Write more thoroughly, give some links
this is not done yet. If you are interested, refer to the links instead. (and don’t expect what I will randomly write here to exceed these professional materials!)
The appendices are intended as a window for some interesting topics, and unfortunately also a storage place for inmature thoughts. So please refer to professional works for further information.
As your ‘classmates’ have mentioned in the conversations, the consistency of a logical system means whether there are conflicts in it. If an arithmetic system (with boolean operations as well as ) asserts ‘2 + 2 = 5’ after some deduction, while keeping other concepts aligned to the elementary textbook, then it is inconsistent. We are also aware now that a system can not be easily judged as consistent.
Concepts like ‘consistency’ are something that describes the whole system. It’s like we praise a car as ‘well-made’ if every screw and gear fit well. Therefore, people are eager to measure such properties.
There are some other ideas like that, one of which is completeness. A system is complete when it can describe everything in its vocabulary. For a logical system, it should be able to prove or falsify any statement.
Consistency still lives in the syntactic world, that is, the system
itself. But completeness can communicate with the semantic world as well.
In fact, a common way of proving incompleteness is to find a statement
that has clear semantic truth value but can not be deduced in the system.
Completeness is thus defined as, “all that is true can be proved in the
system”.
The concept computability and decidability are deeply interwined with computer science, which is usually linked with some kind of problems, just lik SAT. Some problem is computable if we will finally know the answer in finite time. (The undecidable may just end up with a loop.)
Conversely, there is another property, ‘soundness’, saying “all that can
be proved in the system is true“.
People say propositional logic is sound and complete, and consistent. But just is there something that is not?
Well, the not-so-perfect systems are easily built. To be inconsistent, we can just introduce a rule that proves a statement and its negation together, for example A and not A. To be incomplete, we simply remove every syntactic rule in the system. To be unsound, we simply introduce a rule that proves something false.
However, these properties are not independent. We know that everything is provable from a contradiction, which is exactly what inconsistency grants. Therefore, if some logic (which admits ‘everything is provable from a contradiction’) is inconsistent, it is then complete.
Mathematicians used to think that some common logical systems were consistent and complete, until the famous Theorem of Incomplete stood out. It is basically derived from a ‘liar paradox’, where one sentence claims itself to be false. From then on, people understand that if the system is powerful enough, it can not be both consistent and complete.
https://en.wikipedia.org/wiki/Completeness_(logic)
https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems
Read along about reasoning like humankind.