Why I don’t “believe” logic, yet use it all the time.
Logic is school of thought that allows us to state, “If we assume that X, Y, and Z are true then we may derive that W is true as well.” There are many different logics which allow the derivation of different things given the same basis; colloquially we speak of “logic” or “proof” as utilizing whatever logic we happen to need. I might speak at some length on the differences of various logics, but that is not my purpose today.
Suppose I were to present a proof that depends, as one of its assumptions, on whether you are tired. Are you tired? Can you always clearly identify yourself as either tired or not tired? What magic happens the instant you switch between those two states? Perhaps “is tired” isn’t a good predicate to use in boolean logic.
So let’s assume I re-work my proof using a continuum logic and include a numerical representation of your tiredness in the proof. How tired are you? Let’s assume we pick a few representative moments to suggest the scale we are using: your tiredness is 0 during a brisk shower after a morning jog and it’s 17 when trying to stay awake during a boring lecture. But now we find another problem: isn’t the trying to stay awake when bored kind of tiredness different than the fully engaged but sleep deprived kind of tiredness? Lumping them both onto the same scale seems dishonest.
So maybe tiredness is some kind of multi-dimensional measure with a weary dimension and a bored dimension? But that doesn’t work either; what about the differences between boredom from repetitive events versus events that don’t make sense? Or weary from too much exercise, or not enough sleep, or sunburn? And how do we tell if it’s a different kind of tired or just a different cause for the same kind of tired? Maybe there is really a single cut-off between tired and not tired: “when you are tired you don’t notice your name spoken at three decibels above ambient sound” or something like that.
When people ask me if I’m tired there are rare cases I can say “yes” or “no” with confidence, but most of the time I’m left, in all honesty, with “I don’t know.” Maybe they mean something different from it than I do. Maybe I’ll cross whatever the line is before I finish giving the answer. Maybe I think I’m tired but I’m not. I can’t even give an answer like “38% chance I am tired” or “weary 10 ± 3, bored 2 ± 5.” I mean, honestly, how should I know if I’m tired?
Logic is a beautiful and functional tool. Once you get into its domain of crisply defined truth, be it discrete, continuous, probabilistic, or any of the stranger domains, logic works, it works like a charm, and it works every time. But finding the input truths to fuel this machine practically impossible.
I sometimes think logic ought to mimic the warning labels on consumer electronics, “opening case voids warranty”: “applying in real world invalidates truthfulness.” Logic is incredibly brittle. Given any false assumption you can prove anything else you want. This is only true of sufficiently expressive logics. But since people usually lump all logics into one big family “logic” that’s not much of a restriction. It’s like a car fuel tank with a posted placard “Warning: inserting a single molecule of water in this tank will cause the plant to implode.”
I love logic. It’s a wonderful tool. You can ask it, “if I were to accept x as true, what else would I also have to accept because of that?”—you can ask, and it answers. But I’d never suggest it as the way to understand the universe. Perhaps later I’ll discuss other not-suggested ways to understand the universe…. That’s just not what this machine is for.