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## Here's the Solution to this Question

1.The sentence is false under all circumstances.

Proof;

(A⊃J ~B) interpreting this part we have ~B which means (not True) yielding false

A is true, Thus it becomes (T ⊃ F) = False

((A⊃J ~B) ⊃J~C) on this we have intepreted the first part, now ~C means (not true) yielding False,

hence it reduces to (false ⊃ false) = True

[((A⊃J ~B) ⊃J~C) ⊃J~X] on this ~X means (not false) yielding True, hence it reduces to ( True ⊃True) yielding True.

now  ~ [((A⊃J ~B) ⊃J~C) ⊃J~X] reduces to  ~True which becomes False

thus we may conclude that the sentence is false.

2.The sentence is True under all circumstances.

proof

(A⊃J B) , since both A and B are True, this part becomes True

((A⊃J B) ⊃J X), since the first part became true and X is false, this part reduces to (True ⊃ False) which yields False

((A⊃J B) ⊃J X) ⊃J Z) ,since the first part became false and Z is false, this part reduces to( False ⊃ False) which yields True

[(((A⊃J B) ⊃J X) ⊃J Z) ⊃J Y],since the first part is True and Y is false, this part reduces to (True ⊃ False) which yields False

(X ⊃J Z),since X is True and Z is False, this reduces to ( True ⊃ False) which yields False

((X ⊃J Z) ⊃J A), since the first part is False and A is True, this reduces to ( False ⊃ True),which yields True

J [((X ⊃J Z) ⊃J A) ⊃J X],since the first part is True and X is False, this reduces to (True ⊃ False) which yields False

From the above  [(((A⊃J B) ⊃J X) ⊃J Z) ⊃J Y] ⊃J [((X ⊃J Z) ⊃J A) ⊃J X] reduces to ( False ⊃ False) which yields True.

Hence we conclude that the sentence is True.

3. The sentence is true under all interpretation.

proof

(A∨B), since both A and B are True, this part reduces to True.

((G∨H) ⊃A), since A is True and from the relation of ⊃ (implication) where when the first part is unknown/false but the second part is True ,then the result becomes True. Hence ((G∨H) ⊃A) becomes True.

(A∨B) ≡((G∨H) ⊃A), since the first part part became True and also the second part is True, this reduces to ( True ≡ True) which yield True.

Hence we conclude that the sentence is True.

4. The sentence is True under all interpretations.

proof

(H· A),since A is true but H is unknown, this relation reduces to False.

(G ≡≡ (H· A)),since the second part is False and G is unknown, from the relation (≡≡) which is True when ( both the first part and second part are True) or when ( both the fist part and the second part are False), (G ≡≡ (H· A)) reduces to True.

(X⊃J Z), since both X and Z are False, this part reduces to ( False ⊃ False) which yields True.

J «~H V I) ⊃J (X⊃J Z)), since the second part as shown above is True and from the relation of (⊃),this part reduces to True

(G ≡≡ (H· A)) ⊃J «~H V I) ⊃J (X⊃J Z)), From the above, this reduces to ( True ⊃ True) which yields True.

Hence ,we conclude that the sentence is True .