[Haskell-cafe] FOL

[Tony Morris]

I would like to know if all 16 possible functions that accept two
boolean arguments have names in First-Order Logic. I know they have
Haskell function names (e.g. \p -> \_ -> id p, \_ -> \_ -> const True),
but I'm hoping there is a more general name that is recognised by anyone
with a feel for logic, but not necessarily Haskell.

I have listed all sixteen of these functions below using Haskell (named
a to p) along with the name of those that I am aware of having a name as
a comment.

Thanks for any tips.

{-

p q | a b c d e f g h i j k l m n o p
0 0 | 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1
0 1 | 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
1 0 | 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
1 1 | 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

-}

a :: Bool -> Bool -> Bool
a _ _ = False

b :: Bool -> Bool -> Bool -- conjunction AND
b False _ = False
b _ q = q

c :: Bool -> Bool -> Bool
c False _ = False
c _ q = not q

d :: Bool -> Bool -> Bool
d p _ = p

e :: Bool -> Bool -> Bool
e False q = q
e _ _ = False

f :: Bool -> Bool -> Bool
f _ q = q

g :: Bool -> Bool -> Bool -- exclusive disjunction XOR
g False q = q
g _ q = not q

h :: Bool -> Bool -> Bool -- disjunction OR
h False False = False
h _ _ = True

i :: Bool -> Bool -> Bool -- negation of disjunction NOR
i False False = True
i _ _ = False

j :: Bool -> Bool -> Bool -- biconditional
j False q = not q
j _ q = q

k :: Bool -> Bool -> Bool
k _ q = not q

l :: Bool -> Bool -> Bool
l False q = not q
l _ _ = True

m :: Bool -> Bool -> Bool
m p _ = not p

n :: Bool -> Bool -> Bool -- implication
n False _ = True
n _ q = q

o :: Bool -> Bool -> Bool -- negation of conjunction NAND
o False _ = True
o _ q = not q

p :: Bool -> Bool -> Bool
p _ _ = True

-- 
Tony Morris
http://tmorris.net/


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