This question is maybe too broad and probably incredibly trivial. I feel like there must be some resources which answer me on the web but for a long long time I couldn't really find an answer to my question and after asking it to chat, I decided to make a post out of it. If this question is not appropriate, please try to edit.
My question is: Wikipedia says: 'A formula is satisfiable if it is possible to find an interpretation (model) that makes the formula true.' How is 'true' in this sentence defined? I know what a model is and the definition of the satisfaction relation etc. But the perspective 'we have some symbol set and created a syntax and now we are interpreting it and if the interpretation of the purely syntactical thing/formula is true in the interpretation(?), then the model satisfies the formula'. So in my mind, we inherit a definition of the truth from some theories which are already build (for example groups, real numbers etc.) and we are using them to define our 'fundamentalist truth'. And a side note, we are in first-order logic (and to be honest, I don't know in which degree this information is relevant).
And for example I don't understand principle of excluded middle because I don't understand in which setting it says a proposition true. I spend hours to understand it, and didn't even come close. I hope that this is a good justification to ask such a question. I hope that someone out there understand my problem and hits me with an explanation. I would really appreciate it.
EDIT: I want to explain my mind. I picture it like this: We have a syntax developed and we define a sequent calculus, and if we start with a list of formulas (taking them as axioms) then we deduce/prove other formulas. I picture this process as "sudden": (Picture)
(1) We have an alphabet and a language, so there is a universe where all the possible combinations lie, empty word and so on.
(2) We define what a formula is, this makes only some words, let's say blue. Now we have some blue things and some colorless.
(3) We choose some blue's as axioms and this is also suddenly make them green.
(4) Then we define a sequent calculus and this makes everything which are reachable from green words suddenly red. Axioms are also now red.
(5) We have done so far just syntactical things. We can for example define consistency and for example if everything is red if and only if the words which were green some time ago were not consistent.
(6) Now this was just one universe with a specific "base" (greens). We can picture now a multiverse structure, for each choose of a base (any subset of blue words), we have a universe. If we choose empty set, then what we get is the smallest universe, which contains "no matter what logically correct statements". If we have too much (like in the example (5)), then suddenly everything becomes red, so lot's of choices of green words makes whole universe blueless (red + colorless). This blueless'ness is equivalent to inconsistency of the base (which prove for a formula $\phi$, both $\phi$ and $\neg \phi$.
(7) I assume that what I wrote is correct (probably not rigorously, but as a "picturing".
(8) Now, some of the universes are bad, for example, everything is red. We want to choose a specific base, and we have some reds, some blues, and we want to also interpret formulas in a logically correct way, and we look if in this universe, what is red and what is blue correspond to the what we know about that mathematics.
(9) The risk is, we could just say that what in our mind blue/red is, should be blue/red, and that is it. This without logic, but we want to minimize the human error, and therefore we restrict to the mathematics we assume to the just bases and rules of sequent calculus.
(10) By the way, is there a theory, where this all things doesn't suddenly happen and there is a notion of "laziness"? I obviously don't know but, and this probably gibberish but maybe take a kind of type theory with a notion of laziness like in haskell? This is a side question, doesn't need to be answered.
(11) We prove some relative consistency results about some specific bases and there is lots of other things we can do of course.
(12) I honestly, don't see how can we break the circularity. We are taking sets as domains to the structures and we are creating set theory. We have functions already, but we are trying to define them properly. Is this actually a circularity? I don't want to ask circularity questions in this sense, because it is not the topic of the question, but still an answer would be appreciated.
(13) We choose a specific universe, than interpret the language in a good, structured way, definition of the satisfaction rule for example is also a piece of mathematics we "assume". We assume that this kind of perspective is good/strong, maybe someone can deny that and that's why there is alternatives to first order logic, like type theory.
(14) Debates about LEM are debates about what kind of a sequent calculus to choice.
(15) If the interpretation is good enough to satisfy the base, then we choose it as a model of our theory, and we are done.
(16) Interpretation part of the theory, make some syntax "true", and sequent calculus part of it make is "provable", and Gödel's first completeness theorem says that they correspond: If we had a green base, and now have some blues/reds, then a formula is in reds (provable) if and only if every interpretation which satisfies the green base also satisfies to this red formula.
(17) Now, let's go back to the LEM. Firstly, a proposition is what we called formula above. So let $\phi$ be a formula, instead of true/false, what should we use to denote false? Is it same thing to be unprovable? I am just about to grasp it, an answer to this questions should finish my endeavors.
So there is two side questions (10) and (12) and two main questions (17) and "Is my thought process correct? Is what I wrote above correct?". Don't consider side questions and this question consist of two questions which are similar. Sorry for the insufficient question in the beginning and I hope that now this question makes sense and opens.
Again, thank you!