Math Problem Statement

Prove that the following argument is valid: p →∼ q, r → q, r `∼ p.

Solution

To prove that the argument pq, rq, r  pp \rightarrow \sim q, \ r \rightarrow q, \ r \ \vdash \ \sim p is valid, we will use natural deduction to show that the conclusion p\sim p follows logically from the premises.

Given premises:

  1. pqp \rightarrow \sim q (Premise)
  2. rqr \rightarrow q (Premise)
  3. rr (Premise)

To prove:

p\sim p (i.e., the negation of pp).

Proof:

  1. pqp \rightarrow \sim q (Premise 1)
  2. rqr \rightarrow q (Premise 2)
  3. rr (Premise 3)
  4. From 2 and 3, by Modus Ponens: qq (If rqr \rightarrow q and rr is true, then qq must be true).
  5. q\sim q (Negation of qq) must be true because of premise 1 (if pp were true, then qq must be false, since pqp \rightarrow \sim q).
  6. Assume pp is true temporarily (Assumption for Indirect Proof or Reductio ad Absurdum).
  7. From pp, by premise 1 (pqp \rightarrow \sim q), we get q\sim q (from 1 and 6).
  8. But, from step 4, we already know that qq is true.
  9. This leads to a contradiction because qq and q\sim q cannot both be true.
  10. Therefore, the assumption that pp is true must be false.
  11. Hence, p\sim p (by Negation Introduction, or Reductio ad Absurdum).

Conclusion:

The conclusion p\sim p follows from the premises. Therefore, the argument is valid.


Do you want more details or have any questions about this proof?

Here are 5 related questions to explore:

  1. What is the Modus Ponens rule in logic?
  2. How does Reductio ad Absurdum work in proving logical statements?
  3. Can we use a truth table to verify this argument’s validity?
  4. What is the role of assumptions in natural deduction proofs?
  5. How would the proof change if rr were false?

Tip: In proofs by contradiction, always aim to show that assuming the negation of the desired conclusion leads to a contradiction.

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Math Problem Analysis

Mathematical Concepts

Propositional Logic
Natural Deduction
Modus Ponens
Reductio ad Absurdum

Formulas

Modus Ponens: If p → q and p are both true, then q must be true.
Negation Introduction: If assuming p leads to a contradiction, then ∼ p is true.

Theorems

Modus Ponens
Negation Introduction (Reductio ad Absurdum)

Suitable Grade Level

College-level logic or advanced high school