Assume that P , Q , R , Z are complex well-formed formulas in the language of pr

Question

Assume that P,Q,R,Z are complex well-formed formulas in the language of propositional logic. Next, suppose we want to see whether or not P,Q,R semantically entails Z In other words, we want to see if "P, Q, R therefore Z" is a valid argument in the language of propositional logic. We test to see if this is the case by using the truth-tree method and this method reveals a tree with at least one completed open branch.

1. What does this completed open branch tell us about whether P,Q,R semantically entails Z ? That is, does it tell us P,Q,RZ or P,Q,RZ ?
2. What does this completed open branch tell us about the truth values of {P,Q,R,Z} ?

Explanation / Answer

(2) A closed tree indicates that there is no way of consistently assigning truth valuesto propositional letters so that every proposition in the branch is true (including thosein the stack). Thus, there is no interpretation that would make P, Q, R, Z true.

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