Wow. Sven went over a lot about the closure principle today. I have to admit, I myself was a little confused at times. He uses a lot of technical language, doesn't he? Fortunately, he did say that it's just most important that we understand the most basic form of the closure principle, known as Closure of Knowledge under Entailment:
1.I know P (I know that Roy is my brother).
2. P entails Q (If Roy is my brother, then Roy is a boy).
3. I know Q (I know that Roy is a boy).
This is the simplest version of the closure principle. Sven also gave us a slightly different version known as Closure of Knowledge under Known Entailment, which looks like this:
1. I know P (I know that I ate a whole cake).
2. I know that P entails Q (I know that if I ate a whole cake that I consumed calories).
3. I know Q (I know I consumed calories).
What’s the difference? In the first case, I do not need to be aware of the fact that P entails Q in order to also know Q. In the second case, I must know that P entails Q in order to know Q.
The Belief Problem. Here is an example where the closure principle is shown to be false: Frank knows that it’s snowing outside (P). He also knows that snow can make driving dangerous (he knows P entails Q). However, if Frank is being thoughtless and doesn’t stop to think, he can fail to realize that the driving could be dangerous right now (he fails to know Q). Because Frank does not believe currently that the snow will make his driving dangerous, the closure principle has failed.
C2 (Closure Principle 2): If Sam (1) knows P and (2) knows that P entails Q and Sam believes Q, then Sam knows Q. Here is an example:
1. Sam knows P (Sam knows that it is raining).
2. Sam knows that P entails Q (Sam knows that if it rains, then the floors get slippery).
3. Sam believes Q (Sam believes that the floors are slippery).
4. Therefore, Sam knows Q (Sam knows the floors are slippery).
What’s the difference? In the original version of the closure principle, someone does not need to believe the second proposition/statement. They only need to know that the first statement (P) entails or makes necessary the second statement (Q). C2 requires that a person also actively believes the entailed statement in order to know it.
Justification Problem. There is also the claim that one not need just believe Q in order to know Q under entailment. One must also have good reason to believe Q.
C3: If Sam knows P and knows that P entails Q and also has a justified belief in Q, then Sam knows Q. here is an example of C3:
1. Sam knows P (Sam knows that I got my hair cut in a salon).
2. Sam knows that P entails Q (Sam knows that if I cut my hair in a salon, then I paid money for it).
3. Sam is justified when he believes Q (Sam has good reason to think I paid money for my haircut because I showed him the receipt or I told him how much it cost).
4. Therefore, Sam knows Q (Sam knows I spent money).
Problem of Triviality. In the new version of the closure principle, then we are just saying that if Sam knows P and Sam also has a justified belief in Q, then Sam knows both P and Q. In other words, in order to know that Q because Q is entailed by P then I must have a justified belief in Q. But if I must have a true, justified belief in Q then it seems that I would already know Q independently of whether it is entailed by P.
There is much more to say about the closure principle, but this post here includes the basics!