In order to study the internal logical relationship of simple propositional sentences, we need to decompose simple propositions and use single words and predicates.

Words and quantifiers are used to describe them, and the internal relationship and quantitative relationship between individuals and the whole are studied. This is predicate logic or first-order logic.

An object (subject, object, etc.) that can exist independently in an atomic proposition. In a sentence) is called a single word. And used to

Predicates describe the nature of objects or the relationship between objects.

A single word can be divided into two types, a single constant and a single variable, both of which take values in a single field.

Let d be a non-empty single field and define it

(indicating that all n individuals take values on the individual field D) The values of n elements are on {0, 1}.

A function called an n-ary propositional function or an n-ary predicate is expressed as p (x 1, x2, ..., xn). Where the individual variables x 1, x2, ..., xn ∈ d.

A predicate with 1 indicating a specific attribute or relationship is called a predicate constant.

Predicates that represent abstract or general attributes or relationships are called predicate variables.

If Tong Wang is a "three-good" student, then her academic performance must be very good.

Let S(x):x be a good student, H (x): X get good grades, A: Tong Wang,

Then the proposition is symbolized as: S(a) → H(a)

Li Xinhua is Li Lan's father, and Li Lan and Zhang San are classmates.

Let F(x, y):x is Y's father, M (x, y): X and Y are classmates, B: Li Xinhua, C: Li Lan, D: Zhang San.

Then the proposition is symbolized as: F(b, c) ∧ M(c, d).

Full name quantifier (? X): all x; Any x; X of all things; Every x;

Existential quantifier (? X): some x; There is at least one x; Some x; There is x;

Where x is called the action variable. Generally, its quantifier is added before its predicate and recorded as (? x)F(x)，(？ X)F(x). At this time, F(x) is called the category of universal quantifier and existential quantifier.

The unified individual domain is the total individual domain, and the range of individual variables in each sentence is represented by unary characteristic predicates. When this characteristic predicate is added to the propositional function, it must follow the following principles:

For the full-name quantifier (? X), characterize the characteristic predicate of its corresponding individual domain, and add it as an implied antecedent.

For the existential quantifier (? X), characterize the characteristic predicates of the corresponding individual domain and add them as conjunctions.

If p (x 1, x2, then

Expressions that meet the following conditions are called well-formed formulas /wff, or formula for short.

Given a compound G, if argument X appears in the range of quantifiers using arguments, it is said that argument X appears as a constraint argument, and argument X at this time is called a constraint argument. If the appearance of X is not restrained, it is called free appearance, and the argument X at this time is called free argument.

Let g be an arbitrary formula. If there are no individual variables in G, G is called a closed formula.

In propositional logic, every formula has its equivalent paradigm, which is a unified expression form. Paradigm plays an important role in studying the characteristics of a formula, such as eternal truth and eternal fallacy. For the formula of predicate logic, there are also paradigms, in which the toe-in paradigm is equivalent to the original formula, and other paradigms have only a weak relationship with the original formula.