The object considered by the proposition is called an individual

An individual is something that exists independently. Individuals can be specific, such as 5, 3, 2, and Zhang San can also be abstract, such as people.

Specific and specific individuals are called individual constants

Uncertain individuals are called individual variables

When discussing individuals, it is usually necessary to specify the scope of the individual discussion, which is called the individual domain, represented by D. It is generally assumed that D is not empty

We take all imaginable objects in the world, such as all animals, all plants The set composed of objects, all letters, all numbers, etc. is called the total individual domain, simply Called the global domain, it is the largest individual domain.

Words that express individual properties and relationships between individuals are called predicates. predicate is relation

The predicate that expresses the properties of an individual is called a 1-element predicate, which expresses the properties of n individuals. The predicate of the relationship between is called n-ary predicate.

For any n-ary predicate, in order to express the predicate and its arity at the same time, Like expressing an n-ary function, it is expressed in the form of P(x1, x2, · · · , xn).

And G(x, y) : x > y, it is a function about propositions, called propositional functions

The selection of predicate is related to the individual domain

Another way to make P(x) a proposition is to quantify the individual variables x

Words that express individual quantitative characteristics are called quantifiers

The current quantification is only performed on individuals and not on predicates, so it is called first-order predicates. word logic

The quantifier must be followed by Volume variables, such as ∀x, ∀y, · · · , ∀δ, ∃x, ∃y, · · · , ∃δ. Therefore, ∀x, ∃x is a overall. The individual variables following the quantifier are called guide variables

If all individual variables in the propositional function are quantified, we get a proposition question

The role or jurisdiction of the quantifier ∀x or ∃x is called the scope or scope of ∀x or ∃x, and the individual variable x within the scope is called the constraint variable.

To express "Zhang San's father", "the sum of the squares of two numbers", etc., we need to use functions Numbers are commonly called functions in predicate logic.

Predicate formula (predicate formula) is referred to as formula, which is the same as the understanding of proposition formula. In this way, as long as it is a correctly written symbol string or expression with clear meaning (including predicates) is the predicate formula

Corresponds to any natural number n, n-ary predicate P and n arbitrary individuals t1,t2, · · · ,tn, P(t1,t2, · · · ,tn) is a predicate formula.

A is a predicate formula, then ¬A is a predicate formula.

If A and B are predicate formulas, then A ⋆ B is a predicate formula, where ⋆ is a binary Logical connectives.

If A is a predicate formula, then ∀xA, ∃xA are predicate formulas.

The symbol string obtained by using (1)(2)(3)(4) above a finite number of times is the only predicate formula.

If A and B are predicate formulas, then A ⋆ B is a predicate formula, where ⋆ is a binary Logical connectives.

The steps to symbolize propositions in predicate logic are as follows

There are infinitely many interpretations of the predicate formula, and each interpretation (interpretation) I consists of 5 Composed of parts,

Logical Equivalence of Two Eliminating Quantifiers

A predicate formula that is true under any interpretation is called a permanent true or valid formula

There is both an interpretation of 1 and a predicate that has an interpretation of 0 The formula is called neutral or accidental

The neutral predicate formula cannot be determined within a finite number of steps; the ever-true (or ever-false) predicate formula can be determined in Determination within limited steps.

Suppose H1, H2, · · · , Hn and C are propositional formulas. If H1, H2, · · · , Hn are all true, It can be concluded that C is necessarily true, then it is said that the inference of C is derived from H1, H2, · · · ,Hn The form is valid (valid argument form), denoted as H1, H2, · · · ,Hn ⇒ C.

Suppose H1, H2, · · · ,Hn and C are propositional formulas, then the filling of H1,H2, · · · ,Hn ⇒ C The necessary condition is that H1 ∧ H2 ∧ · · · ∧ Hn → C is a permanent formula, That is, H1 ∧ H2 ∧ · · · ∧ Hn ⇒ C

The following logical implication holds: (1) ∀xA(x) ∨ ∀xB(x) ⇒ ∀x(A(x) ∨ B(x)). (2) ∃x(A(x) ∧ B(x)) ⇒ ∃xA(x) ∧ ∃xB(x).

Assume A is a predicate formula, if A = Q1x1Q2x2 · · · Qnxn(· · · B · · ·)(n ≥ 0), Where Qi is ∀ or ∃, and there is no component in B, it is called A = Q1x1Q2x2 · · · Qnxn(· · · B · · ·) is the forward normal form of A

1. Reduce the logical connectives to predicate formulas containing only ¬, ∧, ∨.

2. Use the following two equivalent expressions to move the negative connective inwards. (1) ¬∀xA(x) = ∃x¬A(x) (2) ¬∃xA(x) = ∀x¬A(x)

3. Use equivalent expressions to move all quantifiers to the front, using renaming techniques if necessary.

Assume A and B are predicate formulas. If A and B have the same value under any interpretation, Then A and B are said to be logically equivalent, denoted as A = B.

The necessary and sufficient condition for A = B is that the predicate formula A ↔ B is always true.

¬∀xA(x) = ∃x¬A(x).

¬∃xA(x) = ∀x¬A(x).

∀x(A(x) ∧ B) = ∀xA(x) ∧ B

∀x(A(x) ∨ B) = ∀xA(x) ∨ B

∃x(A(x) ∧ B) = ∃xA(x) ∧ B

∃x(A(x) ∨ B) = ∃xA(x) ∨ B.

∀x(A(x) ∧ B(x)) = ∀xA(x) ∧ ∀xB(x)

∃x(A(x) ∨ B(x)) = ∃xA(x) ∨ ∃xB(x)

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