Can Logical Inference Rules Prove This Propositional Argument?

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Discussion Overview

The discussion revolves around the validity of a propositional argument involving several premises and a conclusion. Participants are tasked with providing a step-by-step proof using logical inference rules, exploring the application of these rules in the context of the given premises and conclusion.

Discussion Character

  • Homework-related

Main Points Raised

  • One participant presents a series of steps attempting to derive the conclusion ¬t ∨ w from the premises, starting with u → ¬p and ¬p → r ∧ ¬s.
  • Another participant outlines a sequence of implications leading to the conclusion, including the use of u ∨ w and ¬w to derive u and subsequently r ∧ ¬s.
  • A question is raised regarding the permissibility of re-using the premise ¬w in the proof process.
  • Another participant responds affirmatively, stating that a premise can be reused in the argument.

Areas of Agreement / Disagreement

There is no consensus on the validity of the steps taken in the proofs, and the discussion includes questions about the rules of inference and the reuse of premises.

Contextual Notes

Participants have not resolved the implications of reusing premises, and there may be assumptions about the application of inference rules that are not explicitly stated.

MarcL
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Homework Statement


For each of the premise-conclusion pairs below, give a valid step-by-step argument ( proof ) along with the name of the inference rule used in each step

premise { ¬ p → r ∧ ¬ s , t → s , u → ¬p , ¬w , u ∨ w } conclusion : ¬t ∨ w

Homework Equations



All the inference rules, Modus ponens, Modus tollens, etc...

The Attempt at a Solution



[/B]I tried by using the w term but it didn't work so I did this:

1) u → ¬ p Prmise
2) ¬p → r ∧ ¬s ass
3) u → r ∧ ¬ s

However I seem stuck
 
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  1. u⇒¬ p
  2. ¬ p⇒r ∧ ¬ s
  3. ∴u ⇒ r ∧ ¬ s
  4. t⇒s
  5. ∴¬ s⇒¬ t
  6. u ∨ w
  7. ¬w
  8. ∴u
  9. ∴r ∧ ¬ s
  10. ∴¬ s
  11. ∴¬ t
  12. ¬w
  13. ∴¬t ∨ w
 
you're allowed to re-use ¬w>?
 
MarcL said:
you're allowed to re-use ¬w>?
Why not? A premise is a premise.
 

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