Self-study
I am having a lot of trouble with practice problems on sufficient and necessary conditions. Each practice problem I attempt concerning the contrapositive, I end up missing the mark (especially on the translating into Lawgic formulas). Does anyone have any tips on how I can master this topic?
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3 comments
you are not alone this is hard....
Keep in mind the point of formalism is to gain intuitive understanding. You can also pull it out when you are stuck. It is a tool! Mechanics use tools, but they don't immediately jump to a wrench when they see any problem.
Do you think you have any gaps in how you understand necessity and sufficiency in daily conversation? For example, if I told you a car cannot move unless it has wheels, would you understand that wheels are necessary, but not sufficient for a car to move?
In other words, a car can't move without wheels. Wheels are necessary for a car to move. If moving car, then wheels. No wheels, no moving car (contrapositive). Lacking wheels, a car cannot move. For a car to move it needs wheels. The fact that the car was moving indicates that it has wheels. Wheels are an integral part of every moving car.
However, at no point does having wheels mean that a car can move. Maybe cars also need engines and transmissions. That is fine. What we know is that without wheels, she ain't moving. We can never prove the car moves based on wheels being necessary.
Those transformations in your head have to be automatic.
If you wanted to be formal, you could start with thinking about it positively:
If moves, then wheels. If A, then B.
Contrapositive:
Not wheels, not move. Not B, Not A.
Really sit with this and think about it. Why does this make sense? Why can we say no wheels means no moving? If all we know is that if something moves it has wheels, how do we get a contrapositive?
Let's go through the scenario. All moving things have wheels. So if something DOESN'T have wheels, can it move? Uhh... No, because if it moves, it would have wheels. But this thing doesn't have wheels. So... it must not move.
Better yet, imagine there is a closed box in front of you. I tell you, "If Sam is brain dead, there will be a squirrel in the box." If A, then B.
So now what do you do? You open the box, of course. When you look in, there is NO SQUIRREL. NOT B is reality. What can you conclude from that?
Well, Sam must not be brain dead. If he WAS brain dead, there would have been a squirrel, but I looked in the box and the reality is there is no squirrel. We do not live in that universe where Sam is braindead; there was no squirrel.
Hey! I wrote a response to a related question here. Hopefully that helps! In case the link doesn't work, I've pasted my comment below.
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Yes, you're right about the example you gave -- being an apple is sufficient (i.e. enough to guarantee) that something is a fruit. In other words, being a fruit is necessary (i.e. required) for something to be an apple. At a conceptual level, we can also think of this as "If something isn't a fruit, it isn't an apple." That's called reasoning by the contrapositive: since apple --> fruit, then /fruit --> /apple. I'd say it could be helpful to get comfortable contraposing logical relationships -- first in the real world (e.g. dog/animal, chair/furniture, water/liquid, etc.), then with more abstract symbols. After that, practice representing sufficient and necessary conditions symbolically using circles. Sticking with the apple/fruit example, "apple" would be a circle that is surrounded by the bigger circle of "fruit." These circles show you something important: it is possible to be within the necessary condition (i.e. be a fruit) without being inside the sufficient condition (i.e. being an apple) and likewise that being outside the outer circle ("fruit" circle) guarantees that you are also outside of the inner circle ("apple" circle). Visually, sufficient conditions are inner circles and necessary conditions are outer circles. I think knowing both the symbolic and the visual ways to represent sufficiency and necessity is really useful groundwork for understanding a lot of LSAT question types and logical flaws.
And as an additional challenge, you can start working with several conditions at once to see how "sufficiency" and "necessity" are relative terms that depend on the conditions you are referring to. For example, "person" is sufficient for "organism," which is sufficient for "living." Therefore, "living" is necessary for "organism," which is necessary for "person." So /living --> /organism --> /person.
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