Formal Logic (Some & Most)

[sjlsat]

Hi Team,

I'm currently cracking down on the 11 principles of making formal logical inferences (thanks so much for such great lessons, btw!) and had a quick question about some and most statements.


I understand that the some relationship goes both ways, as in A <---> B (with an s on top). This would mean some A's are B's and some B's are A's.

If we have a most relationship, as in most A's are B's, A ---> B (with an m on top), would this mean that some B's are A's since "most" has "some" underneath it on the logic ladder?

Or does this only apply to the starting variable, as in some A's are also B's? This is where I'm confused - if the latter is true, does that not mean some B's are A's because the some relationship goes ways?

Please let me know if I'm not making sense!!! Thank you!!

[Adam Tyson]

That makes total sense, sjlsat, and you are correct that a "most" relationship in one direction implies a "some" relationship in the opposite direction. If most As are Bs, then at least some Bs are As. Well done!

[Desperatenconfused]

Does this mean that you can go from a most statement to a some are not statement?

As in, if most dogs are black, I can say some dogs are not black, and some black things are not dogs?

Then, if you can go from an all statement to a most statement: all dogs are black, which means most dogs are black, then can't we say that some dogs are not black as well?

Which is super confusing, because if all dogs are black, then how are some dogs not black?

[Jeff Wren]

Hi Desperatenconfused,

No, you definitely cannot go from a "most" statement to a "some are not" statement.

You likewise cannot go from a "some" statement to a "some are not" statement.

The key to understanding "most" is that it literally means anything that is more than half, all the way up to and including "all." In other words, if you want to think of "most" in terms of percentages, it would be 51-100%. The important point to understand is that most does not guarantee or imply that "some are not."

In your example, based solely on the wording "most dogs are black," it is possible that "all dogs are black," so we do not know from this statement that some dogs are not black. (Obviously in the real world, we do know this, but not based on the wording of the most statement.)

In the same manner, a "some" statement means "at least one," all the way up to and including "all." In other words, if you want to think of "some" in terms of percentages, it would be 1-100%. The important point to understand is that "some" does not guarantee or imply that "some are not."

If you had the statement that "some dogs are black," it is possible that they are all black, so you do not know that some dogs are not black based on the wording alone.

Note that the words "some" and "most" are used differently in formal logic than they are commonly used in everyday life. In everyday life, if I said "some dogs are black," you probably wouldn't assume that it would be possible that all of them are black.