Arguments for the Closure Principle.
Deductive Knowledge Argument: We can always reason to new knowledge if we know our premises and we also know that our premises entail, or make necessary, other statements. Deduction like this is a way of extending knowledge. Deduction is the most secure way of extending our knowledge! Because of this, the closure principle is also a good way to extend knowledge.
Knowledge and Assertion Argument: Well, it just seems really silly that if someone says that they know (1) If it’s snowing, there is frozen water in the sky and (1) it’s snowing and yet denies (3) there is frozen water in the sky. In other words, to deny the closure principle is to deny the basic laws of logic.
Really, both types of arguments are just making the point that the closure principle is founded in basic laws of logic, so to deny the closure principle seems like the same thing as denying the laws of logic, which is irrational!
Arguments against Closure:
Regret Closure Argument: Knowing is a way of relating to a statement. Regretting is another way of relating to a statement. Dretske thinks that if entailment doesn't work for some kinds of relating to propositions, then it does not apply to knowing, either! If Sam regrets getting a haircut and Sam also knows that getting a haircut entails spending money, then Sam must regret having spent money. But this seems silly. Another example of regret closure: If I regret eating a whole chocolate cake and I know that eating a cake means consuming calories, then under the closure principle it seems I must regret consuming calories. But I know that I have to consume SOME calories in order to live! So it seems silly that just because I regret eating a whole cake that I also regret consuming calories. Dretske says that if the closure principle fails for regret then it also fails for knowledge.
Entailment Fails for Other Ways of Extending Knowledge: Dretske also notes that other ways of extending our knowledge are cases where the principle of closure fail. Because of this, we have reason to think that the principle of closure also fails for knowledge. Our memories, our perceptions or the statements of others are not instances where we can have knowledge under entailment. This means that entailment should not hold for knowledge, either.
Some philosophers think that a result of the closure principle is that in order to know something (P), then we must know that P on only P really is the case. In other words, we must exclude all other alternatives in order to know P. For example, in order to say that we know that an animal is a zebra, we must know that it is not a painted mule, an alien changeling or a hologram.
RAT and Restriction of Closure. RAT: Relevant Alternative account Theory. Proponents of RAT think that the closure principle is too strict. If in order to know P, then I must also be able to eliminate all alternatives to P, then it seems that we never have knowledge. We never eliminate all alternatives. But many times, skeptical alternatives (like holograms and aliens) are irrelevant. According to RAT, in order to know P, one must only rule out relevant alternatives. In this case, the closure principle does not hold in general (because the closure principle requires that we exclude all other possibilities).