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Question one:

  1. (C>D) & (D>B)
  2. (B>D) & (E>C)
  3. (D>C)
  4. BvE ∴ DvB
  5. ?
  6. ?
  7. ?
  8. ?
  9. DvB

I'm fairly sure this questions has constructive dilemma at the end, but after four hours of working on these two problems I feel like I can't make any more progress. Here's what I solved so far!

  1. DvC 2,4, Constructive dilemma
  2. C>D 1, Simplification
  3. B>D 2, Simplification
  4. ?
  5. DvB ?

Question two:

  1. (~K&~N)>[(~P>K)&(~R>G)]
  2. K>N
  3. ~N&B
  4. ~Pv~R ∴ ~RvG
  5. ?
  6. ?
  7. ?
  8. ?
  9. ?
  10. ?
  11. ?
  12. ?

This second problem is more involved than the first, but I genuinely don't understand how to get the answer.

  1. ~N 3, Simplified
  2. ~K 2,5, Modus tollens
  3. ~k&~N 5,6 Conjunction
  4. (~P>K)&(~R>G) 1,7 Modus ponens
  5. KvG 4,8 Constructive dilemma
  6. ?
  7. ?
  8. ~RvG ?

Literally any type of help or hints to solving either or would mean the world to me!

  • Already asked in MSE. – Mauro ALLEGRANZA Feb 10 at 11:05
  • What are the symbols "+" and "/" supposed to be? – Eliran Feb 10 at 17:00
  • Since this question was closed on Math SE without any answers I assume it may be worth answering here. I also assume that "+" refers to conjunction and "/" separates the premises from the conclusion. – Frank Hubeny Feb 10 at 23:29
1

Wikipedia describes constructive dilemma in the following way:

Constructive dilemma is a valid rule of inference of propositional logic. It is the inference that, if P implies Q and R implies S and either P or R is true, then either Q or S has to be true. In sum, if two conditionals are true and at least one of their antecedents is, then at least one of their consequents must be too. Constructive dilemma is the disjunctive version of modus ponens, whereas, destructive dilemma is the disjunctive version of modus tollens.

The OP starts a proof for the first question by noting that one could use constructive dilemma to obtain "DvC".

DvC 2,4, Constructive dilemma

One can write "DvC" as "CvD". That might be the next line. Then one can use that line and line 1 with constructive dilemma to reach the desired conclusion: "DvB".


For the second problem the OP has reached line 9 with "KvG", but on line 6 one has "~K". By the use of disjunctive syllogism, one can derive "G".

Wikipedia describes disjunctive syllogism in the following way:

If we are told that at least one of two statements is true; and also told that it is not the former that is true; we can infer that it has to be the latter that is true. If P is true or Q is true and P is false, then Q is true.

Having "G" one can use disjunction introduction to obtain the desired conclusion "~RvG".

Wikipedia describes disjunction introduction in the following way:

Disjunction introduction or addition (also called or introduction) is a rule of inference of propositional logic and almost every other deduction system. The rule makes it possible to introduce disjunctions to logical proofs. It is the inference that if P is true, then P or Q must be true.


"Constructive dilemma" Wikipedia https://en.wikipedia.org/wiki/Constructive_dilemma

"Disjunction introduction Wikipedia https://en.wikipedia.org/wiki/Disjunction_introduction

"Disjunctive syllogism" Wikipedia https://en.wikipedia.org/wiki/Disjunctive_syllogism

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