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Question 1204396: INSTRUCTIONS: Use an ordinary truth table to answer the following problems. Construct the truth table as per the instructions in the textbook.
Given the argument:
K ⊃ (M ∨ ∼ H) / M ⊃ H / M ⊃ K // K ⊃ H
This argument is:
Group of answer choices
Invalid; fails in 1st line.
Invalid; fails in 2nd line.
Valid.
Invalid; fails in 4th line.
Invalid; fails in 3rd line.
Found 2 solutions by Edwin McCravy, math_tutor2020:
Answer by Edwin McCravy(20055)   (Show Source):
You can put this solution on YOUR website!
Since I can't see your textbook, I can only guess what the author wants.
Under the K's you write TTTTFFFF
Under the M's you write TTFFTTFF (there are M's in the conclusion, so we skip.
Under the H's you write TFTFTFTF
Since the conclusion is a conditional statement, it's the only thing we
need to consider". Only "T ⊃ F" fails.
K ⊃ (M ∨ ∼ H) / M ⊃ H / M ⊃ K // K ⊃ H
T T
T F <-- fails because T cannot imply F.
T T
T F <-- fails because T cannot imply F.
F T
F F
F T
F F
It is not valid because it fails in lines 2 and 4.
Edwin
Answer by math_tutor2020(3816)  (Show Source):
You can put this solution on YOUR website!
This is one way to form the truth table.
There are ways to condense things, but I prefer the more expanded out version to see how each piece is formed.
Feel free to go with the condensed table if you want.
| | | | | Premise | Premise | Premise | Conclusion | | K | M | H | ~H | M v ~H | K -> (M v ~H) | M -> H | M -> K | K -> H | | T | T | T | F | T | T | T | T | T | | T | T | F | T | T | T | F | T | F | | T | F | T | F | F | F | T | T | T | | T | F | F | T | T | T | T | T | F | | F | T | T | F | T | T | T | F | T | | F | T | F | T | T | T | F | F | T | | F | F | T | F | F | T | T | T | T | | F | F | F | T | T | T | T | T | T |
T = true
F = false
Here is a review of various truth table rules
https://www.algebra.com/algebra/homework/Conjunction/truth-table1.lesson
The row I've marked in red, line 4, represents the situation where we have all true premises, but they lead to a false conclusion.
This happens when: K = true, M = false, H = false
Because we have all true premises leading to a false conclusion, we have an invalid argument.
Answer: Invalid; fails in 4th line.
Side note: line 2 comes close but not all premises are true here.
Another example of an invalid argument is found in problem 3 of this link
https://www.algebra.com/algebra/homework/Proofs/Proofs.faq.question.1204272.html
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