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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