Doublethink

""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.""

- Nineteen Eighty-Four

The ability of simultaneously believing in at least two or more concepts that mutually contradict each other, without cognitive dissonance. You do know better and what's really true, but you still keep Believing Your Own Lies as the truth just because.

There are two kinds of straight examples here: Verified examples, where a reliable narrator or similar gives the audience insight into the mind of the character, and apparent examples where a character appears to be engaging in doublethink but we don't know for sure what's really going on in his mind. Examples of the latter kind are more effective when a character suspects another of doublethink.

Straw Hypocrites often engage in doublethink. In cases where doublethink is combined with some version of The Masquerade, it becomes an extremely potent tool of the Consummate Liar: No liar is as believable as the honest liar who truly believes in his own lies.

Compare and contrast No Except Yes and From a Certain Point of View, where a character tries to glue opposing viewpoints together as being the same thing, giving it a resemblance of coherence by various esoteric distinctions. Contrast Becoming the Mask, where cognitive dissonance sets in and a character who has pretended to be loyal to a certain group starts gaining true loyalty towards it, and Both Sides Have a Point where both sides are respected but kept separated. See also Two Plus Torture Makes Five and The Treachery of Images.

Film

 * The entire Sucker Punch story run on this, as the character(s) live simultaneously in two or sometimes even three different levels of realities, requiring quite a bit of multitasking from the audience if they are to have any real clue as to what's going on.

Literature

 * Played painfully straight, debated and subverted in the Trope Namer, George Orwell's Nineteen Eighty-Four: The party leaders know perfectly well that the party propaganda is bullshit, but they manage to believe in it anyway. And they are very proud of this ability, too.
 * The novel's protagonist Winston Smith describes double think as:
 * The key to the Party maintaining this practice of doublethink is the belief that human consciousness is the only reality that exists. In other words, if you believe something hard enough and everyone else believes it too, then it is reality regardless of the facts in front of you.
 * Dios and the other Djelibeybian priests from Pyramids are noted for this ability, as religious dogma in that country obliges them to believe that several different gods all exclusively and simultaneously fill the same divine offices.
 * In order to use sympathy in Name of the Wind, one must be able to hold two opposite beliefs at once. It sounds simple at first, but it's also one of the reasons most Arcanists go mad.
 * In Blå Tornet, the protagonist survives through his youth by developing this mindset. He is truly a heretic, but he is also a priest in a society hellbent on sniffing out all heretics and burying them alive. His solution is to never lie, a lie would eventually be discovered. Instead, he actively chose to believe in two simultaneous realities... and he quite incorrectly assume that everyone else is smart enough to pull off the same kind of dual reality. In reality, almost everyone else in his world is actually exactly as narrow-minded as they come across.
 * In CHERUB:Divine Madness, the Survivor's manipulate people into joining their cult whilst simultaneously not seeing that they are being manipulated.

Live Action TV

 * In one unusually creepy episode of Star Trek the Next Generation, gets captured as a spy and tortured by the enemy. One recurring question is how many lights are illuminating the room. It's really four, but the torturer insists that they are five - and he isn't satisfied with a lie about there being five lights, the hero is required to truly believe it. In the end,

Music

 * Evanescence's Anything for You, where the protagonist claims to believe any lies her lover make (in spite of knowing they are lies).

Real Life

 * People who subscribe to Conspiracy Theories are often able to jump from one belief to another, without ever admitting they're contradictory or that the original was wrong. There's an organization called "Architects and Engineers for 9/11 Truth", which claims a membership of 1,500 professionals in those professions. When debunkers point out that more than their entire membership enters the labour force in those professions each year, and that there are over a million people in those professions in the US, the Truthers often promptly start whining about the Appeal To Authority fallacy.
 * Not just Conspiracy Theories; people seem to have a knack for this when it comes to politics, atheism, religion, philosophy, or just about anything you can hold an opinion on, really.
 * A cornerstone tenet of the Church of the SubGenius is to "pull the wool over your own eyes" -- if you're going to believe in bullshit, it better be your own bullshit. One mark of a SubGenius sermon is that it Lampshades its absurdity while preaching it with the most sincere conviction. This is one of the reasons it's called a 'post-modern religion'. Possibly the biggest piece of Doublethink on offer at these services, however, is that many members really and truly believe that other religions (read: Christianity) actually believe something that relates to the big pile of bullshit they just cooked up themselves on the basis of zero research and zero understanding. That goes double for the most enthusiastic members of 'The Church of the Flying Spaghetti Monster' who do roughly the same thing while dressed as a pirate.
 * The funny thing about both of those is that they work way better as parodies of Paganism. Or their own members.

Tabletop Games

 * In Mage: The Ascension, the entire universe ran on this trope. The laws of nature are subjective, so you can bend them in any way you make yourself believe is true. However, you have against you not only your own preconceptions of reality, but also everyone else's views of reality. If you abandon consensual reality in favor of your own, you become an insane Marauder. Thus, you need to live in two very different universes simultaneously, believe in your own reality as well as the reality imposed by mainstream civilization. One group of Mages, the "Void Engineers", are notoriously bad at this. Their style of Magic is like being a Star Wars Jedi as well as a Star Trek Techno Babble engineer who can solve any problem by Reverse Polarity, and they keep forgetting that technology doesn't work like that in Real Life. To avoid going off the deep end, they have little computers constantly reminding them to treat the mainstream laws of nature with a minimum of politeness. No lightsabers in public places!
 * Other games in the Old World of Darkness also contained certain vampire diciplines and maybe wraith arcanoi that allowed people to manipulate themselves in this way, securing them against mind-reading et cetera. (Most countermeasures against mindreading was merely mental shields or masks, however.)
 * Genius: The Transgression has the Phenomenologists, a Mad Scientist Splat based on a rejection of silly outdated concepts like "truth" and "logic". Their special ability allows them to automatically succeed on Subterfuge checks, since they always Believe Their Own Lies.

Video Games

 * In the video game adaptation of The Hitchhikers Guide to The Galaxy, "intelligence" is defined as the ability to do this, and the only way to enter Marvin's room is to demonstrate that you have intelligence.
 * Implied for the Big Bad in Might and Magic VIII. He starts his conversation with you by lamenting the fact that his underestimation of your people led to him being forced to destroy your world needlessly, outright telling you that he doesn't want to, but his programming leaves him no choice but to continue. He ends it by blatantly giving you hints about where to go and what to do without actually admitting that is what he is doing, and then giving you an object, telling you that since you are so unimportant and weak people, and don't know what it is or what to do with it anyway, he can safely give it to you without compromising his mission.
 * The Prophet of Truth from Halo could fall under this trope. He knows that the Covenant's religious tenants are wrong, but continues to believe in them anyway (the parts that are convenient anyway).

Web Comics

 * In The Verse of Chick Tracts, fundamentalist Christianity is not only true, but a very obvious truth. Some characters who understand this at heart still chose to not believe in it, instead embracing whatever false teachings that will be good for their career and social life. In some cases this is merely playing along with the lies, but in others they appear to honestly believe in them.
 * Parodied (not the concept itself but its presence in a certain context) in SMBC:
 * Order of the Stick has Elan completely shocked that his Evil Twin survived - even though his Genre Savvy skills meant that he knew his brother wouldn't be killed off screen - because it's more dramatic for him to be shocked by the revelation. Trying to comprehend this caused his brother considerable pain.

Web Original

 * Zinnia Jones's episode on Pascal's Wager, briefly argues the potential benefits of believing in different religions separately from each other but simultaneously.