Phi 270 F02 test 4 in pdf format
Analyze the sentences below in as much detail as possible, providing a key to the non-logical vocabulary you use. Notice the special instructions for 2.
1. Only bears performed.
[answer]
2. If everyone cheered, the elephant bowed. [In this case, restate your answer using an unrestricted quantifier.]
[answer]
3. No one laughed at any performers except clowns.
[answer]
Synthesize an English sentence with the following logical form:
4. (∀x: Px ∧ Cxt) Ctx
[C: λxy (x called y); P: λx (x is a person); t: Tom]
[answer]
Use derivations to establish the validity of the following arguments. You may use attachment rules.
5.
∀x Fx
∀x ¬ (Fx ∧ Gx)
∀x ¬ Gx
[answer]
6.
∀x (∀y: Fy) Rxy
(∀x: Fx) ∀y Ryx
[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 divides one of the derivation's open gaps.
7.
∀x Rax
(∀x: Rbx) ¬ Rxa
∀x ¬ Rbx
[answer]
Phi 270 F02 test 4 answers
1.

Only bears performed

(∀x: ¬ x is a bear) ¬ x performed

(∀x: ¬ Bx) ¬ Px
[B: λx (x is a bear); P: λx (x performed)]
2.

If everyone cheered, the elephant bowed

everyone cheered → the elephant bowed

(∀x: x is a person) x cheered → the elephant bowed

(∀x: Px) Cx → Be
∀x (Px → Cx) → Be
[B: x bowed; C: x cheered; P: x is a person; e: the elephant]
Incorrect:
(∀x: Px) (Cx → Be) or: ∀x (Px → (Cx → Be))
these say: If anyone cheered, the elephant bowed
3.

No one laughed at any performers except clowns

all performers except clowns are such that (no one laughed at them)

(∀x: x is a performer ∧ ¬ x is a clown) no one laughed at x

(∀x: x is a performer ∧ ¬ x is a clown) (∀y: y is a person) ¬ y laughed at x

(∀x: Fx ∧ ¬ Cx) (∀y: Py) ¬ Lyx
[C: λx (x is a clown); F: λx ( x is a peformer); P: λx (x is a person); L: λxy (x laughed at y)]
Incorrect:
(∀y: Py) ¬ (∀x: Fx ∧ ¬ Cx) Lyx
says: No one laughed at all performers who weren't clowns
4.

(∀x: x is a person ∧ x called Tom) Tom called x

(∀x: x is a person who called Tom) Tom called x

everyone who called Tom is such that (Tom called him or her)

Tom called everyone who called him
5.
│∀x Fx a:2
│∀x ¬ (Fx ∧ Gx) a:3
├─
│ⓐ
2 UI ││Fa (4)
3 UI ││¬ (Fa ∧ Ga) 4
4 MPT ││¬ Ga (5)
││●
│├─
5 QED ││¬ Ga 1
├─
1 UG │∀x ¬ Gx
6.
│∀x (∀y: Fy) Rxy b:3
├─
│ⓐ
││Fa (4)
│├─
││ⓑ
3 UI │││(∀y: Fy) Rby a:4
4 SB │││Rba (5)
│││●
││├─
5 QED │││Rba 2
│├─
2 UG ││∀y Rya 1
├─
1 RUG │(∀x: Fx) ∀y Ryx
7.
│∀x Rax a:3,b:4,c:5
│(∀x: Rbx) ¬ Rxa c:6,a:7,b:8
├─
│ⓒ
│││Rbc (6)
││├─
3 UI │││Raa (7)
4 UI │││Rab
5 UI │││Rac
6 SB │││¬ Rca
7 SC │││¬ Rba
│││
│││││¬ Rbb
││││├─
│││││○ Rbc,Raa,Rab,Rac,¬Rca,¬Rba,¬Rbb ⇏ ⊥
││││├─
│││││⊥ 9
│││├─
9 IP ││││Rbb 8
│││
││││¬ Rba
│││├─
││││○ Rbc,Raa,Rab,Rac,¬Rca,¬Rba ⇏ ⊥
│││├─
││││⊥ 8
││├─
8 MCR │││⊥ 2
│├─
2 RAA ││¬ Rbc 1
├─
1 UG │∀x ¬ Rbx
Counterexample presented by tables Counterexample presented by a diagram
range: 1, 2, 3  
abc
123
 
R 1 2 3
1 T T T
2 F F T
3 F F F
Grayed values are not required to divide either gap;
the value for R22 is not required to divide the 2nd gap