I need to convert the following sentences to Linear Logic formulas-:
1) Bob can spend $1 to purchase a bottle of water or a bag of chips (Bob's choice). (D means Bob has a dollar; W means Bob has a bottle of water; C means Bob has a bag of chips.)
2) You can exchange a ten-dollar bill for two five-dollar bills. (T means you have a ten dollar bill and F means you have one five dollar bill.)
3) If Irene has a water bottle, she can refill it with water as many times as she wants. (B means Irene has a water bottle; W means Irene has water.)
4) If you give a man a fish, he'll eat for a day (once). If you teach a man to fish, he'll eat for the rest of his life (as many times as he wants). (F means you have a fish, E means the man eats, T means you have time to teach the man to fish.)
5) If you flip a coin it will come up heads or tails (you have no control over which). (F represents the action of flipping the coin, H means it comes up heads, T means it comes up tails.)
6) If you have a headache, taking ibuprofen will cure your pain. (H means you have a headache, I means you have ibuprofen.)
Here are the answers I have till now, could you please confirm these and help me with the ones not completed?
1) D ⊸ W & C
2) T ⊸ F ⊗ F
3) B ⊸ !W
4) No clue
5) F ⊸ H ⊕ T
6) No clue
Your answers for 1), 2), 3) and 5) are correct.
The answer for 4) is: \begin{align} (F ⊸E) \otimes (T ⊸\,\, !E) \end{align} Indeed, sentence 4) is the conjunction ($\otimes$) of two linear implications ($⊸$).
The answer for 6) is: \begin{align} (H \otimes I) ⊸ H^\perp \end{align} Indeed, sentence 6) is a linear implication ($⊸$) where the antecedent is a conjunction ($\otimes$). I interpret "curing your pain" as ''you don't have a headache".