Use rules of inference to show that the hypotheses “If it does not rain or if it is not foggy, then thesailing race will be held and the lifesaving demonstration will go on,” “If the sailing race is held, then thetrophy will be awarded,” and “The trophy was not awarded” imply the conclusion “It rained.”
Question
Use rules of inference to show that the hypotheses “If it does not rain or if it is not foggy, then thesailing race will be held and the lifesaving demonstration will go on,” “If the sailing race is held, then thetrophy will be awarded,” and “The trophy was not awarded” imply the conclusion “It rained.”
Solution
To solve this problem, we can use the rules of inference in propositional logic. Let's denote:
- p: It does not rain
- q: It is not foggy
- r: The sailing race will be held
- s: The lifesaving demonstration will go on
- t: The trophy will be awarded
The hypotheses can be translated into the following logical statements:
- (p ∨ q) → (r ∧ s)
- r → t
- ¬t
We are asked to prove that ¬p (It rained) is true.
From the second and third hypotheses, we can use Modus Tollens (which states that if p → q is true and q is false, then p must also be false) to infer that ¬r (The sailing race was not held) is true.
Now we have a new statement, ¬r. We can use this with the first hypothesis in a Modus Tollens inference again. The first hypothesis is a conditional statement with a conjunction (r ∧ s) in the result. We can rewrite this statement as two separate conditional statements:
- (p ∨ q) → r
- (p ∨ q) → s
Using ¬r with the fourth statement in a Modus Tollens inference, we can conclude that ¬(p ∨ q) is true. According to De Morgan's laws, this is equivalent to ¬p ∧ ¬q.
Therefore, it must be true that ¬p (It rained). So, the conclusion "It rained" is indeed implied by the given hypotheses.
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Hi everyone! Today, we're going to talk about what I learned about making inferences. An inference is a guess or conclusion that we make using clues or evidence from the text or pictures.Now, let's practice making inferences together. I'm going to show you a picture of a girl holding an umbrella and walking outside. What do you think the weather is like? Hmm, well, I see that the girl is wearing a raincoat and holding an umbrella, so I think it's probably raining outside. See, by using the clues in the picture, we can make an inference about the weather.Making inferences is like being a detective and using clues to figure things out. Keep practicing and soon you'll be a pro at making inferences. Thanks for watching and see you next time!
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