Consider the following premises. For all x_20, if x_20 writes a letter, then x_20 is a carpenter. If it rains, then James is running. If Patricia were a carpenter, then Elisabeth is a mechanic. If James goes running, then Patricia is writing a letter. If Elisabeth were a mechanic, then it is raining. Can we infer the following from them? If is not true that Patricia is writing a letter if and only if Elisabeth is a mechanic.
 From the fact that for all x_20, if x_20 writes a letter, then x_20 is a carpenter we can infer that if Patricia writes a letter, then Patricia is a carpenter. From the fact that if Elisabeth were a mechanic, then it is raining, and that if it rains, then James is running we can infer that if Elisabeth were a mechanic, then James is running. From the fact that if Patricia writes a letter, then Patricia is a carpenter, and that if Patricia were a carpenter, then Elisabeth is a mechanic we can infer that if Patricia writes a letter, then Elisabeth is a mechanic. From the fact that if Elisabeth were a mechanic, then James is running, and that if James goes running, then Patricia is writing a letter we can infer that if Elisabeth were a mechanic, then Patricia is writing a letter. Finally, from the fact that if Patricia writes a letter, then Elisabeth is a mechanic, and that if Elisabeth were a mechanic, then Patricia is writing a letter we can infer that Patricia is writing a letter if and only if Elisabeth is a mechanic, which contradicts that If is not true that Patricia is writing a letter if and only if Elisabeth is a mechanic. Therefore, the answer is no.