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.

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

There is an elf who neglects no one. [Do not use ∀ in your analysis of this; that is, use ∃ in your analysis of any quantifier phrases.]

Everyone watched a movie. [On one way of understanding this sentence, it would not be true unless everyone watched the same movie. Analyze it according to that interpretation.]

Someone sang to someone else.

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.

Rudolph guided the sleigh that flew.

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.

A pair of sentences φ and ψ entails a sentence χ (in symbols, φ, ψ ⊨ χ) if and only if ...

Bill called Carol and mentioned his father to her.

Dave found a coin

A coin is such that (Dave found it)

(∃x: x is a coin) Dave found x

There is an elf who neglects no one

Something is an elf who neglects no one

∃x x is an elf who neglects no one

∃x (x is an elf ∧ x neglects no one)

∃x (x is an elf ∧ ¬ x neglects someone)

∃x (Ex ∧ ¬ someone is such that (x neglects him or her))

∃x (Ex ∧ ¬ (∃y: y is a person) x neglects y)

Everyone watched a movie

some movie is such that (everyone watched it)

(∃x: x is a movie) everyone watched x

(∃x: Mx) everyone is such that (he or she watched x)

(∃x: Mx) (∀y: y is a person) y watched x

The alternative interpretation Everyone is such that (he or she watched a movie) could be true even if there was no one movie that everyone watched

Someone sang to someone else

Someone is such that (he or she sang to someone else)

(∃x: x is a person) x sang to someone else

(∃x: Px) someone other than x is such that (x sang to him or her)

(∃x: Px) (∃y: y is a person ∧ ¬ y = x) x sang to y

Using Russell’s analysis:

Rudolph guided the sleigh that flew

the sleigh that flew is such that (Rudolph guided it)

(∃x: x is a sleigh that flew ∧ only x is a sleigh that flew) Rudolph guided x

(∃x: (x is a sleigh ∧ x flew) ∧ (∀y: ¬ y = x) ¬ (y is a sleigh ∧ y flew)) Grx

also correct: (∃x: (Sx ∧ Fx) ∧ ¬ (∃y: ¬ y = x) (Sy ∧ Fy)) Grx
also correct: (∃x: (Sx ∧ Fx) ∧ (∀y: Sy ∧ Fy) x = y) Grx

Using the description operator:

Rudolph guided the sleigh that flew

[ _ guided _ ] Rudolph the sleigh that flew

Gr(Ix x is a sleigh that flew)

Gr(Ix (x is a sleigh ∧ x flew))

A pair of sentences φ and ψ entails a sentence χ if and only if there is no possible world in which both φ and ψ are true and χ is false

or

A pair of sentences φ and ψ entails a sentence χ if and only if χ is true in every possible world in which both φ and ψ are true

Bill called Carol and mentioned his father to her

Bill and Carol are such that (he called her and mentioned his father to her)

[x called y and mentioned x’s father to y]_xy Bill Carol

[x called y ∧ x mentioned x’s father to y]_xybc

[Cxy ∧ Mx(x’s father)y]_xybc