figured out the reasoning behind this, and my logic instructor confirmed it.

This leaves us with a pathetically stripped down version of the square of opposition. The only time we can use all of the traditional relationships between propositions is if we know in advance that instances of the subject actually exist. If not, the only relationship that holds true is contradiction.

But there’s one problem with Boole’s interpretation: It uses two different senses of “true” and “false.”

Take the argument against subalternation: it can be true that all unicorns have horns, but still be false that some unicorns have horns. But the sense in which “some unicorns have horns” is false is different from the sense in which “all unicorns have horns” is true. “All unicorns have horns” is true in the same sense in which two and two make four. That two and two make four is true in an abstract sense; whether or not you actually add two and two, it’s still true. It simply follows from the nature of two and two that they make four, whatever particular two things you’re talking about. Similarly, “all unicorns have horns” is a claim about the nature of unicorns, regardless of any particular unicorns.

But “some unicorns have horns,” as Boole interprets it, is a statement about individual unicorns, not about the nature of unicorns.

Now, compare the criterion of falsehood for each of these propositions. The statement “all unicorns have horns,” being a statement about the nature of unicorns, is false if the nature of unicorns is not such that they have horns. But the statement “some unicorns have horns” is false if no unicorns exist regardless of whether the nature of unicorns is such that some of them might have horns and some might not. So when we say that an A statement is false, we mean something different from what we mean when we say that an I statement is false.

Or to put it another way: Try turning the propositions into conditional statements. “All unicorns have a horn” would be like “If a given thing is a unicorn, then it must necessarily have a horn.” What would be the subaltern of that? I should think it would be, “If a given thing is a unicorn, then it might have a horn.” It obviously follows that if it’s true that unicorns must have horns, then it’s also true that they might have horns.

But how would you turn Boole’s interpretation of I propositions into conditionals? The only way I can think of is, “If a given thing is a unicorn, then it might have a horn. By the way, some given things actually are unicorns.” This is actually two statements, not one. So when someone says that “some unicorns have horns” is false because there are no unicorns, he is not denying the first, conditional statement, but the second, existential one. The universal proposition is false only when the implied conditional does not hold, but the particular proposition, which is actually two propositions, is false when the (supposedly) implied existential statement is false even if the implied conditional holds. Again, Boole is using two different senses of truth and falsity.

Or another way to see the problem in Boole’s thinking: Boole says that a particular proposition can only be true if there’s at least one actual instance of the subject class. But remember the conditional form of an I proposition: “If a given thing is a unicorn, then it might have a horn.” If Boole says that statements of this form are only true if there is at least one [whatever term is used in the place of “unicorn”], then he’s essentially saying that a conditional with “might” in the conclusion can only be true if the condition comes true at least once. Which I find ironic, because Boole also asserted that a conditional with a false condition is always true. I also find it unbelievable, and I also don’t see why it doesn’t apply to universals. If my interpretation of Boole’s thinking is right, then the statement “If I go to Mars without a space suit I will die” can be true even if I never actually go to Mars without a space suit, but the statement “If I go to Mars without a space suit I might die” can only be true if I actually go to Mars without a space suit. I don’t know about you, but that sounds pretty arbitrary and absurd to me.

And in fact, the situation might even be more bleak than I make it out to be. I said earlier that the criterion for “all unicorns have a horn” to be false is if it is not in the nature of unicorns that they must have horns. But apparently, that’s not what Boole would have said. According to that website I cited earlier, Boole would just have said that any universal statement about things that don’t exist is true. So apparently all unicorns are hippies, all elves are prune juice, all ogres are Communists, and all female quarians are hot. However, at the same time, it is also true that no unicorns are hippies, no elves are prune juice, no ogres are Communists, and no female quarians are hot. (And you can’t say that that situation is impossible because A’s and E’s are no longer contrary.) When a guy says that all that is true, you gotta wonder what he means by “true.”

So what interpretation should one follow? Well, I would go with what Peter Kreeft says in his book Socratic Logic, namely that no proposition carries existential import. Propositions in logic are basically equivalent to conditionals, where the universals contain “must necessarily” in the conclusion and the particulars contain “might” in the conclusion. A conditional statement doesn’t imply that its condition ever comes true at any time, and similarly a logical proposition doesn’t imply that its subject actually exists.

And for heaven’s sake, a conditional with a false antecedent is not necessarily true. I don’t see how anyone can think it makes sense to say it’s true that if Mars is green, then I don’t need glasses.

EDIT: I just realized why Boole thought that A’s and E’s are not contrary if there are no instances of the subject class. To see why, you have to look at Venn diagrams. A Venn diagram, as you probably know, looks like this:

Venn1

The leftmost region represents things that are unicorns but are not things that have horns. The middle region represents things that are unicorns and are also things that have horns. The rightmost region represents things that have horns but are not unicorns. The way these things work in modern logic is, you fill in a region to show that nothing of that kind exists, put an X in a region to indicate that at least one thing of that kind exists, and leave a region blank to indicate that you have no way of knowing whether or not things of that kind exist. So how would you graph the statement, “All unicorns have horns”? Well, in modern logic, you would graph it like this:

Venn2

If all unicorns have horns, then you can be sure that there’s no such thing as a unicorn that doesn’t have a horn. But A propositions don’t have existential import, so you can’t assert that there actually are unicorns. So the middle regions stays blank. Now, how would you diagram “No unicorns have horns”? That would look like this:

Venn3

Again, if no unicorns have horns, you can be sure that no unicorns that have horns exist, but you can’t be sure that any unicorns actually exist, so the leftmost region stays blank.

But what if no unicorns exist?

Venn4

The leftmost region is filled in, just as with the A proposition. But the middle region is also filled in, just as with the E proposition. So there you go―the A and the E are both true. And every universal statement about things that don’t exist is true because every such statement really boils down to a negative existential statement. So since all such negative existential statements must be true, all universal statements about non-existent things are true.

I asked my logic teacher about this today, and his answer seemed to pretty much confirm this.

Needless to say, I don’t agree with this interpretation, and I already have some idea how I’ll go about arguing against it. But since I can’t put that up yet due to homework, I just wanted to make it clear that I understand the argument.