What is my fallacy? LSAT Reasoning Question: Titanium Ink

Question

I have a question regarding an LSAT Reasoning question and it drives me crazy

Question is:

Until recently it was thought that ink used before the sixteenth century did not contain titanium. However, a new type of analysis detected titanium in the ink of the famous Bible printed by Johannes Gutenberg and in that of another fifteenth-century Bible is known as B-36, though not in the ink of any of numerous other fifteenth-century books analyzed. This finding is of great significance, since it not only strongly supports the hypothesis that B-36 was printed by Gutenberg but also shows that the presence of titanium in the ink of the purportedly fifteenth century Vinland Map can no longer be regarded as a reason for doubting the map’s authenticity.

The reasoning in the passage is vulnerable to criticism on the ground that:

(A) The results of the analysis are interpreted as indicating that the use of titanium
as an ingredient in fifteenth-century ink both was, and was not, extremely
restricted.

(B) If the technology that makes it possible to detect titanium in printing ink has
only recently become available, it is unlikely that printers or artists in the
fifteenth-century would know whether their ink contained titanium or not

(C) It is unreasonable to suppose that determination of the date and location of a
document’s printing or drawing can be made solely on the basis of the
presence or absence of a single element in the ink used in the document

(D) Both the B-36 Bible and the Vinland Map are objects that can be appreciated
on their own merits whether or not the precise date of their creation or the
identity of the person who made them is known.

(E) The discovery of titanium in the ink of the Vinland Map must have occurred
before titanium was discovered in the ink of the Gutenberg Bible and the
B-36 Bible.

Official Answer is:

The author first concludes that the presence of titanium in the ink of the Gutenberg and B- 36 Bibles supports the theory that Gutenberg printed both. Okay, this hinges on titanium’s being a rare ingredient in 15th-century ink and therefore characteristic of Gutenberg. Then the author concludes that the presence of titanium in the ink of the Vinland Map suggests that the map dates from the 15th century. But that conclusion hinges on titanium’s being typical of 15th-century ink. Clearly, both conclusions can’t be true: If titanium was typical of 15th-century ink, then the B-36 Bible could have been printed by anyone.

I don't understand why the author supposedly concluded that the presence of titanium in the ink of the Vinland Map suggests that the map dates from the 15th century and that this conclusion hinges on titanium’s being typical of 15th-century ink, since I don't find that in the text. The following is my reasoning, so please follow it and tell me where my error it:

I am not a native speaker, but my interpretation of the second part of the following sentence:

"This finding is of great significance since it not only strongly supports the hypothesis that B-36 was printed by Gutenberg but also shows that the presence of titanium in the ink of the purportedly fifteenth century Vinland Map can no longer be regarded as a reason for doubting the map’s authenticity.

is:

There was strong evidence that the Vinland Map was from the 15th century, but the fact that it contained titanium was a reason to doubt it, since it was assumed that the probability to find titanium in ink from that century was smaller than the probability that the map was against all evidence from that century.

Meanwhile, numerous books from the 15th century had been analyzed, and they found titanium in at least two. Hence, the fact itself that it contained titanium was no longer a reason to doubt it since now the probability that it was a fake was much smaller than the probability to find titanium in ink from the 15th century

However, since it was still unlikely, it supports the hypothesis that B-36 was printed by Gutenberg

Let's make that clear with an example:

100 bibles from the 15th had been found. It is known that 10% of them had been printed by Gutenberg (G). Hence, the prior probability that a random bible (B-36) was printed by Gutenberg is P(G)=0.1.

Now, let's assume Gutenberg used titanium (Ti) for every second print p(Ti|G) = 0.5 and 30 bibles had been analyzed and titanium had been found in two of the bibles of which one was known to be from Gutenberg. Hence, the estimated probability to find titanium in a random bible is p(Ti)=0.07 +/- error (random sample 30 of 100 bibles of which 1 or more are Gutenberg bibles)

Then, p(G|Ti) = 0.5 * 0.1 / 0.07 = 0.714 +/- error. Accordingly, the finding strongly supports the hypothesis (0.1 -> 0.71).

