Question

Please provide the following line into direct symbolization

Provided that Alice but not the Hatter is cross unless mad, the Hatter, but not Alice, is mad unless he frowns

__________________________
P: Alice is cross
Q: Alice frowns
R: Alice is mad
S: The Hatter is cross
T: The Hatter frowns
U: The Hatter is mad
V: The Dormouse dozes
W: The Dormouse frowns
X: The Dormouse is mad

Answer 1

To symbolize the statement 'Provided that Alice but not the Hatter is cross unless mad, the Hatter, but not Alice, is mad unless he frowns', we use the logical constructs for implication and negation to create two expressions: (¬R → P) ↑ (U → ¬S) and (U → T). The symbols P, R, S, T, and U correspond to Alice being cross, Alice being mad, the Hatter being cross, the Hatter frowning, and the Hatter being mad, respectively.

Step-by-step explanation:

The student has asked for help in symbolizing a complex statement about the characters in Lewis Carroll's Alice's Adventures in Wonderland. To translate the given line into direct symbolization, we need to carefully adhere to the definitions of the propositional symbols provided. The statement 'Provided that Alice but not the Hatter is cross unless mad, the Hatter, but not Alice, is mad unless he frowns' can be broken down into two parts:

• Alice is cross unless she is mad.
• The Hatter is mad unless he frowns.

Using the given symbols, we can express these two parts as:

1. (¬R → P) ↑ (U → ¬S): Alice is cross if she is not mad, and the Hatter is not cross if he is mad.
2. (U → T): The Hatter is mad if he does not frown.

We have used the logical symbols where '→' stands for 'if...then' (implication), '↑' stands for 'and', and '¬' denotes 'not'. Each symbol (P, Q, R, S, T, U, V, W, X) represents a specific statement about Alice, the Hatter, and the Dormouse, according to the information given in the question.