Truth Tables for the Ternary Logic With Null by Sobociński [71, p.70][79]

Internally, we represent nullable propositional formula as a Maybe of a propositional formula (in src/propositions/NullPropositions.hs):

type NullableFormula a = Maybe (PropositionalFormula a)

So, a nullable formula is either null (represented by the value Nothing) or a propositional formula p (represented by the value Just p). We write ⊤ for true and ⊥ for false.

¬
_________________
Nothing | Nothing
Just ⊥  | Just ⊤
Just ⊤  | Just ⊥


∧
_________________________
Nothing Nothing | Nothing
Nothing Just ⊥  | Just ⊥
Nothing Just ⊤  | Just ⊤
Just ⊥  Nothing | Just ⊥
Just ⊥  Just ⊥  | Just ⊥
Just ⊥  Just ⊤  | Just ⊥
Just ⊤  Nothing | Just ⊤
Just ⊤  Just ⊥  | Just ⊥
Just ⊤  Just ⊤  | Just ⊤


∨
_________________________
Nothing Nothing | Nothing
Nothing Just ⊥  | Just ⊥
Nothing Just ⊤  | Just ⊤
Just ⊥  Nothing | Just ⊥
Just ⊥  Just ⊥  | Just ⊥
Just ⊥  Just ⊤  | Just ⊤
Just ⊤  Nothing | Just ⊤
Just ⊤  Just ⊥  | Just ⊤
Just ⊤  Just ⊤  | Just ⊤


⇒
_________________________
Nothing Nothing | Nothing
Nothing Just ⊥  | Just ⊥
Nothing Just ⊤  | Just ⊤
Just ⊥  Nothing | Just ⊤
Just ⊥  Just ⊥  | Just ⊤
Just ⊥  Just ⊤  | Just ⊤
Just ⊤  Nothing | Just ⊥
Just ⊤  Just ⊥  | Just ⊥
Just ⊤  Just ⊤  | Just ⊤


⇔
_________________________
Nothing Nothing | Nothing
Nothing Just ⊥  | Just ⊥
Nothing Just ⊤  | Just ⊥
Just ⊥  Nothing | Just ⊥
Just ⊥  Just ⊥  | Just ⊤
Just ⊥  Just ⊤  | Just ⊥
Just ⊤  Nothing | Just ⊥
Just ⊤  Just ⊥  | Just ⊥
Just ⊤  Just ⊤  | Just ⊤