The other way round, the Vinland Map might contain some special features (e.g. manufacturing techniques ...) which are found on only 0.01% of the maps outside the 15th century while the probability to find titanium in ink was considered to be much smaller: p << 0.01 Hence, this was a valid reason to doubt that it is from the 15th. However, after updating p(Ti) to 0.1 the probability that the Vinland Map was from the 15th century increased to 90% and thereby the fact that the ink contained titanium can no longer be regarded as a reason for doubting the map’s authenticity.

I checked several forums where this question was discussed and almost all people agreed that the official answer is without any doubts correct.

What is my fallacy?

Answer 1

Answer

Both CraiglCraigl and Bumble have done a good job pointing out the two important issues in flaws in your reasoning, and I am going to approach the problem from a slightly different angle.

For starters, it is important to see that there is no Bayesian reasoning presented conceptually in the text. So, what you are looking for is feedback on how your approach to the problem erred. You obviously have mathematical training, and this LSAT question is not testing your ability to formally reason, but rather your ability to use informal logic. Informal logic forces you to read between the lines. In this case, what you must understand is the importance of detecting the enthymeme which in a modern interpretation is applied to an argument with implied or implicit propositions. So, let's expand on CraiglCraigl's comment.

The entire text reduces to the issue of the implications that trace-Ti ink has on document authenticity. The passage can be rewritten as two syllogisms utilizing the deep-structure of the text:

P1. The G bible and B-36 used trace-Ti ink.
P2. The frequency of trace-Ti ink in contemporaneous books is 0.
C1. Therefore, it is likely that B-32 was printed by Gutenberg.

P3. The V map used standard-Ti ink.
P4. The frequency of standard-Ti ink in contemporaneous books is greater than 0.
C2. Therefore, it is likely that the V map is authentic.

Note the trap the author of the question has written using P2 and P4! The author does NOT explicitly make the propositions, but yet they are necessary for reasoning to the two distinct conclusions, AND they seem to contradict each other. This apparent contradiction should create cognitive dissonance in a broader sense. This is the fallacy of equivocation using enthymemes and is meant to determine whether you can use natural language like a lawyer does. How exactly? In the context of the G and B-36 bibles, CONTEMPORANEOUS_BOOKS_1 doesn't actually include the G and B-36 bibles, whereas in the second case with the V map (CONTEMPORANEOUS_BOOKS_2), it does. This ultimately leads the passage to a bad conclusion, because the difference between CONTEMPORANEOUS_BOOKS_1 and CONTEMPORANEOUS_BOOKS_2 is one of opposites. Also, to ensure notice, the author has written the passage to conflate reasoning involving trace-TI detected by the new test and standard-TI ink present in the V-map. How does one draw a new conclusion about the frequency of the V map when the standard-Ti ink was completely irrelevant in the authenticity to begin with? Apples and oranges, as we say.

Ultimately, the lesson to be learned here is that the skills needed to complete this question are informal logical skills using the natural language of English, and not formal logic skills in the artificial language of mathematics. With the emphasis on formal reasoning in math and logic, it is quite easy to try to apply the tools of formality where the language skills dominate because of the relative simplicity of the former in comparison with the complexity of the latter. Just remember, lawyers, ply their trade much more in rhetoric than computation.

Addendum

Skim in reference for the following argument:

Alright, on lunch, I took a few moments and reviewed your claim that there was synonymy between the sentence in the passage:

This finding is of great significance since it not only strongly supports the hypothesis that B-36 was printed by Gutenberg but also shows that the presence of titanium in the ink of the purportedly fifteenth century Vinland Map can no longer be regarded as a reason for doubting the map’s authenticity.

and yours:

There was strong evidence that the Vinland Map was from the 15th century, but the fact that it contained titanium was a reason to doubt it, since it was assumed that the probability to find titanium in ink from that century was smaller than the probability that the map was against all evidence from that century.

