grammar - Sentence ending じゃよ

Maruko:「おとおさんはちょっとは遠慮{えんりょ}しながら食{た}べてよ」
Dad:「いいんじゃよ」

Grandad:「そうじゃよ、みんなたくさん食べればいいよ」

Maruko is telling dad to hold back a bit while eating (because the food is a way to thank her grandad).

I don't understand whether じゃ is a positive or negative predicate. I assume it's short for じゃない and hence negative. Both options seem to lead to a logical contradiction though.

Positive:

Dad should hold back a little while eating
That's okay
That's right. Everyone should eat alot

Negative:

Dad should hold back a little while eating
That's not good
That's wrong. Everyone should eat alot

I guess if it's negative and both grandad and dad are replying to Maruko's statement then it makes sense i.e. they both agree that what Maruko said is wrong. I find these negated endings like いいんじゃない　really confusing as to whether there positive, negative, rhetorical etc.

Answer

This is called 「[役割語]{やくわりご}」 = "role language".

役割語 is often used in fiction to "help" the readers/viewers envision certain age/gender/profession/social status, etc. groups. It is mostly imaginary and it relies on the stereotypes people have about others.

(This is why I keep telling Japanese-learners to be careful with children's stories as the largest amount of 役割語 is indeed found in children's stories.)

「じゃよ」 here is clearly the 役割語 for 「だよ」 in Standard Japanese. As you know, Maruko's family lives in Shimizu, Shizuoka and they speak pretty "standard" except for the occasional use of 役割語. 「じゃよ」 is the affirmative sentence-ender that people "expect" older characters to use in fiction (if they grow up in the Japanese language). It does not really matter whether or not that is actually a common sentence-ender for older people in Shimizu in real life. This is fiction, not a documentary.

One also needs to know that many 役割語 often happen to be actual dialectal words borrowed from all over Japan. That is because 役割語 still has to make sense. You rarely create completely new words no one would be familiar with. Everything a character says must sound at least somewhat familiar.

Not sure how you end up thinking 「じゃよ」 would be negative. It is affirmative all the way (just as 「だよ」 is. You may have a tendency to let your own "translated words" get in your way of understanding.

Maruko: "You should slow down, Dad." ← in regards to how Dad is eating.

Dad: "It's Ok (for me to eat like this)." ← affirms his own action and intention to keep doing the same.

Grandad: "That's right; We should all eat a lot." ← affirms Dad's statement.

You probably thought Dad was denying Maruko's statement. He actually is but the words he actually uses are positive - "Yes, it's OK for me to eat like this!"

And then, Grandad affirms Dad's statement by saying "That's right!" = 「そうじゃよ」.