Kenneth Rosen Edition 6th Exercise 1.1 Question 43 (Page No. 20)

Question

Fuzzy logic is used in artificial intelligence. In fuzzy logic, a proposition has a truth value that is a number between 0 and 1, inclusive.A proposition with a truth value of 0 is false and one with a truth value of 1 is true. Truth values that are between 0 and 1 indicate varying degrees of truth. For instance, the truth value 0.8 can be assigned to the statement “Fred is happy,” because Fred is happy most of the time, and the truth value 0.4 can be assigned to the statement “John is happy,” because John is happy slightly less than half the time. Use these truthvalues to solve below exercise.

The nth statement in a list of 100 statements is “Exactly n of the statements in this list are false.”

1. What conclusions can you draw from these statements?
2. Answer part (a) if the nth statement is “At least n of the statements in this list are false.”
3. Answer part (b) assuming that the list contains 99 statements

Comments

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https://gateoverflow.in/131789/%23propositionlogic-%23rossen#a131803

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N=99

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https://gateoverflow.in/132684/%23rosen_chapter1

https://gateoverflow.in/131789/%23propositionlogic-%23rossen

https://gateoverflow.in/122512/rosen-proposition-logic

Answer 1

a) Consider 100 statements each of which say : 

  1) Exactly 1 statement is false.
  2) Exactly 2 statements are false.
  3) Exactly 3 statements are false. 
  ...
  99) Exactly 99 statements are false.
  100) Exactly 100 statements are false.

In this list the 99th statement is true and the rest of the statements are false. i.e. (1-98 and 100 are false)

b) Now this statements are modified as:

  1) Atleast 1 of the statement is false.
  2) Atleast 2 of the statement are false.
  ...
  49) Atleast 49 of the statements are false.
  50) Atleast 50 of the statements are false.
  ...
  99) Atleast 99 of the statments are false.
  100) Atleast 100 of the statements are false.

   We can see that 1 through 50 are all true and statements 51 through 100 are all false.

c) Consider a  list of 3 statements, 

1) Atleast 1 of the statement is false. 
2) Atleast 2 of the statements are false.
3) Atleast 3 of the statements are false.

Now, consider each statement and see whethere this can be true. None of the statements can be true and it's a logical paradox. This can be extended to any odd number of statements and hence, this remains a logical paradox for 99 statements.