Consider the following premises. For all x_12, x_12 is a carpenter or x_12 will play a game. For all x_12, x_12 will drink water or x_12 will climb a mountain. For all x_12, if x_12 were a carpenter, then x_12 is writing a letter. There is at least one x_12 for which if x_12 plays a game, then x_12 is writing a letter. For all x_12, x_12 will not drink water. Can we infer the following from them? If we can, name the inference rule being used: For all x_12, either x_12 will write a letter or x_12 will climb a mountain are not true.
 From the fact that for all x_12, if x_12 were a carpenter, then x_12 is writing a letter, that there is at least one x_12 for which if x_12 plays a game, then x_12 is writing a letter, and that for all x_12, x_12 is a carpenter or x_12 will play a game we can infer that there is at least one x_12 for which x_12 will write a letter via existential disjunction elimination. From the fact that for all x_12, x_12 will drink water or x_12 will climb a mountain, and that for all x_12, x_12 will not drink water we can infer that for all x_12, x_12 will climb a mountain via universal disjunctive syllogism. Finally, from the fact that there is at least one x_12 for which x_12 will write a letter, and that for all x_12, x_12 will climb a mountain we can infer that there is at least one x_12 for which x_12 will write a letter and x_12 will climb a mountain via existential conjunction, which contradicts that For all x_12, either x_12 will write a letter or x_12 will climb a mountain are not true. Therefore, the answer is no.