Sufficient and Necessary Forum

[SingleLadies]

Hi all,

As you can see this is my first post (long time listener first time caller).

I am working through the Power Score Logic Games Bible, and am having difficulty with Sufficient and Necessary conditional rules. I can't seem to understand the basic rules which allow me to properly identify and diagram the "original diagram" and "the counter positive" when setting up my answers (i am still on linear questions).

for example how do I diagram the following;
"if P is scheduled for Monday, then V and X must be scheduled for Saturday." and
"R is not scheduled for Thursday unless L is scheduled for Monday"?
I am hoping that someone can help me to this down.

Thanks in advance.

[sngilbert]

For your example I would diagram the rules like this:

Pm -->Vs+Xs (on paper '+ Xs' would be below Vs)

and

Rth-->Lm

(also, would diagram the contrapositives)

Anytime there was an additional constraint on the variable in the conditional statement (day of the week, placement order, etc), I would always place a subscript along with the variable. For conditional statements that include 'and' &/or 'or' in the sufficient &/or necessary conditions as well as an additional constraint (like the first one above), and especially if there is more than one, diagrams run the risk of becoming too big and clumsy when you start diagramming contrapositives. However, for me, the subscript really helped me remember the rule. Hope that helps.

[SingleLadies]

thanks for the quick and helpful response.

I am looking for some rules that will allow me to properly order the sufficient and the necessary. On this example as with many others my tendency is to reverse the two. I mapped mine identically to yours except backwards.

[sngilbert]

The way I remembered the order was like this:

Post diagram, I thought of all conditional statements as they were in the If...then form. Sometimes they'll throw words such as when, any, all, every, etc for the sufficient condition but when I diagram it's simply 'If.' And the same with necessary conditions: they'll often give you only, only if, must, etc. I thought of these simply as 'then.'

Since the sufficient 'triggers' the necessary condition, it must come first. The first rule describes this perfectly: Pm -->Vs+Xs and can be thought of exactly as it's written

"if P is scheduled for Monday, then V and X must be scheduled for Saturday"

The second rule is more confusing because of the unless. Not sure if you have gotten to the part in the Bible but it does a good job explaining the formula for words like unless: Negating what comes before the unless, thus becoming the sufficient condition. And keeping intact what comes after the unless, which becomes the necessary.

[trudat15]

For unless, an easy way I found to do it was change UNLESS to IF NOT.

So it would be : If not L on Monday, Then R is not scheduled for thursday. Diagramming that would give you the contrapositive of sngilbert's original diagram, which are of course, equal.

[SingleLadies]

brilliant! i think i have it now.

[ikari_ningen]

Here are some examples in English if they help the diagrams or concepts make more sense:

1. If you are a mother, then you are a woman.

Being a mother is sufficient for being a woman, or, equivalently, being a woman is necessary for being a mother.

2. If you are a woman, then you are a mother.

While 1 is always true, 2 isn't because being a woman isn't sufficient for being a mother, or, equivalently, being a mother is not necessary for being a woman.

Sufficient conditions tell you requirements that guarantee a thing is in a certain state (like being a woman or being a mother), but by themselves may or may not be required to satisfy that state. Being a mother guarantees that a person is a woman (since mothers are just women with kids), but someone can be a woman without being a mother.

Necessary conditions tell you requirements that must be satisfied in order to be in a certain state (like being a woman or being a mother), but by themselves may or may not be enough to guarantee being in that state. Being a woman is a requirement for being a mother, but by itself it isn't enough, since a woman needs to have a kid in order to count as a mother.

Some conditions can be both sufficient and necessary:

3. A shape is a triangle if and only if it is a polygon with exactly three sides.

Being a triangle is sufficient and necessary for being a polygon with exactly three sides, or, equivalently, being a polygon with exactly three sides is sufficient and necessary for being a triangle. This just means that having a triangle guarantees that I have a three-sided polygon, and having a three-sided polygon guarantees that I have a triangle (this is using the sufficient condition terminology). Alternatively, this just means that having a triangle is a required state if I have a three-sided polygon, and having a three-sided polygon is a required state if I have a triangle (this is using the necessary condition terminology).

[Precessional]

Funny. I'm exactly where you are in the PS Games Bible, and was thinking it through this afternoon.


After some ponderin', I came to this understanding:


--ImageRemoved--

This is a picture of an area enclosed by dotted lines.




--ImageRemoved--

Block B is necessary for covering the area.

It covers part of it, but not all of it. But it's not sufficient for describing all of the area--When I randomly point to an area in the dotted lines, I may or may not hit Block B.




--ImageRemoved--

Block A is sufficient. It covers all possibilities within the dotted line.





So, if A then B.


If I have the larger area, covered by Block A, I also have the smaller area, covered by Block B:





In other words, satisfying the sufficientA guarantees that the necessary B is also satisfied.



Example:

If A represents all of Smurf village, then B might be a particular member, say, Smurfette. If we have, in our evil grasp, Smurf Village, then we also have Smurfette.

Smurf Village --> Smurfette



Or, considering the contrapositive: If not B, then not A.

If we don't have Smurfette, then there's no way in hell we have all of Smurf Village.

Not Smufette --> Not Smurf Village

[SingleLadies]

^ great posts

love the smurf example. ha.