Conditional Disjunction

Conditional Disjunction
Conditional Disjunction
	; Venn diagram
Definition
Truth table
Normal forms
Disjunctive
Conjunctival
Polina Zhegalkina
Belonging to pre-complete classes
Saves 0	Yes
Saves 1	Yes
Monotone	Not
Linane	Not
Self-dual	Not

Conditional disjunction is a ternary (having 3 operands ) logical operation introduced by Alonzo Church ^[1] . The result of a conditional disjunction is similar to the result of a more general ternary conditional operation ( if o1 then o2 else o3 ), which is used in one form or another in most programming languages as one of the ways to implement branching in algorithms. For the operands p , q , and r , which determine the truth of the judgment , the value of the conditional disjunction [ p , q , r ] is determined by the formula:

[p,q,r]~\leftrightarrow ~(q\rightarrow p)\land (\neg q\rightarrow r).

{\ displaystyle [p, q, r] ~ \ leftrightarrow ~ (q \ rightarrow p) \ land (\ neg q \ rightarrow r).}

{\ displaystyle [p, q, r] ~ \ leftrightarrow ~ (q \ rightarrow p) \ land (\ neg q \ rightarrow r).}

In other words, the entry [ p , q , r ] is equivalent to the entry: “If q , then p , otherwise r ”, which can be rewritten as “ p or r , depending on q or not q ”. Thus, for any values of p , q and r, the value of [ p , q , r ] is p if q is true, and is equal to r otherwise.

In combination with constants denoting each true value, the conditional disjunction is functionally complete for classical logic . ^[2] Its truth table is as follows:

**Conditional Disjunction**
$a$ ${\ displaystyle a}$ $a$	$b$ ${\ displaystyle b}$ $b$	$c$ ${\ displaystyle c}$ $c$	$[a,b,c]$ ${\ displaystyle [a, b, c]}$ ${\ displaystyle [a, b, c]}$
0	0	0	0
0	0	one	one
0	one	0	0
0	one	one	0
one	0	0	0
one	0	one	one
one	one	0	one
one	one	one	one

In addition to conditional disjunction, there are other functionally complete ternary operations.

Notes

↑ Church, Alonzo. Introduction to Mathematical Logic. - Princeton University Press, 1956.
↑ Wesselkamper, T., “A solely sufficient operator”, Notre Dame Journal of Formal Logic , Vol. XVI, No. 1 (1975), pp. 86-88.

[1] Church, Alonzo. Introduction to Mathematical Logic. - Princeton University Press, 1956.

[2] Wesselkamper, T., “A solely sufficient operator”, Notre Dame Journal of Formal Logic , Vol. XVI, No. 1 (1975), pp. 86-88.

Conditional Disjunction
Venn diagram
Definition	$(q\rightarrow p)\land (\neg q\rightarrow r)$ ${\ displaystyle (q \ rightarrow p) \ land (\ neg q \ rightarrow r)}$ ${\ displaystyle (q \ rightarrow p) \ land (\ neg q \ rightarrow r)}$
Truth table	$(01000111)$ ${\ displaystyle (01000111)}$ ${\ displaystyle (01000111)}$
Normal forms
Disjunctive	${\overline {p}}{\overline {q}}r+p{\overline {q}}r+pq{\overline {r}}+pqr$ ${\ displaystyle {\ overline {p}} {\ overline {q}} r + p {\ overline {q}} r + pq {\ overline {r}} + pqr}$ ${\ displaystyle {\ overline {p}} {\ overline {q}} r + p {\ overline {q}} r + pq {\ overline {r}} + pqr}$
Conjunctival	$({\overline {q}}+p)(q+r)$ ${\ displaystyle ({\ overline {q}} + p) (q + r)}$ ${\ displaystyle ({\ overline {q}} + p) (q + r)}$
Polina Zhegalkina	$p\oplus qr\oplus r$ ${\ displaystyle p \ oplus qr \ oplus r}$ ${\ displaystyle p \ oplus qr \ oplus r}$
Belonging to pre-complete classes
Saves 0	Yes
Saves 1	Yes
Monotone	Not
Linane	Not
Self-dual	Not

$a$ ${\ displaystyle a}$ $a$	$b$ ${\ displaystyle b}$ $b$	$c$ ${\ displaystyle c}$ $c$	$[a,b,c]$ ${\ displaystyle [a, b, c]}$ ${\ displaystyle [a, b, c]}$
0	0	0	0
0	0	one	one
0	one	0	0
0	one	one	0
one	0	0	0
one	0	one	one
one	one	0	one
one	one	one	one

$a$ ${\ displaystyle a}$ $a$	$b$ ${\ displaystyle b}$ $b$	$c$ ${\ displaystyle c}$ $c$	$[a,b,c]$ ${\ displaystyle [a, b, c]}$ ${\ displaystyle [a, b, c]}$
0	0	0	0
0	0	one	one
0	one	0	0
0	one	one	0
one	0	0	0
one	0	one	one
one	one	0	one
one	one	one	one

Conditional Disjunction

Notes

More articles:

$a$ ${\ displaystyle a}$ $a$	$b$ ${\ displaystyle b}$ $b$	$c$ ${\ displaystyle c}$ $c$	$[a,b,c]$ ${\ displaystyle [a, b, c]}$ ${\ displaystyle [a, b, c]}$
0	0	0	0
0	0	one	one
0	one	0	0
0	one	one	0
one	0	0	0
one	0	one	one
one	one	0	one
one	one	one	one