I was pondering about logical implications today and I decided to share my thoughts. Logic in the English language is quite different from in formal logic. For instance, when someone asks if you would like coffee or tea, he or she might be taken aback if you said both. This is because what they meant was the logical exclusive or, instead of the logical or. Now, I would like to share my thoughts about implication in logic and in the English language.
Consider the following commands:
1) Do this if you want to.
2) Do this only if you want to.
Suppose we let the event “do this” be A, and “you want to” be B, then the first sentence is equivalent to A if B, ie B implies A, and the second sentence is equivalent to A implies B.
We know that A implies B is only invalid when A occurs but not B. Both propositions are valid under the outcome that A and B both occur, and under the outcome that both A and B do not occur. That is, you want to do it and you do it, and you don’t want to do it and you don’t do it.
Interestingly, the third case that is acceptable by both proposition differs. In the first scenario, A and not B is acceptable, ie, you don’t want to do it and you do it. In the second scenario, not A and B is acceptable, ie, you want to do it and you don’t do it.
So when someone says the first sentence, logically, it means that it’s ok to do it regardless of whether you want to or not. Hence, it can be seen as a preference for the recipient to do it. Similarly, in the second sentence, it is ok to not do it, and so a preference to not do it is insinuated.
Like the exclusive or case, most people will just interpret both sentences to mean the bi-implication. That is, do this if and only if you want to do it. So if you want to do it, you should do it, and if you don’t want to do it, you shouldn’t do it.
However, I believe there is a possibility that the subconscious works out these logical implications and craft the sentence intuitively depending on the preference of the speaker. So if I perceive say, studying to be good, and say, eating junk food to be bad, I would say accordingly, “Go study if you want to”, and “Eat junk food only if you want to” to hint my preference.