Consider an argument (∑, A) where ∑ is the set of premises B1, B2, Bn and A is the conclusion. Assume this argument is not valid (syntactically or semantically). Explain whether it is possible or impossible for the following statements or sets of statements to be true or false, and explain why they can or cannot be true or false:
(a) Every premise in E is true, and A is true.
(b) A is false, but every premise in Σ except B2 is true.
(c) Epistemic agent X fully believes every premise in Σ to be true, it's very important to X whether or not belief in A (belief that A is true) is rationally defensible, X does not believe A (X does not believe that A is true), X knows that the argument is valid, and X is not being irrational in any way.
(d) A person can consistently believe all of the premises while not believing the conclusion to be true.
(e) Every B₁ EΣ is true in an interpretation I, but A is not true in interpretation I.
(f) There is an interpretation I such that all of the premises in Σ are true and A is false.
(g) There is an interpretation I such that all of the premises in Σ are true and A is true.
(h) There is no way to derive A from the set of premises, E, using the rules of natural deduction.
(i) There is some sequence of correct applications of derivation rules leading from the (for- malized translations of the) premises to the (formalized translations of the) conclusi
(j) There is a countermodel to the argument.
(k) The argument is sound.
(l) The argument is unsound.