Logic Games Tips | Conditional Reasoning

LSAT Blog Logic Games Tips Conditional ReasoningThe following two rules are extremely common in Grouping: In-Out / Selection games, but they give test-takers a lot of trouble.

These rules come up in other types of Logic Games, too. Make sure you can recognize them at a glance and instantly know what they mean.

Please note: "~" is a symbol meaning "not." I'd use a slash ("/" ) through a letter in the below examples to indicate "not."

However, it doesn't come out well when typed, so I'm not using any slashes in this blog post. If you prefer the slash (I do), use it instead.


Rule #1:

Positive Variable -> Negative Variable = + --> -



Original: X -> ~Y
Contrapositive: Y-> ~X

Meaning: at least one does NOT occur, and maybe both will not.


Example:

If I eat cookies, then I don't eat donuts.

If I eat donuts, then I don't eat cookies.

Therefore, I cannot eat at least one of them, but perhaps I'll eat neither.


Whenever you see a positive sufficient condition (the one before the arrow), and a negative necessary condition (the one AFTER the arrow), this means you can never select both, so at least one will not be selected.

(See LSAT Logic: Necessary vs. Sufficient Conditions)

In other words, you must always lack at least one of the two. However, there's no reason you can't lack both.


LSAT Examples:

PrepTest 33, Game 2 - birds in the forest (December 2000 - in Next 10 Actual, page 177):

Original: Harriers -> ~Grosbeaks
Contrapositive: Grosbeaks-> ~Harriers

Meaning: The forest cannot contain both Harriers and Grosbeaks - it will always lack at least one of the two, and maybe it will lack both.


PrepTest 36, Game 1 - fruit stand (December 2001 - in Next 10 Actual, page 278):

Original: Kiwis -> ~Pears
Contrapositive: Pears -> ~Kiwis

Meaning: The fruit stand cannot carry both Kiwis and Pears - it will always lack at least one of the two, and maybe it will lack both.


Rule #2:

Negative Variable -> Positive Variable = - --> +

Original: ~X -> Y
Contrapositive: ~Y-> X

Meaning: at least one MUST occur, and maybe both will occur.

Example:

If I don't eat peas, then I must eat carrots.

If I don't eat carrots, then I must eat peas.

Therefore, I must always eat at least one of the two, but there's no reason I can't have both.

At this point, students often ask, "Why is it possible to have both?"

Answer: Because the rule has no policy for what happens when you already have one of the two.

This rule only has a policy for what happens if I don't eat one (I must eat the other). It has no policy for what happens when I already ate one (or am going to eat). This is why nothing stops me from eating both. There's simply no rule against it.

The only thing this rule means is I can't LACK both.

LSAT Examples:

PrepTest 33, Game 2 - birds in the forest (December 2000 - in Next 10 Actual, page 177):

Original: ~Jays -> Shrikes
Contrapositive: ~Shrikes -> Jays

Meaning: The forest cannot lack both Jays and Shrikes - it will always have at least one of the two, and maybe it will have both.



PrepTest 36, Game 1 - fruit stand (December 2000 - in Next 10 Actual, page 278):

Original: ~Tangerines -> Kiwis
Contrapositive: ~Kiwis -> Tangerines

Meaning: The fruit stand cannot lack both Kiwis and Tangerines - it will always have at least one of the two, and maybe it will have both.


For the birds in the forest game, I would summarize the rule as:

J
S
JS

Every valid list of birds in the forest will fall into one of those three categories - one is in, the other is in, or both are in.


For the fruit stand game, you might summarize the rule as:

K
T
KT

Every valid list of fruits in the fruit stand will fall into one of those three categories - one is in, the other is in, or both are in.

***

You can also see my approach to Birds in the Forest Logic Game.

Photo by _sk / CC BY-NC-SA 2.0



13 comments:

  1. Suppose you're given a linear game where the variables can repeat themselves (suppose we're talking about people using a vending machine in a certain order). Sometimes when this happens, you'll be given a rule that says "every purchase B made was made before any of F's". But then you might also be given a rule that says, "at least one of C's purchases was made before D's." I understand the distinction, but I can never quite figure out how to diagram it

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  2. How about:

    B-F

    C1 - D

    (I would make the 1 a subscript - a small #1 - instead of a full-sized #1. This would indicate that (at least) the first C occurs before any D.

    Alternatively, you could put a "greater-than or equal to" sign, then a 1C then a dash, then D.

    It would look *something* like this (there's no way to type a "greater-than or equal to" sign:

    > 1C - D
    -

    Hope this helps!
    Steve

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    Replies
    1. I had this exact question; came across the boxes game in PT 63 Game 4 with the green ball that's placed lower than any red balls rule (second rule). Bless JB for asking here! And thanks, Steve for your response.

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  3. This was really helpful. It makes that somewhat hairy (feathery??) "birds in a forest" game so much more manageable....

