Phi 270
Fall 2013
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Phi 270 F99 test 4

F99 test 4 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.

1. Sam invited every vertebrate to the party, but only people accepted his invitation
answer
2. Tom didn’t send anything to the printer
answer
3. No game that every child liked was complete
answer

Synthesize an English sentence whose analysis would yield the following form.

4. (∀x: Px) (∀y: Ry ∧ Txy) Sy
P: [ _ is a person]; R: [ _ is a room]; S: [ _ was reserved]; T: [ _ thought of _ ]
answer

Use derivations to establish the validity of the following arguments. You may use attachment rules.

5.
∀x (Fx → Gx)
∀x Fx → ∀x Gx
answer
6.
∀x ∀y (Fyx → ¬ Py)
∀x (Px → ∀y ¬ Fxy)
answer

Use a derivation to show that the following argument is not valid and describe a structure (by using either a diagram or tables) that is a counterexample lurking one of the derivation’s open gaps.

7.
∀x ∀y (Fy → ¬ Rxy)
∀x Rxx
∀x ∀y ¬ Rxy
answer

F99 test 4 answers

1.

Sam invited every vertebrate to the party, but only people accepted his invitation

Sam invited every vertebrate to the party ∧ only people accepted Sam’s invitation

every vertebrate is such that (Sam invited it to the party) ∧ only people are such that (they accepted Sam’s invitation)

(∀x: x is a vertebrate) Sam invited x to the party ∧ (∀x:¬ x is a person) ¬ x accepted Sam’s invitation

(∀x: Vx) Isxp ∧ (∀x:¬ Px) ¬ Ax(Sam’s invitation)

(∀x: Vx) Isxp ∧ (∀x: ¬ Px) ¬ Ax(is)

A: [ _ accepted _ ]; I: [ _ invited _ to _ ]; P: [ _ is a person]; V: [ _ is a vertebrate]; i: [ _ ’s invitation]; p: the party; s: Sam

2.

Tom didn’t send anything to the printer

everything is such that (Tom didn’t send it to the printer)

∀x Tom didn’t send x to the printer

∀x ¬ Tom sent x to the printer

∀x ¬ Stxp

S: [ _ sent _ to _ ]; p: the printer; t: Tom

3.

No game that every child liked was complete

No game that every child liked is such that (it was complete)

(∀x: x was a game that every child liked) ¬ x was complete

(∀x: x was a game ∧ every child liked x) ¬ Cx

(∀x: x was a game ∧ every child is such that (he or she liked x)) ¬ Cx

(∀x: Gx ∧ (∀y: y was a child) y liked x) ¬ Cx

(∀x: Gx ∧ (∀y: Dy) Lyx) ¬ Cx

C: [ _ was complete]; D: [ _ was a child]; G: [ _ was a game]; L: [ _ liked _ ]

4.

(∀x: x is a person) (∀y: y is a room ∧ x thought of y) y was reserved

(∀x: x is a person) (∀y: y is a room x thought of) y was reserved

(∀x: x is a person) every room x thought of was such that (it was reserved)

(∀x: x is a person) every room x thought of was reserved

everyone is such that (every room he or she thought of was reserved)

every room anyone thought of was reserved

5.
│∀x (Fx → Gx) a:3
├─
││∀x Fx a:4
│├─
││ⓐ
3 UI │││Fa → Ga 5
4 UI │││Fa (5)
5 MPP │││Ga (6)
│││●
││├─
6 QED │││Ga 2
│├─
2 UG ││∀x Gx 1
├─
1 CP │∀x Fx → ∀x Gx
6.
│∀x ∀y (Fyx → ¬ Py) b:5
├─
│ⓐ
│││Pa (8)
││├─
│││ⓑ
│││││Fab (7)
││││├─
5 UI │││││∀y (Fyb → ¬ Py) a:6
6 UI │││││Fab → ¬ Pa 7
7 MPP │││││¬ Pa (8)
│││││●
││││├─
8 Nc │││││⊥ 4
│││├─
4 RAA ││││¬ Fab 3
││├─
3 UG │││∀y ¬ Fay 2
│├─
2 CP ││Pa → ∀y ¬ Fay 1
├─
1 UG │∀x (Px → ∀y ¬ Fxy)
7.
│∀x ∀y (Fy → ¬ Rxy) a:4,b:5
│∀x Rxx a:6,b:7
├─
│ⓐ
││ⓑ
││││Rab (11)
│││├─
4 UI ││││∀y (Fy → ¬ Ray) a:8, b:9
5 UI ││││∀y (Fy → ¬ Rby) a:12, b:13
6 UI ││││Raa (10)
7 UI ││││Rbb (14)
8 UI ││││Fa → ¬ Raa 10
9 UI ││││Fb → ¬ Rab 11
10 MTT ││││¬ Fa
11 MTT ││││¬ Fb
12 UI ││││Fa → ¬ Rba 15
13 UI ││││Fb → ¬ Rbb 14
14 MTT ││││¬ Fb
││││
││││││¬ Fa
│││││├─
││││││○ ¬Fa,¬Fb,Rab,Raa,Rbb ⊭ ⊥
│││││├─
││││││⊥ 16
││││├─
16 IP │││││Fa 15
││││
│││││¬ Rba
││││├─
│││││○ ¬Fa,¬Fb,Rab,
│││││          Raa,Rbb,¬Rba ⊭ ⊥
││││├─
│││││⊥ 15
│││├─
15 RC ││││⊥ 3
││├─
3 RAA │││¬ Rab 2
│├─
2 UI ││∀y ¬ Ray 1
├─
1 UI │∀x ∀y ¬ Rxy
 

The counterexample below lurks in both gaps: