Proofs and Symbolization in TFL and FOL

Question 1. Proofs

The following is a derivation from ¬ (A ? B) to ¬A  ¬B. Fill in all the gaps in the derivation (there are five gaps, indicated by roman numbers, i, ii, iii, iv, v). Some of the gaps require you to add citations to the proof, others require you to fill in sentences. You can figure out what is required by looking at where the gap appears in the proof, looking at what comes before and what comes next. 2 points for each answer.

1 ¬ (A ? B)

2 ¬(¬A  ¬B) for reduction

3  ¬B for reduction

4       ¬A  ¬B (i)__________

5  ¬(¬A  ¬B) R 2

6    B (ii) _________

7   (iii)______ For reductio

8 ¬A  ¬B I 7

9 ¬(¬A  ¬B) R 2

10 A ¬E 7-9

11 A ? B (iv)_________

12 ¬ (A ? B) (v)_________

13 ¬A  ¬B ¬E 2-12

Provide proofs which show the following.

a) (¬C ? B) → A, ¬A ? B → C

b)A  C ? (¬A ? B) → (C ? B)

c)P → (P → (P  Q)) is a theorem.

d)P and ¬¬P are provably equivalent.

e)¬R → Q and ¬(R  Q) are provably inconsistent.

Suppose that X and Y are provably equivalent. Does it follow that X → Y is a theorem? Explain your answer. Note that ‘X’ and ‘Y’ are variables here, they are not particular sentences of TFL. 6 points

Use the translation key below to symbolize the following sentences of English in FOL. 4 points each

domain: cities

a: Vienna

b: Berlin

c: Paris

Fx: __x is a French city.

Gx: __x is a German city.

Exy: __x is east of __y

Wxy: __x is west of __y

a) Paris is a French city that is west of Berlin.

b) All cities that are east of Vienna are also east of Paris.

c) Some German city is east of Paris.

d) Only French cities are west of Paris.

e) Not all German cities are east of Berlin.

f) For any two cities, if the first is west of the second, then the second is not west of the first

Use the translation key below to translate the following sentences of FOL into English. 

domain: historical events

Ix: __x is an important event

Px: __x was inspired by philosophical ideas

Gx: __x was inspired by Greek philosophy

Axy: __x occurred after __y

Bxy: __x occurred before __y

f: The French Revolution 

a: The American Revolution

c: The American Civil War

(a) (Pa ? Pf) ? ¬Pc [use “but”]

(b) x Bxa

(c) ¬x (Ix ? Bxa)

(d) x (Ix → Axf)

(e) x ((Py ? Bxa) → Gx)

(f) x y Byx 

Answer the following about challenges symbolizing English sentences. 

(a) Why would it be a mistake to symbolize ‘Some winters are long and hard’ as x Lx ?∃y Hy (with the domain of winters, and the obvious key for Lx and Hx)? How should it be symbolized? 

(b) Why would it be a mistake to symbolize ‘Mae wishes that it wasn’t raining’ as ¬ Wm (Wx: —x wishes it was raining; and m: Mae)? How could it be symbolized? (You will need to define a new predicate.)