Examples of Logical Problems (Math Problem Sample)

I.
1 ∀x(Ax → Bx)
2 ∀x (Ax→Cx)
3 Ad→Bd 1, ∀Elim
4 Ad→Cd 2, ∀Elim
5 Ad Assumption for CP
6 Bd 3,5 →Elim
7 Cd 3,4 → Elim
8 Bd & Cd 6,7 & Intro
9 Ad → (Bd &Cd) 5,-8 →Intro
10 ∀x[Ax→(Bx&Cx)] 9, ∀Intro
1 ∀x(¬Cx →¬Ax)
2 ∃xCx → (∃x¬(¬Bx V ¬ Dx)
3 ¬(¬∃xAx V ∃xBx) Assumption for IP
4 ¬Cd → ¬Ad 1, ∀ Elim
5 Cd Assumption for IP
6 ∃xCx 5, ∃intro
7 ∃x¬(¬Bx V ¬Dx) 2,6 → elim
8 ¬(¬Ba V ¬Da) Assumption for 7, E-elim
9 ¬Ba Assumption for IP
10 ¬Ba V ¬Da 9, V intro
⊥ 8, 10 ⊥ intro
¬¬Ba 9, 10-11 ¬ intro
Ba 12, ¬ elim
∃xBx 13, ∃-intro
∃xBx 8, 9-11, ∃-elim
¬∃xAx V ∃xBx 12, V intro
⊥ 3, 13 ⊥ intro
¬Cd 5, 6-14 ¬intro
¬Ad 4, 16 → ∃ elim
∀x¬Ax 16, ∀ intro
∀x¬Ax 8, 9-20, ∃ elim
Ab Assumption for IP
¬Ab 21, ∀ elim
⊥ 22, 23 ⊥ intro
¬∃x Ax 24, ⊥ elim
¬∃x Ax V ExBx 25 V intro
⊥ 3, 26 ⊥ intro
¬¬(¬∃x Ax V ∃x Bx) 3, 4-27 ¬intro
¬∃x Ax V ∃x Bx 28, ¬ Elim
1 ¬∃x Ax V ∀x (¬Bx V Ex)
2 ∃xBx → ∀ x¬Ex
3 ¬∃x Ax Assumption for V elim
4 Ab & Db Assumption for CP
5 Ab 4, & Elim
6 ∃x Ax 5, ∃ intro
7 ⊥ 3, 6 ⊥ intro
8 ¬Bb 7, ⊥ elim
9 Ab & Db → ¬Bd 4-8 → intro
10 ∀ x[(Ax&Dx) → ¬Bx] 9, ∀ intro
∀ x(¬Bx V Ex) Assumption for V elim
¬Bd V Ed 11, ∀ elim
¬Bd assumption for V elim
Ab & Db → ¬Bd 4-13, → intro
∀ x[(Ax & Dx) → ¬Bx] 14, ∀ intro
Ed assumption for V elim
Dd 4, & elim
∃x Dx 17, ∃ intro
∀ x¬Ex 2, 18 → elim
¬Ed 19, ∀ elim
⊥ 16, 20 ⊥ intro
¬Bd 21, ⊥ elim
Ab & Db → ¬Bd 4-22 →intro
∀ x[(Ax & Dx) → ¬Bx] 23, ∀ intro
∀ x[(Ax & Dx) → ¬Bx] 12, 13-15, 16-24, V elim
∀ x[(Ax & Dx) → ¬Bx] 1, 3-10, 11-25 V elim
1 ∀ xAx ↔∃x (Bx & Cx)
2 ∀ x (Cx →Bx)
3 ∀ xAx assumption for → intro
4 ∃x (Bx & Cx) ↔ elim
5 Bd & Cd 4, ∃ elim
6 Cd 5, & elim
7 ∃x Cx 6, ∃ intro
8 ∃x Cx 4, 5-7 ∃ elim
9 ∀ xAx →∃x Cx 3-7 → intro
10 ∃x Cx assumption for IP
Cb 9, ∃ elim
Cb →Bb 2, ∀ elim
Bb 10, 11 → elim
Bb & Cb 10, 12 & intro
∃x (Cx & Bx) 13, ∃ intro
∃x (Cx & Bx) 10, 11-15 ∃ elim
∀xAx 1,16↔ elim
∃x Cx → ∀ xAx 9, 17 → intro
∀xAx ↔ ∃x Cx 8, 18, ↔ intro
1 ∃x (¬Ax →Bx)
2 ∃x Ax → Vx(Cx→Bx)
3 ∃x Cx
4 ¬ExBx assumption for Ip
5 Cd 3, ∃ elim
6 ¬Ae→Be 1, assumption for ∃ elim
7 ¬Ae assumption for IP
8 Be 6, 7 → elim
9 ∃x Bx 8, ∃ intro
10 ∃x Bx 1, 6-9, ∃ Elim
⊥ 4, 10 ⊥ intro
¬¬Ae 7, 8-11 ¬ intro
∃x Ax 12, ∃ intro
Vx(Cx→Bx) 2, 13 → elim
Cd→Bd 14, ∀ elim
Bd 5, 15 → elim
∃x Bx 16, ∃ intro
⊥ 4, 17 ⊥ intro
¬¬∃x Bx 4, 5-18 ¬ intro
∃x Bx 19, ¬ elim
II.
W is a world in which all senators are republican and in which all democrats are not governors
D={a,b,c}
S={a,b}
D{c}
G{a}
III.
D={a,b}
A={a,b}
B{0-empty}
IV.
1 D=(c,d,b}
F={0-empty]
2 D={c,d,b,e}
K={,b,e>}
3 D={c,d,e}
F={c,d,e}
G={c,}
4 D={c,d,e}
F={0-emopty}
G={c,d,e}
5 D={c,d,e}
F={c,d}
Q={e}
V. Not clear what above statement is and the requirement for an answer to this question
VI.
Statement 3 is not a logical truth because Txy can be given an interpretation on which the sentence is false: Txy – x likes y.
In a world in which Bob likes Jane and Jane likes Tom but in which Bob does not like Tom the sentence wo...