A duck named Donald Duck says “All ducks always lie.” Based only on the information above, which one of the following statements can be logically inferred with certainty?
Donald Duck’s statement is false.
The question presents a classic logical paradox involving a statement made by Donald Duck. We need to determine, based solely on this information, which conclusion can be drawn with absolute certainty.
Donald Duck's Statement: A Logical Puzzle
Donald Duck, who is a duck, makes the statement: “All ducks always lie.”
To logically infer the certainty of this statement, we must consider two possibilities: either Donald Duck's statement is true, or it is false.
Statement Analysis: Case by Case
Let's examine what happens if we assume the statement is true and what happens if we assume it is false.
- Case 1: Assume Donald Duck's statement is true.
If the statement “All ducks always lie” is true, then every single duck, including Donald Duck himself, must always lie.
However, if Donald Duck always lies, then the very statement he just made (“All ducks always lie”) must be a lie.
If his statement is a lie, then it means the statement is false.
This creates a contradiction: The statement cannot be both true (our initial assumption) and false simultaneously.
Therefore, the initial assumption that Donald Duck’s statement is true leads to a logical inconsistency. This case is impossible.
- Case 2: Assume Donald Duck's statement is false.
If the statement “All ducks always lie” is false, it means that it is not true that all ducks always lie.
This implies that there must be at least one duck that does not always lie; meaning, there is at least one duck that tells the truth (at least sometimes).
Donald Duck made the statement. If his statement is false, it means Donald Duck is lying when he says “All ducks always lie.”
If Donald Duck is lying, then his statement “All ducks always lie” is indeed false.
This scenario leads to no contradiction. It is perfectly consistent for Donald Duck to be a duck who sometimes lies (and is lying in this instance) and for the statement “All ducks always lie” to be false (because not all ducks always lie).
Therefore, the initial assumption that Donald Duck’s statement is false is logically consistent.
Inference from Logical Analysis
Based on the detailed analysis of both cases, the only logically sound conclusion is that Donald Duck's statement must be false.
Option Examination for Certainty
Let's evaluate each provided option based on our logical inference:
- Donald Duck always lies.
While Donald Duck is lying in this specific instance (because his statement is false), we cannot infer with certainty that he "always" lies. The falsity of his statement only means that "not all ducks always lie" (i.e., at least one duck tells the truth, or perhaps ducks sometimes tell the truth). Donald Duck could be a duck who sometimes lies and sometimes tells the truth. The statement does not provide enough information to conclude he always lies.
- Donald Duck always tells the truth.
As shown in Case 1, if Donald Duck always told the truth, his statement would be true, leading to a contradiction. Therefore, this option is certainly false.
- Donald Duck’s statement is true.
As demonstrated in Case 1, assuming the statement is true leads to a direct contradiction. Thus, this option is certainly false.
- Donald Duck’s statement is false.
As demonstrated in Case 2, assuming the statement is false leads to no contradiction and is logically consistent. This is the only possible outcome derived from the paradox.
Therefore, the only statement that can be logically inferred with certainty is that Donald Duck’s statement is false.
Important Questions from Instructions
Given below are two statements followed by two conclusions. Assuming these statements to be true, decide which one logically follows.
Statement:
I. All film stars are playback singers.
II. All film directors are film stars.
Conclusions:
I. All film directors are playback singers.
II. Some film stars are film directors.
All people in a certain are either ‘Knights’ or ‘Knaves’ and each person knows every other person’s identity. Knights NEVER lie, and knaves ALWAYS lie.
P says “Both of us are knights”, Q says “None of us are knaves”.
Which one of the following can be logically inferred from the above?Fact: If it rains, then the field is wet.
Read the following statements:
(i) It rains
(ii) The field is not wet
(iii) The field is wet
(iv) It did not rain
Which one of the options given below is NOT logically possible, based on the given fact?A smart city integrates all modes of transport, uses clean energy and promotes the sustainable use of resources. It also uses technology to ensure the safety and security of the city, something which critics argue, will lead to a surveillance state.
Which of the following can be logically inferred from the above paragraph?
(i) All smart cities encourage the formation of surveillance states.
(ii) Surveillance is an integral part of a smart city.
(iii) Sustainability and surveillance go hand in hand in a smart city.
(iv) There is a perception that smart cities promote surveillance.The binary operation □ is defined as a □ b = ab + (a + b), where a and b are any two real numbers. The value of the identity element of this operation, defined as the number x such that a □ x = a, for any a, is_______.
