Analogical

Virgil - "Sorry to ruin that for you."

Logan, immediately - "You didn't ruin anything."

— Virgil and Logan, Fitting In (Hogwarts Houses!) | Sanders Sides. Analogical is the slash ship between Logan "Logic" Sanders and Virgil "Anxiety" Sanders from the Sanders Sides fandom.

Canon
Virgil and Logan are two of Thomas' sides. Virgil is a former "Dark Side", now Core Side (also referred to as "Light Side"), who embodies Thomas' anxiety, and Logan is one of the Core Sides who embodies Thomas' logical side. When Virgil was first introduced as the antagonist, Logan held a distaste towards him, believing Virgil making Thomas anxious was exclusively a bad thing. However, overtime, Logan is shown to grow relatively friendly with Virgil, especially after "ACCEPTING ANXIETY, Part 2/2: Can Anxiety Be Good?" where Virgil was formally accepted as a side.

A New Year of Lying to Myself... In song!!
In "A New Year of Lying to Myself... In Song!! | Sanders Sides", the two are shown to have an rocky start, with Logan complaining about being interrupted by the other sides, stating "Unbelievable, this was totally supposed to be [his] video.", with Virgil replying "Get over it, [he's] the least popular character and [he] know it.".

Losing My Motivation | Sanders Sides
In "Losing My Motivation | Sanders Sides", Logan accuses Virgil of being the antagonist behind Thomas not being motivated to make an video. Virgil reflects the accusation, and continues to be in the video, contributing to Logan's theory when Patton is unable to. A few back-and-worth banter is shown throughout the video.

My Personality Q&A
In "My Personality Q&A", when asked to compliment the other Sides, Virgil says, directing towards Logan, "[He] understand[s] reality better than the other guys, and that is comforting". Logan and Virgil consider themselves more grounded in reality than the others, and while Virgil is inherently anxious about that very reality, there is a mutual understanding between the two.

Am I ORIGINAL?
When Logan is shushed by Thomas, he looks at Virgil, and in response Virgil shows agreement, rolling his eyes. Here, Logan calls Virgil an defeatist.

My negative thinking
While Logan and Virgil had a full video dedicated to them arguing, ("My NEGATIVE Thinking"), Logan admits that "While [Virgil] may be wrong about a lot of things, [he] do[es]n't necessarily mind [his] company", telling Virgil that he had enjoyed the debate the two had, and adding how "The other two can bring in a whole lot of sunshine, and that can be unbearable. And [he] can't imagine having a debate with either one of [them]". In response, Virgil, though hesitant, admits that he's surprised, saying that "[he] thought [he] didn’t like [him], especially after last time when [he] called [him] a defeatist." referencing to the episode before.

In references to the episode stated above, Virgil insults Logan saying "[He] would write an angsty sonnet illustrating [his] contempt for [him] if [he] actually cared enough about what [he] were saying right now" at the beginning of the episode. A sonnet isn't a form of random poetry, it is almost exclusively a love poem. Though unknown if it was intended, Virgil told Logan he'd write him, in modern terms, an emo song about pining for someone.

The Sanders Sides 12 days of Christmas
In "The Sanders Sides 12 DAYS OF CHRISTMAS!", the only other time Logan actually sang, apart from about Crofter's, was that reworked line to make Virgil feel included. It was the word 'friend', as part of a harmony, but regardless is seen as the logic side showing an soft spot towards Virgil.

In the end card of said episode, Logan gets Virgil for his secret santa. He hands him a gift card, stating "Virgil, [he] had [him], so [he] thought an appropriate gift would be this gift card... uh, for some audiobooks that [he] could listen to on those large headphones that [he] own... to calm [himself] down in moments of excessive alarm.", Virgil, in response lets a small smile grow on him face before shutting it down immediately choosing to respond, "Oh, that's... that's alright, [he] guess[es]."

Moving on
In "MOVING ON, Part 2/2: Dealing With a Breakup | Sanders Sides", when other sides are looking back on memories and significant objects, Virgil has an anxiety attack due to the pressure of all the positive energy and energy of the mind-palace. There, Logan is the first one to notice, asking him, " Virgil, are you okay? ". Virgil nods to him for a moment, and throughout the video Logan is shown to look towards Virgils direction ever so often, assumingly to check up on how the latter is doing.

Have I grown? - Five years later
In the episode "Have I Grown? - Five Years Later | A Sanders Sides Special" when the sides are bring up how they feel of Virgil, Logan replies "We could not function without [him], as we, saw", referencing to "Accepting Anxiety: Part one", acknowledging Virgil's job as anxiety and who he is as an individual.

Quotes
" Virgil, are you okay? "

— Logan to Virgil, MOVING ON, Part 2/2: Dealing With a Breakup | Sanders Sides.

Virgil - " Can I take a stab? "

Logan - " Who gave him a knife!? "

— Losing My Motivation | Sanders Sides.

" That was dark even for me "

— Virgil in response to Logans spiral, Losing My Motivation | Sanders Sides.

Fanon
On AO3, it's the second most popular ship for both Virgil and Logan, with Prinxiety and Logicality coming up as first.

The ship between them started picking up steam after "My NEGATIVE Thinking", with fans exclaiming that the relationship is relatively healthy, with the two balancing each other and encouraging the other in the ship. Their dynamic is about understanding and patience, as seem from both of their behavior throughout the series. Logan agrees with the majority of what Virgil has to say and noticeably seems to stand up for him, stating things such as "you're right" or "he's got a point" when argument rises between the four of them. Fans of the ship appreciate and often point out how the relationship between the two are not just "you're adorable" or them looking "fondly" at each-other but it's also them saying acknowledging each other and focus on communication, which ends majority of their interaction to end with a note of "wow, that's a good point, let's come to an agreement on the issue and better Thomas", which makes Analogical stand out from the other ships in Sander Sides.

In fan-works, the two are often shown to balance each-other out and supporting their flaws, often having the trope "Opposites Attract" used. Instances of the two's dynamic written through websites such as AO3 and Wattpad include Logan supporting Virgil through panic attacks, commended him for his small victories, with Virgil, in turn, shown to also support Logan in the ways he best could, helping him out with small social interactions, telling him when the right time is to keep quiet, and adjusting things to fit Logan's liking.

Analogical is often shipped alongside Royality and Dukeceit.

Fandom
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Trivia

 * Common symbols used for this ship include: 💙🖤💜, 🌩👓, and 🤖👿.
 * Occasionally, Logan will call Virgil "Virge" as an nickname, with Virgil in return calling Logan "L", and later on "Lo".
 * Logans iconic phrase, "FALSEHOOD!" was first said during an episode with Logan and Virgil having a debate against each other, said out of frustration in response to Virgil's taunting. In a later episode (Have I Grown? - Five Years Later), Virgil looks back on this, stating "[He] also regret[s] starting Logan off on that whole "Falsehood" bit."