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In 1984 this is a term for someone who could, knowing all the facts, still believe a falsehood to be true.

Is there a word for someone who uses a misunderstood fact to support their argument?

According to Nineteen Eighty-Four by George Orwell, doublethink is:

To know and not to know, to be conscious of complete truthfulness while telling carefully constructed lies, to hold simultaneously two opinions which cancelled out, knowing them to be contradictory and believing in both of them, to use logic against logic, to repudiate morality while laying claim to it, to believe that democracy was impossible and that the Party was the guardian of democracy, to forget whatever it was necessary to forget, then to draw it back into memory again at the moment when it was needed, and then promptly to forget it again, and above all, to apply the same process to the process itself—that was the ultimate subtlety: consciously to induce unconsciousness, and then, once again, to become unconscious of the act of hypnosis you had just performed. Even to understand the word—doublethink—involved the use of doublethink.

The power of holding two contradictory beliefs in one's mind simultaneously, and accepting both of them… To tell deliberate lies while genuinely believing in them, to forget any fact that has become inconvenient, and then, when it becomes necessary again, to draw it back from oblivion for just as long as it is needed, to deny the existence of objective reality and all the while to take account of the reality which one denies—all this is indispensably necessary. Even in using the word doublethink it is necessary to exercise doublethink. For by using the word one admits that one is tampering with reality; by a fresh act of doublethink one erases this knowledge; and so on indefinitely, with the lie always one leap ahead of the truth.

E.g. North Korean citizens, think, that they live in a free country and at the same time, they accept the dictatorship of Kim. Thus, they are unaware of the fact that they are free in captivity. This is a peculiar and real-life example of Doublethink.

Wikipedia article sum it up as: https://en.wikipedia.org/wiki/Doublethink

Doublethink is the act of simultaneously accepting two mutually contradictory beliefs as correct, often in distinct social contexts. Doublethink is related to, but differs from, hypocrisy and neutrality. Also related is cognitive dissonance¹, in which contradictory beliefs cause conflict in one's mind. Doublethink is notable due to a lack of cognitive dissonance—thus the person is completely unaware of any conflict or contradiction.

If you read the same book carefully, George Orwell coined a word and a phrase to represent doublethink in simple words. They can be used in same sense.

thoughtcrime

and

two plus two equals five (2 + 2 = 5)

Other closely related phrase which became very famous during 2017 in U.S. is,

Alternative facts

Which means, "Alternative facts are not facts, but falsehoods."

i.e. we are forced to believe [something] which is not true.

This phrase is very much consider as an Orwellian². There is a Wikipedia entry on it: https://en.wikipedia.org/wiki/Alternative_facts

_{1: Cognitive dissonance differs from doublethink. In cognitive dissonance, contradicting belief causes conflict in the mind, whereas, in doublethink, we learn to accept contradicting ideas.}

_{2: Refers to George Orwell, author of the book Nineteen Eighty-Four}

"Deluded person" would be the closest term I can think of to your meaning.

While this doesn't strictly specify that the false belief is due to a misunderstanding of facts, it does mean a rather dogged false belief. This would not be the clinical definition--which discounts faulty information--but a quick Google search or corpus search turns up uses for people who simply misunderstand something but are resistant to being corrected.

If you, however, you meant someone who knows a particular fact is popularly misunderstood and is taking advantage of this to promote an idea, exploiting this misunderstanding, this would be 'intellectually dishonest' and a person who employs such clever, but false reasoning could well be described as a 'sophist'.

i like misguided- but white nationalists do this all the time. while i believe they have been severely misguided, i feel like this word even with its modifying adverb, isn't powerful enough.

off-base (in American) US

2. Slang taking a position or attitude that is unsound or in error

phrase [usually verb-link PHRASE] If you say that someone is off base,
you mean that they are wrong. [mainly US, informal] Am I wrong? Am I
way off base? Because I want you to set me straight if you think I'm
wrong.
https://www.collinsdictionary.com/us/dictionary/english/off-base

off base Incorrect; mistaken or misinformed; not aligned with reality.
An allusion to baseball, in which baserunners are only "safe" if their
foot is touching a "base." Hyphenated if used before a noun. Primarily
heard in US. Your arguments would be compelling if your entire premise
wasn't completely off base. You need to stop making these off-base
assumptions about how the process works.
https://idioms.thefreedictionary.com/off+base

i feel like the baseball analogy really brings it home for me. (ha). but seriously, they think they're on, but the ump says they're out !

**wait, now i feel like i answered this incorrectly because i was distracted by other answers. knowing all the facts and still believe a falsehood to be true. that's like, me knowing that cigarettes are proven to have negative health impacts, but i smoke organic american spirits and think i'm above empirical statistical facts.

i think what you are referring to may be cognitive dissonance.

Definition Psychologist Leon Festinger first proposed a theory of
cognitive dissonance centered on how people try to reach internal
consistency. He suggested that people have an inner need to ensure
that their beliefs and behaviors are consistent. Inconsistent or
conflicting beliefs lead to disharmony, which people strive to avoid.

In his 1957 book, A Theory of Cognitive Dissonance, Festinger
explained, "Cognitive dissonance can be seen as an antecedent
condition which leads to activity oriented toward dissonance reduction
just as hunger leads toward activity oriented toward hunger reduction.
It is a very different motivation from what psychologists are used to
dealing with but, as we shall see, nonetheless powerful."
https://www.verywellmind.com/what-is-cognitive-dissonance-2795012

.

cognitive dissonance noun. (psychology)
an uncomfortable mental state
resulting from conflicting cognitions; usually resolved by changing
some of the cognitions https://www.dictionary.com/browse/cognitive-dissonance

one more:

Cognitive dissonance refers to a situation involving conflicting
attitudes, beliefs or behaviors. This produces a feeling of mental
discomfort leading to an alteration in one of the attitudes, beliefs
or behaviors to reduce the discomfort and restore balance.
https://www.simplypsychology.org/cognitive-dissonance.html

I've heard "solecist" used to describe a person who does that.

Definition of solecism
1 : an ungrammatical combination of words in a sentence
also : a minor blunder in speech
2 : something deviating from the proper, normal, or accepted order
3 : a breach of etiquette or decorum
Merriam-Webster

As an example, M-W uses a quote from a New Yorker article: "In 1964, Petula Clark had a hit with “Downtown.” But the songwriter Tony Hatch has said he was inspired by Times Square—a solecism forgivable from a Brit. In native parlance, the word denotes Manhattan below Fourteenth Street."

M-W also includes the derivation:

The city of Soloi had a reputation for bad grammar. Located in Cilicia, an ancient coastal nation in Asia Minor, it was populated by Athenian colonists called soloikos (literally "inhabitant of Soloi"). According to historians, the colonists of Soloi allowed their native Athenian Greek to be corrupted and they fell to using words incorrectly. As a result, soloikos gained a new meaning: "speaking incorrectly." The Greeks used that sense as the basis of soloikismos, meaning "an ungrammatical combination of words." That root in turn gave rise to the Latin soloecismus, the direct ancestor of the English word solecism. Nowadays, solecism can refer to social blunders as well as sloppy syntax.

It sounds like "double talk" (sometimes "doublespeak"). They are more often used with deceptive intention, but can be applied in the example you give.

They don't understand the truth in the matter, so they use half-truths
to support their "double talk".

https://www.thefreedictionary.com/double+talk

https://idioms.thefreedictionary.com/doublespeak

https://www.thefreedictionary.com/doublespeak