Define the idea of a sentence φ implying a sentence ψ by completing the following with a definition in terms of truth values and possible worlds:

Suppose that φ and ψ are mutually exclusive (i.e., that φ ▵ ψ) and also that χ and ψ are mutually exclusive (i.e., that χ ▵ ψ). Can you conclude that φ and χ are equivalent (i.e., that φ ≃ χ)? Say why or why not in a way that makes use of the definitions of mutual exclusiveness and equivalence.

In the right circumstances, a tautology like Dave is Dave can convey genuine information—i.e., can convey information that rules out some possible worlds. (i) Use the definition of tautology to explain why this information cannot come from what is said—i.e., from the proposition expressed by the tautology. And (ii) use the definition of implicature to explain how such information might be conveyed as a suggestion.

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.

Al saw the meteor go by and Bill did, too; but Cal actually saw it land

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.

φ implies ψ if and only if there is no possible world in which φ is true and ψ is false (or: if and only if ψ is true in each possible world in which φ is true)

No. It does not follow that φ ≃ χ; that is, there are sentences φ and χ as described that are not equivalent. While φ and χ are each false in any possible world in which ψ is true, this still allows them to have different truth values in a possible world in which ψ is false. And, if they do have different truth values in any possible world, they are not equivalent. (For example, It’s very hot and It’s very cold each exclude The temperature is moderate and thus each forms a mutually exclusive pair when taken together with the latter sentence, but the first two sentences are clearly not equivalent.)

(i) A tautology cannot convey genuine information by way of the proposition it expresses because there is no possible world in which it is false, so the proposition expressed by a tautology rules out no possibilities. (ii) However, if the tautology is appropriate in only certain circumstances, using the tautology will suggest that such circumstances do occur. This suggestion will be false if the circumstances do not in fact occur, so it rules out some possibilities and provides genuine information. (For example, if someone asked the question Why did Dave do that?, the answer Dave is Dave would tend to suggest that Dave might be expected to do that sort of thing. This may be because the tautology could not do much that is relevant to answering the question apart from calling attention to features of Dave—such as his personality, character, or habits—so it would be appropriate as an answer only if such things sufficed to explain Dave’s action.)

Al saw the meteor go by and Bill did, too; but Cal actually saw it land

Al saw the meteor go by and Bill did, too ∧ Cal actually saw the meteor land

(Al saw the meteor go by ∧ Bill saw the meteor go by) ∧ Cal saw the meteor land

[Although there would be nothing really wrong with leaving the word actually in C, this term serves merely to emphasize the contrast between this sentence on the one hand and A and B on the other, so it loses its point once C is considered independently.]

(Al told Lew about Paris ∧ Al told Sue about Paris) ∧ (Al told Hugh about Venice ∧ Al told Hugh about Florence)

Al told Lew and Sue about Paris ∧ Al told Hugh about Venice and Florence

Al told Lew and Sue about Paris, and he told Hugh about Venice and Florence

[Or you might say … but he told Hugh … if the variation in the places (or the number of places) Al told various people about deserved emphasis.]