Each of the following arguments in English may be similarly translated, and for each, a formal proof of validity (using only the nine elementary valid argument forms as rules of inference) may be constructed. These proofs vary in length, some requiring a sequence of thirteen statements (including the premises) to complete the formal proofs. The suggested abbreviations should be used for the sake of clarity. Bear in mind that, as one proceeds to produce a formal proof of an argument presented in a natural language, it is of the utmost importance that the translation into symbolic notation of the statements appearing discursively in the argument be perfectly accurate ; if it is not, one will be working with an argument that is different from the original one, and in that case any proof devised will be useless, being not applicable to the original argument.

If Nihar buys the lot, then an office building will be constructed; whereas if Payton buys the lot, then it will be quickly sold again. If Rivers buys the lot, then a store will be constructed; and if a store is constructed, then Thompson will offer to lease it. Either Nihar or Rivers will buy the lot. Therefore either an office building or a store will be constructed. (N —Nihar buys the lot; O —An office building will be constructed; P—Payton buys the lot; Q—The lot will be quickly sold again; R —Rivers buys the lot; S —A store will be constructed; T—Thompson will offer to lease the store.)