Technical Reasoning/Structure of Arguments

Arguments Generally

Notice that arguments are how we demonstrate the necessary truth of some conclusion, from a set of assumptions and accepted inferences rules. We will use the words "argument" and "proof" interchangeably.

In the above proof, we assumed the existence of two points A and B.

With the help of some other assumptions and inference rules, we were allowed to infer the existence of a circle centered at A and running through B.

After a long enough sequence of propositions like these, we eventually ended at the proposition "therefore $\mathrm{\Delta }ABC$ is equilateral". This was the conclusion, and it ended the argument for the theorem at the beginning.

In the abstract, this is always the structure of a proof:

• A collection of basic assumptions, called the "premises" of the argument.
• A sequence of propositions, such that every next one is a premise or inferred from earlier propositions (using an allowed inference rule, of course).
• The last proposition in the sequence is the conclusion. The "goal" of the argument is to arrive in this way at the conclusion.