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Math Help - Symbolic Logic Help

  1. #1
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    Symbolic Logic Help

    Hey guys,

    I've been studying this book "Introduction to Logic" and have come across 3 tough questions I was hoping I could get explained to me as I have no clue where to start. They deal with the rules of inference and the rules of replacement.

    1. (D⊃F) • (P⊃N)
    2. D v P
    3. (D ⊃ ~N) • (P⊃ ~F)
    4. (D ⊃ F) / F ≡ N

    (I started off here simplifying the 1st line, but that's as far as I got)
    ---------------------------

    1. (Q v ~T) v S
    2. ~Q v (T • ~Q) / T ⊃ S

    -------------------------

    and

    1. (C v B) ⊃ (T • L)
    2. ~C ⊃ (K ⊃ ~K)
    3. ~T / ~K

    I'd honestly appreciate any help whatsoever, even some hints. Thanks again.
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  2. #2
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    Quote Originally Posted by johnnyboy1 View Post
    Hey guys,

    I've been studying this book "Introduction to Logic" and have come across 3 tough questions I was hoping I could get explained to me as I have no clue where to start. They deal with the rules of inference and the rules of replacement.

    1. (D⊃F) • (P⊃N)
    2. D v P
    3. (D ⊃ ~N) • (P⊃ ~F)
    4. (D ⊃ F) / F ≡ N

    (I started off here simplifying the 1st line, but that's as far as I got)
    ---------------------------

    1. (Q v ~T) v S
    2. ~Q v (T • ~Q) / T ⊃ S

    -------------------------

    and

    1. (C v B) ⊃ (T • L)
    2. ~C ⊃ (K ⊃ ~K)
    3. ~T / ~K

    I'd honestly appreciate any help whatsoever, even some hints. Thanks again.
    The name of the author might be helpful.

    Anyway, looking at the first sequent, the fourth premiss looks redundant to me.
    On top of that it's not a valid sequent.
    Consider the valuation: v(F) = T, v(N) = F, v(D) = T, v(P) = F.

    However, replace the conclusion by its negation; then the sequent is valid and a derivation is pretty straight forward.
    The big steps would be two applicatons of the rule of inference formulation of complex constructive dilemma.
    At that point, you'll have (F V N) & (~F V ~N). (Think definition of exclusive disjunction.) The rest is busy work.
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  3. #3
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    I've been studying this book "Introduction to Logic" and have come across 3 tough questions
    So what do the questions ask to do?

    4. (D ⊃ F) / F ≡ N
    At first I thought, Interesting, they invented a new propositional connective: division. I am wondering what its truth table, axioms or inference rules are. Then I realized that this is a task to derive one formula from several others:

    1. (D⊃F) • (P⊃N)
    2. D v P
    3. (D ⊃ ~N) • (P⊃ ~F)
    4. (D ⊃ F)
    ----------------------------
    F ≡ N

    Also, I assume that the bullet • denotes conjunction, right?

    Are you supposed to use truth tables or inference rules?
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  4. #4
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    all 18 inference rules. yep, • is a conjunction. (i.e Mary went shopping and Gary played video games = M • G)

    I got the last 2 questions btw:

    (Q v ~T) v S, ~Q v (T • ~Q) : T ⊃ S

    1. (Q v ~T) v S; Premise
    2. ~Q v (T • ~Q); Premise
    3. Q v (~T v S); 1 Association
    4. Q v (T ⊃ S); 3 Implication
    5. (~Q v T) • (~Q v ~Q); 2 Distribution
    6. ~Q v T; 5 Simplification
    7. ~Q v ~Q; 5 Simplification
    8. ~Q; 7 Tautology
    9. T ⊃ S; 4, 8 Disjunctive Syllogism


