Logical Consistency of Belief (or the Lack Thereof)

Last post, I showed that it is possible to believe two contradictory statements at the same time. Now I'm going to go a step farther and show that none of the rules of logic necessarily apply to beliefs at all.


First, an informal, verbal argument; there's a more rigorous proof in prop calc below. We discussed before the difference between believing a statement is false, and not believing the statement. The fundamental difference can be expressed thusly: When you believe that a statement is false, you are in a state of belief -- namely, a state of believing the statement's negation. When you don't believe the statement, you are not in a state of belief.

It is possible to believe two contradictory statements, but impossible to believe and not believe the same statement. This is because in the former case, you're in a state of believing the first statement, and in a state of believing the second statement. There's nothing inherently contradictory about being in a state of believing one statement and simultaneously being in a state of believing another statement. However, to believe and not believe the same statement would require you to be in a state, and also not be in the same state, which is inherently contradictory.

So, let's assume that the laws of logic apply to beliefs: that if you believe a statement, you also believe its logical consequences. (Obviously, since we've already established that it's possible to believe contradictory statements, the law of non-contradiction is excepted.) Well, now we've got a problem, because through a simple proof with the delightful name "Principle of Explosion," you can show that any and every statement follows from a logical contradiction. In other words, if beliefs always behave logically, then anyone who believes two contradictory statements must therefore believe everything.

Now, none of us has ever met a person who literally believes everything. But maybe that just proves that, while it's logically possible to believe two contradictory statements, for some reason nobody ever does? Except for one problem: plenty of people do believe contradictory statements, in the form of religious mysteries. "God is three" and "God is one", for example. "God is all-powerful, all-knowing, and all-loving," and "Suffering exists." The most common argument given by believers (not theologians, but believers "on the ground") is that logic does not apply to God. However, in order to believe in a being that transcends the rules of logic, ones beliefs must also transcend the rules of logic!

In short, over the last two posts we've established that beliefs need not be logically consistent with one another. That will be important in the next post, which I hope to get up this weekend -- and I guarantee that one will be entirely prop-calc free!

Rigorous proof:

The first step is to translate into symbols the statement "Beliefs obey the rules of logic." Now, all rules of inference ultimately take the form p -> q -- if one statement is true, then another statement must also be true. For example, the rule Disjunctive Syllogism can be stated as

((a V b) * ~a) -> b

((a V b) * ~a) is the p, and b is the q.

Any particular logical proof is going to consist a series of steps that take the form:

([Premise(s)] * [Rule of Inference]) -> [Conclusion]

Note that we're referring to the premises and conclusion of the step here, not the premises and conclusion of the entire proof. Using the generic p -> q form of the rules of inference, we can see that one way of stating "the rules of logic apply" is:

(p * (p -> q)) -> q

Unfortunately, we can't quite use this for beliefs. The reason is that the above assumes that the syllogism

Premise: p
Premise: q
Conclusion: p * q

holds. It does for logic, but does it necessarily hold for belief? We honestly have no idea, at this stage, whether (B(p) * B(q)) = B(p * q). Hence, we can't use (B(p) * (p -> q)) * B(q) as a way of saying "logic applies to belief" because it doesn't cover rules of inference with multiple premises, like Modus Ponens and Disjunctive Syllogism.

For those rules of inference, we need to use the form (B(p) * B(q) * ((p * q) -> r)) -> B(r). As we'll see below, this actually covers all the rules of inference. We'll begin with two premises. These create our "test believer" -- we're imagining a person who believes an arbitrary statement p as well as its negation ~p, and who also does not believe an arbitrary statement q. (Remember, although we proved B(p) * B(~p) is not a contradiction, B(p) * ~ B(p) still is one!)

Absurdly, we will have to take a break partway through to demonstrate that (p * q) -> (p * q). For some reason, p -> p is not one of the variants of the rule of inference Tautology. I've never understood why. And since (p * q) -> (p * q) both requires fewer steps to prove, and is the statement we actually need later in the proof, I took the detour.

1. B(p) * B(~p) , Premise (from prior proof)
2. ~B(q), Premise
3. (B(p) * B(q) * ((p * q) -> r)) -> B(r), Assumption
4. B(p) * (p -> r), Assumption
5. p -> r, 4, Simplification
6. (p*p) -> r, 5, Tautology
7. B(p), 4, Simplification
8. B(p) * B(p), 7, Tautology
9. B(p) * B(p) * ((p*p) -> r), 6, 8, Conjunction
10. B(r), 3, 9, Modus Ponens
11. (B(p) * (p -> r)) -> B(r), 4, 10, Induction
12. p * q, Assumption
13. q * p, 12, Commutation
14. p * q, 13, Commutation
15. (p * q) -> (p * q), 12, 14, Induction
16. B(p) * B(~p) * ((p * q) -> (p * q), 1, 15, Conjunction
17. B(p * ~p), 3, 16, Modus Ponens
18. p * ~p, Assumption
19. p, 18, Simplification
20. p V q, 19, Addition
21. ~p, 18, Simplification
22. q, 20, 21, Disjunctive Syllogism
23. (p * ~p) -> q, 18, 22, Induction
24. B(p * ~p) * ((p * ~p) -> q) , 17, 23, Conjunction
25. B(q), 11, 24, Modus Ponens
26. ~B(q) * B(q), 2, 25, Conjunction
27. ~((B(p) * B(q) * ((p * q) -> r)) -> B(r)), 3, 26, Disproof by Contradiction

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).

On the Glory of Meat

Is there anything meat can't do?

Different kinds of meat can run, crawl, swim, or fly. They can breathe water or air. They can grow in Antarctica and in the depths of deep-sea volcanic vents, from the bottom of the ocean to the upper canopies of the rainforests. Meat can jump hundreds of times its own height or run at 80 miles an hour. It can build a coral reef or chew apart a giant redwood tree.

Nobody debates these facts. We see meat do amazing things every day. A pump of meat can keep blood flowing through a man for decades, and legs of meat can carry him for just as long.

So why do people have such a hard time believing that meat can think? That it can talk? Why do they insist on positing some mystical outside force that enables meat to do so? Given all the amazing properties of meat, is meat that talks and ponders so much of a surprise?

I don't think so.