Question

Assuming that 1+1=2 is necessary, then it holds under any and all conditions. So, doesn’t that mean that anything is relevant to whether 1+1=2? That is, you can’t speak of some sentence U that is seemingly unrelated to 1+1=2 without having access to that 1+1 in fact is equal to 2. So, U implies 1+1=2.

Still, classical and intuitionistic conditionals seem to produce odd results. For example, suppose ~A="The axiom of choice is unprovable" and C="The continuum hypothesis is provable." It seems that ~A->(C->~A) is unwarranted since, though the continuum hypothesis is unprovable if the axiom of choice is unprovable, it seems to be the case that C->~A isn’t an example of an actual implication, even if it is valid in a given proof system. But, I am willing to accept that in such a proof system any attempt at refuting C->~A leads to absurdity. What do you think are some properties of a natural-language/intuitive interpretation of implication?

Answer 1

Conditional statements express logical relations between propositions. Universal statements are equivalent to conditionals. Intuition is involved in the interpretation of implications in natural language.

Step-by-step explanation:

Conditional statements express the logical relations between two propositions. Universal statements are logically equivalent to conditionals. In the case of the equation 1+1=2, its truth is so clear that it can be considered an example of intuition. Intuition operates in other realms besides mathematics and can be based on definition or commonly accepted knowledge. There are various properties of a natural-language/intuitive interpretation of implication, such as necessity, sufficiency, and the ability to craft effective arguments using conditionals.