Williamson writes that some may “deny outright when Mars is a borderline case that it is either dry or not dry, and therefore to answer the original question in the negative. However, if one answers the original question in the negative this means that Mars is neither dry nor not dry. Another way to put this is that Mars is both not dry and not not dry, which is a contradiction. I thought that this could be avoided if one simply rephrased the original question from “Was Mars always either dry or not dry?” to (the new original question) “Was Mars always either dry or wet?” Thus to answer the new original question in the negative would be to say that Mars is both not dry and not wet, which has no contradiction.
1. If the new original question means the same as the original question then the original question can be changed to “Was Mars always either dry or wet?”
2. The new original question means the same as the original question.
3. Therefore the original question can be changed to “Was Mars always either dry or wet?”
4. If the original question can be changed to “Was Mars always either dry or wet?” then answering the original question in the negative is not a contradiction.
5. Therefore answering the original question in the negative is not a contradiction.
Williamson may be able to respond by denying premise 2 of the argument. To show how Williamson could respond more clearly I will change the original question to something easier to quantify. I will change it to “Was Mars always either below zero degrees or not below zero degrees?” Let us assume that Mars does reach above and below zero degrees. To answer this question in the negative would be to say that Mars is both not below zero degrees and not not below zero degrees, which is a contradiction. Thus I will change the question from “Was Mars always either below zero degrees or not below zero degrees?” to “Was Mars always either below zero degrees or above zero degrees?” These two questions are not the same because the second question has a middle, namely zero degrees. Therefore X and its opposite are not the same as X and ~X. Below zero or not below zero is smaller than 0 or greater than or equal to 0, while below zero or above zero is smaller than 0 or greater than 0. Some may deny this line of argument for a term such as ‘dry’, or say that ‘dryness’ can not be quantified.
1. If the new original question means the same as the original question then the original question can be changed to “Was Mars always either dry or wet?”
2. The new original question means the same as the original question.
3. Therefore the original question can be changed to “Was Mars always either dry or wet?”
4. If the original question can be changed to “Was Mars always either dry or wet?” then answering the original question in the negative is not a contradiction.
5. Therefore answering the original question in the negative is not a contradiction.
Williamson may be able to respond by denying premise 2 of the argument. To show how Williamson could respond more clearly I will change the original question to something easier to quantify. I will change it to “Was Mars always either below zero degrees or not below zero degrees?” Let us assume that Mars does reach above and below zero degrees. To answer this question in the negative would be to say that Mars is both not below zero degrees and not not below zero degrees, which is a contradiction. Thus I will change the question from “Was Mars always either below zero degrees or not below zero degrees?” to “Was Mars always either below zero degrees or above zero degrees?” These two questions are not the same because the second question has a middle, namely zero degrees. Therefore X and its opposite are not the same as X and ~X. Below zero or not below zero is smaller than 0 or greater than or equal to 0, while below zero or above zero is smaller than 0 or greater than 0. Some may deny this line of argument for a term such as ‘dry’, or say that ‘dryness’ can not be quantified.
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