Introduction to Logic - Exercise 2.2 (c) help | Coursera Community

# Introduction to Logic - Exercise 2.2 (c) help

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The section is on Evaluation and I get the wrong answer, can someone explain it for me?

p=1, q=0, r=1

Evaluate:

p ^ q => r

My solution:

1 ^ 0 => 1
0 (because a conjunction is only true if both conjuncts are true) =>1
Therefore false
(but according to the grading this is incorrect)
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Best answer by hamster 25 January 2019, 20:27 Okay, so lets start this by saying its been a while since I did formal logic. But hopefully I can help you out.

The above image is a truth table. We can clearly see that P ^ Q (p AND q) is true if and only if P is true and Q is true.

Now lets look at implication (P => Q) Notice that on the third line is says that:

• P => Q is true when:
• P is false.
• Q is True
To understand this, its important that you realise that implication is not causation! The reason for this is because causation is not "truth functional" and by this we simply mean that if both P and Q are true it is still unclear whether P cause Q. for example:

• let P equal "The world is round"
• let Q equal "Salmon is pink"
• Therefore: Salmon is pink because the shaped of the world caused them to be pink.
In this case P and Q are clearly true, but the conclusion in nonsense. Or in more general terms, we cannot know for certain whether "P cause Q" is true when given true premises.

Basically, the entire point of deductive logic is that IF the truth status of the premises are known then we can be 100% certain the the conclusion is correct. And this holds for any P and for any Q. In the case of causation the conclusion is not certain to follow from the premises. And so therefore deductive logic abandons the concept.

While I'm here the forth line is (false => false = true) is the idea is that anything logically follows from a falsehood. (see: https://en.wikipedia.org/wiki/Principle_of_explosion ). Thats not relevant to your problem but I thought I'd flag it up.

So thats the first thing you need to know to solve this homework problem. The next thing you need to know is:

1. "scope"
2. that letters can represent complex functions.
For example if I have "P v Q" I can set P to be anything I like. P could be short for "R v T" and Q could be short for ¬X and ¬P. In which case P v Q actually means:

• R v T v ¬X ^ ¬ P
Now comes the question of "scope". In logic, just as in math there are rules which dictate the order of operations. for example:

• (4 + 4) * 0
• 4 + (4 * 0)
Brackets tell us the order in which we should calculate the sum. When there are no brackets you assume "bodmas". Logic works in the same way:

• (¬P) v Q
• ¬ (P v Q)
So, if I remove the brackets from the above wff what do you suppose the meaning is? Whats the convention?

Okay, with everything I have said here I believe you should be able to solve the homework problem. Good luck!
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### 4 replies

Hi @Confusean. Have you posted your question in your course discussion forum? For course assignments, it's best to ask there first, as this community has learners taking all different courses. +1
I'm unable to find any link to the course discussion forum for this course. Possibly because the course isn't scheduled to start until April 1st? It's also very likely that I'm missing something obvious
Yes, I believe that's it – discussion forums coincide with course sessions, so check for them closer to April 1. You'll find the heading "Discussion Forums" in the left sidebar when you're viewing your course. In the meantime, perhaps there's another community member who has taken this course or who otherwise has the knowledge to answer your question!
Userlevel 6 +4 Okay, so lets start this by saying its been a while since I did formal logic. But hopefully I can help you out.

The above image is a truth table. We can clearly see that P ^ Q (p AND q) is true if and only if P is true and Q is true.

Now lets look at implication (P => Q) Notice that on the third line is says that:

• P => Q is true when:
• P is false.
• Q is True
To understand this, its important that you realise that implication is not causation! The reason for this is because causation is not "truth functional" and by this we simply mean that if both P and Q are true it is still unclear whether P cause Q. for example:

• let P equal "The world is round"
• let Q equal "Salmon is pink"
• Therefore: Salmon is pink because the shaped of the world caused them to be pink.
In this case P and Q are clearly true, but the conclusion in nonsense. Or in more general terms, we cannot know for certain whether "P cause Q" is true when given true premises.

Basically, the entire point of deductive logic is that IF the truth status of the premises are known then we can be 100% certain the the conclusion is correct. And this holds for any P and for any Q. In the case of causation the conclusion is not certain to follow from the premises. And so therefore deductive logic abandons the concept.

While I'm here the forth line is (false => false = true) is the idea is that anything logically follows from a falsehood. (see: https://en.wikipedia.org/wiki/Principle_of_explosion ). Thats not relevant to your problem but I thought I'd flag it up.

So thats the first thing you need to know to solve this homework problem. The next thing you need to know is:

1. "scope"
2. that letters can represent complex functions.
For example if I have "P v Q" I can set P to be anything I like. P could be short for "R v T" and Q could be short for ¬X and ¬P. In which case P v Q actually means:

• R v T v ¬X ^ ¬ P
Now comes the question of "scope". In logic, just as in math there are rules which dictate the order of operations. for example:

• (4 + 4) * 0
• 4 + (4 * 0)
Brackets tell us the order in which we should calculate the sum. When there are no brackets you assume "bodmas". Logic works in the same way:

• (¬P) v Q
• ¬ (P v Q)
So, if I remove the brackets from the above wff what do you suppose the meaning is? Whats the convention?

Okay, with everything I have said here I believe you should be able to solve the homework problem. Good luck!