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All questions concern our propositional logic and material covered in Chapter 3 of the course text. The marks for each question are shown in brackets.
Hence by TRANSITIVITY 3 times, and EQUANIMITY, we have that .
Hence by TRANSITIVITY 3 times .
SOLUTION In state (p,q,r)=(T,F,F) the sentence evaluates to F. However is complete, and so the sentence cannot be a theorem.
SOLUTION Assume . However , and substitution with p:= P gives that . However by Metatheorem ``Redundant True'' and Equanimity we have that . Metatheorem 3.7 states that any two theorems are equivalent, and in particular that . Leibniz with E being gives that , and so by equanimity, .