No mathematician today would flatly refuse to accept the results of an enormous computation as an adequate demonstrat...
teddyteddtedFebruary 21, 2018
Why SN Arrows Away from Quantifier Statements
Hey there! I have this written down in my notes but can't find it in the lesson review.
Why when combining QS and SN statements does the arrow always have to be facing away from the QS statement or always to the right?
This is mentioned in the video explanation and I've accepted it as fact, but wanted to know the logic behind it.
Thank you!
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The logic is that when the arrow points away, the variable the statements have in common is sufficient.
For example:
A ==> B C-some-A
Notice that "A" is what these statements share in common and A is sufficient in our S & N statement.
We combine as follows:
C-some-A ==> B (notice the arrow is pointing away) C-some-B
If the arrow points to the quantifier as opposed to away, the variable in common is necessary but we know that the existence of the necessary condition tells us nothing so we can't make a valid deduction in this scenario.
For example:
A ==> B B-some-C
Notice "B" is the variable in common but B is necessary in our S & N statement so we know nothing else. If you tried to combine them it would look like this:
A ==> B-some-C (notice the arrow is pointing to)
No valid deduction
Hope that helps! Please let us know if you have any other questions.
Brett-LindsayJuly 16, 2020
I think a simple example from real life will help this a little. I seem to recall Mehran gave a similar example on the video:
Some fruits are mangoes. F --some-- M All mangoes are sweet. M --> S
From that we can deduce that some fruits are sweet. F --some-- M --> S F --some -- S
From this, we know that some fruits (mangoes at least) are sweet. We know nothing about any other type of fruit.
We couldn't go the other way, though:
Some fruits are sweet. F --some-- S All lemons are fruits. L --> F
L --> F --some-- S
This wouldn't work. Just because lemons are fruits, does not guarantee that lemons are among the sweet fruits. So, you wouldn't be able to conclude that some lemons are sweet.
I'm not sure this is any easier to understand, but sometimes different explanations can help a little.
