Objectivity and Logic


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


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 "title": "Truth". As usual, I think an operational definitions can save us from groping around in the dark.


By Aumann's agreement theorem, if two people are both trying to come to a mutual understanding and they're both rational, then they should only disagree if their priors are different "title": "Aumann's agreement theorem". The fact people's beliefs often don't converge at the end of conversations implies that most people are (shockingly) not terribly rational and/or honest.

Moreover, the fact that most people go their entire lives without reading a single meta-analysis and the fact that even meta-analyses are far from perfect implies that most people don't have terribly accurate empirical knowledge of social science.

So, it should be pretty obvious that most people's beliefs don't match reality - that is most people for most issues are neither rational nor knowedgable enough for this theorem to apply, which begs the question why the theorem matters at all. The tangential answer is that Scott Aaronson has reaped a wealth of real-life lessons from it it "title": "Common Knowledge and Aumann's Agreement Theorem".


The relevant answer is that I believe Aumann's theorem gives a good grounding for objectivity. Namely, it implies that the only reason people disagree at the end of a conversation is because of insufficient time or 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 logical uncertainty [todo: cite]. 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, reasonable differences in priors will be overcome by sufficient 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. Since most people haven't read a single meta-analysis or literature review on most topics, the vast majority times people disagree the answer is simply that neither of them is sufficiently knowledgable.


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 logical should not be rigorously applied, but I want to push back against the idea that logic can not be applied. Since I use logic a great deal later on, I need 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 "title": "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 false.

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 weird 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 $Q$" is true
  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 be 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 might not be the only valid mode of reasoning. You might be able to reach true conclusions without logic - using common sense, emotions, etc. - but those conclusions better not contradict logic, because otherwise they are either false or just incoherent.

Works Cited [show]