Go quantify yourself.
- Jul 15, 2016
- Advice on Logical Reasoning, LSAT
Some LSAT students fail to learn quantifiers. No one who fails to learn quantifiers has mastered the LSAT.
If you’re scared already, fear not. It’s worth knowing your way around some, most, and all statements and the inferences you can and can’t draw from them. And while it’s worth just memorizing what you can and can’t do with quantifiers, it’ll be easier to memorize the valid inferences if you understand how they work.
Let’s start with a “some” statement: some competitors in the Tour de France take performance-enhancing drugs. That just means that there’s at least one person who does both those things. So it’s equally accurate to say that some performance-enhancing drug users compete in the Tour de France. So the diagram could look like:
TDF -s- PED
PED -s- TDF
It’s a dash, not an arrow, because it’s reversible. Always. Now let’s take a conditional statement: All competitors in the Tour de France shave their legs. That diagram looks like:
TDF —> SL
Now what can we get from these two statements? Again “some” just means “at least one,” though it could be most or even all. But there has to be one. Let’s give him a name and call him Lance. Lance is a Tour de France competitor who uses performance-enhancing drugs. Since Lance is in the Tour de France, what else must be true about him? You guessed it — he shaves his legs. So that’s at least one person, namely Lance, who uses PEDs and shaves legs. So the valid conclusion is:
PED -s- SL
Could that be reversed? You bet — “some” statements are always reversible. And notice that we used the term mentioned in both statements, competing in the Tour de France, to tie together the other terms in the conclusion.
Now let’s take a different conditional statement: everyone on the Russian Olympic team uses performance-enhancing drugs:
ROT —> PED
Now back to Lance. He uses performance-enhancing drugs. But we have no clue whether he’s on the Russian Olympic team. So given the statements:
TDF -s- PED
ROT —> PED
There’s no valid conclusion to draw. Lance doesn’t sound like a Russian name, after all. What’s the difference between that one and the first one that worked? It’s all about the “all” statement. Keep in mind that “some” statements are always reversible, so it doesn’t matter what’s on what side of the dash. In the first example, the thing the “some” and “all” statements had in common was TDF, and it was on the left hand, sufficient side of the conditional statement. In the second example, they had PED in common, and that was on the right hand, necessary side of the conditional statement. That’s the critical difference. To draw an inference from quantifiers, remember this:
The shared term must be on the sufficient side of the stronger statement.
That goes not only for “some” and “all” but also for the other valid inferences you can make from quantifiers.
Suppose I told you that actually, most competitors in the Tour de France use PEDs. Sounds plausible. We’d diagram it like this:
TDF -m-> PED
This is manifestly not reversible. There are way too many roided-out weightlifters in gyms all over who would be quite angry to hear that you thought they would put on spandex, shave their legs, and race a bike all over France. So “most” statements, in general, aren’t reversible.
Given the same two “all” statements from before, what conclusions could we draw? It turns out that, in this case, turning “some” into “most” doesn’t change a thing. We can still prove that some PED users shave their legs — we still have Lance and his cohort. But there are lots more people who shave their legs but aren’t Tour de France competitors, as well as plenty of people who take PEDs without going anywhere near a bicycle. So we can’t say that most who shave their legs use PEDs, nor that most PED users shave their legs. The valid conclusion stops at “some.”
We also, still, can’t draw any connection between the Tour de France and Russian Olympians. There are plenty of drug-using bicycle racers but we don’t know where they’re from.
But what if I told you that everyone who uses PEDs fears a urine test:
PED —> FUT
Most Tour de France competitors use PEDs. As soon as you use PEDs, we now know that you fear a urine test. So the valid conclusion is:
TDF -m-> FUT
There’s one more valid conclusion: What if most of the pizza has anchovies and most of the pizza has pineapple:
PZ -m-> A
PZ -m-> P
“Most” means more than half. It could be just barely more than half, but it has to be more than half. Visualize it: Even if we tried to keep the anchovies and pineapple separate, they have to overlap at least a little. So there’s a valid conclusion: some of it has anchovies and pineapple.
A -s- P
That’s the only way to get something out of two “most” statements. You can’t chain “most” together transitively. For example, most residents of Alaska are US citizens, and most US citizens live in Alaska. Both true statements. Clearly, we can’t conclude that residents of Alaska don’t live in Alaska. No valid conclusion.
Quantifiers only come up on a couple questions per LSAT. But they’re worth knowing well regardless. Why? Those couple questions are pretty hard if you don’t know your quantifiers well and you have to figure it out on the spot. But if you know all the valid inferences like the back of your hand, those questions become pretty predictable and not terribly stressful. You can then get them right without much sweat and have more time and energy for the other stuff. So take the time to get to know them. It’s worth both having them flat-out memorized and understanding how they work.
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