In mathematics when a "P implies Q" statement is true it means that every time P is true, Q is true also. What about everyday usage? For example consider the statement: "If it is raining, then I am inside the house." Assume that is true that it is raining and that I am inside the house. I think no one will argue that this holds in general. So this conditional is only true for the time we make the statement? But how outside of mathematics can someone know when to understand such if-then statements as temporary statements? Also when we speak about our habits we usually say "When I watch TV, I eat popcorn." Does the speaker mean that he does this every single time?
In the world of formal, mathematical logic, statements are treated as either true or false. In the world of natural language, all kinds of statements are uncertain or ambiguous. The same thing applies to conditional statements, "if A then B", which may likewise be true, false, or uncertain, ambiguous, and equivocal. The attempts to extend formal mathematical logic to these kinds of statements have not been generally successful.
The three valued logic of Lukasiewicz is possibly the most direct attempt to deal with uncertainty. This has not been generally accepted in part because important logical laws fail, although this failure can be traced to the behavior of conditional statements with the third logical value. This has much in common with the fuzzy logic of Zadeh, and again, the case of doubtful conditionals causes difficulties.
- " If it is raining, then I am inside the house", as it is used in ordinary language, implies a quantification over time :
for all time t , if it is raining at time t, then I am inside the house at time t.
In other words and equivalently :
there is no time t such that (1) it is raining at t, and (2) I am not inside the house at time t.
Suppose I live in a country where it never rains, there is no time t such that it is raining at t. So, a fortiori, there is no time t such that (1) it is raining at t, and (2) I am not inside the house at time t. So, the sentence is true.
Suppose I live in a country when it rains 364 days out of 365. On the non-raining day, I am outside. The sentence is still true : there is no time t such that (1) it is raining at t, and (2) I am not inside the house at time t.
So, the natural language understanding of " if... then " is not always opposed to the mathematical one.