# Religion and the closed world assumption

008 August 11, 2013 --

Among the various arguments used to invalidate the existence of God, and implicitly religion, the one that atheists seemingly consider to be fundamental is closely related to Occam's razor: as there has been no proof of God's existence until now, it is superfluous to state that God exists. The argument is so strong that theists have no other option but to simply ignore it, an uncivilized attitude that is however the healthiest approach for their own good.

Occam's razor being nothing more than a principle based on common sense, the argument doesn't stand by itself. Rather, it is based on a formalized model of the world called the closed world assumption. One can guess from its name that it is also more of a principle than an actual truth, since it poses other hard questions such as, if the universe is closed, then is it possible for it to be finite? If it's finite, then what is "outside" of the universe? Is there a naked singularity? And so on, and so forth.

The notion of "closed world" can be easily explained using set theory (or category theory) and logic. Let $$U$$ be a possibly infinite set of objects, say, logical predicates, which we will use for simplicity, without sacrificing the model's validity. More exactly, $$U$$ describes the set of predicates which hold in a given, possibly infinite, universe. The closed world assumption states that all other predicates except those in $$U$$ are considered to be false. Therefore if we have a set $$U = \{p_1, p_2, p_3\}$$ and another predicate $$p_4$$, then $$p_4$$ will be necessarily false in $$U$$. This might seem trivial, but it isn't.

The opposite assumption is that of an "open world", a set $$U$$ split in three disjoint sets $$U_t, U_f, U_u$$. The first two contain predicates that are necessarily true or false respectively. The third set contains predicates whose truth values are yet unknown. Trivial? Definitely not. An observation is that the open world assumption is more closely related to agnosticism than to pure theism.

The advantage of the closed world assumption is that not only is it simpler, but it's also more convenient to use in practice. For example, the animal brain uses it to discriminate useful information from noise, an approach without which it would certainly overload. Prolog uses negation as failure to construct a Turing machine based on first-order logic, thus using the boolean false as a default value for undefined variables, which is essential for theorem proving.

However, the closed world assumption is not sufficient for a big part of real-world applications, specifically those that deal with partial information. In such situations, representing unknown information in the model can lead to much better solutions: for example, no one can know what each person on Earth is thinking right now. Assuming that they aren't thinking about anything (or that they aren't thinking at all) can be useful in some situations and disastrous in others, such as language interpretation, the typical example here being search engine optimization. Moreover, some models use uncertainty as a measure for the lack of information, particularly statistical and probabilistic models.

So maybe asking whether God exists isn't quite the right question from the purest scientific point of view, since it's now pretty clear that it's not something one can find out even with state of the art tools. The question should, I believe, rather be relaxed to "is God really necessary?", or "how much God should we allow into our lives?". It makes much more sense, as God is mainly a social construct, and it reevaluates religion not from a rigid, scientific point of view, but from an utilitarian one. It is also a proven method, given that statistically speaking, God is much more popular in undeveloped countries, while educated people prefer ditching it (or Him, or Her, or whoever that is) altogether.

Then again, in our mystical/spiritualistic conception of religion, we've deprived the concept of God of its worship-related semantics, or it's just that we're worshiping our own gods without calling them "God". But this is another story.