Is logic circular at the basic level? 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."
 A: 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.
