Wednesday, August 19, 2015

Consistency and the material conditional

This is part of my series on debugging the ontological argument.

In the previous post of this series, I introduced Gödel's Ontological Argument (GOA) by discussing all the things Gödel got right.  I also use this discussion as a means of gradually breaking down the argument without throwing all of it in your face at once.1
The three major parts of the GOA are:
  1. God is consistent (proven in earlier steps, to be discussed in next post).
  2. If something is consistent, then it is possible.
  3. If God is possible, then God is necessary (for reasons already discussed).
 Here I will discuss the second step.  At first it is highly counter-intuitive, but upon understanding the logic, you will find that the reasoning is valid, and even trivial, but still unsatisfying.

Material consistency

The intuitive definition of consistency is that there are no contradictions.  To say proposition S is consistent is to say that S does not lead to any contradictions.  But there's a lot of work done by the phrase "lead to".  In logic, we might translate this to logical implication. $$\lnot \exists Q ( S \Rightarrow (Q \wedge \lnot Q) )\tag{1}\label{1}$$ In English, this is "There does not exist a proposition Q such that S implies both Q and not-Q."  Here, "implies" is logical implication, also sometimes called the "material conditional".

The material conditional is notorious for causing countless errors in students of math and logic.2  The statement "If S, then R" means that either R is true, or S is false (or both), and it says nothing about the conceptual connection between R and S.  For instance, I can say, "If Mars is a giant egg, then the moon is made of cheese." and it would be literally true, because Mars is not in fact a giant egg.

The material conditional makes for a particular definition of consistency, which I will call material consistency

If S is materially consistent, then S must be true!

The reasoning is actually quite trivial.  The only way for a material conditional "If S, then R" to be false, is if S is true and R is false.  Thus, if S does not imply a contradiction, then S must be true.  In fact, if there is anything whatsoever that S does not imply, then S must be true.

You should find this a rather unsatisfying definition of consistency.  Literally everything that is untrue is materially inconsistent.  It is inconsistent for me to live in Southern California.  It is inconsistent for you to be standing next to me.  It is inconsistent for my pen to be out of ink.  Is that really what we want to say?

Strict consistency

The major reason that so many people find the material conditional counterintuitive is that we are accustomed to so many other kinds of conditionals in natural language.  For example, I am much entertained by these examples documented by Language Log:
If you want to know, 4 isn't a prime number.
If Eskimos have dozens of words for snow, Germans have as many for bureaucracy.
It's all perfectly normal — if troublesome to varying degrees.
Language Log refers to these as the biscuit conditional, bleached conditional, and concessive conditional respectively.  In my personal experience, it's a standard joke among math and logic enthusiasts to naively interpret a conditional statement using the material conditional even when it doesn't make sense.

But even when people are using a more logical kind of conditional, they are often thinking of more than just the material conditional.  For example, the statement, "If Mars is a giant egg, then the moon is made of cheese," might mean that Mars being a giant egg might somehow physically cause the moon to be made of cheese.  Or perhaps it means that given the counterfactual universe where mars is a giant egg, then the moon would also be made of cheese.

Since this series is primarily concerned with what can be translated to symbolic logic, we will take a particular conditional called the "strict conditional", also called entailment.  Symbolically, I'll distinguish between implication and entailment by using different kinds of arrows: $$S \Rightarrow R\tag{2}\label{2}$$ $$S \rightarrow R \tag{3}\label{3}$$ Statement \ref{2} means "S implies R", while statement \ref{3} means "S entails R".  S is said to entail R if S implies R in all possible worlds.  We've already built modal logic to make sense of the concept of possibility, so we might as well use it.

The strict conditional leads to another definition of consistency, which I will call strict consistency.  "S is strictly consistent" means $$\lnot \exists Q ( S \rightarrow (Q \wedge \lnot Q) )\tag{4}\label{4}$$ If S is strictly consistent, then S must be possible.

This follows trivially, since if S does not entail a contradiction, then there must be at least one possible world where S does not imply a contradiction.  As argued before, if S does not imply a contradiction in some possible world, then S is true in that possible world.

Why this is unsatisfying

In my history of arguing about the ontological argument, I find that some people find it "obvious" that God is consistent.  There's simply no contradiction to be had in the definition of a perfect being.  And then, you hardly need the rest of the GOA, you just assert that God therefore exists.

The problem I have with this argument is, what kind of consistency do you think is so obvious?  The very idea of strict consistency has nothing whatsoever to do with the definition of God, so it doesn't matter how sensible (or insensible) the definition of God is.  All that matters is whether God exists in a possible world.  If not, then God implies a contradiction, because using the material conditional, God implies absolutely everything.

Furthermore, the idea of strict consistency relies on the idea of possibility, which in turn relies on our choice of modal logic semantics.

For example, consider the statement, "If the moon is made of cheese."  Let's consider "possibility" to refer to all possible pasts and futures.  I don't believe that the moon is made of cheese in any possible past or future, therefore, the very idea is strictly inconsistent.

Now, let's consider "possibility" to instead refer to all universes with the same physical laws.  I believe it's physically possible to have a moon with the same chemical makeup as cheese, and thus we would say that the idea is strictly consistent.  But which is it?  Is a moon made of cheese consistent or inconsistent?

In the next post, I will discuss how God's consistency is proven in the context of the GOA.


1. If for some reason you really do want the entire argument thrown at you all at once, I wrote up Gödel's ontological argument step by step in 2009.

2. A good exercise, if you've never seen it, is to try the Wason selection task.  The task is, given a number of cards in front of you, to decide which cards need to be flipped to verify a particular hypothesis about them.