Objectivity and Logic

But every error is due to extraneous factors (such as emotion and education); reason itself does not err.
Kurt Godel
Logic is justly considered the basis of all other sciences, even if only for the reason that in every argument we employ concepts taken from the field of logic, and that ever correct inference proceeds in accordance with its laws.
Alfred Tarski


What does it mean to be objective? Wikipedia says Objectivity (philosophy)

Objectivity means the state or quality of being true even outside of a subject's individual biases, interpretations, feelings, and imaginings.

Of course, this definition just passes the buck to what it means for something to be true, and this has a wide array of proposed definitions Truth. As usual, I think an operational definitions can save us from groping around in the dark.


By Aumann's agreement theorem, if two rational people are both honestly trying to come to ascertain the truth, then they should only disagree if their priors are different Aumann's agreement theorem.. Not only that, their convergence mutual agreement should happen quickly The Complexity of Agreement. The fact people's beliefs often don't converge (never mind quickly!) implies that most people are (shockingly) not terribly rational, dishonest, or have shockingly strong priors.

I believe Aumann's theorem gives a good grounding for objectivity. Given that priors (beliefs without evidence) should be weak, the theorem strongly suggests the only real reason people disagree at the end of a conversation is because of dishonesty and/or poor rationality. So, we can define something as objective if

A claim is objective if and only if the vast majority of sentient beings would agree that it is true if only they were intelligent, rational, and experienced enough.

We have to include the "intelligence", because (unlike ideal reasoners) people suffer from "bounded rationality", which is neccessary for ideal probabilistic reasoning. We have to include "experienced", because people should update their beliefs based on evidence (see next chapter) - not just conversations. Finally, we have to include "vast majority" because reasoners can differ in their priors. Fortunately, priors should generally be weak and easily overcome by good evidence.

Preempting Bad Feelings

This kind of definition may rub some people the wrong way. After all, this means if I disagree with you, I'm implying that the reason is that you're just not intelligent, rational, or experienced enough - well, that, or that I'm not intelligent, rational, or experienced enough - but who thinks that about themselves?

However, there are, in my opinion, good reasons this shouldn't evoke a negative reactions. First, when two people disagree, it should never be in a binary way. That is, Alice should never think X is simply true while Bob thinks it is false. Instead, if Alice and Bob are rational, they'll "merely" assign different probabilities to X being true.

Moreover, when most people disagree, it's often the case that neither is knowledgable enough to reliably discern the truth. For instance, if two people are arguing about global warming but neither has read a meta-analysis on global warming, in what sense is their conversation actually about the truth? Ditto for gun control, tax cuts, and the majority of policy issues - not to mention things as straightforward as whether or not you should floss, take multivitamins, or even whether smoking cigarettes is harmful. Indeed, even experts often disagree due to lack of knowledge in their own fields. In other words, a enormous chunk of disagreement is caused simply by neither party having enough knowledgable - intelligence and rationality don't even get a chance to come into play.

Finally, these implications are what everyone really implies anyways when they think someone else is wrong - that they lack the intelligence, reason, or knowledge to come to the correct conclusion - that or they're biased and/or lying. It's just most people don't don't come out and say it.

There's also value-disagreement, but that's a separate (though related) discussion.


On a related note, I sometimes hear from others that there are certain subjects you can't apply logic to. I agree that there are often situations in which logic shouldn't be rigorously applied, but I want to push back against the idea that logic can't be applied. Since I use logic a great deal later on, I want to justify using it in all contexts, but first I need to make a distinction.

When most people say "logic", what they really mean is "a hybrid of accepted rules of reasoning and common sense". This is very different from what mathematicians mean by the term. In mathematics, logic is basically a collection of reasoning rules that are true by definition.

For example, the most famous rule is modus ponens, which states that if $P$ is true and "$P$ implies $Q$" is true, then this means $Q$ is true. There are a plethora of other rules List of rules of inference., but this should suffice as an example of why mathematical logic is undoubtedly valid - in all subjects.

Mathematicians define the word "implies" using something called a truth table:
$P$$Q$$P \Rightarrow Q$

In the above truth table, we wrote down all combinations of true and false for $P$ and $Q$, and then we wrote what the true and false values would be for $P \Rightarrow Q$ (i.e. "$P$ implies $Q$").

You can tell by looking at the table that if both $P$ and $P \Rightarrow Q$ are true, then our universe must be #2, which means $Q$ must be true.

Now, at this point, you might be wondering if our definition of "implies" could be wrong. The answer is, and this is important, that definitions can't be wrong - they can only be unintuitive.

Let me clarify. I'm not saying you can define things however you want and then say whatever you want and claim that your listener is only misinterpreting you. Shared definitions are an extremely useful convention that allow society to function.

What I'm saying is that within a conversation, you and your partner(s) can agree to adopt definitions that are convenient inside the conversation but might deviate from social norms - after all, mathematicians in English-speaking countries agree with those in French-speaking ones.

What this means is that even if you think we defined "implies" wrong, you still can't reject the line of reasoning - though you may, of course, request we use some other word instead. Alas, hundreds of years of mathematical convention are against you.

Now, some people feel that this isn't a good enough demonstration. Maybe, we made a misstep in our reasoning. On some level, this kind of critique is impossible to respond to: we have to assume that we aren't completely insane. On another level, I do have a response: the reason modus ponens works is because we'll never say "$P$ implies $Q$" unless modus ponens would work.

To elaborate, if you disagree with modus ponens, then you believe there are some statements, $P$ and $Q$ such that

  1. $P is true$
  2. $P$ implies $Q$
  3. $Q$ is false

If you found such an example, then I would just respond that your second assumption is clearly false. Since $Q$ is false and $P$ is true, it can't bee the case that $P$ actually implied $Q$.

From this point of view, logic is merely a feature of language. The mere fact that words have distinct meanings requires there to be relationships between words. If we define all humans to be mammals and all mammals to be animals, we must define all humans to be animals. To not do so would be to reject any coherent language in lieu of random sounds.

[You can, of course, say "but what if humans are an exception and aren't animals?". To this, I'd reply, then clearly not all mammals are animals, so you made an incorrect assumption. As usual, logic isn't wrong, you are.]

Ultimately, logic may not be the only valid mode of reasoning. You might be able to reach true conclusions without logic - using common sense, intuition, emotions, statistics, tarot cards, etc. - but those conclusions better not contradict logic, because otherwise they are either false or just incoherent.

Aumann's agreement theorem. (2017, March 12). In Wikipedia. Retrieved December 10, 2017, from https://en.wikipedia.org/w/index.php?title=Aumann%27s_agreement_theorem&oldid=770007863 List of rules of inference. (2017, June 3). In Wikipedia. Retrieved 12:14, June 16, 2017, from https://en.wikipedia.org/w/index.php?title=List_of_rules_of_inference&oldid=783611660 Objectivity (philosophy). (2017, June 24). In Wikipedia. Retrieved 19:07, July 4, 2017, from https://en.wikipedia.org/w/index.php?title=Objectivity_(philosophy)&oldid=7873607363 The Complexity of Agreement. Proceedings of the thirty-seventh annual ACM symposium on Theory of computing, 634-643. https://www.scottaaronson.com/papers/agreestoc.pdf Truth. (2017, June 10). In Wikipedia. Retrieved 11:54, June 16, 2017, from https://en.wikipedia.org/w/index.php?title=Truth&oldid=784917633