Propositional Logic

phil 3: homework 5Due 2/5 at 11:59pm.Late homework is not accepted, but remember that your lowest 3 scores will be dropped. Please turn it in on Gauchospace in .doc or .pdf format. Also please do not include your name on the assignment; it will be graded blindly.Exercise 1. Use a truth table to determine whether the following sentences are a tautology, contingency, or contradiction:a) P ? (Q? P )b) (P ? Q)? Pc) (P ? Q) & (P & ? Q)d) (P ? (Q & R))? (P ? R)For Exercises 2 and 3 use these sentence letters.• P : She wears high heels.• Q: I wear sneakers.• R: She is cheer captain.• S: I am on the bleachers.Exercise 2. Consider the following argument: If she wears high heels and I wear sneakers, then I am on the bleachers. She does not wear high heels. Therefore, if I do not wear sneakers, then I am not on the bleachers.a) Translate this argument into the language of propositional logic. (Put it in either standard or sequent form.)b) Use truth tables to say whether or not the argument is valid. Explain your answer!Exercise 3. Consider the following argument: Either she is cheer captain or I am on the bleachers. She is cheer captain only if I wear sneakers. If I am on the bleachers, then I wear sneakers. Therefore, I wear sneakers.a) Translate this argument into the language of propositional logic. (Put it in either standard or sequent form.)b) Use truth tables to say whether or not the argument is valid. Explain your answer!Exercise 4. Find a well-formed formula which does not use the symbol ? and has the following truth table. (Hint: Think about the symbols ? and ?. Note that you can check your answer using truth tables. . . )