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I've been having thoughts for a while about what constitutes a proof. Formal logic usually consists of incredibly detailed steps and, as such, is usually not utilized that often in everyday life.

However, we often ask people to do things such as "prove that you were not at place X at time T". While some people may use purely objective logic to do this (such that it's undeniable to everyone that the proof is presented to), there may be places in which a subjective proof is more practical. When I say subjective, I mean that the proof is a conversation between the prover and the person who the proof is being presented to. At the end of such a proof, the validity need only be undeniable to the subject of the proof. I would say that the Turing Test is an example of a subjective proof, since an observer could just be watching two random sentence generators and mistake them to be humans. The determination of the validity of the proof is left to the person who the proof is being presented to. The proof is also mainly intended for that person/subject.

I haven't found anything on this, so are there more proofs of this nature? Does the field of logic allow subjective proofs or are all proofs strictly objective (if you prove something for someone, it must be provable to everyone)? Any help on the subject is greatly appreciated.

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If I dare flip that on its head, "formal" proofs are not entirely as objective as one thinks, they're just objective enough that very very few people in the world have the mathematical background to identify the holes (unless you choose to define the word "objective" around them, of course). Tarski was one such individual. He tackled issues such as attempting to provide a formal definition for "true" and identifying limits of formal language.

There are actually many variants of "proofs" out there. One I find is a statistical proof known as a "zero information proof." The example most often given is that you see two openings to cave systems. You haven't found a connection between them. Someone claims they are connected, and offers to show you how for $1000. However, they don't trust you with the secret - they want to be paid up front. You declare you wont trust them until they've at least proven such a path exists. What a conundrum!

The zero information proof is a proof that is designed to give no information while it is being undertaken, a very useful thing for cryptography. In this example, you turn around, and they randomly enter one cave entrance (left or right). They go out of sight, and then you shout at them either "left" or "right" randomly. They are expected to be able to come out of the requested entrance. If they know the way, they can do it 100% of the time. If they are lying, then they can only do it 50% of the time. One can repeat this process multiple times to be arbitrarily sure they know their stuff before forking over cash!

It's worth noting that the zero information proof doesn't proves something is true, it proves something is statistically likely to be true. A proof which proves something is true is much more difficult, and is what "formal proofs" went after. However, even they run into limitations with their language, because formal proofs are traditionally provided in a formal language with a given interpretation (such as First Order Logic with Tarski's interpretation)

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It seems that you re mixing two meanings of the term "proof" : demonstration and evidence.

Regarding what constitute a valid demonstartion, I think that we cannot have "subjective demonstrations".

See Yuri Manin, A Course in Mathematical Logic for Mathematicians (2010), page 45 :

A proof becomes a proof only after the social act of “accepting it as a proof.” This is as true for mathematics as it is for physics, linguistics, or biology. The evolution of commonly accepted criteria for an argument’s being a proof is an almost untouched theme in the history of science. In any case, the ideal for what constitutes a mathematical demonstration of a “nonobvious truth” has remained unchanged since the time of Euclid: we must arrive at such a truth from “obvious” hypotheses, or assertions that have already been proved, by means of a series of explicitly described, “obviously valid” elementary deductions.

Thus, the method of deduction is a method of mathematics par excellence.

[...] Every proof that is written must be approved and accepted by other mathematicians, sometimes by several generations of mathematicians. In the meantime, both the result and the proof itself are liable to be refined and improved.

The historical "variation" of what are the criteria for an "acceptable" proof does not imply that mathematics and proofs are supra-human : they are human (and social) activities; thus their "objectivity" is basically inter-subjectivity.

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