SOLUTION: Prove using Inference and Replacement Rules:
1). Q -> R
2). R -> S
3). ~S
Therefore, Q • ~R
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Question 1204566: Prove using Inference and Replacement Rules:
1). Q -> R
2). R -> S
3). ~S
Therefore, Q • ~R
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!
I'll use the ampersand symbol & in place of the center dot.
Unfortunately this argument is invalid, as indicated by the truth table below. Focus on the bottom row highlighted in red.
This is where we have all true premises, but they lead to a false conclusion.
| | | | Premise | Premise | Premise | Conclusion |
| Q | R | S | ~R | Q --> R | R --> S | ~S | Q & ~R |
| T | T | T | F | T | T | F | F |
| T | T | F | F | T | F | T | F |
| T | F | T | T | F | T | F | T |
| T | F | F | T | F | T | T | T |
| F | T | T | F | T | T | F | F |
| F | T | F | F | T | F | T | F |
| F | F | T | T | T | T | F | F |
| F | F | F | T | T | T | T | F |
Review these truth table rules
https://www.algebra.com/algebra/homework/Conjunction/truth-table1.lesson
As such, it is impossible to form a logical derivation of an invalid argument.
The invalid argument happens when:
Q = false
R = false
S = false
Those three items will make all of the premises true but they lead to a false conclusion.
Some more practice with invalid arguments can be found here
https://www.algebra.com/algebra/homework/Conjunction/Conjunction.faq.question.1204396.html
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