Define the idea of two sentences being mutually exclusive by completing the following with a definition in terms of truth values and possible worlds:

Suppose you are told about some sentences φ and ψ and the tautology ⊤ that φ ⊨ ⊤ and ⊤ ⊨ ψ. (i) What does this tell you about the possible truth values of φ? And (ii) what does it tell you about the possible truth values of ψ? In each case, explain your answer by reference to the definitions of a tautology and of implication (i.e., entailment).

Give an example of three sentences where the first implies (i.e., entails) the second, the second implicates the third (i.e., has the third as an implicature), but the first does not implicate the third. (It may be easiest to choose the second and third sentences as an example of an implicature—and it doesn’t have to be a new example—and then look for a sentence that implies one without implicating the other.) Be sure to say enough about the context of your sentences for me to be able to see that what you claim about them is so.

Analyze the sentence below in as much detail as possible, presenting the result using symbolic notation and also English notation (i.e., using both … and). Be sure that the unanalyzed components of your answer are complete and independent sentences (and give a key to the abbreviations you use for them); also try to respect any grouping in the English.

Although Al took the first turn, he missed the second; but he found his way to the meeting.

Synthesize an English sentence that has the analysis below. Choose a simple and natural sentence whose organization reflects the grouping of the logical form.

Use derivations to check whether each of the claims of entailment below holds. If an entailment fails, present a counterexample by providing a table in which you calculate the truth values of the premises and conclusion on an assignment of truth values that divides an open gap. (Your table should indicate the value of any compound component by writing this value under the main connective of the component.) Do not use the rule Adj in the first derivation, but you may use it in the second.

φ and ψ are mutually exclusive if and only if there is no possible world in which φ and ψ are both true (or: if and only if, in each possible world, at least one of φ and ψ is false)

(i) It tells you nothing about the possible truth values of φ. The fact that φ ⊨ ⊤ tells you that φ is not true in any world in which ⊤ is false; but, since ⊤ is a tautology, there are no worlds in which it is false, so you are told nothing about the truth value of φ in any possible world.

(ii) It tells you that ψ is true in every possible world—i.e., it, too, is a tautology. The fact that ⊤ ⊨ ψ tells you that ψ is true in any world in which ⊤ is true; but, since ⊤ is a tautology, it is true in every possible world.

Although it isn’t the only way to find an example, an easy way of doing so is to think about a yes-but answer in the case of a true sentence with a false implicature. Such an answer will imply the sentence (since it says it is true) but not have the implicature (because it explicitly rejects it). For example, in a context where There’s a cooler in the trunk implicates There’s beer in the cooler, the sentence There’s a cooler in the trunk but there’s no beer in it will imply the first but won’t implicate the second.

Although Al took the first turn, he missed the second; but he found his way to the meeting

Although Al took the first turn, he missed the second ∧ Al found his way to the meeting

(Al took the first turn ∧ Al missed the second turn) ∧ Al found his way to the meeting

Al wrote to Bob ∧ (Al spoke to Carol ∧ Al spoke to Dave)

Al wrote to Bob ∧ Al spoke to Carol and Dave

Al wrote to Bob, and [or: but] he Al spoke to Carol and Dave

[Notice that, while reporting the same facts, you might choose and if you were asked whether Al reached these people and but if you were asked how he reached them and were asked this in a context where the difference between writing and speech was important.]