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

Consider the following two items of information about sentences φ, ψ, and χ:

Which of (i) and (ii) indicates stronger logical relations among these sentences by ruling out more patterns of truth values for the three? And which further patterns does it rule out?

Suppose you know that φ implicates ψ and that ψ implies (i.e., entails) χ. Can you tell whether φ implicates χ? (Be sure to explain you answer by reference to the definitions of implication and implicature.)

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.

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 from §2.4.

φ and ψ are contradictory (φ ⋈ ψ) if and only if there is no possible world in which φ and ψ share the same truth value (or: if and only if, in each possible world, φ has a truth value different from the one ψ has).

Both (i) and (ii) rule out certain values for φ and ψ when χ is false. While (ii) says that φ and ψ cannot be both true in this sort of case, (i) says more. It says of each of φ and ψ that it cannot be true when χ is false, so it says, on top of what (i) says, that neither the pattern φ true and ψ false nor the pattern φ false and ψ true can occur when χ is false. So (i) rules out three of the eight possible patterns of truth values for three sentences while (ii) rules out only one of the eight.

The basic idea is that ψ says part of what φ suggests, and χ says part of what ψ says. So χ says part of what φ suggests and is therefore an implicature of φ. This can be explained in terms of the definitions as follows. To say that φ implicates ψ is to say that ψ is true whenever φ is appropriate. But if ψ implies χ, then χ must be true whenever ψ is, so it must be true whenever φ is appropriate. Therefore, φ does implicate χ.

B: the first baseman went after the ball; F: the first baseman fumbled the ball; S: the runner went on to second