You should spend time thinking about these problems yourself. I hope you have done so.
I do not think you are asking for a truth table “proof” of these. Truth table is too straightforward for anyone to have problem with. You are here asking for a deduction using inference rules.
The first problem
The second problem
1 (1) (~F & G) -> H Premise
2 (2) ~I -> G Premise
3 (3) ~I & ~F Premise
3 (4) ~I 3, &-Elimination
2,3 (5) G 2,3 modus ponens
3 (6) ~F 3, &E
2,3 (7) ~F&G 5,6 &-intro
1,2,3 (8) H 1,7 ->-elimination
1,2,3 (9) H v J 8, v-intro
1 (1) K v L Premise
2 (2) ~K Premise
3 (3) (L v L)-> (~M v N) Premise
4 (4) ~N Assumption (assume the antecedent of the conclusion)
1,2 (5) L 1,2 Disjunctive syllogism (can be proved separately if need be)
1,2 (6) L v L 4, v-Intro
1,2,3 (7) ~M v N 3,5 ->-Elimination
1,2,3,4 (8) ~M 4,7 Disjunctive syllogism
1,2,3 (9) ~N->~M 4,8 ->-intro
The disjunctive syllogism proof. A v B, ~B, therefore A
1 (1) A v B Premise
2 (2) ~B Premise
3 (3) ~ A Assumption
4 (4) A Assumption
5 (5) B Assumption
2,5 (6) contradiction 2,5 ~-elimination
2,5 (7) ~~A 3,6 ~-intro
2,5 (8) A 7, double negation
1,2 (9) A 1,4,4,5,8 v-elimination
For problem three and four, I am uncomfortable with the premises of the examples you gave. They have serious flaws in them. For example, being admitted to heaven requires at least these things: believing in the Absolute Oneness of God, the Almighty AND being morally virtuous AND doing good deeds AND receiving mercy from God, the Almighty.
Is there a way to create a logic proof using the variables given and solve it using that statement even if it hold false assumptions?
Yes, there is a way. I will give my examples which are in the same logical form as the examples you gave and work from there.
Problem three revisited
If Bob works hard, then he will succeed. But if Bob does not work hard, then he will do things that he enjoy. On the other hand if Bob does not succeed, then he will not do things that he enjoy. Therefore, Bob has to work hard.
M-Bob works hard
L-Bob does things that he enjoys
To prove: M->H, ~M->L, ~H->~L therefore H
Here, we proceed by reduction ad absurdum, i.e. we assume the negation of the conclusion and derive a contradiction with the premises.
1 (1). M->H Premise
2 (2) ~M->L Premise
3 (3) ~H->~L Premise
4 (4) ~H Assumption
3,4 (5) ~L 3,4 ->Elimination
2,3,4 (6) ~~M 2,5 Modus Tollens
2,3,4 (7) M 6, double negation
1,2,3,4 (8) H 1,7 ,->Elim
1,2,3,4 (9) contradiction 4,8 ~Elim
1,2,3 (10) ~~H 4,9 ~Intro
1,2,3 (11) H 10, Double negation
Problem four has issues as well. I have problems with these assumptions. Our ability to choose and God being The All-Knower is logically consistent and sound.
Freedom of choice implies absence of external coercion on the individual will. And knowledge is knowledge. How could the fact that God, the All-Knower knowing what one will chose determine that choice? For example, I might see a student, who does not study at all and he/she is always inattentive in class and does not do his homework. Then in my limited knowledge, I know that that student is going to fail. Does my knowledge determines that the student fails? I would not say that. If that student had listened to the numerous advice and encouragements of his teachers and done his/her homework and pay attention in class, he/she might have had a more brilliant future. It is in fact the student choice of not studying that determines his/her failure!
Now, God, the Almighty is the All-Knowing; there is nothing that escapes His knowledge and He has power over everything. And in no way does God’s knowledge of the future determines the choices that we make, because if we were compelled to do anything by anyone, we would not have freedom/choice. So God, the Almighty knows everything, including the future; but still, God gives us the ability to choose and the choices we make are still our choices and are not imposed on us.
Again this is a sensitive issue and too philosophically important to be taken lightly. So, I will give my examples and work from there.
Problem four revisited.
If my teacher believes on Monday that I will miss class on Tuesday, then either I make my teacher’s past beliefs false, or I miss class on Tuesday. I will not make my teacher’s past beliefs false if either my teacher is right or the past is unalterable. The past is unalterable. It follows that if my teacher believes on Monday that I’ll miss class on Tuesday, then I will miss class on Tuesday.
B=my teacher believes that I will miss class on Tuesday
F= I make my teacher’s pass belief false
R=I do not miss class on Tuesday.
I=my teacher is right
P=the past is unalterable
To prove: B->(Fv~R), (I v P)->~F, P therefore B ->~R
1 (1) B->(F v ~R) Premise
2 (2) (I v P) –>~F Premise
3 (3) P Premise
4 (4) B Assumption (assume antecedent of conclusion)
5 (5) R Assumption
1,4 (5) (F v ~R) 1,4 ->Elimination
6 (6) F Assumption
2,6 (7) ~ (I v P) 2,6 Modus Tollens
3 (8) I v P 3, v-Intro
2,3,6 (9) contradiction 7,8 ~Elimination
2,3,6 (10)~R 5,9 ~Intro
11 (11) ~R Assumption
1,2,3,4 (12) ~R 5,6,10,11,11, v-Elim
1,2,3 (13) B->~R 4,12 ->-intro
A note though on the homework you were given. The last two problems are clear examples of sophistry. Sophism mean a false argument, especially one intended to deceive.
Let us analyze the statement: “I cannot refrain from lying on Tuesday.” In effect, in propositional logic, the meaning of that statement is no different from the statement: “I will not not lie on Tuesday”, i.e. “I will lie on Tuesday”. But clearly these two statements do not convey the same meaning in ordinary discourse, for the latter can be true while the former can be false, i.e. in the case where lying is a choice. If someone lie on Tuesday, it does not imply that such a person could not have done otherwise. But the statement in the problem i.e. “I cannot refrain from lying on Tuesday” intends to remove the choice of the individual in lying. This is a clear cut case of sophistical argumentation.
Moreover, use of words such as “can”, “must”, “cannot”, “possible”, “impossible” belongs to another more sophisticated branch of logic called modal logic. Which is beyond the scope of propositional logic. And that is why when I removed all modal elements of the argument, it lost all its “teeth and claws”!
I hope this helps.