Mastering Symbolic Logic

[The Laughing Buddha]

I'm not sure where this goes, but I'll post it here, as it seems to be a quasi-mathematical subject.

 

 

Anyway, I was doing derivations from conclusions using implication rules, and I was quite confused to say the least.

 

The rules of implication are: Modus Ponens: A->B, ~B/~A Hypothetical syllogism: A->B, B->C/A->C Modus Tollens: A->B, ~B/~A Disjunctive syllogism: AvB, ~A/B AvB, ~B/A Simplification: A&B/A A&B/B Addition: A/AvB B/AvB Conjunction: A, B/A&B Constructive dilemma: (A->B)&(C->D), AvC/BvD

 

Now, when I applied these to some problems, I had numerous difficulties. I'm not sure where to begin. Can someone help me out?

 

Here were some of the questions:

 

derive S&N from:

 

1. (V<->N)&S

 

2. (GvS)->(V&N)

 

3.

 

4.

 

 

 

derive ~Z from:

 

1. G

 

2. (G&D)->~Z

 

3. G->~(Zv~M)

 

4. (Zv~M)vD

 

5.

 

6.

 

derive A from:

 

1. R->T

 

2. RvA

 

3. F

 

4. F->(~T&K)

 

5.

 

Now my purpose in posting these sample problems is to figure out how to do this stuff, as my skill at symbolic logic is weak enough as it already is. Here's a few more:

 

 

derive SvE:

 

1. (D<->Z)->E

 

2. Z&D

 

3. Z->(Xv(D<->Z))

 

4. X->S

 

5.

 

derive GvN:

 

1. ~U

 

2. CvB

 

3. (HvB)->(I&G)

 

4. C->U

 

5.

 

derive K:

 

1. (XvP)&R

 

2. P->Q

 

3. X->Z

 

4. (ZvQ)->(K&J)

 

5.

 

derive ZvU:

 

1. WvA

 

2. V&G

 

3. ~N->Z

 

4. W->~N

 

5. (VvH)->(A->U)

 

6.

 

derive I:

 

1. Tv~P

 

2. ~T&(Q->H)

 

3. (A->T)&(X->I)

 

4. (~T&~P)->(AvX)

 

5.

 

I hope I'm not boring anybody by posting this, but it's because I'm pretty lost right now and I really want to master symbolic logic and need all the help and advice I can get.

 

Also, is there any software that can be used to generate proofs? I've heard of logic proof generators, but I'm not sureas to where to find these. Are there any good books on mastering symbolic logic?I'd also like to talk to people who have expertise in this subject area as well.

Edited January 29, 2012 by The Laughing Buddha

[The Laughing Buddha]

Nobody knows of any software, online resources for generating proofs, such as those used in the propositional calculus?

 

 

 

I'll post solutions to these problems later on today, unless somebody has any suggestions.

 

Here's one solution to one of the problems already listed:

 

The derivation of A:

 

Pr.1. R->T

 

Pr.2. RvA

 

Pr.3. F

 

Pr.4. F->(~T&K)

 

Pr.5. ~T&K 3,4 Modens Ponens

 

Pr.6. ~T 5 Simplification

 

Pr.7. ~R 1,6 Modens Tollens

 

Concl.8. A 2,7 Disjunctive syllogism

[ydoaPs]

Do you know about DeMorgan's rule? What have you tried so far on the questions?

[triclino]

I'm not sure where this goes, but I'll post it here, as it seems to be a quasi-mathematical subject.

 

 

Anyway, I was doing derivations from conclusions using implication rules, and I was quite confused to say the least.

 

The rules of implication are: Modus Ponens: A->B, ~B/~A Hypothetical syllogism: A->B, B->C/A->C Modus Tollens: A->B, ~B/~A Disjunctive syllogism: AvB, ~A/B AvB, ~B/A Simplification: A&B/A A&B/B Addition: A/AvB B/AvB Conjunction: A, B/A&B Constructive dilemma: (A->B)&(C->D), AvC/BvD

 

Now, when I applied these to some problems, I had numerous difficulties. I'm not sure where to begin. Can someone help me out?

 

Here were some of the questions:

 

derive S&N from:

 

1. (V<->N)&S

 

2. (GvS)->(V&N)

 

3.

 

4.

