Final answer:
The given modal logic formulas require corrections. The correct equivalences are ~◊(p & ~q) = □~(p & ~q), and ~◊(~p & ~q) = □~(~p & ~q), upholding the principles of modal logics.
Step-by-step explanation:
When it comes to checking whether formulas in modal logic are correct, we need to understand the basics of modal operators. The modal logic operators include ○ (possibility) and □ (necessity). Let's examine each formula one by one:
- ~○(p & ~q) - This statement can be read as 'It is not possible that both p is true and q is false.'
- ~○(~p & ~q) - This statement says 'It is not possible that both p is false and q is false.'
- ~○(p & ~q) = □~(p & ~q) = □(~p & q) - The equivalencies here are incorrect because the transformation of the expressions inside the modal operators does not hold. The correct transformation should stay consistent with the original negation or affirmation of the individual propositions. Therefore, the last equivalence, which negates both propositions and inverts their relationship, is incorrect.
- ~○(~p & ~q) = □~(~p & ~q) = □(p & q) - Similar to the previous item, the expressions within the modal operators are being transformed incorrectly. The last equivalence proposes that 'It is necessary that both p is true and q is true.' This would only be correct if the original expression was 'It is not possible that p is true and q is true,' which it is not.
The correct equivalences for these modal logic expressions are:
- ~○(p & ~q) is equivalent to □~(p & ~q), meaning 'It is necessary that not both p is true and q is false.'
- ~○(~p & ~q) is equivalent to □~(~p & ~q), meaning 'Not both p must be false and q is false.'
Understanding these equivalences helps reinforce the foundational aspects of logic.