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.

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

Not a crumb was left, but there was a note from Santa. [Do not use ∀ in your analysis of this; that is, use ∃ in your analysis of any quantifier phrases.]

A card was sent to each customer. [On one way of understanding this sentence, it would be true even if no two customers were sent the same card. Analyze it according to that interpretation.]

At most one size was left.

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.

Ann found the note that Bill left.

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 are logically equivalent (in symbols, φ ≃ ψ) if and only if ...

Ann wrote to Bill and he called her.

Someone called Tom

Someone is such that (he or she called Tom)

(∃x: x is a person) x called Tom

Not a crumb was left, but there was a note from Santa

Not a crumb was left ∧ there was a note from Santa

¬ a crumb was left ∧ something was a note from Santa

¬ some crumb is such that (it was left) ∧ something is such that (it was a note from Santa)

¬ (∃x: x is a crumb) x was left ∧ ∃y (y was a note from Santa)

¬ (∃x: Cx) Lx ∧ ∃y (y was a note ∧ y was from Santa)

A card was sent to each customer

each customer is such that (a card was sent to him or her)

(∀x: x is a customer) a card was sent to x

(∀x: Cx) some card is such that (it was sent to x)

(∀x: Cx) (∃y: y is a card) y was sent to x

Some card is such that (it was sent to each customer) would be true only if there was at least one card that was sent to all customers, so an analysis of it would not be a correct answer

At most one size was left

¬ at least two sizes were left

¬ at least two sizes are such that (they were left)

¬ (∃x: x is a size) (∃y: y is a size ∧ ¬ y = x) (x was left ∧ y was left)

also correct: (∀x: Sx) (∀y: Sy ∧ ¬ y = x) ¬ (Lx ∧ Ly)
also correct: (∀x: Sx ∧ Lx) (∀y: Sy ∧ Ly) x = y

Using Russell’s analysis:

Ann found the note that Bill left

the note that Bill left is such that (Ann found it)

(∃x: x is a note that Bill left ∧ only x is a note that Bill left) Ann found x

(∃x: (x is a note ∧ Bill left x) ∧ (∀y: ¬ y = x) ¬ (y is a note ∧ Bill left x)) Fax

also correct: (∃x: (Nx ∧ Lbx) ∧ ¬ (∃y: ¬ y = x) (Ny ∧ Lby)) Fax
also correct: (∃x: (Nx ∧ Lbx) ∧ (∀y: Ny ∧ Lby) x = y) Fax

Using the description operator:

Ann found the note that Bill left

[ _ found _ ] Ann (the note that Bill left)

Fa(Ix x is note that Bill left)

Fa(Ix (x is a note ∧ Bill left x))

Many different orders are possible for the rules used. In particular, NcP could be used before PCh in the second.

A pair of sentences φ and ψ are logically equivalent if and only if there is no possible world in which φ and ψ have different truth values

or

A pair of sentences φ and ψ are logically equivalent if and only if φ and ψ have the same truth value as each other in every possible world

Ann wrote to Bill and he called her

Ann and Bill are such that (she wrote to him and he called her)

[x wrote to y and y called x]_xy Ann Bill

[x wrote to y ∧ y called x]_xyab