    (C v B) -> (T & L), ~C -> (K -> ~K), ~T : ~K

    1. (C v B) -> (T & L) : Premise
    2. ~C -> (K -> ~K) : Premise
    3. ~T : Premise
    4. ~T v ~L : 3 Addition
    5. ~(T & L) : 4 DeMorgan's Theorem
    6. ~(C v B): 1, 5 Modus Tollens
    7. ~C & ~B : 6 DeMorgan's Theorem
    8. ~C : 7 Simplification
    9. ~B : 7 Simplification
    10. K -> ~K : 2, 8 Modus Ponens
    11. ~K v ~K : 10 Implication
    12. ~K : 11 Tautology


    Thanks for the help emakarov and Piper
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  5. #5
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    Quote Originally Posted by johnnyboy1 View Post
    all 18 inference rules. yep, • is a conjunction. (i.e Mary went shopping and Gary played video games = M • G)

    I got the last 2 questions btw:

    (Q v ~T) v S, ~Q v (T • ~Q) : T ⊃ S

    1. (Q v ~T) v S; Premise
    2. ~Q v (T • ~Q); Premise
    3. Q v (~T v S); 1 Association
    4. Q v (T ⊃ S); 3 Implication
    5. (~Q v T) • (~Q v ~Q); 2 Distribution
    6. ~Q v T; 5 Simplification
    7. ~Q v ~Q; 5 Simplification
    8. ~Q; 7 Tautology
    9. T ⊃ S; 4, 8 Disjunctive Syllogism

    This proof is wrong.In line (4) you cannot conclude :Q v (T ⊃ S) by Implication.

    You could conclude that if (~T v S) was on its own and not within another formula

    T
    he rule for implication is : (A=>B)<=>(not(A) v B)

    Which is your A and which is your B in this case??

    The right proof is:

    1. (Q v ~T) v S; Premise
    2. ~Q v (T • ~Q); Premise
    3. Q v (~T v S); 1 Association
    4. ~Q⊃ ( ~Tv S); 3 Implication
    5. (~Q v T) • (~Q v ~Q); 2 Distribution
    6. ~Q v T; 5 Simplification
    7. ~Q v ~Q; 5 Simplification
    8. ~Q; 7 Tautology
    9. ~T v S 4,8 Modus Ponens
    10. T ⊃ S ; 9 Implication
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  6. #6
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    Quote Originally Posted by johnnyboy1 View Post
    all 18 inference rules. yep, • is a conjunction. (i.e Mary went shopping and Gary played video games = M • G)

    I got the last 2 questions btw:

    (Q v ~T) v S, ~Q v (T • ~Q) : T ⊃ S

    1. (Q v ~T) v S; Premise
    2. ~Q v (T • ~Q); Premise
    3. Q v (~T v S); 1 Association
    4. Q v (T ⊃ S); 3 Implication
    5. (~Q v T) • (~Q v ~Q); 2 Distribution
    6. ~Q v T; 5 Simplification
    7. ~Q v ~Q; 5 Simplification
    8. ~Q; 7 Tautology
    9. T ⊃ S; 4, 8 Disjunctive Syllogism


    (C v B) -> (T & L), ~C -> (K -> ~K), ~T : ~K

    1. (C v B) -> (T & L) : Premise
    2. ~C -> (K -> ~K) : Premise
    3. ~T : Premise
    4. ~T v ~L : 3 Addition
    5. ~(T & L) : 4 DeMorgan's Theorem
    6. ~(C v B): 1, 5 Modus Tollens
    7. ~C & ~B : 6 DeMorgan's Theorem
    8. ~C : 7 Simplification
    9. ~B : 7 Simplification
    10. K -> ~K : 2, 8 Modus Ponens
    11. ~K v ~K : 10 Implication
    12. ~K : 11 Tautology


    Thanks for the help emakarov and Piper
    You're welcome.

    Both of your derivations look fine to me.

    (A minor point regarding the second derivation is that line 9 is superfluous.
    There's no need to break out ~B by Simp. since it's not used subsequently.)

    On the other hand, regarding the first derivation, I think it appropriate to emphasize the fact that you've
    correctly applied a rule of replacement (what you've called "implication") to a subwff of the wff at line 3 to obtain the wff at line 4.

    Good luck in your studies.
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