Antidisjunction (Pirs’s arrow, Webb’s function, Dagger’s function )
It
is designated as: A О
B or A
B .
It is mean that result is true, if false that A and B.
It is read as “not A not B” or “A Pirs’s arrow B”.
A |
B |
not A not B |
T |
T |
F |
T |
F |
F |
F |
T |
F |
F |
F |
T |
Summary
In logic formula signs of logic operations are so appreciable, as characters symbolizing partial propositions. The logic structure of the composite proposition thus is visible clearly, and the possible double meanings are eliminated by application of brackets.
The true and lie — there is all multitude, can accept which one statement variable.
If two characters, which coherent with signs of a conjunction or disjunction, implication or equivalention, be considered as statement variables, each such pair will define by itself two-place function of a propositional calculus.
Basic terms of logical algebra
Proposition is language of propositions which can be or true or false. Correspondingly always propositions have logical value T (true) or F (false).
Number 1 as T and number 0 as F use often.
Main symbols of algebra of propositions are:
propositional variables p1, p2, p3, … ;
one-placed bind
and two-placed binds
←
→ ↔;brackets ( ).
Sign of logical operation |
Other signs of logical operation |
Set of true-signs, corresponding to logical operation (for initial sets of binary operations 00; 01; 10; 11) |
Name of logical operation and binds |
To read as: |
–A |
¬A Ā A' ~A |
10 |
|
Not A |
A
|
A & B A • B A B min(A, B) |
0001 |
conjunction, logical multiply, logical “AND” |
A and B |
A
|
A + B max(A, B) |
0111 |
disjunction, logical adding, logical “OR” |
A or B |
A → B |
A
A
|
1101 |
implication, logical consequenting |
if A else B; A implicated B; A inclinate B |
A
|
A + B A ∆ B |
0110 |
sum on module 2, divisioning disjunction, divisioning “OR”, exclusive disjunction, exclusive “or”, strict disjunction, symmetric difference |
A plus B; either A, or B |
A ~ B |
A ≡ B A ↔ B A
|
1001 |
equivalention |
If and only if A then B; A when and only then, when B; A equivalent B |
A | B |
|
1110 |
anticonjunction, Sheffer’s touch |
false that A and B; Sheffer’s touch B |
|
A О B A
|
1000 |
Pirs’s arrow, antidisjunction, Webb’s function, Dagger’s function |
not A not B; Pirs’s arrow B |
B
B
B
B
B
B
B