 

 

 

derive ~Z from:

 

1. G

 

2. (G&D)->~Z

 

3. G->~(Zv~M)

 

4. (Zv~M)vD

 

5.

 

6.

 

derive A from:

 

1. R->T

 

2. RvA

 

3. F

 

4. F->(~T&K)

 

5.

 

Now my purpose in posting these sample problems is to figure out how to do this stuff, as my skill at symbolic logic is weak enough as it already is. Here's a few more:

 

 

derive SvE:

 

1. (D<->Z)->E

 

2. Z&D

 

3. Z->(Xv(D<->Z))

 

4. X->S

 

5.

 

derive GvN:

 

1. ~U

 

2. CvB

 

3. (HvB)->(I&G)

 

4. C->U

 

5.

 

derive K:

 

1. (XvP)&R

 

2. P->Q

 

3. X->Z

 

4. (ZvQ)->(K&J)

 

5.

 

derive ZvU:

 

1. WvA

 

2. V&G

 

3. ~N->Z

 

4. W->~N

 

5. (VvH)->(A->U)

 

6.

 

derive I:

 

1. Tv~P

 

2. ~T&(Q->H)

 

3. (A->T)&(X->I)

 

4. (~T&~P)->(AvX)

 

5.

 

I hope I'm not boring anybody by posting this, but it's because I'm pretty lost right now and I really want to master symbolic logic and need all the help and advice I can get.

 

Also, is there any software that can be used to generate proofs? I've heard of logic proof generators, but I'm not sureas to where to find these. Are there any good books on mastering symbolic logic?I'd also like to talk to people who have expertise in this subject area as well.

 

 

derive S&N from:

 

1. (V<->N)&S

 

2. (GvS)->(V&N)

 

3. S..............................1,Simplification

 

4. GvS...........................3,Addition

 

5. V&N............................2,4,M.Ponens

 

6. N...............................5,Simplification

 

7. S&N...........................3,6,Conjuction

 

 

derive ~Z from:

 

1. G

 

2. (G&D)->~Z

 

3. G->~(Zv~M)

 

4. (Zv~M)vD

 

5. ~(Zv~M)....................................1,3,M.Ponens

 

6. D...............................................4,5,Disnjuctive Syllogism

 

7. G&D............................................1,6,Conjuction

 

8. ~Z.............................................2,7,M.Ponens

 

 

derive SvE:

 

1. (D<->Z)->E

 

2. Z&D

 

3. Z->(Xv(D<->Z))

 

4. X->S

 

5. Z....................................................2,Simplification

 

6. [Xv(D<=>Z)].....................................3,5,M.Ponens

 

7. (X=>S)&[(D<=>Z)=>E].......................1,4,CONJUCTION

 

8. SvE..................................................6,7 ,Constructive dilemma

 

 

 

 

derive GvN:

 

1. ~U

 

2. CvB

 

3. (HvB)->(I&G)

 

4. C->U

 

5. ~C...............................................1,4,M.Tollens

 

6. B..................................................2,5,Disjunctive syllogism

 

7. HvB................................................6,Conjunction

 

8. I&G.................................................3,7,M.Ponens

 

9. G.....................................................8,Simplification

 

10. GvN................................................9,Conjuction

 

 

 

derive K:

 

1. (XvP)&R

 

2. P->Q

 

3. X->Z

 

4. (ZvQ)->(K&J)

 

5.XvP.................................................1,Simplification

 

6. (X =>Z)&(P =>Q)...............................2,3,Conjuction

 

7. ZvQ.................................................5,6, Constructive dilemma

 

8. K&J..................................................4,7,M.Ponens

 

9. K......................................................8,Simplification

 

 

 

derive ZvU:

 

1. WvA

 

2. V&G

 

3. ~N->Z

 

4. W->~N

 

5. (VvH)->(A->U)

 

6. W =>Z...............................................4,3,Hypothetical syllogism

 

7. V.......................................................2,Simplification

 

8. VvH....................................................7,Addition

 

9. A => U................................................5,8,M.Ponens

 

10. (W=>Z)&(A=>U)..................................6,9,Conjunction

 

11. ZvU...................................................1,10, Constructive dilemma

 

 

 

derive I:

 

1. Tv~P

 

2. ~T&(Q->H)

 

3. (A->T)&(X->I)

 

4. (~T&~P)->(AvX)

 

5.

 

i WILL let you do this last one