Truth tables

A truth function is a function whose domain and codomain is a Boolean set, 𝔹 = {true, false}. True and false are sometimes referred to as truth values.

That is, unary truth functions take a Boolean to a Boolean, 𝔹 -> 𝔹, while polyadic truth functions take an n-tuple of Booleans to a Boolean. In fact, all truth functions always return a single Boolean value.

taking a n-tuple of Booleans (truth values) and returns a single truth value, i.e. a single Boolean value (True or False). All truth function always return a single Boolean.

• A unary truth function takes a Boolean (1 var)

  Bool -> Bool

• A binary truth function takes a 2-tuple of Booleans (2 vars)

  (Bool, Bool) -> Bool

• A binary truth function takes a 3-tuple of Booleans (3 vars)

  (Bool, Bool, Bool) -> Bool

and so on.

where:

• v is number of varibles

• c is number of configurations

• f is number of truth functions

If we let single letters variables (e.g. p, q, r, s, t, etc.) stand for values of Boolean type (each one is either T or F), we see that

the number of truth functions is related to the number of variables, n, in this way: (2^2)^n, which is 4^n. One Boolean variable ranges over 2 values so there are 2^m possible configurations:

v is num of configurations: c = 2^v

v	c	f	arity
0	2	2	nullary	constants TRUE and FALSE, α -> 𝔹
1	2	4	unary	p: 𝔹 -> 𝔹
2	4	8	binary	⟨p: 𝔹, q: 𝔹⟩ -> 𝔹
3	8	64	ternary	⟨p: 𝔹, q: 𝔹, r: 𝔹⟩ -> 𝔹
4	16	256	quaternary	⟨p: 𝔹, q: 𝔹, r: 𝔹, s: 𝔹⟩ -> 𝔹
5	25	256	quinary
6			senary
7			septenary
8			octonary
9			novenary
10			denary
11			undenary
12			duodenary
12			tredenary

quaterdenary

An atomic logic variable, e.g. p, can have one of 2 possible values: it can either be true (T, 1) of false (F, 0). A truth function is an n-ary function from n variables to a truth value (T or F).

For any number n, there are 2^2^n possible n-ary truth functions.

One variable

¬ ∧ ∨ → ← ⇔

p p p ταυ ταυ ταυ p ¬p p∧p p∨p p→p p←p p⇔p 1 0 1 1 1 1 1 0 1 0 0 1 1 1

Two variables

← → ≡ ⇔ ⇒ ⊢

   |q←p|p←q|

p q	∧	∨	p→q	q→p	p→(p→q)	p→(q→p)
1 1	1	1	1	1	1 1 1	1 1 1	.
1 0	0	1	0	1	1 0 0	1 0 1	.
0 1	0	1	1	0	0 1 1	0 1 0	.
0 0	0	0	1	1	0 1 1	0 1 1	.
---	-	-	---	---	-------	-------	----------------------------------------------

c a 8 e b d

1000 8 1010 a 1100 c 1110 e

Three variables

─┬─┬─┬───┐0123456789abcdef │ │ │ ├───────────────────────────────────────────────────────────────────── n│4│3│ 2 │ ─├─┼─┼───┤ x│d│c│a b│F∧ ∨T ─┼─┼─┼───┼────────┬──────────────────────────────────────────────────────────── 0│0│0│0 0│00000000│11111111 1│0│0│0 1│00001111│00001111 2│0│0│1 0│00110011│00110011 3│0│0│1 1│01010101│01010101 ─┤ │ └───┼────────┴──────────────────────────────────────────────────────────── 4│0│1 0 0│ 5│0│1 0 1│ 6│0│1 1 0│ 7│0│1 1 1│ 8│1│0 0 0│ ─┤ └─────┼───────────────────────────────────────────────────────────────────── 9│1 0 0 1│ a│1 0 1 0│ b│1 0 1 1│ c│1 1 0 0│ d│1 1 0 1│ e│1 1 1 0│ f│1 1 1 1│ ─┴───────┴─────────────────────────────────────────────────────────────────────

Misc ternary formulas

(p -> q -> r) -> (p -> q) -> p -> r

p ⇒ (q --> r) ⟾ (p ⟶ q) ⇒ (p ⟶ r) 0 0 0 1 0 0 1 1 0 1 0 0 1 0 1 1 1 1 0 0 1 0 0 1 0 1 1 0 1 0 1 0 0 1 0 0 1 1 1 1 ≡ τ ≡