Cyberspace village: truth tellers and liars

[BMAD]

In a strange village in cyberspace the inhabitants are "truthers" or liars depending not only on the day of the week but also on whether the day is fair or rainy.

Three inhabitants are met. It is known that Amy lies on fair Tuesdays, fair Thursdays, and fair Saturdays, and on rainy Mondays, rainy Wednesdays, and rainy Fridays. At all other times she tells the truth.

On the other hand, both Bonnie and Cybil lie on fair Mondays, fair Wednesdays, and fair Fridays, and on rainy Tuesdays, rainy Thursdays, and rainy Saturdays. At all other times they tell the truth.

Amy, Bonnie, and Cybil make the following statements.

Amy: It is raining and today is Tuesday

Bonnie: It is fair or today is Tuesday

Cybil: It was not Wednesday yesterday and it will not be Wednesday tomorrow.

Questions

1. What day of the week is it?

2. Is it fair or is it raining?

[DejMar]

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Amy's statement "It is raining and today is Tuesday" is a logical conjunction. It is true only if both its operands "It is raining" and "today is Tuesday" are true, otherwise it is false. Thus, for her statement to be true, it is a rainy Tuesday; and, as Amy tells the truth on rainy Tuesdays this is a valid option. As Amy lies on rainy Mondays, fair Tuesdays, rainy Wednesdays, fair Thursdays, rainy Fridays, and fair Saturdays, these are also valid options. All other conditions are not.

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Bonnie's statement " It is fair or today is Tuesday" is a logical disjunction. It is false only if both its operands "It is fair" and "today is Tuesday" are false, otherwise it is true. If it were Tuesday her statement would be true, yet she lies on rainy Tuesdays, thus a rainy Tuesday can be eliminated as a possibility, yet a fair Tuesday would be remain a valid option. If it were a fair Tuesday, a fair Thursday, or a fair Saturday, her statement would also be true. As Bonnie tells the truth on these fair days, these also remain valid options. If it were a rainy Monday, a rainy Wednesday, or a rainy Friday, her statement would be false, and as she tells lies on these rainy days, these remain valid options.

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Cybil's statement, like Amy's, is a logical conjunction. It is true only if both its operands "It was not Wednesday yesterday" and "it will not be Wednesday tomorrow" are true. The first operand is true for all days but Thursday, and the second operand is true for all days but Tuesday. The statement is false for the days Tuesday and Thursday. As Cybil tells the truth on fair Tuesdays and fair Thursdays, it can not be on such a day. As she tells the truth on the rainy days of Monday, Wednesday, and Friday, and on fair Saturdays, these remain valid options.

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(1) Therefore, it is a Monday, Wednesday, Friday or Saturday.
(2) The day is fair if it is Saturday, else it is raining.
 

[plasmid]

I got a different answer.

[DejMar]

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Amy's statement "It is raining and today is Tuesday" is a logical conjunction. It is true only if both its operands "It is raining" and "today is Tuesday" are true, otherwise it is false. Thus, for her statement to be true, it is a rainy Tuesday; and, as Amy tells the truth on rainy Tuesdays this is a valid option. As Amy lies on rainy Mondays, fair Tuesdays, rainy Wednesdays, fair Thursdays, rainy Fridays, and fair Saturdays, these are also valid options. All other conditions are not.

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Bonnie's statement " It is fair or today is Tuesday" can be argued that it is intended as an exclusive disjunction. Due to possible ambiguity, in grammar it is often clarified as such with the conjunctive word 'either'. Yet, an assumption that an exclusive or is meant shall be considered for this solution.

For an exclusive or, the statement is false if either both its operands "It is fair" and "today is Tuesday" are true or if both are false, otherwise it is true. Thus, considering only those days that are yet valid for further consideration, Bonnie's statement is false if it were a rainy day of Monday,  Wednesday, or Friday, or a fair Tuesday.  And Bonnie's statement is true if it were a rainy Tuesday, yet as that is a day she lies, it can be also be eliminated.

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Cybil's statement, like Amy's, is a logical conjunction. It is true only if both its operands "It was not Wednesday yesterday" and "it will not be Wednesday tomorrow" are true. The first operand is true for all days but Thursday, and the second operand is true for all days but Tuesday. The statement is false for the days Tuesday and Thursday. As Cybil tells the truth on fair Thursdays, the day can be eliminated.

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(1) Therefore, it is a Saturday.
(2) The day is fair.

Edited November 17, 2014 by DejMar