Comprehending 'some' in diagrams

drecupcake

Can anyone point me to the lesson on diagramming for 'some' ?
For example: Under the Sufficient Assumption & Psuedo-Sufficient Assumption Section, Lesson 6, I need help remembering how to navigate the diagrams that involve "some."
https://7sage.com/lesson/quiz-on-finding-sufficient-assumptions-with-intersection-statements-1-answers/?ss_completed_lesson=11893

We're given premises with one premise missing that helps us arrive at the conclusion. In this case:

F <--some--> U
[find missing premise]
-----------------------------------
U <--some--> /I

For the life of me I'm having a hard time navigating "some" when diagramming. Any help or letting me know which lessons to review would be greatly appreciated!

quinnxzhang

This is one of my absolute biggest gripes with how LSAT instructors teach students to translate "some". In a real formal logic, like classical first-order logic, "Some P's are Q's" is represented by '∃x(P(x) & Q(x))'. In LSAT terms, this would be something like 'Some(P & Q)'.

I think 'Some(P & Q)' has a ton of benefits over your biconditional-like diagram. First, it makes it obvious that "some" takes the widest scope in the sentence, so that the negation is a "not some" sentence, which is equivalent to an "all not" sentence. Second, it uses the conjunction ("and") symbol, so that you get all the properties of "and", such as commutativity (i.e. some P's are Q's implies some Q's are P's). I think the most important property with the "and" is that you can see how existential sentences are nicely equivalent to negated universal sentences, in virtue of De Morgan's laws.

This is what I'd personally recommend, but I imagine others here who subscribe to the "LSAT" method might disagree.

MrSamIam

What is it that you're having trouble with?
Here's a short breakdown of "some" diagrams.

"Some fish are orange" = "Fish, some orange" = F <-Some-> O (or the other way around, it doesn't matter).

If you're trying to draw an inference with a "Some" statement, the "Some" must be on the sufficient side.
"Some fish are orange. All things that are orange are colored" = F <-Some-> O ----> C =
F <-Some-> C (Some fish are colored)

If it were reversed (O ----> C <-Some-> F) no transitive conclusion could be properly drawn.

This is one way to draw an inference with a "some" statement.
-2 Some statements alone = No valid conclusion
-1 All statement and 1 Some statement, where the "Some" is in the necessary condition = No valid conclusion
-1 Most and 1 Some statement = No valid conclusion

You can also draw a valid conclusion with a "some" and "All" statements, if they share the same sufficient condition.

hlsat180

@drecupcake said:
We're given premises with one premise missing that helps us arrive at the conclusion. In this case:

F <--some--> U
[find missing premise]
-----------------------------------
U <--some--> /I

Missing Premise: F --> /I

Completed chain: U <-some-> F --> /I

Mechanical rule: Follow the arrows. The Some relationship transfers only if the arrow is moving AWAY from the Some relationship. Any arrow moving toward the Some relationship does not transfer back the Some relationship. For example:

A --> B <-some-> C

Relationship between A and C is unknown, as per @MrSamlam's very detailed explanation.

drecupcake

Thanks so much for the replies! Sorry I didn't respond sooner but it didn't email me like I thought it would when I received responses. I definitely am more grounded in real world examples so thanks @MrSamIam for your example. And thanks @hlsat180 for the reminder that the relationship only moves AWAY from the some relationship, that's definitely been tripping me up.

Unfortunately @quinnxzhang I haven't taken formal first-order logic so I'm not familiar with your diagramming method but I thank you nonetheless.