Each of the following claims that a deductive relation holds between a pair of sentences. In each case, judge whether the claim is true and, if not, describe a sort of possibility that shows it is not true. Briefly explain your answers. For example, we can say that The package will arrive sometime does not entail The package will arrive next week because the possibility that it will arrive before or after next week is ruled out by the conclusion but not by the premise. In answering, it is safe to understand the sentences below all in the most straightforward way; you will miss the point of the exercise if you try to look for subtle or obscure possibilities.

Some of the following claims about deductive relations hold for any sentence φ, some for no sentence φ, and others hold only if φ is a tautology or only if it is absurd. In each case, say which is so and explain your answer.

The headings at the left of the table give information about the relation of φ and ψ and those at the top give information about the relation of ψ and χ. Fill in cells of the table by indicating what, if anything, you can conclude in each case about the relation of φ and χ. For example, if φ ⊨ ψ and ψ ⊨ χ, we cannot have φ true and χ false, so φ ⊨ χ (this is the transitivity of implication). However, no other patterns for φ and χ are ruled out, so “φ ⊨ χ” is the most we can say on the basis of the given information, and it can be entered in the upper left cell.