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

Al didn’t run into anyone he knew. [Do not use ∀ in your analysis of this; that is, use ∃ in your analysis of any quantifier phrases.]

Every child was visited by someone. [On one way of understanding this sentence, it could be true even though no one person visited all children. Analyze it according to that interpretation.]

Ed’s ship came close to at least two icebergs.

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.

The agent that Ed spoke to spoke to Fred.

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 sentence φ is a tautology (in symbols, ⊨ φ) if and only if ...

Al congratulated himself.

Someone spoke

Someone is such that (he or she spoke)

(∃x: x is a person) x spoke

Al didn’t run into anyone he knew

¬ Al ran into someone he knew

¬ someone that Al knew is such that (Al ran into him or her)

¬ (∃x: x is a person Al knew) Al ran into x

¬ (∃x: x is a person ∧ Al knew x) Al ran into x

The analysis (∃x: Px ∧ Kax) ¬ Rax would say that there was someone Al knew who he didn’t run into

Every child was visited by someone

every child is such that (he or she was visited by someone)

(∀x: x is a child) x was visited by someone

(∀x: Cx) someone is such that (x was visited by him or her)

(∀x: Cx) (∃y: y is a person) x was visited by y

The alternative interpretation Someone is such that (every child was visited by him or her) would not be true unless some one person visited all children

Ed’s ship came close to at least two icebergs

at least two icebergs are such that (Ed’s ship came close to them)

(∃x: x is an iceberg) (∃y: y is an iceberg ∧ ¬ y = x) (Ed’s ship came close to x ∧ Ed’s ship came close to y)

(∃x: Ix) (∃y: Iy ∧ ¬ y = x) (C(Ed's ship)x ∧ C(Ed's ship)y)

Using Russell’s analysis:

The agent that Ed spoke to spoke to Fred

The agent that Ed spoke to is such that (he or she spoke to Fred)

(∃x: x is an agent that Ed spoke to ∧ only x is an agent that Ed spoke to) x spoke to Fred

(∃x: (x is an agent ∧ Ed spoke to x) ∧ (∀y: ¬ y = x) ¬ (y is an agent ∧ Ed spoke to y)) Sxf

also correct: (∃x: (Ax ∧ Sex) ∧ ¬ (∃y: ¬ y = x) (Ay ∧ Sey)) Sxf
also correct: (∃x: (Ax ∧ Sex) ∧ (∀y: Ay ∧ Sey) x = y) Sxf

Using the description operator:

The agent that Ed spoke to spoke to Fred

[ _ spoke to _ ]  the agent that Ed spoke to  Fred

S(Ix x is an agent that Ed spoke to)f

S(Ix (x is an agent ∧ Ed spoke to x))f

A sentence φ is a tautology if and only if there is no possible world in which φ is false

or

A sentence φ is a tautology if and only if φ is true in every possible world

Al congratulated himself

Al is such that (he congratulated himself)

[x congratulated x]_x Al