Have you ever wondered how some people can reject LEM? LEM states that for all propositions P, either P or not P holds. In this post I will convince you that belief in LEM is obviously delusional.
Some propositions are finite and others are infinite. I will say a proposition is infinite if the most naive way of checking its truth potentially requires infinite number of work. Thus for example, the proposition 3^2 + 4^2 = 5^2 is not infinite because you can determine its truth in a finite number of steps. But a proposition like "for all n, the Collatz sequence starting at n eventually reaches n" is an infinite proposition because to check it potentially requires an infinite amount of work. It might be that there is some very large counterexamples, then you could determine its veracity in a finite number of steps, but you don't know that, and from the way it's stated you potentially might need to check forever (if it's true).
Now this distinction is crucial when it comes to LEM. In the end, asserting LEM for a proposition is the same as asserting that the proposition is meaningful and has a definite answer. How can a proposition not have a definite answer? Consider an example given by Ed Nelson. In the fairy tale of Hansel and Gretel, and let us ask the question: “Did Hansel and Gretel drop an even number of bread crumbs or an odd number?”. Would it make sense to claim that they either dropped an even number or an odd number? That it's definitely one or the other? I think obviously not. What we're dealing is an undetermined part of the fairytale. None of the options is better, you can believe either one and you won't be any wiser. There's no way of deducing from the fairy tale the parity of the crumbs that they dropped. The question is simply irrelevant, and indefinite. In other words, it fails to refer.
When a proposition is finite, one asserts LEM for that proposition, because in the end we have a concrete way of determining the truth of the proposition.
Some propositions are finite and others are infinite. I will say a proposition is infinite if the most naive way of checking its truth potentially requires infinite number of work. Thus for example, the proposition 3^2 + 4^2 = 5^2 is not infinite because you can determine its truth in a finite number of steps. But a proposition like "for all n, the Collatz sequence starting at n eventually reaches n" is an infinite proposition because to check it potentially requires an infinite amount of work. It might be that there is some very large counterexamples, then you could determine its veracity in a finite number of steps, but you don't know that, and from the way it's stated you potentially might need to check forever (if it's true).
Now this distinction is crucial when it comes to LEM. In the end, asserting LEM for a proposition is the same as asserting that the proposition is meaningful and has a definite answer. How can a proposition not have a definite answer? Consider an example given by Ed Nelson. In the fairy tale of Hansel and Gretel, and let us ask the question: “Did Hansel and Gretel drop an even number of bread crumbs or an odd number?”. Would it make sense to claim that they either dropped an even number or an odd number? That it's definitely one or the other? I think obviously not. What we're dealing is an undetermined part of the fairytale. None of the options is better, you can believe either one and you won't be any wiser. There's no way of deducing from the fairy tale the parity of the crumbs that they dropped. The question is simply irrelevant, and indefinite. In other words, it fails to refer.
When a proposition is finite, one asserts LEM for that proposition, because in the end we have a concrete way of determining the truth of the proposition.
