Define entailment for the special case of three premises by completing the following: φ, ψ, χ ⊨ θ if and only if … . (Your answer need not replicate the wording of the text’s definitions, but it should define entailment in terms of truth values and possible worlds.)

Suppose that each of φ and ψ is a tautology (i.e., ⊨ φ and also ⊨ ψ). What, if anything can you conclude about the equivalence of φ and ψ? That is, do you have enough information to conclude that they are equivalent (i.e., that φ ≃ ψ)? to conclude that they aren’t equivalent?—or can’t you say for sure? Explain your answer by reference to the definitions of tautologousness and equivalence, using the concepts of possible worlds and truth values that appear in those definitions.

Does a statement always entail everything it implicates? If you answer yes, explain why using the definitions of entailment and implicature? If you answer no, give an example of a case of implicature that isn’t a case of entailment.

Analyze the sentence below in as much detail as possible, presenting the result in both symbolic and English notation. Be sure that the unanalyzed components of your answer are complete and independent sentences; also try to respect any grouping in the English.

Use derivations to check whether each of the entailments below holds. If one fails, confirm a counterexample—that is, provide a table in which you calculate the truth values of the premises and conclusion on an assignment of truth values which is a counterexample lurking in an open gap.

[This question was on a topic not covered this year]

Why is a single dead-end open gap enough to show that an entailment does not hold? That is, why must all gaps close for a derivation to show that an entailment does hold?

φ, ψ, χ ⊨ θ if and only if there is no possible world in which φ, ψ, and χ are true but θ is false

φ and ψ are equivalent. Because each is a tautology, there is no possible world in which either is false, so there is no possible world in which they have different truth values.

No. Any true sentence with a false implicature provides an example of an implicature that isn’t an entailment because no true sentence can entail a false one, so you need only provide an example of a true sentence with a false implicature.

A dead-end open gap is always has a lurking counterexample that also makes the initial resources of the derivation true and its initial goal false. So any dead-end open gap shows that the derivation for which the derivation is constructed can have true premises and a false conclusion.