1. True or false? a A w qb of PL is valid just in case there is some interpretation I such that Vq) : 1. ( ) (b) A w qb of PL is a contradiction just in case for any interpretation I , VI() 7% 1. (c) A set 1" of wffs of PL is consistent just in case for every interpretation I , V;(q5) : 1 for all Q25 6 F. ) (d {A 33,133 0}? (aCjuA) 2. Explain how to add 6 to the semantics of PL. 3. This exercise is meant to help you see the motivation for separating the roles played by the interpre tation and valuation functions. Consider these denitions: A superinterpretation for PL is any function S that takes each wff of PL to either 1 or 0. (When S maps rt to 1, we write '5' I: 415'.) An argument from premises I' to conclusion q is supervalid just in case for any superin terpretation S, if S F 1b for all 1D in I', then S P q. (When the argument from P to 925 is supervalid, we write: '1" i=3 q'.) A w of PL is gt is supertautology just in case for any S , S I: gt. (a) Give an example of a supervalid argument, and explain why it is supervalid. (b) How many supertautologies are there? Explain your answer. (c) Give an example of a valid argument (i.e., valid in PL according to our oicial denition) which isn't supervalid, and explain why it is not supervalid. (cl) How does supervalidity reect, or fail to reflect, the truthfunctional meaning of the connectives? Briey explain in a way that claries why our focus has been on validity rather than supervalidity. 4. What must an interpretation function be like in order to render each of the sentences below true? Supply a semantic derivation for each. In essence1 what you are doing here is showing exactly which ways of assigning truth values to the sentence symbols would result in the whole sentence's being true. (Check out Handout 3 if you aren't sure how to proceed here.) (a) PA(AVA) (b) (PDQ)/\\P (c) (ADB)V((CAD)E)