# Elementary logic. Doubt about interpretation

I have a doubt about the following exercise from Copi & Cohen Intro. to logic book.

I have to determine the validity of the following argument: "Only those who ignore the facts are likely to be mistaken. No one who is truly objective is likely to be mistaken. Hence no one who ignores the facts is truly objective."

I know that "Only S is P" is usually interpreted as "all P is S", as it said in chapter 7 of the same book; but in the same chapter it says that there are other cases in which it can be read as "All S is P". I think that in the case of the first premisse "Only those who ignore the facts are likely to be mistaken," both interpretations are correct, i.e., "all those that are likely to be mistaken, ignore the facts", and " all those who ignore the facts are likely to be mistaken". I think one is biased towards the last setence because it says something that is more or less true; although, in this case, I prefer the first one. What do you think about this?

• Please see the corpus evidence in my answer and re-evaluate your judgement. Feb 24, 2016 at 8:08

We can start from the true general sentence "All Fishes live in Water" that we can translate, according to the "standard" translation of Categorical proposition of "All F are W" as:

(*) for all x, if Fish(x), then Water-living(x).

Consider now "Only Fishes live in Water", that is plainly false: also whales live in water.

If we translate it as "All F are W", we have now that, for the same interpretation, the predicate logic sentence (*) is both true and false, which is impossible.

Thus, the correct transaltion of "Only Fishes live in Water" must be:

for all x, if Water-living(x), then Fish(x)

which is false, as we expected.

In conclusion, "Only S is P" must be read as "All P is S".

Note. The same holds with "None but" in place of "Only".

I've no access to Copy & Cohen textbook: thus, I cannot comment on their explanation (if any).

• Thank you Mauro. I undestand your answer and agree with you. If you speak Spanish, look at the following. La frase correspondiente en español es "Solo pueden equivocarse los que ignoran los hechos", ¿es esta equivalente a "todos los que se equivocan ignoran los hechos" o a "todos los que ignoran los hechos se equivocan"? Feb 23, 2016 at 1:34
• Your answer is only correct for indefinite referents. See my comment below my answer. Feb 23, 2016 at 3:23
1. Only those who ignore the facts[P] are likely to be mistaken[M].
2. No one who is truly objective[S] is likely to be mistaken.[M]
3. No one who ignores the facts[P] is truly objective[S].

Thus the form of this is the second figure:

1. PM
2. SM
3. SP

The premises however switched places and don't match the figure (It's SP, not PS). If we rearrange them we get the following:

1. No one who is truly objective[P] is likely to be mistaken[M].
2. Only those who ignore the facts[S] are likely to be mistaken[M].
3. No one who ignores the facts[S] is truly objective[P].

By substitution of subject and predicate in the first premise, we can convert this to the first figure. When substitution happens in the negative premise that is universal, the premise doesn't change quantitatively to some, like it does when the premise is positive.

1. No M are P
2. All S are M
3. No S are P

Both premises are universal. The second premise is negative. If one of the two premises is negative, then the conclusion is negative. Both S and P are distributed in the conclusion, because conclusion is universally negative, so they must be distributed in the premises as well. The M and P are distributed in the first premise because it is universally negative. S is distributed in the second premise because it is universally positive. The middle term, the subject and predicate are distributed in both premises and the conclusion. The argument is valid.

• Your answer is wrong about the correct interpretation of the "Only those ... are ..." construction in English. Please see my answer. Feb 22, 2016 at 16:56
• Let's take the following sentence into consideration. "Only the good die young." This sentence is translated into "None but the good die young". For all x, if x is good, then x dies young. All good x's die young. (x) (Yx => Gx)
– user19457
Feb 22, 2016 at 16:59
• Wrong. This sentence is translated as ( for every person x, x is good if and only if x dies young ). See my comment below my answer. Feb 22, 2016 at 17:02
• By the way, some native speakers will interpret as ( for every person x, if x dies young then x is good ), totally contrary to your translation. But I think it's the minority.. The majority would say that to convey that we need to say "Only good people die young." Feb 22, 2016 at 17:03
• @user21820 I don't follow you at all on this. Nothing about "only the good die young" implies that all the good die young. Feb 23, 2016 at 15:11

The most dangerous statement in your question is "I think one is biased towards the last sentence because it says something that is more or less true."

The meaning of a sentence is never based on its truth value in logic, rather whether it is true or false is based on its meaning. If you allow sentences to have their most favorable meaning, you'll quickly run into trouble.

It's always tricky and often ambiguous to translate natural language into formal logical sentences, but in this case, I think the only appropriate translation for "Only those who ignore the facts are likely to be mistaken" is "All [Likely to be Mistaken People] are [Those Who Ignore the Facts]."

