ALG conditional reasoning question

[josuecarolina]

I just finished lesson 2, part 2 and I am looking for some clarification.

When common variables of opposite signs occur in the sufficient condition, then at least one of the two necessary conditions must occur.

Okay, got that. However, they can still both occur right? So the example was something like
if S then not T
If not S then R

so we could have not S and R+T or r or t; but If S, could only be R...right?

also, when the common variable w/ opp signs occurs in nec, then both suff cannot occur was diagrammed 'R/T' (either or);
but for com var with opp in suf it was also R/T but could be both...right? So we have to make sure 'in our head' when we mean either/or but NOT both, or if we mean either, or or both when we diagram 'R/T'. is that right?

thanks!

[Nikki Siclunov]

Let's take the hypothetical you described:

S NOT T

NOT S R

Clearly, S either occurs or it doesn't: the two sufficient conditions cannot be met at the same time, and each condition precludes the other. Consequently, at least one of the two necessary conditions must occur. If, say, S occurs, it follows that T cannot occur; in that case, we have no way of knowing whether R occurs or it doesn't. Likewise, if S does not occur, R must occur but we have no way of knowing if T occurs as well.

Thus, all the possible outcomes with S are: SR or S alone.
All the possible outcomes without S are: RT or R alone.

It's also worth noting that "either/or" is synonymous with "at least one." If both conditions cannot occur simultaneously, test-makers specify that "either A or B must occur, but not both." Absent that qualification, you can presume that the occurrence of both is a possibility.

Finally, in the hypothetical above you should also realize that the two conditions can be combined to form the following conditional chain:

T NOT S R

Thus, at least one of S or R must occur at all times (i.e. either S or R must occur, and it is possible that both occur. By contrast, at most one of T or S can ever occur (i.e. either T or S does not occur, and it is possible that neither occurs). And obviously, if T occurs, it follows that R must occur as well.

Hope this helps!

[josuecarolina]

It's also worth noting that "either/or" is synonymous with "at least one." If both conditions cannot occur simultaneously, test-makers specify that "either A or B must occur, but not both." Absent that qualification, you can presume that the occurrence of both is a possibility.

That should be sticky'd or something. That is just great. Thanks!