How can I make diffrence between conditional statement and biconditional statement when the propositions is not clear?

Question

I cannot make difference between conditional statement and biconditional statement I make difference between them when they are used in Natural language but I don't know which one to pick if its not clear. One example is this in first I think that was not biconditional statement but it turns on that is it. Example: “p: If you finish your meal, then q: you can have dessert.” in my head you can finish the meal without eating the desert, is not like "must". But the The Result says "You can have dessert if and only if you finish your meal." But without showing the answer I would never thought that the answer was p ↔ q.

Answer 1

This is an English question, not a math one. In English, if ... then often means if and only if. In math it is clear that an implication is true if the antecedent is false. You have to look at the context. In this case, there is a clear English implication that if you don't finish your meal you cannot have dessert, which makes it biconditional. The "you can have dessert" is permissive, so having dessert is not required even if the meal is finished.

Answer 2

Example: “p: If you finish your meal, then q: you can have dessert.” in my head you can finish the meal without eating the desert, is not like "must".

A simple test: Negate both the antecedent and consequent. If the statement is still true, then you have a biconditional relationship.

Applying this test to your example: Is it true that, if you DO NOT finish your meal, then you CANNOT have dessert? If so, you have a biconditional relationship.

The principle is the same whether it is stated in natural language or symbolic logic.

$$(P \implies Q) \land (\neg P \implies \neg Q) \space \space \equiv \space \space P \iff Q$$