Using mathematics or logic to reason with logic itself, and it is a surprisingly hard thing for many people to do, not just me, because while humans have logic, using formal rules, formal language, formal systems and formal proofs to explain logic itself, makes everything quite abstract and it likely leads to a lot of confusion. Results in metalogic primarily involve demonstrating properties of particular formal systems including consistency and completeness.
This is currently one of my research foci, because this is a pretty hard topic! Meanwhile...
Here are some rules (even if not formally metalogical):
1. Correlation is not
necessarily causation
2. We can never know
everything
3. "Always" and "never" are almost never true. Society may exaggerate it or use it
as a heuristic. This is because there is likely to be at least one
exception in daily life!
You only need one counterexample to disprove an "always true/always
false" argument!
Rather, for instance, it is nearly impossible to attempt a study on the
entire population of Earth!
Other rules: Five more pillars of logic
What strongly seems true can still be false so it’s natural skepticism.
Lack of evidence is not disproof, argumentum ad ignorantiam, so that’s why police still investigate cases that seem to be false. (And lack of disproofs is not evidence.)
Popperian Science says that: scientific knowledge by falsifiability rather than induction, asserting that theories can never be proven true, only refuted. But for some things, I generally disagree, though it’s hard to prove that something cannot be disproven.
(Those 2 are important for criminal investigations.)
Principle of falsification: You only need one counterargument to disproof an “always” argument. For instance, all primes are odd -> 2 is prime and even.
One way to prove an “always” argument true is by exhaustion: each and every case. For instance, to prove that every McDonald's has some yellow in it (though insane): Must go to literally every single one and look at a place that is yellow, e.g. the logo. However, this only works for finite sets, that’s why it’s more complicated for Math. (Instead of going to every McDonald's, you prove:The first McDonald's has yellow.If any given McDonald's (n) has yellow, then the next McDonald's (n+1) must also have yellow by design as without the Mcdonald’s logo, it would not be a Mcdonald’s. Therefore, all n locations have yellow.) Wait… blue mcdonalds disproof? Luckily, the cheese is yellow.
Principle of truthification: You only need one argument to disproof a “never” argument
ECREE (extraordinary claims require extraordinary evidence): The strength of evidence must match the magnitude of a claim.