- $\forall x \bigl( P(x) \Rightarrow Q(x) \bigr)$
- $\forall x \bigl( P(x) \bigr) \Rightarrow \forall x \bigl( Q(x) \bigr)$
In a world where statement 1 is true, statement 2 must be true through natural deduction. Let us assume that P stands for the category of rainy days, and Q stands for the category of cloudy days. Then, according to statement 1, any rainy day must be cloudy. In other words, in a world where statement 1 is true, we can't have rainy days that aren't cloudy.
If statement 1 is true, then statement 2 is true through natural deduction. So, let's now see the different cases that can satisfy statement 2 as true,
A) The antecedent and consequent are true. B) The antecedent is false but the consequent is true. C) The antecedent and the consequent are false.
The third case (C) says we can have the conditional relation of statement 2 as true, when it is false that all the days are rainy and when it is false that all the days are cloudy. If it is false that all the days are rainy, then this means that maybe 50% of the days are rainy and the other 50% aren't. And if it is false that all days are cloudy, then this means that maybe 75% of the days are cloudy and the other 25% aren't. Thus, case (C) allows the possibility of rainy days that aren't cloudy. Because in such situation if you pick one of the rainy days, then it is possible that it won't be cloudy. However, in a world where both statements 1 and 2 are true, we can't have statement 2 as true as dictated by case (C) because statement 1 never allows the possibility of having rainy days that aren't cloudy. In other words, if we take statement 1 as our premise and statement 2 as our conclusion, then statement 2 can't be true as a result of case (C). Because case (C) contains information not supported by the premise, and in a valid argument the conclusion mustn't contain information not supported by the premises. Specifically, case (C) allows the possibility of rainy days that aren't cloudy, and this is something not supported by the premise(statement 1). The premise says that all rainy days must be cloudy.
My question : In a world where statements 1 and 2 are true, is it possible for statement 2 to be true as a result of case (C)?
