A language element which generates a quantification (such as "every") is called a quantifier. (From Wikipedia)
Some of the logic symbols like ∀, ∃ were introduced in high school, so it's quite easy to understand those quantified sentences.
In the implication section, I learned how to write an "if..., then..." sentence in math language.
And it was quite interesting to find out that if my mom told me that
"If you eat vegetables, then you can have dessert."
I might also have dessert, because she doesn't mention the case I don't eat vegetables. (Real life is different...)
Actually, we can write the sentence like: I eat vegetables => I have dessert.
We cannot tell whether the reverse and the negation are True,
so the implication: "I don't eat vegetables => I have dessert" maybe True or maybe False.
However, the contrapositive of a statement is the same as the original sentence.
It's like: "I eat vegetables => I have dessert" equals "I didn't have dessert => I didn't eat vegetables".
They are both True of False.
The most tricky thing in this week must be "only if".
The professor said in the sentence "I'll go if you insist", "you insist" is antecedent, and "I'll go" is the consequent, which is easy to understand.
However, in the sentence "I'll go only if you insist" is on the opposite.
After the discussion with my friend, I finally figured it out.
In the first sentence, we know that You insist => I'll go, but we cannot indicate that if I go, then you must be insisting.
But in the second sentence, "I'll go only in the case that you insist" indicates that if I'll go, then you must be insisting. BUT, I may still not go if you are not insisting, so the reverse (first sentence) is not sure.
It's easy to understand that the phrase "if and only if" equals to "<=>",
so "if P, then Q" means "P => Q" and "only if P, then Q" means "P <= Q"
(It's just my way to understand these kind of tricky sentences, I'm not sure whether they are technically right, though I think they are right.)
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