From [1], Philip’s declarations in [3] and [5] cannot both be false; otherwise he tells the truth on more than one of the days of the week. From [1] and the fact that the declarations in [3] through [5] were made on consecutive days, Philip’s declarations in [3] and [5] cannot both be true; otherwise he tells the truth on more than one of the days of the week. So either Philip’s declaration in [3] is the only true one or his declaration in [5] is the only true one. Suppose his declaration in [3] is the only true one. Then, from his false declaration in [5], he made the declaration in [3] on a Wednesday or a Friday. Then, from [2], his false declaration in [4] was made on a Thursday or a Saturday.
This situation is impossible.
So his declaration in [5] is the only true one. Then, from his false declaration in [3], he made the declaration in [5] on a Monday or a Tuesday. Then, from [2], his false declaration in [4] was made on a Sunday or a Monday. His declaration in [4] would be true. So his declaration in [4] was made on a Monday.
Then from [2], his true declaration in [5] was made on a Tuesday and Philip tells the truth on Tuesday.
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