Is logic circular at the basic level?

Question

When we are studying logic, we sometimes have to Prove certain logical equivalences. But If I use Logic to prove logical equivalences (or maybe some tautology),then that would be kind of strange, because that would mean We have proved a logical statement without even needing of learning mathematical logic. Does formal Logic start from describing certain formal logic ideas informally with Common sense and words?

For example, in science,Life is said to be the thing that is inside a living thing.What is a living thing? It is the thing that has Life.You see , it suddenly becomes a loophole.(This is kind of annoying)

I see a couple of different ways to avoid this "confusion":
(1)Abstract Life from living thing.
(2)Pretend that you know that life is life.

Is there a deeper point, where we can't abstract mathematical logic anymore from Philosophy , and have to accept that you already know in what framework they are talking about as common sense?(I am not talking about Axioms over here)

Edit: Interestingly , I have found a quote on a stack exchange post which is could be related to my question "Even the most robust and well-developed mathematical thought still ultimately rests on underlying primitive notions - base ideas and concepts which are "defined" by an appeal to experience, or "common sense", and upon which a myriad of derived concepts are constructed. While it is desirable for these to be as "primitive" as possible, ultimately, modern mathematicians and philosophers are all acutely aware that these primitive notions are, in the end, essentially arbitrary from a philosophical standpoint."

As usual, theories are built from axioms, which are accepted without discussion and regardless a concrete interpretation. — , Oct 18 '20 at 12:56
I couldn't really think of a good title.Any suggestions for the titile? — Kripke Platek, Oct 18 '20 at 13:02
look up truth tables, it could be helpful in seeing where some formal logic ideas come from — bfff, Oct 18 '20 at 13:02
It seems to me that at some point in the development of any human language, the usage of words like "not", "and", "or", "if", "all" and "some" had to be standardized. Over thousands of years, as social arrangements became more and more complicated, this process of standardization inevitably led to symbolic logic and formalisms. — Dan Christensen, Oct 18 '20 at 14:36
See the answers and comments for these Mathematics Stack Exchange questions. Also of possible interest are my answers to How do I convince someone that $1+1=2$ may not necessarily be true? AND Highly Rigorous Logic Book AND Foundation of Formal Logic. — Dave L. Renfro, Oct 18 '20 at 15:14
Your example with natural language show that it is not a formal language: it allows for circularity (because it is a closed system). To avoid paradoxes, formal languages are not closed system: they cannot express all (see the Liar Paradox). More specifically, a formal language needs rigorous definitions: thus, we cannot define all, simply because we have to start from somewhere, i.e. form (few) undefined concepts. — Mauro ALLEGRANZA, Oct 18 '20 at 15:17
Then is mathmatics nothing but a language which is not closed and need undefined terms (primitive notation?) to avoid paradoxes? — Kripke Platek, Oct 18 '20 at 16:47
And what makes Philosophy different from mathematics (or other formal languages?).Is one more generalized than the other? — Kripke Platek, Oct 18 '20 at 16:52
I've seen, as a joke, things like the following: sociology is just applied psychology, psychology is just applied biology, biology is just applied chemistry, chemistry is just applied physics, physics is just applied math, math is just applied logic, logic is just applied philosophy, philosophy is just applied sociology, $\ldots$ — Dave L. Renfro, Oct 18 '20 at 17:01

score 3 · Accepted Answer · answered Oct 18 '20 at 18:19

There is no perfect way out of this. Carroll discusses this here, but even that isn't the most extreme facet of the issue: how do we guarantee that we can even communicate intelligibly? Given that all you know about how I use language is the general patterns my use of language follows, on what basis can you be confident that you and I assign the same meanings to the words we use?

There has been a lot written on this sort of "super-skepticism" - see e.g. Wittgenstein - and ultimately it's something we simply have to come to terms with: that one of the assumptions we make when communicating with each other is that we have at least a small amount of common understanding of language, and that one of the assumptions we make when reasoning is that we have at least a small amount of common acceptance of logical deduction. For that matter there's another level of "base circularity" in mathematics, namely the ultimate reliance on (at least) finite strings ... which are themselves mathematical objects; this general issue of requiring (some) mathematical objects to ground mathematics has been discussed for example here.

Now the above (in my opinion anyways) is not to say that we can't fruitfully discuss the general topic of the OP or its thematic cousins! It's just a cautionary remark about what sort of resolution we can expect.

If no language is perfect , then what makes formal language different from other languages? — Kripke Platek, Oct 19 '20 at 06:26
@Prithubiswas What? That's a total non sequitur. Just because nothing's perfect doesn't mean that some things aren't better in certain respects than others. — Noah Schweber, Oct 19 '20 at 06:28
What I mean by "no language is perfect" is that every language requires some level of human common sense to be undetstood fully. For example,you cant explain to a ant about truth tables because an ant doesn't have the level of common sense as a human. "Just because nothing's perfect doesn't mean that some things aren't better in certain respects than others " - that is absolutely true.that is why I guess humans (and me) are happy to ask these question and study about math (even if they are not perfect,they are still really interesting).My question was , when will we call a language formal? — Kripke Platek, Oct 19 '20 at 06:43

Is logic circular at the basic level?

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