'Natural' Language (Implication)--Week 2

There are three  implications are so confuse for me to know which is the 'P' and which is the 'Q' in this week's lecture.

1. Successful program requires skill
2. I'll go only if you ins is
3. Don't know it unless you're tried it.

At the beginning, i am so confused about which implies which. For example statement 1, i think it's the skill implies successful program, however after asking many people for help, I understand that skill don't implies successful program (many people have skill but not everyone of them can make successful program) It's the successful program that implies skill.

This week, i find that there are great differences between  Natural language and the mathematical logic. As  beginners, it may takes us some times to know the differences between them. However, as we think more throughly, logic can also become intuitive for us!

Logic Implication VS Natural Language--Week 2

"If you eat your vegetables, then you can have dessert" (Week 2, Lecture 1)

In our daily language, the implication above means that you can have your dessert if and only if you eat your vegetables. There is no way that you don't eat your vegetable and have the dessert still. This is pretty much intuitive and well-accpeted by us.

However, when it comes to logic, the above statement has different approaches.

if you eat your vegetables, you will surely have your dessert. however, if you don't eat your vegetables, you can have the probability that you can still have the dessert. That mean if we see you have dessert, we are not sure whether you eat the vegetables or not.These refers to something called Vacuous Truth. Once the antecedent is false, the statement is automatic true regardless of the consequent. The reason to explain Vacuous Truth is that we can never find any counter-example to false the statement. Therefore, the whole statement is true.

How 'negation' works --week 1

Negation means a negative statement.
Negation can be done easily by put 'not' into the original statement.

for instance:
5+5 equals 10 (original true statement)
5+5 not equal 10 (negation statement)

one should understand that (not any) means none, (not all) means at lease one statement  is false

All and ANY--week1

'All' means every element of the set, 'ANY' means  at least one element of the set.

For instance, we creat a set S and have three statement s1,s2,s3.

To make all(S) true, we have to check every element in S and make sure s1,s2,s3 are all true.

To make all(S) false, we just need to find (at least) one of  the element in S that is false.

To make any(S) true, we need find at least one of the element in S that is true.

To make any(S) False, we need check all elements in S and make sure all s1,s2,s3 are false.

We should understand the differences between each of the cases above and not mix them up