Define entailment by completing the following with a definition in terms of truth values and possible worlds:

Suppose that φ and ψ are contradictory (i.e., that φ ⧖ ψ) and also that ψ implies χ (i.e., that ψ ⊨ χ). Can you conclude anything about the deductive relations holding between φ and χ? That is, does the information given allow you to rule out one or more of the four conceivable patterns of truth values for two sentences (i.e., TT, TF, FT, FF) in the case of φ and χ, or is it consistent with what you are told that φ and χ be logically independent? You should justify your answer in a way that shows you know the definitions of contradictoriness and implication (but you need not provide the name of the relation between φ and χ if you conclude that they are related).

(i) Present a sentence that, when used in a certain context, has an implicature that suggests something beyond what the sentence says literally, and (ii) briefly explain why the sentence has that implicature in the context you describe. In addressing part (i), be sure to show that the implicature is not part of what the sentence says by describing a way that the implicature could be false while what the sentence says literally is true.

Analyze the sentence below in as much detail as possible, presenting the result using symbolic notation and (and present the same analysis also using English notation—i.e., using both … and … to indicate conjunction). 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.

Ann posed the problem, and Bill and Carol each solved it

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

a set Γ of sentences entails a sentence φ (i.e., Γ ⊨ φ) if and only if there is no possible world in which every member of Γ is true and φ is false (or: if and only if φ is true in each possible world in which every member of Γ is true)

We can conclude that φ and χ cannot be both false. Since φ and ψ are contradictory, we know that they cannot be both true or both false—i.e., their truth values must be different—so ψ must be true if φ is false. And, since ψ implies χ, we know that χ cannot be false when ψ is true. Therefore, χ cannot be false when φ is false.

[Although it is not part of what you were asked, note that we cannot conclude anything else about the truth values of φ and χ. In some cases they might exhibit the patterns TF and FT because φ and ψ might exhibit both, and χ might in fact be equivalent to ψ. And they might exhibit the pattern TT since the given information is consistent with this pattern in a case where ψ is false; for example, suppose φ says The glass is not empty, ψ says The glass is empty, and χ says The glass is not full, and consider a case where the glass is half full.]

Here’s a sample answer. (i) If I say, “I have a lot of work to do” in answer to the question “Are you going to the movie?,” the person asking would in most circumstances conclude that I am not going; but it would be perfectly possible for me to have a lot work to do (so what I have said is true) and nevertheless go to the movie (in which case what I implicated would be false). (ii) The implicature arises because my response to the question is appropriate only if it is taken as an answer; and, unless I am known to be someone makes a point of going to the movies when I have a lot of work to do, the answer made most likely by my response is that I am not going.

Ann posed the problem, and Bill and Carol each solved it

Ann posed the problem ∧ Bill and Carol each solved the problem

Ann posed the problem ∧ (Bill solved the problem ∧ Carol solved the problem)

[The function of each in this sentence is to pointedly leave open the possibility that Bill and Carol worked independently; a sentence that said instead that they worked together could not be analyzed as a conjunction of the components B and C above.]