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#59777
Setup and Rule Diagram Explanation

This is an Advanced Linear: Balanced Game.

The setup of this Advanced Linear game is more challenging than the setup of the first game of this section, primarily because although this game appears to have only a few initial inferences, the relationship of the rules is such that most of the diagram can be filled in. With the proper diagram, the questions are relatively easy.

From the game scenario, we know that there are three variable sets: the five archaeological sites, the three archaeologists, and the three centuries. The five archaeological sites and the three centuries each have numerical order, but the archaeological sites are the better choice for the base since each site has a single archaeologist and a single date. This choice creates a linear setup with two stacks, one for the archaeologists and one for the centuries (remember to leave ample vertical space between the two stacks since each row will likely have its own Not Laws):

pt44_o04_g3_1.png
pt44_o04_g3_1.png (6.19 KiB) Viewed 1722 times
Because the rules have so many consequences, let’s examine each rule individually.

Rule #1. This rule is the most straightforward rule of the game, and it can be represented by placing a “9” in second site space of the Century row:

pt44_o04_g3_2.png
pt44_o04_g3_2.png (2.25 KiB) Viewed 1722 times
Rule #2. If neither the fourth nor fifth site was discovered by O, we can place O Not Laws on each space. And, because there are only three archaeologists, with O eliminated from discovering the fourth and fifth sites we can infer that either F or G discovered those sites. This fact can be represented by individual F/G dual-options on each site:

pt44_o04_g3_3.png
pt44_o04_g3_3.png (4.02 KiB) Viewed 1722 times
Rule #3. While the first two rules can be represented directly on the diagram, this rule must be diagrammed separately. The rule is a bit challenging to represent because it contains two separate pieces of information. The first piece of information is that G discovered exactly one site, and the second piece of information is that the site discovered by G dates from the 10th century. There are different ways to represent this rule, but we will use the following diagram:

pt44_o04_g3_4.png
pt44_o04_g3_4.png (892 Bytes) Viewed 1722 times

From the contrapositive of this rule we can infer that any site that dates from the 8th or 9th century was not discovered by G, and must therefore have been discovered by F or O. Also, be careful not to make a Mistaken Reversal and assume that any site that dates from the 10th century was discovered by G; that is not necessarily the case.

Using the information above, we can infer that since the second site dates from the 9th century it was not discovered by G, and must therefore have been discovered by F or O:

pt44_o04_g3_5.png
pt44_o04_g3_5.png (4.64 KiB) Viewed 1722 times
Rule #4. This is another important conditional rule, which can be diagrammed as follows:
pt44_o04_g3_6.png
pt44_o04_g3_6.png (588 Bytes) Viewed 1722 times
Operationally, this rule indicates that every time a site dates from the 8th century, it must have been discovered by O. The contrapositive of this rule reveals that if a site was not discovered by O, then it cannot date from the 8th century and must instead date from the 9th or 10th century. Thus, the fourth and fifth sites, which were not discovered by O, cannot date from the 8th century, and they must date from the 9th or 10th century. We can add this to our diagram with dual-options:

pt44_o04_g3_7.png
pt44_o04_g3_7.png (5.2 KiB) Viewed 1722 times
Rule #5. Although we have already gained a fair amount of information from the first four rules, the final rule will allow us to fill in most of the remaining spaces in the diagram. First, however, a note about the language used in this rule. In most cases, “either…or” on the LSAT means “at least one of the two, possibly both.” For example, if we see the rule “either A or B must attend,” this means that at least one of the two must be there; they are not both required to attend, but they both could attend. In this particular rule, however, we should note that the word “either” is preceded by the conjunction “than” and is used at the end of a comparison. When this is the case, “either” actually means “both.” For example, “He is taller than either of the other two boys on the team” means that he is the tallest of the three, and, “She likes History better than either Biology or Physics” means that History is her favorite of the three classes mentioned. So, the proper interpretation of this rule is that the third site is more recent than both the first and fourth sites.

Thus, when this rule states that the third site dates from a more recent century than either the first or fourth site, the third site cannot date from the 8th century, and the first and fourth sites cannot date from the 10th century. By itself, this information only seems to result in a series of dual-options for the dates of these sites. However, because we have already concluded that the fourth site must date from the 9th or 10th century (this inference was produced by combining the second and fourth rules), and this rule indicates that the fourth site cannot date from the 10th century, we can conclude that the fourth site dates from the 9th century. And, because the fourth site must date from the 9th century, we can use this rule to determine that the third site must date from the 10th century. Consequently, we can fill in the entire century row with either the exact site dates or with dual-options:

pt44_o04_g3_8.png
pt44_o04_g3_8.png (3.79 KiB) Viewed 1722 times
Further, because the fourth site cannot date from the 10th century, from the third rule we can infer G cannot discover the fourth site, and so the fourth site must be discovered by F. We can also use the third rule to determine that the first site was not discovered by G, and must have been discovered by F or O. These inferences allow us to arrive at our final diagram:

pt44_o04_g3_9.png
pt44_o04_g3_9.png (7.72 KiB) Viewed 1722 times
Note that not all of the dual-options are independent. For example, if the first site dates from the 8th century, then according to the fourth rule it must have been discovered by O. The third rule is also active, and during the game you must keep track of the placement of G. Regardless, you now have a powerful setup with which to attack the game.

Final Note: the language used by the test makers here only states that each site dates from a particular century, not that every century must be used (the same situation applies to the archaeologists). Thus, it is possible that some variables may not be used in the game. Each game is different, so always track the language used in a game carefully to ascertain the situation.
 Jdlanpher
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#1479
Having trouble with the inference made from this rule. The rule states
"the site visited 3rd dates from a more recent century then does either the site visited first or that visited fourth"

the inference derived from this rule, according to the book, is that the third site cannot date from the 8th century(which makes perfect sense) and the first AND fourth sites cannot date from the 10th century(which is where i am confused). Shouldn't the inference be that EITHER the first OR fourth site cannot date from the 10th century??

