The Nature of Truth

[A-Man]

What's true to me is what convinces me it is (with sufficient "proof").

(without getting very philosophical)

The word "proof" is different in different contexts. I like to think there's a hierarchy of classes of "proofs".

• My mother is suddenly yelling at me for stealing the cookies. I see from the corner of my eye my an evil grin on my brother's face. That's "proof" he stole the cookies. That's (probably) reasonable. But not absolutely correct, as this kind would fail me if my brother was grinning for some other unrelated reason.
  
• In a court of law, in a reasonably small list of possibilities, you being the only suspect without an alibi is "proof" you committed the crime; say court learns the committer was wearing a  red shirt, and you happen to be the only one who was wearing a red shirt that day. That's reasonable. But not absolutely correct, as this kind would fail if you're a victim of a plan to incriminate you/police did a sloppy work/unlucky/..etc.
  
• In science, getting positive results for a theory in an experiment again and again is "proof" (appropriately, evidence) the theory is correct (works). That's reasonable, but note absolutely correct. There's the known quote by Einstein: "No amount of experimentation can ever prove me right; a single experiment can prove me wrong."
  
• In formal mathematics and logic, proofs rely on solid foundation that's the closest to "the metal" as you can get. Reasonable of course, and people tend to think you can't go wrong here. But that's not absolutely correct either. There's a lot to say about this.
  

I feel bad for making a list like that, because there are always things in between. Even in mathematics, sometimes certain proofs are favoured over the others.

So, sometimes in maths, if you are familiar with the context, you can allow semi-correctness.
For example, the probability that a 2x2 matrix with randomly generated real coordinates has a determinant that's exactly zero is 0%. But this doesn't mean that there is no matrix with determinant zero (in fact, there are infinitely many. Just let three of the coordinates be zero, and the last can be anything you like). What you can do in regard the matrices with zero determinant as "a degenerate case", and allow the assumption that all matrices have non-zero determinants.

Another example is if you pick a random real number between 0 and 1, then the probability that this number is rational is 0% (so if you want to integrate a function that is not defined for all rational numbers, you can just dismiss all the undefined values as "having measure 0" and proceed with integration).

Now, for a long time, mathematicians accepted this, because it was all "under their watch". Moreover beneath all of this, mathematicians had to agree on  a set of "axioms" or "truths" that they'd just believe in and work upon. But it's no big deal. Until you learn that the foundation is not really that solid.

So we talk a little bit about Mathematical logic. One would imagine that the foundation of mathematics would be consistent (non-contradictory); it would be contradictory if you can prove that A is true, and B is true, but (A and B) is not true, or if you can prove that A is true and that A is not true. You would also want the system to be complete (anything true in the system should be provable within the system).
For example, natural languages like English is a system that is not consistent; suppose A said "I'm lying". If you prove A is saying the truth, then he's lying, but then if he's lying, then he's telling the truth => inconsistent. Hodgepodge. (I think English would also be considered incomplete since it doesn't describe everything it should; you need to update the language to add new words ..etc after every number of years/decades/centuries).

So the goal of mathematicians was to ensure that the axiom system that underlies mathematics is both consistent (you can't construct sentences like "I'm lying") and complete (everything true is provable). They thought they had done so (most popularly with the ZFC axioms). But soon their bubbles would be popped when a guy called Godel proved that any axiom system that allows us to do basic arithmetic, for example, make statements like "After every natural number is another natural number", is incomplete. Such system would also necessarily be unable to prove its own consistency (Godel's Incompleteness theorems ). So now we just accept the incomplete, semi-consistent ZFC axioms as a set of axioms that can prove almost everything in maths (intuitionism).

So yes, logic in its current state is not absolute for even relatively unsophisticated systems, and so is the nature of truth if you look at it as a set of all provable true statements. That's why we need intuition to make some calls. The thing is that the moment we allow intuition to participate in the judgement, things get fuzzy, and different people will have different "thresholds" to what's convincing enough. While that makes things blurry, having a very big threshold as to considering very weak evidence makes you gullible, while having a very small threshold to not consider relatively strong evidence makes you ignorant. In both cases, your judgement would be unreasonable.

Without further complication, what's true to me is what has got what has fallen within my threshold of sufficient evidence. Or just, "what makes sense".