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.
(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:
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:
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:
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?
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.