Why couldn't, for example the 3rd site date from the 10th century as well as the 4th, and the first dating from an earlier century? The rule would still hold in that the 3rd dates from an earlier century than either 1st or the 4th.

My main problem is trying to understand why the inference assumes both 1st and 4th site but the rule says either, or. :|
 Jdlanpher
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#1480
in the third paragraph i meant the third still dates from a *more recent century
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 Dave Killoran
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#1481
Thanks for the message. Let’s take a close look at your question, because you’ve recognized an issue that a very few others have seen, and it is a feature of the English language that LSAC is exploiting. I plan to add a note about this in the next edition of the book [note: and later did], but I’m happy to address it with you now. Here are a couple of thoughts on the issue:

First, and most importantly, from an English standpoint, in most cases, “either…or” means “one or the other.” For example, if we see the rule “either A or B must attend,” this means that at least one of the two must be there; they are not both required to attend. In this particular case, however, we should note that the word “either” is preceded by the conjunction “than” and is used at the end of a comparison. Where this is the case, “either” actually means “both.” For example, “He is taller than either of the other two boys on the team” means that he is the tallest of the three, and “I like History better than either Biology or Physics” means that History is my favorite of the three classes mentioned. In the same sense, if I graduated more recently than either of my brothers, I must be the most recent graduate. So, the explanation in the book uses this "both" interpretation.

Second, from LSAC’s viewpoint, the “both” interpretation is clearly what they intended, because without this interpretation, some of the “correct” answers would no longer be provably correct. For example, in question #13, answer choice (B) would suddenly be a possibility if the first site could date from the 10th century. However, knowing that LSAC intended this interpretation doesn’t help us during a game because obviously we don’t know the answers to the game prior to doing it! I only mention it as being indicative of what they ultimately wanted the interpretation to be, in part because they are the arbiters of what is right and wrong in the LSAT world.

Third, could they have been clearer? No doubt they could have. But, I’ve discovered over time that there are fair number of situations where I wish they had stated it more clearly (like saying “both” in this case). I think that sometimes they aren’t interested in perfect clarity, but rather whatever they feel is minimally defensible in case someone asks about it.

Finally, the good news is that situations like this have only appeared occasionally on the LSAT, so thankfully you are not likely to have to face an identical situation on your test. Regardless, good eye on seeing that issue—it is a great sign for your future success on the LSAT.

Please let me know if that answers your question. Thanks!
 Jdlanpher
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#1483
Thanks Dave,

Although I have always interpreted "either, or" to mean one or the other, I now see how it could mean both. Also, I agree with you about the unclarity of this sentence, and I still think many people would read it and interpret "one or the other" and not necessarily be wrong. However, I understand that it is up to the test makers, so I am glad to be prepared for this in the future if it should happen to come up on the test.

Thanks again!
John
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 Dave Killoran
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#1484
Hi John,

Glad I could help!

It's a tricky one, no doubt. Grammatically, they do have a valid defense, and the key is seeing the "than" before the "either"—a point that is so subtle it's almost unfair. But, it's their world, and we just live in it.

Thanks!
 quiz555
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#1784
Hey Dave,

Thank you so much for clearing that up. I remember a couple of weeks back when I first did the game in the Logical Reasoning Bible and just couldn't understand how certain questions were provable. I subsequently searched for explanations to the game in the same book, but could not understand why so many inferences were deduced from an "either or" statement. The explanation in the book interprets the rule as "both" without really giving any reason why it must be "both". After consulting a former logic professor and two other English professors, I realized that the "than" undoubtedly transforms it into a "both" rule. Something that is rarely talked about in logic or English grammar.

HOWEVER, I have found my self in a very similar situation were the use of the word "than" is not used in an "either or" statement and yet the statement must still be interpreted as a "both" rule in order for certain questions to be provable.

This game is in the December 2008 logic game section. Test 4, game 4 in my PowerScore virtual coursework book. The rule states:

The site that includes Quinn must take place before any site visit that includes either Rodriguez or Taylor.

May you please shed some light on why the rule should be interpreted as a "both" and not an "either or"?

Thank you!!
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 Dave Killoran
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#1789
Hi Quiz,

Yes, as you can see from the above discussion, the "than either" point is a subtle one, which is why I'm glad we have a forum such as this one to clear it up. I will say, though, that such a construction is pretty rare on the test and the rule that spawned this discussion is the only instance of it I can remember off the top of my head on this test.

Turning to the rule you ask about--"The site visit that includes Quinn must take place before any site visit that includes either Rodriguez or Taylor"--the issue creating the "both" is the phrase "before any site visit." That combination of the "any" and the "either/or" sets up a situation where Q must be ahead of any site with R, and any site with T. If you think about this language for a moment, let's say R visited a site ahead of Q (but Q visited a site before T). Would that meet the stipulation in the rule? No, the "Any site" language makes it universal for Q to be ahead.

Does that help make sense of this rule? Please let me know. Thanks!
 quiz555
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#1793
Perfect, got it!

Thanks Dave, I really appreciate your help!
 KathrynJ
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#3466
In section 3, the analysis of the rules of the game of questions 13-17 (archeological sites), it states, regarding rule 5,
"The last rule states that the third site dates from a more recent century than either the first or fourth site. Thus, the third site cannot date from the 8th century, and the first and fourth sites cannot date from the 10th century."
Since the rule says :The site visited third dates from a more recent century than does either the site visited first or that visited fourth," does that mean that "either...or" leads directly to not-laws for the two sites for the 10th century? "either...or" means not-law for both?

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