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# IF AND ONLY IF

Member
edited September 2013 45 karma
Can someone explain to me the implication of this statement? I know it sets up a two way street between the two terms but why is that the case???
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• #### If and only ifFor statements with if and only if, does it matter which variable I write first? Example: V is selected if and only if P is selected the if and…

• Member
45 karma
For ex-- Grace helps move the sofa if but only if Heather helps move the recliner. I just am not seeing the english to logic relevance I guess.
• Alum Member
183 karma
It is simply saying that something is sufficient and necessary for something else. For example, perhaps Grace has decided in her head that: "If Heather helps move the recliner then she (Grace) will help move the sofa." But Heather moving the recliner is also a necessary condition for Grace to move the Sofa - so Grace continues to think in her head: "but I will help move the sofa ONLY IF heather moves the recliner." You see what's going on here. Heather moving the recliner is sufficient for Grace to help move the sofa but it is also necessary for Grace to help move the sofa. If, for instance, Grace thought ONLY: "If heather helps move the recliner then i'll move the couch," the absence of Heather moving the recliner does not necessarily mean that Grace won't help move the sofa (negating a sufficient condition does nothing). For example, maybe Heather gives Grace \$10 and this was sufficient for Grace to move the couch. However, when Grace tacks on that she'll move the couch ONLY IF heather moves the recliner then no matter what else happens between Grace and Heather (whether or not Heather gives Grace \$10 or not) if Heather does not move the recliner then Grace will not move the couch. On the flip side, if Grace only said that Heather moving the couch was a necessary condition for her moving the sofa, then Heather moving the recliner doesn't guarantee that Grace will do anything (the precense of a necessary condition alone tells you NOTHING). The dual nature of if and only if statements captures subtle thought processes that always go on in people's heads. It only brings them to a written logical expression. For instance say I will score only one point at a time in basketball if and only if a shoot a free throw. If i shoot a free throw and make it I get one point. Also I get one point per one made shot only if i'm shooting a free throw. Otherwise, (if i am not shooting a free throw) i do not get one point per one made shot. I would get two points or three points per shot if i made it.

Hope that helps
• Member
45 karma
Thanks!! You explained it way better than kaplan lol they are the ones that confused me in the first place. the basketball analogy nailed it. so now on when i see if and only if it goes both ways so the contrapositives are mutual back and forth unlike when you have a if _ then _ statement where you can flip and negate once this way if there's no heather moving recliner there's no grace, if there's no grace then theres no heather.
• Alum Member
183 karma
I can tell you personally that nothing has been more intuitive than 7sage. I looked at a couple books (not 7sage) and they take the most simple things that can occur such as sufficient and necessary conditions and seemed to obscure the shit out of them.

For the example above, you essentially have two implications; (1) if grace moved the sofa then heather moved the recliner (2) If heather moves the recliner then Grace moves the sofa. Simply put - and I've not gotten through all of the conditional lessons in 7 sage - in the this example if Heather did not move the recliner then Grace did not move the couch; also if grace did not move the couch then Heather didn't move the recliner. Symbolically, (/ equals "not") (H stands for Heather moving the recliner and G stands for Grace moving the couch; -> means "implies") /H->/G and /G->/H.

specific to this question only I highly recommend going through 7sage's lawgic lessons if you have not. But also generally, they give great instruction for any section of the LSAT i have thus far encountered.
• Member
6 karma
So If AND Only If, are basically two different formal logic statements squished together (bi conditional)? If that's true, then both must be either true/possible/able, or false/not true/not possible/unable. That I understand.

However, what's the contrapositive? is simply negating both terms the CP? For example as you say, if Hr If And Only If Gs, that means Hr <--->Gs. But is the CP /Hr <---->/Gs? and if that's the case, then does that mean that there are only two possible outcomes, or can one happen and one not happen?
• Member
6 karma
And one more question, where can I find a list of terms that I can memorize for these formal logic conditions, (e.g. If/then, if and only if, if but only if, if yet only if, unless, not unless, etc.etc)
• Alum Member
edited November 2014 180 karma
There are two kinds of biconditionals; Always together and always apart.

Always together = A <->B. This means there are four possibilities. If A then B, if B then A, If not A then not B and if not B then not A.

Always Apart = A <-> Not B. This means if we have A then we won't have B and vice versa.

There are only four logical indicators for biconditionals
1. If and/but only if (always together indicator)... but usually means and
2. either or, but not both (always apart)
3. but not otherwise (always together)
4. except (always apart)

Check out 7sage's biconditional explanations (not sure if free user's can access it :S)