This study aimed to design a task in an everyday context to support high school students' exploration of the truth values of propositional conditionals and to understand their difficulties. The results showed that students could identify that the truth value of a propositional conditional is determined by two variables (the truth values of the antecedent and the consequent) while organizing the propositional conditionals generated from the open sentence and that a social contract situation can help students understand why the truth value of a conditional with a false antecedent is true. The study also revealed that students' difficulties in determining the truth values of conditionals are closely related to the time variable in an everyday context and the confusion between conditionals and causality. The findings from this study on the strategy and difficulties exhibited by students when determining the truth value of everyday conditional sentences can inform future efforts to teach formal mathematics in real-world contexts.