## Wednesday, November 11, 2009

### A Simple Argument From Theism And Truth-Functionality To Bivalence

Classical logicians hold that statements can relate to truth in exactly two ways--'T' and 'F.' On the other hand, various deviant logics embody the assumption that the range of possible truth-values is wider than this. Perhaps "neither" or "both" is an option, or perhaps statements sometimes have a third truth-value that isn't best thought of us either the joint absence or the joint presence of the two traditional options, but as something else entirely. Maybe, when we figure all of this out, there are really five truth-values, or twelve, or....

Actually, no. Don't worry, gentle reader, I've come up with a devastatingly convincing deductive proof that this is not the case, that there are in fact two and only two real truth-values. In terms of logical machinery, this most excellent of all proofs relies on Modus Ponens alone, which is part of the overlapping consensus between all sorts of different logics, so no questions are begged. (Indeed, I often read claims in the literature that one of the conditions for a connective → in some system "counting" as actually being "a conditional" is that it satisfies Modus Ponens. This is terribly convenient for my purposes, so let's put aside any nit-picking issues about whether this is entirely reasonable given some solutions to various paradoxes involving implication and so on, and just assume that this claim is exactly right as stated.) As far as substantive assumptions, I'll assume truth-functionality--the truth-value of complex formulas is a function of the truth-values of their atomic components--which seems plausible enough, regardless of what you take the alethic options to be. Even less controversially, I'll assume that God exists and is just. Which is, like, obvious.

I mean, seriously, haven't you ever seen a beautiful sunset? Or a cute puppy? You have? Well, how do you explain those things without postulating an omnipotent and omnibenevolent being that created all things? Yeah, that's what I thought. You can't.

What? Some of you still aren't convinced? Really? For God's sake, haven't you people read any Plantinga? This could not be more straightforward. If I believe that God exists on the basis of the sunsets-and-puppies-argument, then I have every reason to be confident that my divinely-designed cognitive faculties got the job done and I'm not missing any possible objections. On the other hand, if you believe that God does't exist, and as such that your cognitive faculties arose naturalistically without any sort of benevolent entity supernaturally intervening to make sure they were reliable, then you can prattle on all you like about your "evidence" and your "arguments" against theism, and those of us who know we were created by God can just point and laugh. "Look at the evolved thing trying to come up with arguments with its puny little monkey brain! Haha!" And so on.

I'm sorry I had to slap you down like that, atheist readers, but I get so impatient with your failure to recognize the obvious truth of theism. So, that's enough of that, eh? We can assume the existence of a gloriously powerful and benevolent creator deity and go from there?

Good.

Now, assume there are only two truth-values. That means that for any formula with two propositional variables in it, the truth table will be four lines long. For example, take a simple truth table for conjunction.

α ∧ β
T T T
T F F
F F T
F F F

Nice, simple, clean little truth table there. Now, add another propositional variable.

(α ∧ β) ∧ ∂
T T T T T
T T T F F
T F F F T
T F F F F
F F T F T
F F T F F
F F F F T
F F F F F

Still only eight lines long. Takes about twenty seconds to write it all out long-hand. Pretty straightforward.

Now, assume that there's even one more possible truth-value. Since this argument should apply to any and all many-valued proposals, it doesn't matter exactly how we understand this third option--gappy, glutty, undecided, partially true and partially false, something else entirely--so we'll just write it as "O" for "other." Since "O" could be all sorts of things, and different ways of filling it in could have different consequences for how it impacts the truth-values of larger formulas O-valued statements enter into, and we want to be absolutely general here, we'll err on the side of extreme caution and just write down a ? for every formula with an O-valued component. So, here's a 3-valued version of the same truth table we just did.

(α ∧ β) ∧ ∂
T T T T T
T T T ? O
T T T F F
T ? O ? T
T ? O ? O
T ? O ? F
T F F F T
T F F ? O
T F F F F
O ? T ? T
O ? T ? O
O ? T ? F
O ? O ? T
O ? O ? O
O ? O ? F
O ? F ? T
O ? F ? O
O ? F ? F
F F T F T
F F T ? O
F F T F F
F ? O ? T
F ? O ? O
F ? O ? F
F F F F T
F F F ? O
F F F F F

Now, you want to know what the difference is between this truth table and previous one? They were for the same formula, but that second one was a huge pain in the ass to write up. And that's even typing it up, when you can copy and paste chunks and then go back to change bits. Imagine being a student in an introductory symbolic logic class in a world where some three-valued logic had replaced classical logic as the orthodox, establishment choice taught to beginning students with Hurley-type textbooks. You're given a quiz, and for something as simple as a fucking three-way conjunction, you have to write out that whole thing, long hand, and keep track of it all?

Yikes.

So, I submit the following obvious truth, which we can call the Principle of Divine Justice:

PDJ: A just God would arrange the universe in such a way that people could accurately represent its logical structure without going through the hassle of writing twenty-seven line truth tables for simple three-variable statements.

Now, since we know that God is just, it follows that God has arranged reality in such a way that the only ways that statements can be are "true" and "false." There are no other options.

QED.