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 Mr. Jones is the manager’s next-door neighbor, then Mr. Jones’s annual earnings are exactly divisible by 3. If Mr. Jones’s annual earnings are exactly divisible by 3, then $40,000 is exactly divisible by 3. But $40,000 is not exactly divisible by 3. If Mr. Robinson is the manager’s next-door neighbor, then Mr. Robinson lives halfway between Detroit and Chicago. If Mr. Robinson lives in Detroit, then he does not live halfway between Detroit and Chicago. Mr. Robinson lives in Detroit. If Mr. Jones is not the manager’s next-door neighbor, then either Mr. Robinson or Mr. Smith is the manager’s next-door neighbor. Therefore Mr. Smith is the manager’s next-door neighbor. ( J —Mr. Jones is the manager’s next-door neighbor; E — Mr. Jones’s annual earnings are exactly divisible by 3; T —$40,000 is exactly divisible by 3; R —Mr. Robinson is the manager’s next-door neighbor; H —Mr. Robinson lives halfway between Detroit and Chicago; D —Mr. Robinson lives in Detroit; S —Mr. Smith is the manager’s next-door neighbor.)