Analyze the sentences below in as much detail as possible, providing a key to the non-logical vocabulary you use. State your analysis also in a form that expresses any generalizations using unrestricted quantifiers.

Every door was locked.

Only people who had witnessed the event were able to follow the description of it.

[It is possible for the scope of only to change with emphasis; although varying interpretations are less likely with this sentence than with others, you may choose whichever scope seems most plausible to you.]

No key opened every door.

[You should understand this sentence to leave open the possibility that some key opened some door.]

Synthesize an English sentence with the following logical form; that is, find a sentence that would have the following analysis:

(∀x: Px ∧ Nxa) (Dxm ∨ Axm)

A: [ _ was acted on at _ ]; D: [ _ was discussed at _ ]; N: [ _ was on _ ]; P: [ _ was a proposal]; a: the agenda; m: the meeting

Use derivations to show that the following arguments are valid. You may use any rules.

Use a derivation to show that the following argument is not valid and present a counterexample by describing a structure that divides an open gap. (You may describe the structure either by depicting it in a diagram, as answers in the text usually do, or by giving tables.)

Every door was locked

Every door is such that (it was locked)

(∀x: x is a door) x was locked

only people who had witnessed the event were able to follow the description of it

only people who had witnessed the event are such that (they were able to follow the description of it)

(∀x: ¬ x is a person who had witnessed the event) ¬ x was able to follow the description of the event

(∀x: ¬ (x is a person ∧ x had witnessed the event)) ¬ Fx(the description of the event)

Other possible (though less likely) interpretations:

Not a possibility:

No key opened every door

No key is such that (it opened every door)

(∀x: x is a key) ¬ x opened every door

(∀x: Kx) ¬ every door is such that (x opened it)

(∀x: Kx) ¬ (∀y: y is a door) x opened y

Although there are equivalent analyses, one that differs only in the location of ¬ is likely to be wrong. In particular, (∀x: Kx) (∀y: Dy) ¬ Oxy rules out the possibility that some key opened some door.

(∀x: Px ∧ Nxa) (Dxm ∨ Axm)

(∀x: x was a proposal ∧ x was on the agenda) (x was discussed at the meeting ∨ x was acted on at the meeting)

(∀x: x was a proposal on the agenda) (x was discussed or acted on at the meeting)

Every proposal on the agenda is such that (it was discussed or acted on at the meeting)

Every proposal on the agenda was discussed or acted on at the meeting

Counterexample presented by a diagram

Counterexample presented by tables