    I just stumbled across a conditional rule -- albeit not one in a selection in/out game -- that could be ripe grounds for misinterpretation. This would be the rule in preptest 23, game 4. The rule states:

    Any candidate who speaks fifth must speak first at at least one of the other meetings.

    So: 5th --> 1st.

    My first reaction when looking at both rules together was to say: ok, so, if someone goes fifth they must also go first. But the setup of the game allows for people to speak twice in the 5th timeslot...so that you could have a situation where you have three different people going first and then just two different people going fifth. That still wouldn't violate the rule, and is exactly what they test in question #23.

    I guess, as always?, the lesson is that you really need to be vigilant to stick very closely to what the rules are telling you logically -- and to never jump to assuming that the fruit stand can't lack BOTH kiwis and pears, say, when you just know kiwis --> notpears.

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  4. Great explanation. I found a rule game quite challenging. I don't recall the game but the sufficient condition was negative, and the following condition had two premises such as:

    If Yews are out, then oaks or laurels are in, but not both.

    It was the contrapositive that tripped me up and of course it was tested. Can you please explain why in the contrapositive- if both oaks and laurels are out, MUST yews be in?

    Thanks a million for all your help Steve!

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  5. Osmel, try breaking it down into its constituent parts:

    If Yews are out, then Oaks OR Laurels are in:
    ~Y ---> L or O
    ~(L and O) ---> Y

    If Yews are out, then both CANNOT be in:
    ~Y ----> ~(L and O)
    L or O -----> Y

    Remember that the contra of an AND statement is OR, and vice versa.

    So in this problem, if you have either L or O in the group, Y must be there. Also, if you know one of them or both of them are not in, Y must be there (this is an intuitive leap you need to make).

    ReplyDelete
  6. Hopefully somebody sees this much delayed comment and could respond as I've just joined this blog.

    Tom, I'm confused about your reasoning. In your first rule you say ~Y --> L or O. Fine, lets say you have ~Y, ~O, and L. That meets that rule.
    Then, looking down at your last rule you say L or O --> Y. So, since in my hypothetical we have ~Y, ~O, L, we see that L being in means we have to have Y as well. Now we have both ~Y and Y. How can that be?

    ReplyDelete
  7. Following up on my anonymous Yews game question to Tom, above, I think the trouble is when you split up the rule in question to two parts you lose the sense of what's going on with the "but not both" part of the rule.

    Keep in mind I haven't seen the game, just the rule ~Y --> L or O but not both. I'm assuming that there are only two categories, included and ~included. If there are more than two categories my analysis would be slightly different.

    So I would write this symbolically as:

    ~Y --> (L or O) AND not (L and O)

    If you write the contrapositive from the rule as written above, note that the negative of "not (L and O)" is "(L and O)".

    However, since I'm used to changing all my "and"s to "or"s and vice versa, and changing all my includeds to not includeds and vice versa when writing contrapositives, I like rewriting the original rule here to allow us to do that.

    So Rule as Rewritten:

    ~Y --> (L or O) and (~L or ~O). Do you see that this is the same rule as the original just restated?
    So the contrapositive is:

    (~L and ~O) or (L and O) --> Y.

    Luckily, however, we can get to the same place much much easier.

    Note that the original rule REALLY says if not Yews then L and O are split up between included and ~included. In other words, if ~Y, then L or O is included and (because of the "not both" part of the rule) the other one must be not included.

    So the contrapositive would be that if L and O are together in the same group (i.e. not split up) then Y is included.

    So if you have O and L, then you would have Y included. Or if you have ~L and ~O then you also have Y included.

    So hopefully that easily explains why if both Laurels and Oaks are out, then Yews are in.

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    Replies
    1. I hit this problem from Feb 2000 and spun.
      Your explanation is quite good. The Last condition
      of that problem reads: "If it is not the case that
      the park contains BOTH laurels and oaks, then it
      contains furs and spruces." And the analysis given states, "There are three circumstances - first if the park has laurels but not oaks, the second ifthe park has oaks but not laurels and the third if the park has neither. Conversely if the park doesnt have firs and spruces it must have both laurels and oaks." They didnt convert the "and."
      I would greatly appreciated your showing how u might diagram this.

      Thank you

      Delete
  8. Steve, or any other helpful person:
    If the condition states:
    "If the stand carries watermelons, then it carries figs or tangerines, or both."
    Then do I write and/or in the diagram between the two letters, and if so, does it change when I diagram the contrapositive?

    Can someone please help me on this? Thank you.

    N.

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  9. Ignore the above post. It's ok, I got it from another page. Thanks anyway.

    ReplyDelete
  10. amazing! thank you for having this blog. with gratitude from vancouver bc canada

    ReplyDelete
  11. CA$H EXPLANATION PAPO

    ReplyDelete