One strategy for looking at formal language is using what we know in linguistics to consider such claims of equivalency. The diagram above is a technique (rarely taught these days) in elementary school to help us visualize your claim.

Note that the text from the passage is a complex sentence because it has two predicates linked to the subject "This finding". Through a series of steps including eliminating a clause, subsuming the presence of the ink into the finding, dealing with the negation, and ultimately turning the claim into a simple predicate with a transitive verb we wind up with a claim:

The finding confirms authenticity. C(f,a)

Now, immediately, you start your claim with:

There was evidence. ∃p(e)

So, the conclusion is unavoidable: 'Finding trace-ink confirms document-authenticity' simply doesn't contain or entail the proposition 'There was evidence'. This of course can be restated in plain language.

There is no implicit or explicit proposition in the original passage that claims there was strong evidence (or any evidence) to draw a conclusion about V before the new test. Or, the passage merely asserts the conclusion but does not describe the reasoning to arrive at it.

purportedly fifteenth century Vinland Map

Who purports? Why? When did this purporting take place? And most importantly, how did they do so? With strong evidence? Really? Where does the passage say that directly or indirectly? The fact is it simply does not. The sentence you drew from claims only that "the finding confirms V authenticity", and does NOT address the methodology of the previous claim to the contrary. So where did it come from?

You. You created the claim based on your understanding of the philosophy of science. It's not an outrageous presupposition, but your claim is irrefutably a hypothetical claim. It could be there was strong evidence, or it could be the evidence was weak. It could be that the evidence was non-existent, and the original claims were mere speculation without empirical grounding. We simply aren't told anything about the evidence regarding V prior to the new test.

So, there's another fallacy in your reasoning, According to Damer in his Attacking Faulty Reasoning, this is a fallacy of missing evidence, and more specifically the Fallacy of the Contrary-to-Fact Hypothesis:

This fallacy consists in making a poorly supported claim... [by] treating hypothetical claims as if they were statements of fact.

And there you have it! In a typical US state's statute on evidence and discovery, your expert testimony as a logician regarding this matter would be objectionable because it assumes facts not in evidence. Good luck!

Answer 2

You say: "There was strong evidence that the Vinland Map was from the 15th century..." but this is not actually given in the question. We are only told that the map's authenticity has been challenged on the grounds that its ink contains titanium and this was previously believed to be unknown in the 15th century.

The discovery that two 15th century bibles contain titanium leaves us with two plausible hypotheses:

1. Titanium ink was extremely rare in the 15th century, and possibly Gutenberg was the only printer using it.
2. Titanium ink was fairly common in the 15th century, and maybe only a few examples of printed works using it have survived.

If we assume 1, the evidence against the authenticity of the Vinland map remains. If titanium ink was extremely rare in the 15th century then this generates a prior presumption that any document found to contain it is probably not from the 15th century. It would be fair to say that if there is some other extremely good evidence that puts the Vinland map in the 15th century, then we might still accept it, but we are not given any such information.

If we assume 2, there is no reason to doubt the authenticity of the Vinland map, but equally there is no reason to connect B-36 to Gutenberg, since any 15th century printer might have made it.

The quoted author is trying to have it both ways, so A is the correct answer.

Answer 3

There is no fallacy the reasoning is valid.

First, yes both Bibles printed with rare ink formulation suggests their common origin -- that part is simple.

For the map, it's just a bit more complicated. Here is the original claim:

".. the presence of titanium in the ink of the purportedly fifteenth century Vinland Map can no longer be regarded as a reason for doubting the map’s authenticity"

It doesn't say it implicitly but it sounds like there must have been strong reasons supporting the map's authenticity. Otherwise, the presence of titanium ink, an impossibility, would be regarded as proof of its being a fake, rather than merely as "a reason for doubting". And with the recent discovery, that reason is gone -- a valid statement.

Answer 4

You know sometimes official tests ask questions that aren't very good and we just have to be ok with it unfortunately. Nobody and nothing is perfect including standardized tests. I scored fairly well on standardized tests in high school and myself encountered several "bad questions."