Using multiple neccessary conditions for inferencing

[ckgonzalez12]

Powerscore Logic Games Bible 2015

I'm going through a practice LSAT (LSAT Superprep Test B game #2 for anyone who's referencing!) and having issues with inferencing in a grouping game.

In the grouping game, a rule is

L/ and O/ F and S.

The section of the LSAT Bible on Multiple sufficient and neccesary conditions mentions no inferencing that can be done directly from this, but on page 283, they inference a similar situation out into details.

would it be correct, then, to inference the formula above into
L F
L S
O F
O S

I can't find any explanation on the details of this. Thank you!

[ckgonzalez12]

I've come to a self-conclussion that these are INCORRECT inferences, but I am just not sure why, if it was able to be done on page 283. Is it because of the and/or?

[Robert Carroll]

ck,

No Double Not-Arrow inferences result from this because of the specific nature of the rule and its contrapositive.

First, the conditional itself:

L or O F and S

Note that the sufficient condition is different from the way you diagrammed it. "Not laurels and not oaks" differs from "not both laurels and oaks"; the first statement says that both are absent, so you connect the respective negations of each type with "and". whereas the second statement says that it's not the case that both are in; this means that one or the other or both are not in. The negation of a statement connected by "and" is produced by negating each constituent of that statement and changing the "and" to "or" as I did above.

The contrapositive is then:

F or S L and O

The sufficient condition follows from the rule about negating a statement that includes "and", while the necessary condition follows from the rule about negating a statement that contains "or" and canceling the double negations thus produced.

As no pair of types excludes each other, given these conditionals, no Double Not-Arrow results.