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.