Moore's Paradox and Contradictory Belief

Warning: Propositional calculus ahead. If this bothers you, don't worry -- my posts are normally going to be written entirely in English. In this case, though, my reasoning was sufficiently complex that I had trouble keeping track of it myself, and using prop calc helped me make sure each step was valid.

Moore’s Paradox basically breaks into two statements: “It is raining” and “I believe it is not raining.” Does it really entail a paradox – are these two statements really contradictory?

First of all, the way Moore’s paradox is written, there are really three ways to approach it. The first is to consider the statements themselves, in isolation from the circumstances. In that case, all we are really saying is that there exists some state, and I believe it doesn’t exist. In other words, that I believe something false. No paradox there – I am not omniscient, and therefore there is nothing self-contradictory about me believing something that’s false.

Next we consider the fact that I am uttering these statements. Is that paradoxical? Well, again, no. I can utter any string of sounds I can pronounce.

The only non-trivial approach is to consider the claim that I am truthfully asserting both that it is raining and that I believe it isn’t raining. Now we have to consider what is meant by “belief” and “truthful”.

We might argue that “truthful” is synonymous with “true”. In that case, consider the case where it is raining, but I honestly believe it isn’t. First, for whatever reason, I say a statement which I mistakenly believe to be false: “It is raining.” In actuality, that statement is true, however. Thus, it is a truthful assertion. Then I say a statement which I believe to be true: “I believe it is not raining.” Once again, this statement is true. Thus, there is no paradox – I can truthfully assert both statements while believing only the second.

However, what if we define “truthful” as being about my mental state, rather than the truth? (In other words, that a truthful statement is the opposite of a lie, not a false statement.) In that case, we may have a paradox if we assume that two things are true: first, that truthfully asserting something implies that you believe that something, and second, that it is impossible to believe something while also believing its opposite. Written in logical notation* below:

A1) T(x) -> B(x)
A2) ~(B(x) * B(~x))

Let’s start with examining whether the first is true. That largely depends on how we “believe”. If we define it narrowly, such that belief implies some kind of emotional investment or deep conviction, then no, (A1) is not necessarily true. I can think that something is the case without being particularly affected by it. On the other hand, “belief” can also be more broadly defined as synonymous with thinking something’s true. In that case, (A1) becomes necessarily true: Under the definition of “truthfully” we are using, I cannot truthfully assert something without thinking that it’s true.

What, however, of (A2)? Can it be demonstrated to be true or false?

Well, to determine this, we’re going to need to put a premise out there: namely, that not believing a statement is not the same as believing the statement is false. For example, “I don’t believe in God” is not the same as “I believe there’s no God”. Let’s also declare as one of our premises that I believe something – we don’t have to specify what. Finally, we’ll test out (A2) from above by adding it as an assumption and see if we arrive at a contradiction.

1) ~(~B(p) = B(~p)), premise
2) B(p), premise
3) ~(B(x) * B(~x)), assumption
4) ~((~B(p) -> B(~p)) * (B(~p) -> ~B(p))), 1, material equivalence
5) ~(~B(p) -> B(~p)) V ~(B(~p) -> ~B(p)), 4, De Morgan’s
6) ~B(p) V ~B(~p), 3, De Morgan’s
7) B(~p) -> ~B(p), 6, material implication
8) ~(~B(p) -> B(~p)), 5, 7, disjunctive syllogism
9) ~(B(p) V B(~p)), 8, material implication
10) ~B(p) * ~B(~p), 9, De Morgan’s
11) ~B(p), 10, simplification

Step (11) contradicts premise (2), and therefore our assumption (3) must be false. It is possible for me to believe two contradictory statements at the same time**, and therefore Moore’s supposed paradox is not a paradox at all.

*My keyboard cannot easily support the logical notation scheme with which I am most familiar, so here’s a quick rundown of my attempt to construct something that looks like it from easily accessible characters:

T(x) I truthfully assert x
B(x) I believe x
~p negation of p
p * q p and q
p V q p or q (inclusive)
p -> q if p, then q
p = q p is logically equivalent to q

In proofs, each numbered statement will be followed by a comma, then its justification in italics -- the numbered statement(s) from which it's derived, and the rule of inference used.

**I’m not ready to give the proof, but intuitively I would say that this means the following is also not true: (B(p) * B(q)) -> B(p * q).