Consider the following premises. For all x_14, x_14 will climb a mountain. For all x_14, if x_14 were a carpenter, then x_14 is an artist. For all x_14, if x_14 climbs a mountain, then x_14 is a carpenter. For all x_14, x_14 is not sad. There is at least one x_14 for which x_14 is sad or x_14 will write a letter. Can we infer the following from them? If we can, name the inference rule being used: For all x_14, either x_14 will write a letter or x_14 is an artist are not true.
 From the fact that for all x_14, if x_14 climbs a mountain, then x_14 is a carpenter, and that for all x_14, x_14 will climb a mountain we can infer that for all x_14, x_14 is a carpenter via universal modus ponens. From the fact that there is at least one x_14 for which x_14 is sad or x_14 will write a letter, and that for all x_14, x_14 is not sad we can infer that there is at least one x_14 for which x_14 will write a letter via existential disjunctive syllogism. From the fact that for all x_14, if x_14 were a carpenter, then x_14 is an artist, and that for all x_14, x_14 is a carpenter we can infer that for all x_14, x_14 is an artist via universal modus ponens. Finally, from the fact that there is at least one x_14 for which x_14 will write a letter, and that for all x_14, x_14 is an artist we can infer that there is at least one x_14 for which x_14 will write a letter and x_14 is an artist via existential conjunction, which contradicts that For all x_14, either x_14 will write a letter or x_14 is an artist are not true. Therefore, the answer is no.