Have you ever used predicate calculus to prove the validity or invalidity of an argument? It's fun, in a nerdy kind of way, and a lot less terrifying and tedious than it sounds. Let's give it a try! How about we test the argument "the irreducibly complex nature of the eye proves God's existence"?
Aw, come on. What would Jesus do?
Leaving aside the abject pseudo-scientific fatalism of declaring (usually without a shred of evidence other than the claimant's own bewilderment) any naturally occurring structure as beyond naturalistic explanation, and also the question of whether the eye really is irreducibly complex (for our purposes it makes no difference), let's approach this from a strictly logical point of view. Is the argument logically valid? Essentially we can boil it down to this:
- The eye evolved or was created, but not both
- If it did not evolve then it was created
- If it was created then there is a creator
- The eye is irreducibly complex and thus did not evolve
- Therefore there is a creator
Seems fairly sound, on the face of it. But that's why we have formal systems of logical analysis: natural language can sometimes fool us into thinking things that only seem to be correct. So as a first step, let's assign a label to each of the claims it makes (known as "predicates"):
| Label |
Predicate |
| a |
The eye evolved |
| b |
The eye was created |
| c |
A creator exists |
Now we need to connect those predicates together to restate the argument in a formal way. We'll need four of the symbols defined by predicate calculus to do this, and those symbols are:
| Symbol |
Meaning |
 |
And (true if both predicates are true) |
 |
Exclusive-or (true if one predicate is true and the other false, but not both) |
 |
Not (true if the predicate is false) |
 |
Implies (we'll get to this in a moment) |
The meanings of these symbols, or "operators," follow pretty closely to the "natural" sense we use them. The only slightly tricky part is (implies), which has one somewhat counter-intuitive aspect until you think about it for a bit. The statement "a b" is considered false if a is true and b is false, but true in all other circumstances. This is the mathematical meaning of "implies" rather than the natural one: the only time an implication can be proven false is when the hypothesis (a) is true but the conclusion (b) is not. Makes sense, right?
Okay, so this is where the real nerdiness begins. Let's take those predicates (a, b and c) and the operators and begin stating the argument formally, line by line:
a b
The eye evolved (a) or the eye was created (b) but not both.
a b
If the eye didn't evolve then it was created.
b c
If the eye was created then there is a creator (c).
a
The eye did not evolve.
Since all these have to be true in order for the final line (the conclusion – that a creator exists) also to be true, we join them together with ands:
(a b) (a b) (b c) (a)
The parenthesis around them is to clarify which operators are affecting which predicates. There are rules to it but in this case it's pretty straightforward, since I've already gone through it step by step.
Next we join this (the hypothesis) with the conclusion using another (implies):
((a b) (a b) (b c) (a)) c
In other words, we're saying that the hypothesis is logically equivalent to the conclusion, since that's another meaning of . If the two do turn out to be logically equivalent then we say the argument is logically valid. If they're not, it isn't.
That's all well and good (and deliciously nerdy), but what do we do now? Well, we could deduce an elegant formal proof using the fundamental axioms of predicate calculus, but I don't have the faintest idea how to do that and, let's be honest, nor do you, so let's take the brute-force approach: we'll construct what's called a truth table.
Truth tables allow us to test all combinations of true and false for all the predicates in the statement. The truth table for (not) looks like this:
| a |
a |
| T |
F |
| F |
T |
Thus when a is true, a is false, and when a is false, a is true. Let's look at the truth table for (and) as another example:
| a |
b |
a b |
| T |
T |
T |
| T |
F |
F |
| F |
T |
F |
| F |
F |
F |
Easy, yes? You betcha. But it gets even easier! We can use instead what's called an abbreviated truth table, which represents the same information more compactly. Again for it looks like this:
| a |
|
b |
| T |
T |
T |
| T |
F |
F |
| F |
F |
T |
| F |
F |
F |
As you can see, the third column of the "full" truth table is now the second column of the abbreviated version. We simply write the outcome of applying the operator to the two predicates underneath the operator itself.
Remember how I said "this is where the real nerdiness begins"? Well, this is where we get into the totally cool shit, which is actually demonstrating the validity or invalidity of the argument! I know you're just as excited as I am.
Now we make another truth table, except this one represents the entire argument as constructed above. It might look a little daunting, and you have two options: try to follow along with the truth values (and my explanation is, admittedly, probably not adequate for that, but with this primer at hand you should have no problems) or scroll past in horror and trust I'm honest. Either works for me, though I recommend the latter.
So here it is, in all its majesty:
| (( |
a |
|
b |
) |
|
( |
|
a |
|
b |
) |
|
( |
b |
|
c |
) |
|
( |
|
a |
)) |
|
c |
| T | F | T | | F | | F | T | T | T | | F | | T | T | T | | F | | F | T | | T | T |
| T | F | T | | F | | F | T | T | T | | F | | T | F | F | | F | | F | T | | T | F |
| T | T | F | | T | | F | T | T | F | | T | | F | T | T | | F | | F | T | | T | T |
| T | T | F | | T | | F | T | T | F | | T | | F | T | F | | F | | F | T | | T | F |
| F | T | T | | T | | T | F | T | T | | T | | T | T | T | | T | | T | F | | T | T |
| F | T | T | | T | | T | F | T | T | | F | | T | F | F | | F | | T | F | | T | F |
| F | F | F | | F | | T | F | F | F | | F | | F | T | T | | F | | T | F | | T | T |
| F | F | F | | F | | T | F | F | F | | F | | F | T | F | | F | | T | F | | T | F |
The things I do for you.
Okay. Now what? Look at the column highlighted in yellow (the second from the right). The rules say that when testing logical equivalence in this way, if that column is all true, then the hypothesis is logically equivalent to the conclusion. And in this case, indeed, we can see it is.
So wait. The argument is logically valid? Yup. Weren't expecting that, were you?
But that says nothing about the truth of the argument. Formal logic is silent on that matter: all it can tell you is that the conclusion is true if and only if all the predicates in the hypothesis are true as well.
That is to say, provided a and b above are both true (in the real, everyday sense) then c is also true. In this argument, however, that's not the case. The problem lies with this part of the hypothesis:
(a b) (a b)
Remember, this is saying "the eye evolved (a) or the eye was created (b) but not both, and if the eye didn't evolve then it was created."
But a moment's thought (which is a moment longer than most creationists and supporters of "intelligent design" give it) reveals this is not true. Even if the eye didn't evolve, that doesn't necessarily mean it was created. What this amounts to, as Richard Dawkins (amongst many others) says, is nothing more than falling back on a default: god. But the default was chosen arbitrarily. I could just as easily say "if the eye didn't evolve then it was brought to Earth in the belly of a faster-than-light fruit bat by Derek the generous but slightly shy hacksaw." The argument would still be logically valid, but it would also remain utterly arbitrary and thus preposterous to anyone with half a brain.
If creationists (or even proper scientists, for that matter) do ever find something in the realm of biology that is genuinely irreducibly complex, i.e. which could not have evolved by gradual steps (and this is looking less and less likely with every passing news report of yet another discovery that reinforces evolutionary theory), all they'll have proven is that evolution is wrong, not that creationism (nor, sadly, Derekism) is right. There's a big difference.
Despite being absolutely convinced that not a single one of you is still reading, I'll sum up by saying that, first of all, logic is a beautiful, beautiful thing, but its beauty is no guarantee against it being abused by the demented. Also, yes, I'm fully aware that I need to get out more.