"All [Those Who Ignore the Facts] are [Likely to be Mistaken]" seems plausible from our real world knowledge, but it doesn't actually capture the stated meaning of the original.

• Thanks for your answer Chris. When I said "biased" (I think that's the right word in english) I made a kind of "psychological commentary", not relevant to logic, but relevant to my understanding of one friend of mine (wrong) tendency to translate that sentence as "All those who ignore the facts are likely to be mistaken. I agree with the answer that you and Mauro gave me. Feb 23, 2016 at 15:28
• @Evangelion045 - Yes, both Mauro and I are saying the same thing in different ways. If you agree with those answers, you can vote them both up and accept one of them. Feb 23, 2016 at 15:30
• @Evangelion045: While Chris's paraphrasing as "All [Likely to be Mistaken People] are [Those Who Ignore the Facts]." is actually correct when interpreted in English, it does not translate into a quantified implication as implied by Mauro or user19457 but still translates into what I stated in my answer. His claim that "All [Those Who Ignore the Facts] are [Likely to be Mistaken]" does not capture the meaning of the original also happens to be correct, but not for the reasons he says at all. Feb 23, 2016 at 17:16

"Only Xs are P" never means "All Xs are P", but rather "The only things that are P are Xs", which is equivalent to "All those that are P are Xs". However, "Only those Xs are P" does mean "All and only the Xs are P". This is because "those" somehow makes the phrase refer to the whole group of "Xs", and so none are left out when asserting that they are "P".

In this case, therefore, "Only those who ignore the facts are likely to be mistaken." means "Those who ignore the facts, and no one else, are likely to be mistaken." This is of course false in the real world, since there are people who do not ignore the facts but are mistaken anyway because they are not aware of the facts. But it is exactly what is stated by the English sentence. You then see that the argument is valid, but unsound. So you can say that this question is just a matter of understanding English, not logic.

Those who disbelieve my answer can check the Corpus of Historical American English for themselves. To prevent bias, I provide here the first 20 results for the search string ". Only those" in the past 50 years or so. Almost all of them provide unambiguous evidence for the semantics I claimed. The rest should also be interpreted in the same manner.

Only those who have been self-supporting for a year are eligible for relief.

Only those who have worked for their money ever acknowledge, in their deepest consciousness, the [...]

Only those people are going to be hard to know.

Only those West Berliners who have immediate relatives in the Communist part of the city may pass [...]

Only those forms fittest for their particular environment survived this process of natural selection, the rest [...]

Only those who work the miracle know what the hazards are, and they are not likely to [...]

Only those vibrational distortions of the molecule which give rise to a change of dipole moment interact [...]

Only those who've locked themselves out of society by barricading their doors with research books or [...]

Only those who have suffered ever come to have a heart that is wise.

Only those with gun permits would be allowed to buy ammunition.

Only those near the door knew at first there was a shooting.

Only those in the most extreme isolation, and these are atypical, fail to have an [...]

Only those suffering from that plague called liberalism can say that these restrictions put by a legitimate [...]

Only those who have denied their being yearn to play at it.

Only those who have been through the process can truly understand how far superior is an education [...]

Only those were safe who never loved and did not love. = Only those who never loved and did not love were safe.

Only those local organizations can do the job of combining neighborhood resources and talents with the financial [...]

Only those changes are held to constitute progress which directly or indirectly tend to heighten human happiness [...]

Only those in the cockpit and the first few rows of seats were spared.

• So in this case the correct interpretation would be the conjunction of the interpretations I gave? Feb 22, 2016 at 16:53
• @Evangelion045: Right; an equivalence is the same as the conjunction of the implications in both directions. Feb 22, 2016 at 16:54
• @Evangelion045: By the way, the same phenomenon occurs with other determiners like "the/this/that/these/all/some/a few/...", not just "those". In general "Only X is/are Y" has to be interpreted according to whether "X" is definite or indefinite. I actually have an explanation that accounts for the most of the usages of the copulative verb, but it is English, not logic, and your language intuition is good enough most of the time. =) Feb 22, 2016 at 17:01
• I don't buy the equivalence of "only those" with "all." Can you cite some supporting evidence? In my experience, that doesn't actually match how the construction is commonly used in natural language. Feb 23, 2016 at 15:20
• @ChrisSunami: Then I'm afraid you do not really understand natural language. Your own answer shows your misunderstanding, because while you interpret the sentence as "All [Likely to be Mistaken People] are [Those Who Ignore the Facts].", you fail to realize that "[Those Who Ignore the Facts]" is definite and hence your interpretation necessarily implies that the two groups are the same. I do not have the time to find and cite supporting references but I am a native speaker with some training in linguistics and natural language processing. If you disagree, show an actual counter-example! Feb 23, 2016 at 17:06