Which one of the following is the contrary of \( (\forall x)Px \)?
Show Hint
- \( (\forall x) \sim Px \)
- \( \sim (\forall x) \sim Px \)
- \( (\exists x) \sim Px \)
- \( \sim (\exists x) Px \)
The Correct Option is A
Solution and Explanation
Step 1: Understanding "contrary".
In classical logic, the contrary of a universal affirmative proposition \( (\forall x)Px \) is the universal negative: \[ (\forall x) \sim P(x) \] This asserts that no individual in the domain satisfies \(P(x)\), which is directly contrary to every individual satisfying it.
Step 2: Difference from contradiction.
Note that the contradiction of \( (\forall x)Px \) is \( \sim (\forall x)Px \equiv (\exists x)\sim Px \), but the contrary here is another universal statement: \( (\forall x)\sim Px \), asserting the opposite property universally.
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