Analyze the following sentences in as much detail as possible, providing a key to the items of non-logical vocabulary (upper and lower case letters apart from variables) that appear in your answer. Notice the special instructions given for each of 1, 2, and 3.

Sam saw a supernova.

[Give an analysis using a restricted quantifier, and restate it using an unrestricted quantifier.]

None of the flights Al was on were delayed.

[Do not use ∀ in your analysis of this; that is, use ∃ in your analysis of any quantifier phrases.]

Someone ate every cookie.

[On one way of understanding this sentence, it would be false if the cookies were eaten by several people. Analyze it according to that interpretation.]

Fred had to make at least two connections.

Analyze the sentence below using each of the two ways of analyzing the definite description. That is, give an analysis that uses Russell’s treatment of definite descriptions as quantifier phrases as well as one that uses the description operator to analyze the definite description.

Al opened the package.

Use a derivation to show that the following argument is valid. You may use any rules.

Use a derivation to show that the following argument is not valid, and use either a diagram or tables to present a counterexample that divides an open gap of your derivation.

Complete the following to give a definition of entailment in terms of truth values and possible worlds:

A set Γ entails a sentence φ (i.e., Γ ⊨ φ) if and only if ...

Analyze the sentence below using abstracts and variables to represent pronominal cross reference to individual terms (instead of replacing pronouns by such antecedents). A letter standing for an individual term should appear in your analysis only as often as the individual term appears in the original sentence.

Al called both Bill, who called him back, and Carol, who didn't.

Sam saw a supernova

A supernova is such that (Sam saw it)

(∃x: x is a supernova) Sam saw x

None of the flights Al was on were delayed

¬ some flight Al was on was delayed

¬ some flight Al was on is such that (it was delayed)

¬ (∃x: x is flight Al was on) x was delayed

¬ (∃x: x is a flight ∧ Al was on x) x was delayed

The analysis (∃x: Fx ∧ Nax) ¬ Dx would say that Al was on at least one flight that wasn’t delayed (i.e., that not all the flights he was on were delayed)

Someone ate every cookie

someone is such that (he or she ate every cookie)

(∃x: x is a person) x ate every cookie

(∃x: Px) every cookie is such that (x ate it)

(∃x: Px) (∀y: y is a cookie) x ate y

The alternative interpretation Every cookie is such that (someone ate it) would be true even if the cookies were eaten by several people (i.e., even if no one person ate all of them)

Fred had to make at least two connections

at least two connections are such that (Fred had to make them)

(∃x: x is an connection) (∃y: y is an connection ∧ ¬ y = x) (Fred had to make x ∧ Fred had to make y)

Using the description operator:

Al opened the package

[ _ opened _ ]  Al  the package

Oa(Ix x is a package)

A set Γ entails a sentence φ if and only if there is no possible world in which φ is false while every member of Γ is true

or

A set Γ entails a sentence φ if and only if φ is true in every possible world in which every member of Γ is true

Al called both Bill, who called him back, and Carol, who didn't

Al, Bill, and Carol are such that (the first called both the second, who called him back, and the third, who didn't)

[x called both y, who called x back, and z, who didn't call x back]_xyz Al Bill Carol

[x called y, who called x back ∧ x called z, who didn't call x back]_xyzabc

[(x called y ∧ y called x) ∧ (x called z ∧ ¬ z called x)]_xyzabc