One More Thing: Boolean Conditionals


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.


Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s