A quick research around the topic of proof is already sufficient to come across the word logic. Both logic and proof are used around premises and statements. Both a premise and a statement can be true or false, and that is what logic and proof are all about. Truth is the motivation.
It is not a long way from logic to thinking. Logic lives within our thinking, we talk about logical thinking. Logic is defined as the study of the laws of thought or correct reasoning. When we investigate logic, we are within an area of our thinking power. The studies of logic are very old, the laws of logic have been observed and written down a long time ago.
Logic is the study of correct reasoning. Logic studies arguments, which consist of a set of premises that leads to a conclusion. Premises and conclusions express propositions or claims that can be true or false. Arguments can be either correct or incorrect. An argument is correct if its premises support its conclusion. Deductive arguments have the strongest form of support: if their premises are true then their conclusion must also be true. Formal logic ist interested in deductively valid arguments and is widely used in mathematical logic. Necessity and sufficiency are terms used to describe a conditional or implicational relationship between statements. A necessary condition is one that must be present in order for another condition to occur. But the other condition may require more conditions to be present before it can be produced. When all required conditions are met, then we have sufficient conditions. For example, water is necessary and sufficient to quench my thirst. Water is also necessary to bake a bread, but it is not sufficient. I need also flower and heat. And if you can make a bread that deserves the name and does not need water, water would no longer be a necessary condition. This logic is important when describing concepts.
We use logic all the time, every day countless times, to navigate the world around us. With logic we can predict the result of a situation that we have never experienced before. We can also use previous experiences and transfer them to become useful for new tasks. Logic can be used to make conclusions based on one or more premises. An example for this can be found in the second paragraph of this article, the fourth sentence after the quote. However, we cannot prove logic itself. And we do not need to, because logic is a part of our nature and does not require proof. An article about logic and proof is basically also an article about thinking. And our thinking we do not need to prove either. We just need to observe it. To do so, we need to first bring it forth ourselves. Because of this, thinking is very close to us, and with extensive observation of thinking no open questions remain.
From the perspective of proof one could say that proving stops where thinking begins. With a proof we want to make a statement directly accessible for our thinking. The correctness of the laws of thinking cannot be proven. The logic is a part of this. Still, we use logic to make a proof. We experience that it works. The power of thinking is given to us same as the physical world is given to us. Both logic and proof rely on our thinking to be fundamentally correct. This refers to the rules of thinking, not to the content of our thoughts. Who questions the thinking itself will not find solid ground in neither logic nor proof. A close observation of thinking will reveal that we have full access to it, as we can both bring it forth and observe it ourselves. As a result we can find a stable foundation in our thinking. We can build a solid worldview on that foundation. Thinking is the connection, the bridge between us and the world. Thinking processes our observation. Doing so, it adds the concepts that are not given to us as part of our sensual observation. Thinking goes beyond the sensual perception, it can perceive concepts.
A proof makes a statement accessible for our thinking. The easiest examples for this are within mathematics. Mathematics is the scientific area that uses proof the most. In physics we usually do not use the word proof, but experiment.
A mathematical proof uses the laws of logic and math to make the correctness of a statement obvious. For example, if a formula can be rearranged or simplified so that it says “1=1” then it is proven correct.
Modern physics measures a success of a theory by determining its validity or invalidity. The term proof is avoided here, because the validity is demonstrated at a certain time, in a certain place and under certain conditions. The experiment does usually not allow the conclusion that a theory is universally proven independent of space and time. If it is repeatable when the same conditions are met, the theory can count as valid until it comes into conflict with other observations.
In a formal proof we have a set of premises that we assume are correct, and then we can make conclusions with logic. If we consider all rules of logic when doing so, the conclusion is formally proven. As soon as one premise is denied, the formal proof collapses and becomes useless. Nothing can be said about the correctness of the conclusion anymore.
Mathematics enjoy the luxury of having universally valid proof, but it is in a unique position there. Also the formal logical proof stands and falls with its premises. Without logic we loose orientation, but logic will only give us truth when we use it reasonably. We must know the beginning and the end of our logical chain of arguments, we must think “all the way through”. And if we want to understand the world, we must first observe it without prejudice. If we believe in seemingly conclusive chains of arguments, and elevate them to the level of proof, we can quickly block our access to reality.