Checking logical entailment with truth tables has the merit of being conceptually simple. However, it is not always the most practical method. The number of truth assignments of a language grows exponentially with the number of logical constants. When the number of logical constants in a propositional language is large, it may be impossible to process its truth table.
Proof methods provide an alternative way of checking logical entailment that addresses this problem. In many cases, it is possible to create a proof of a conclusion from a set of premises that is much smaller than the truth table for the language; moreover, it is often possible to find such proofs with less work than is necessary to check the entire truth table.
We begin this lesson with a discussion of linear reasoning and linear proofs. We then move on to hypothetical reasoning and structured proofs. Once we have seen both linear and structured proofs, we show how they are combined in the popular Fitch proof system, and we provide some tips for finding proofs using the Fitch system. We finish with definitions for soundness and completeness - the standards by which proof systems are judged.