Phi 270
Fall 2013
site index
(Site navigation is not working.)

Phi 270 F02 test 5

F02 test 5 questions

Analyze the following sentences in as much detail as possible, providing a key to the non-logical vocabulary (upper and lower case letters) appearing in your answer. Notice the additional instructions given for the first.
1. Al received a card that made him laugh [Give this analysis also using an unrestricted quantifier.]
answer
2. There is a toy that every child wanted
answer
3. Santa left at least two packages
answer
Analyze the sentence below using each of the two ways of analyzing the definite description the battery. That is, analyze it using Russell’s analysis of definite descriptions as quantifier phrases and then analyze it again using the description operator.
4. The battery is dead
answer
Use derivations to show that the following argument is valid. You may use any rules.
5.
∃x (Fx ∧ Gx)
∃x (Gx ∧ Fx)
answer
Use a derivation to show that the following argument is not valid and use either tables or a diagram to describe a counterexample lurking in an open gap.
6.
∃x ∃y Rxy
∃x Rax
answer
Complete the following to give a definition of entailment in terms of truth values and possible worlds:
7. A set Γ entails a sentence φ (i.e., Γ ⊨ φ) if and only if …
answer
Complete the following truth table by calculating the truth value of the sentence on the given assignment. Show the value of each component by writing it under the main connective of that component, and circle the truth value of the sentence as a whole.
8.
ABCD(A→B)∧¬(C∨¬D)
TFFT
answer
Give at least two restatements of the following sentence as an expansion on a term appearing in it (i.e., as an abstract applied to such a term):
9. Raba
answer

F02 test 5 answers

1.

Al received a card that made him laugh

some card that made Al laugh is such that (Al received it)

(∃x: x is a card that made Al laugh) Al received x

(∃x: x is a card ∧ x made Al laugh) Rax

(∃x: Cx ∧ Lxa) Rax
∃x ((Cx ∧ Lxa) ∧ Rax)

C: [ _ is a card]; L: [ _ made _ laugh]; R: [ _ received _ ]; a: Al

2.

There is a toy that every child wanted

Something is a toy that every child wanted

Something is such that (it is a toy that every child wanted)

∃x x is a toy that every child wanted

∃x (x is a toy ∧ every child wanted x)

∃x (Tx ∧ every child is such that (he or she wanted x))

∃x (Tx ∧ (∀y: y is a child) y wanted x)

∃x (Tx ∧ (∀y: Cy) Wyx)

C: [ _ is a child]; T: [ _ is a toy]; W: [ _ wanted _ ]

3.

Santa left at least two packages

at least two packages are such that (Santa left them)

(∃x: x is a package) (∃y: y is a package ∧ ¬ y = x) (Santa left x ∧ Santa left y)

(∃x: Px) (∃y: Py ∧ ¬ y = x) (Lsx ∧ Lsy)

L: [ _ left _ ]; P: [ _ is a package]; s: Santa

4.

using Russell’s analysis:

The battery is dead

The battery is such that (it is dead)

(∃x: x and only x is a battery) x is dead

(∃x: x is a battery ∧ (∀y: ¬ y = x) ¬ y is a battery) x is dead

(∃x: Bx ∧ (∀y: ¬ y = x) ¬ By) Dx
or: (∃x: Bx ∧ (∀y: By) x = y) Dx

using the description operator:

The battery is dead

D the battery

D(Ix x is a battery)

D(Ix Bx)

B: [ _ is a battery]; D: [ _ is dead]

5.
│∃x (Fx ∧ Gx) 1
├─
│ⓐ
││Fa ∧ Ga 2
│├─
2 Ext ││Fa (6)
2 Ext ││Ga (5)
││
│││∀x ¬ (Gx ∧ Fx) a:4
││├─
4 UI │││¬ (Ga ∧ Fa) 5
5 MPT │││¬ Fa (6)
│││●
││├─
6 Nc │││⊥ 3
│├─
3 NcP ││∃x (Gx ∧ Fx) 1
├─
1 PCh │∃x (Gx ∧ Fx)
6.
│∃x ∃y Rxy 1
├─
│ⓑ
││∃y Rby 2
│├─
││ⓒ
│││Rbc
││├─
││││∀x ¬ Rax a:4, b:5, c:6
│││├─
4 UI ││││¬ Raa
5 UI ││││¬ Rab
6 UI ││││¬ Rac
││││○ Rbc,¬Raa, ¬Rab,¬Rac ⊭ ⊥
│││├─
││││⊥ 3
││├─
3 NcP │││∃x Rax 2
│├─
2 PCh ││∃x Rax 1
├─
1 PCh │∃x Rax
 
7. A set Γ entails a sentence φ if and only if there is no possible world in which every member of Γ is true but φ is false (or: if and only if φ is true in every possible world in which all members of Γ are true)
8.
ABCD(A→B)∧¬(C∨¬D)
TFFTFⒻTFF
9. Up to the choice of variables, the possibilities are the following:
[Rabx]xa, [Rxba]xa, [Rxbx]xa, [Raxa]xb