How do increases to INT affect languages known.

[Jelatinator]

I've seen some old posts on this subject but I couldn't find any FAQ or any dev's weighing in on this topic. There seems to be arguments from two sides and I could see the reasoning for both. I've also included some CRB snippets that add to the confusion. I know there are some situations in Pathfinder where similar things happen, but lets try to keep the discussion to only Starfinder.

1. Increased INT is retroactive and gives you additional languages known.

2. Increased INT is not retroactive and the only way past first level to gain new languages is through putting ranks into the culture skill.

CRB 21:
The number of bonus languages your character knows at the ***start*** of the game. Even if this modifier is a penalty, your character can still use her starting languages unless her Intelligence score is lower than 3. See page 40 for more about languages.

CRB 26:
If an ability score increase results in a change to an ability modifier, don’t forget to adjust any statistics that rely on that modifier, such as attack bonuses, saving throws, total skill bonuses, Resolve Points, Stamina Points, and the DCs of class features and spells. Note that ability score increases are effective ***retroactively;*** when your character’s ability score increases, it increases his total number of ability-based statistics—things like Resolve Points, Stamina Points, or skill ranks—as if he had the higher value at previous levels as well.

CRB 40:
A character ***begins*** play speaking and reading Common, her racial tongue if any), and the language of her home planet (if any). She can also choose a number of bonus languages equal to her Intelligence bonus from the lists below.

[BigNorseWolf]

Pathfinder used the exact same language and the increase was retroactive. The idea is to be able to make a character or look at a character sheet without having to take your socks off and figure out what someone's int was 12 levels ago. Instead you just add up everything on the sheet in front of you.

Linky

Intelligence: If my Intelligence modifier increases, can I select another bonus language?

Yes. For example, if your Int is 13 and you reach level 4 and apply your ability score increase to Int, this increases your Int bonus from +1 to +2, which grants you another bonus language.
Technically, Int-enhancing items such as a headband of vast intelligence should grant a specific language (in the same way they do for skill ranks).

If you are at the start of the game, then your intelligence modifer adds to your languages known.

A 5th level character is not at the start of the game

Therefore his intelligence modifier at 5 doesn't matter.

If A then B
Not A

Therefore not B

This is Denying the Antecedent. Which is kind of annoying, because as a social convention if you specify something you imply, but not say, that the specificity is important. If you have a sign in a bar that says "no purple dragons" that suggests the ban is specific to purple dragons, so green dragons are OK.

But logically the patronage of green dragons in that bar is unknown, not allowed.

[Garretmander]

Yep, it's retroactive for that exact reason. Ease of use. Also known as the reason negative levels no longer remove feats/skill points/spells known and hit dice like they used to in the distant past.