Linking common elements in formal logic

[summer20]

I work through formal logic questions with good accuracy. Whenever this comes up in a question, it slows me down:

How do you link two chains with a common sufficent condition? For Ex:

Roses always provide a stunning display of color: Roses Stunning display of color

Some roses have no scent: Roses Scent

I can make an inference: stunning display of color scent

but besides that, how can I link the two statments in one continuous chain?

I also have the same question with regard to two statements that have a necessary condition in common. For Ex:

Every true work of art is obscene: True work of art Obscene

Every sculpture in the Museum is obscene: Sculpture Museum Obscene

There is no way to link these two statements in one chain and no inference that can be drawn from those two statements, right?

Lastly on a related note, I know that when two sufficient conditions have opposite necessary conditions, the two sufficient conditions cannot co-exist. For Example:

A B

C B

Inference: A C

But what if there was a "some"/ "most" in the second statement, for instance:

Example 1:
A B
C B

Example 2:
A B
C B

No inference in either of these examples, right?

Thank you all so much, this forum and your books have been invaluable.

[Dave Killoran]

Hi Summer,

Thanks for the questions! I'll go through these in order.

You actually have connected the Roses inference by simply making the inference, but if you wanted to draw it out, it would appear as:

• Scent Roses Stunning display of color

As I note in the book, you also don't have to draw that in a straight horizontal line. It could be vertical, diagonal, whatever.

Same goes with the Obscene relationship:

• True work of art Obscene Sculpture Museum

They are linked there in a chain, but still no inference results.


With the two sufficients with opposite necessary conditions, yes, you are correct


And last, you are also correct about B/B situation--no inferences can result because in some/most relationships there's no way to turn B into B (which is not the case with full arrow statements because you can take the contrapositive).

Please let me know if that helps. Thanks!

[summer20]

Dave,

This helps tremendously. Thank you so so much!