EDIT
 Compressing Reality

Objectivity and Logic

Truth

What does it mean to be objective? Wikipedia says "title": "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 "title": "Truth". As usual, I think an operational definitions can save us from groping around in the dark.

Aumann

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 "title": "Aumann's agreement theorem". Not only that, their convergence mutual agreement should happen quickly "title": "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 and/or honest.

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.

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.

Finally, this is really what we always imply when we think someone else is wrong - we just don't come out and say it. Otherwise, we'd think all of our disagreements are merely differences in subjective experience - if that were the case, they wouldn't be disagreements.

Logic

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 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 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:
 $P$ $Q$ $P \Rightarrow Q$ 1 T T T 2 T F F 3 F T T 4 F F T

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 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.