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Question

For each of the following pairs of formulas, can you find a model and a world in it which distinguishes them, i.e. makes one of them true and one false? In that case, you are showing that they do not entail each other. If you cannot, it might mean that the formulas are equivalent. Justify your answer. (a) \( \square p \) and \( \square ? p \) (b) \( \square \neg p \) and \( \neg \diamond p \) (c) \( \square(p \wedge q) \) and \( \square p \wedge \square q \)

Answer

Based on the given pairs of formulas, statement a is true, statement b is false, and statement c is true. The explanation is as follows: The term "formula" refers to a mathematical expression used to represent a quantity. In Microsoft Word, you can display a formula's value by selecting it and pressing F9. If you link two cells in Microsoft Excel using the Text number format, the formula will be shown instead of the updated value. Statement a is not true; both formulas are equivalent, expressing that p is true in all possible worlds. Statement b is true; while the first formula expresses that p is necessarily true, the second formula expresses that p is possibly necessarily true. In a world where p is true, the first formula is true and the second formula is false. Statement c is not true; both formulas express that p and q are true in all possible worlds.