Development course and manifestation

Augustus de morgan first discovered the following relations in propositional logic:

Not (p and q) = (non-p) or (non-q)

Not (p or q) = (non-p) and (non-q)

Augustus de Morgan's discovery influenced george boole's research on the algebraic solution of logic problems, which consolidated Augustus de Morgan's position as the discoverer of this law, although Aristotle also noticed a similar phenomenon, which was also well known by logicians in ancient Greece and the Middle Ages (quoted from Bocheński's History of Formal Logic).

The expression of this law in formal logic;

Eg(P wedge Q)= (

V shape

E.g. q)

eg(P vee Q)=(

Wedge (

E.g. q)

In set theory:

(first time to participate in b) c = a c cup b c

(a glass of b) c = a c hat b C.

In the extension of classical propositional logic, this duality still holds (that is, we can find its duality for any logical operator). Because it exists in the identity of the prescribed negative relation, people always introduce another operator of Augustus de Morgan duality as an operator. This leads to an important property of logic based on traditional logic, that is, the existence of negative paradigm: any formula is equivalent to another formula, in which negation only appears when it acts on non-logical atoms in the formula. The existence of negative normality promotes many applications, such as manipulating logic gates in digital circuit design. In formal logic, this property is a necessary condition for finding the conjunctive normal form and disjunctive normal form of formulas. Computer programmers use them to convert complex statements like IF ... and (... operational research ...) and then ... into its equivalent form; They are also commonly used in the calculation of elementary probability theory.

We define any propositional operator p (p, q, ...) based on the basic proposition p, q is:

Such as mbox^d (

eg p，

eg q，...).

This concept can be extended to logical quantifiers, for example, full-name quantifiers and existential quantifiers are dual:

For all x, P(x) equations.

Eg has x,

For example, P(x),

"P(x) holds for all x" is equivalent to "there is no x, so P(x) does not hold";

There is an x, P(x) equation.

For example, for all x,

Such as P(x).

"the existence of x makes P(x) true" is equivalent to "not all x, P(x) is not true".

In order to describe the duality of these quantifiers to De Morgan's law, a model with a few elements in its definition domain D is established, such as

D = {a，b，c}。

rule

For all x, P(x) is equal to P(a) wedge-shaped P(b) wedge-shaped P(c).

"P(x) holds for all x" is equivalent to "P(a) holds" and "P(b) holds" and "P(c) holds"

and

There is x, P(x) equivalent P(a) vee P(b) vee P(c).

The existence of X makes P(x) hold, which is equivalent to P(a) hold or P(b) hold or P(c) hold.

However, applying De Morgan's law,

Wedge wedge

For example (

E.g. P(a) vee

E.g. P(b) vee

eg P(c))

"P (a) holds and P (b) holds and P (c) holds" is equivalent to "none (P (a) holds or P (b) holds or P (c) holds)".

and

equivalent

For example (

Wedge

Such as a P(b) wedge

E.g. P(c)),

"P (a) holds" or "P (b) holds" or "P (c) holds" is equivalent to "None (P (a) does not hold, P (b) does not hold)"

Duality of quantifiers in test model.

Therefore, the duality of quantifiers can be further extended to block and diamond operators in modal logic:

Box p device

Such as diamonds

eg p，

Diamond equipment

Eg frame

eg p。

When counting, you must be careful not to repeat or miss. In order to prevent overlapping parts from being counted repeatedly, people have developed a new counting method. The basic idea of this method is to calculate the number of all objects contained in a certain content without considering overlap, and then eliminate the number of repeated calculations when counting, so that the calculation results are neither missing nor repeated. This counting method is called inclusion and exclusion principle.