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.
Top Questions on Philosophy
- Whose philosophy is the basis for Socially Useful Productive Work (SUPW)
- The Book 'Democracy and Education' authored by
- W.V.O. Quine, in his paper, “Two Dogmas of Empiricism”, rejects the boundary between synthetic and analytic statements. Which among the following correctly depict(s) his proposal of ‘Empiricism without the Dogmas’?
- Gottlob Frege, in his paper, “On Sense and Reference”, analyses the statements, ‘a = a’ and ‘a = b’, where the reference of ‘a’ is the same as that of ‘b’. Which among the following statement(s) correctly explain(s) his conception of sense and reference?
- Which among the following statement(s) is/are in conformity with Immanuel Kant’s notion of ‘synthetic a priori knowledge’?
Questions Asked in GATE XH- C4 exam
Here are two analogous groups, Group-I and Group-II, that list words in their decreasing order of intensity. Identify the missing word in Group-II.
Abuse \( \rightarrow \) Insult \( \rightarrow \) Ridicule
__________ \( \rightarrow \) Praise \( \rightarrow \) Appreciate
- A car started from city P at 9:40 AM. The time taken for the car to reach city Q is 4 hours and 50 minutes. The time of arrival of the car at city Q is:
- Consider a square field ABCD. The diagonal AC is 50 meter. The cost of laying grass in the field is Rs. 5 per square-meter. The total cost for laying grass in the field ABCD is Rs. ________ (rounded off to two decimal places).
In the following figure, four overlapping shapes (rectangle, triangle, circle, and hexagon) are given. The sum of the numbers which belong to only two overlapping shapes is ________
- Many countries are facing water shortage crises in the past few years. A report of the United Nations has named India among the worst countries for poor quality of water. The report ranks 122 countries according to the quality of their water as well as their commitment to improve the situation. Some countries in Europe are considered the worst because of the quality of its groundwater. Rain failed in some parts of India in the past. The vast areas of Rajasthan, Madhya Pradesh, and Andhra Pradesh were affected by severe drought. People without water turn desperate and violent. Consequently, the food godowns were attacked in some of the states. Based on the details given in the passage, which of the following option(s) is/ are correct statement(s)?