First case
Propositional logic accepts that the proposition A∨-A is necessarily true. This is called the law of the excluded middle. Intuitionist logic differs in that it denies this axiom.
Suppose that Joe is a believer in propositional logic (but also reserves some credence for intuitionist logic). Joe also believes a set of other propositions, whose conjunction we’ll call X, and has total certainty in X.
One day Joe discovers that a contradiction can be derived from X, in a proof that uses the law of the excluded middle. Since Joe is certain that X is true, he knows that X isn’t the problem, and instead it must be the law of the excluded middle. So Joe rejects the law of the excluded middle and becomes an intuitionist.
The problem is, as an intuitionist, Joe now no longer accepts the validity of the argument that starts at X and concludes -X! Why? Because it uses the law of the excluded middle, which he doesn’t accept.
Should Joe believe in propositional logic or intuitionism?
Second case
Karl is a theist. He isn’t absolutely certain that theism is correct, but holds a majority of his credence in theism (and the rest in atheism). Karl is also 100% certain in the following claim: “If atheism is true, then the concept of ‘evil’ is meaningless”, and believes that logically valid arguments cannot be made using meaningless concepts.
One day somebody presents the problem of evil to Karl, and he sees it as a crushing objection to theism. He realizes that theism, plus some other beliefs about evil that he’s 100% confident in, leads to a contradiction. So since he can’t deny these other beliefs, he is led to atheism.
The problem is, as an atheist, Karl no longer accepts the validity of the argument that starts at theism and concludes atheism! Why? Because the arguments rely on using the concept of ‘evil’, and he is now certain that this concept is meaningless, and thus cannot be used in logically valid arguments.
Should Karl be a theist or an atheist?
Third case
Tommy is a scientist, and she believes that her brain is reliable. By this, I mean that she trusts her ability to reason both deductively and inductively. However, she isn’t totally certain about this, and holds out a little credence for radical skepticism. She is also totally certain about the content of her experiences, though not its interpretation (i.e. if she sees red, she is 100% confident that she is experiencing red, although she isn’t necessarily certain about what in the external world is causing the experience).
One day Tommy discovers that reasoning deductively and inductively from her experiences leads her to a model of the world that entails that her brain is actually a quantum fluctuation blipping into existence outside the event hole of a black hole. She realizes that this means that with overwhelmingly high probability, her brain is not reliable and is just producing random noise uncorrelated with reality.
The problem is, if Tommy believes that her brain is not reliable, then she can no longer accept the validity of the argument that led her to this position! Why? Well, she no longer trusts her ability to reason deductively or inductively. So she can’t accept any argument, let alone this particular one.
What should Tommy believe?
— — —
How are these three cases similar and different? If you think that Joe should be an intuitionist, or Karl an atheist, then should Tommy believe herself to be a black hole brain? Because it turns out that many cosmologists have found themselves to be in a situation analogous to Case 3! (Link.) I have my own thoughts on this, but I won’t share them for now.
Joe doesn’t have a logic problem; he has an epistemology problem.
Karl, if he thinks that the problem of evil invalidates theism, is simply claiming that theism is internally inconsistent. Whether or not he thinks evil is consistent with anything outside of the theistic model is irrelevant.
Tommy has made a profound mistake regarding representation, but that is not her real problem. Her belief is like solipsism – completely atheoretical. It cannot be reliable or unreliable because it is predicated on things being just as they are, always.
In Tommy’s case, it seems like what’s happening is more like the kind of case where: someone initially thinks they’ve found a theory that matches the evidence they’ve received well so they update in favor of it but after reflecting on the implications of the theory more they realize it doesn’t actually explain the evidence well so they initially update incorrectly. In other words, it doesn’t seem like anything more mysterious is going on then someone realizing they made an epistemic mistake because they didn’t fully think through the implications of their beliefs.
Joe’s case is more puzzling to me. I think it is because more generally I don’t really know what’s the rational way to deal with uncertainty in the proper way to reason deductively or uncertainty in the proper way to update your beliefs inductively. For example, imagine Joe is certain of X, has a .7 credence in the law of the excluded middle, has a .1 credence in a proposition A, and then he realizes he can use the law of the excluded middle to prove A from X. What should his new credence in A be? Should this realization alter his credence in the law of the excluded middle as well?
Similarly, I think my puzzlement about Karl’s case may be connected to a more general confusion about how to rationally update beliefs when you have uncertainty about whether a statement is meaningless.
My intuition about these cases though says that Joe should believe in Intuitionist logic and Karl should be an atheist (assuming the problem of evil was providing evidence against all forms of theism) because they learned that the alternatives were inconsistent with things they are certain of.
Agreed about Joe and Karl, but not about Tommy. More later 🙂
At one point Karl is a hundred percent confident of mutually exclusive propositions, their conjunction implies atheism but their conjunction also implies theism. (If xyz about evil then it’s a meaningful concept [or a meaningless one that CAN be used in a valid argument?] then modus ponens plus tax: theism) He only noticed the first implication, once he becomes aware of the second one he should become 100% convinced of the negation of a conjunction of beliefs that he used to have 100% confidence in. So Karl should be a theist assuming that his brief career as atheist didn’t avail him of a powerful reason to motivate a continued visit. I think something subtle is going on in Joe’s situation, maybe the best way to translate X is into a intuitionist system with its own connotation for ‘truth values’ that aren’t in the scope of the excluded middle that he once accepted. If so, he should return to his previous stance on logic. At any rate the relationship of various logics to truth values is an issue that Joe should think about if he wants to makes sense of his views, or he could just take a break from thinking about logic. Tommy should be more confident of her interpretations of the external world given her experience, those are the type of things people use to argue for and against… uh… thermodynamic solipsism. I like this problem in the abstract, it’s like a bad trip to the dmv, A sends you to B and B won’t process you so you have nowhere to land.
“I like this problem in the abstract, it’s like a bad trip to the dmv, A sends you to B and B won’t process you so you have nowhere to land.”
HAHA! Perfect. Yes, that’s exactly what I’m getting at here.