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First-order logic formula
First-order logic and first-order theory
First-order logical reasoning theory
Discrete mathematics first-order logic
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Describing a mathematical theory with first-order logic will first involve the objects discussed in this theory, the functions defined on these objects, and the relations or properties between these objects. The object discussed in mathematical theory is called individual, and the non-empty set composed of individuals is called universe or individual domain. According to the definition in general mathematics, N-tuple function is the mapping from the set of all N-tuples of an individual in universe A to A. N-tuple (α 1, α 2, ..., α n) is mapped to the individual in A, which is expressed as F (α 1, α 2, ..., α n). Function increases personal expression. People also consider which N-tuples in universe A satisfy the relation R, that is, which N-tuples (α 1, α2, …, αn) in A make R(α 1, α2, …, αn) true. R(α 1, α2, …, αn) at this time is a proposition.
In all kinds of relations, equal relations are often used. Because it is often necessary to know whether the expressions of different individuals refer to the same object. For example, whether 3+3 and 2×3 represent the same number.
Propositional conjunctions can be used to form more complex relationships or propositions. It is necessary to introduce quantifiers when describing the properties of some infinite objects. For example, when "for any natural number, there is a prime number greater than it", the quantifiers "all individuals" and "existing individuals" are introduced to form a more complicated relationship or proposition through quantifiers. "All individuals in the universe" is called the universal quantifier, which forms the proposition that "all individuals in the universe have certain properties", which is true when all individuals in the universe have this property, otherwise it is false. "Individuals in the universe" is called existential quantifier, and the proposition that "individuals in the universe have certain properties" is true when some individuals in the universe have such properties, otherwise it is false.
In "all individuals" and "existing individuals", quantifiers are added to individuals in the universe, which is called first-order quantifiers. Quantifiers used in first-order logic are limited to first-order quantifiers. All functions, existential functions, all relations and existential relations are second-order quantifiers. In addition, there are high-order quantifiers. Correspondingly, there are second-order logic, higher-order logic and so on.
According to the general principle of establishing formal system (see logical calculus), the formal system of first-order logic should include its language, that is, first-order language, as well as logical axioms and reasoning rules.
The symbols of first-order languages include the following categories.
(1) single variable x, y, z, ...
② Function symbol (indicating function)? ，g，h，…； Individual symbols (representing individuals in the universe) α, B, с, …; And predicates (indicating relationship) p, q, r, ... Among them, there is a binary predicate =, which is called equation (indicating identity relationship).
(3) Propositional conjunctions ∧, ∨, →, quantifier first-order logic.
(existential quantifier), first (full name quantifier).
Words such as ① and ③ are called logical symbols, and other symbols, namely ② symbols other than these words are called non-logical symbols.
Inductive definition of terms and formulas of first-order languages is also called formation rules. Definition of project:
(1) independent variables and individual symbols are terms.
② If t 1, t2, …, tn are terms,? It is an n-ary function symbol. (t 1, t2, …, tn) is an item.
The formula can be defined as:
(1) If t 1, t2, …, tn is an item and p is an n-ary predicate symbol, then p(t 1, t2, …, tn) is a formula, also called an atomic formula.
② If A is a formula, then A is a formula; If a and b are formulas, then A∧B, A∨B, A→B and a ∨ b are formulas.
(3) If A is a formula, first-order logic
XA, XA is a formula.
If the parameter x appears in Formula A, it looks like first-order logic.
The part of xB or xB is called the constraint occurrence of X in A; Otherwise, it is called the free appearance of X in A, such as in the first-order logic of formula x=0∨.
x(x & gt； 0), the first x is a free reference, and the second and third x are constrained references. A formula without free parameters is called a closed formula.
Predicate calculus, as a formal system, can specify its interpretation. Given a universe, individual symbols, function symbols and predicates appearing in predicate calculus are interpreted as individuals in the universe and functions and relationships defined in this universe in turn. This universe and its explanation of formal symbols in predicate calculus are called a structure or model of this calculus. From the interpretation of single symbol and function symbol, it can be seen that a term can be interpreted as a compound function and refers to an individual. The atomic molecular formula P (T 1, T2, ... TN) is interpreted as the individual referred to by T 1, T2, ..., TN satisfies the n-ary relation p. If formula A(x) expresses the relation, then xA(x) is interpreted as all individuals in the universe satisfy the relation A, the first-order logic.
XA(x) is interpreted as the existence of individuals satisfying the relation A in the universe.
The inference rules of predicate calculus can be stipulated as follows: first-order logic
The logical axioms of predicate calculus state the nature of logical symbols, which can be divided into three categories:
(1) Propositional axiom is a formula obtained by changing the proposition appearing in tautology (see propositional logic) into any formula in predicate calculus in Yuan Dynasty;
② First-order logic of identity axiom x=x and equality axiom
③ Substitution axiom Axα→ First-order logic
XA and xA→Axα, where Axα represents all degrees of freedom of x in modern formula a. ..
The axiom of predicate calculus, that is, logical axiom, does not define specific functions or relationships, but only deals with the general properties of logical terms. In other words, its interpretation of individual symbols, functional symbols and predicates can be any individual, function and relationship in any universe. This abstract nature of predicate calculus is very important to the development of model theory in recent years.
Under the framework of predicate calculus, the formal system of different mathematical theories can be obtained by expressing mathematical axioms in formal language (not necessarily completely). This formal axiom describes the properties of some specific illogical symbols and is called illogical axiom. For example:
There is only one non-logical symbolic binary predicate ≤ 1 in the formal system of total order theory. In addition to logical axioms, it also has illogical axioms: ① X ≤ Y ∧ Y ≤ Z → X ≤ Z; ②x≤y∧y≤x→x = y； ③x≤x； ④ x ≤ y ∨ y ≤ X. The set of natural numbers and its order relation is a model of total order theory.
There are only two illogical symbols in the formal system of group theory: individual symbol 1 and binary function symbol. Its illogical axioms are: ① x (y z) = (x y) z; ②x 1 = x； 1x = x； ③ First-order logic
Y(x y= 1∧y x= 1). Any group is its model.
The illogical symbols in the formal system of number theory are: single symbol 0, unary function symbol S and two binary function symbols+sum. The axioms of number theory (or piano arithmetic) are: ① s(x)=0, ②s(x)=s(y)→x =y, ③x+0=x, ④ x+s(x)=s(x+y), ⑥ X.0 = 0, ⑥ X.
first order logic
first order logic
The operation of natural numbers is one of its models.
It is stated that in a first-order language, all formal theorems (see logical calculus) derived from logical axioms, illogical axioms and inference rules are called first-order theories and denoted as T. In order to distinguish different first-order theories T, it is enough to point out illogical symbols and axioms in T language. Any first-order theory contains predicate calculus as its subsystem.
A formula that is true in any model of predicate calculus is called a formula that is always true or valid. For example, the formula A(x, y)∨ A(x, y) is an effective formula, but x ≤ y ≤ y ≤ x is not. Because in the fully ordered structure, the explanation of this formula holds for any value of x and y in a single domain. In a semi-ordered structure, for example, the universe of the structure is the set of all subsets of a set, and ≤ is interpreted as the inclusion relation of the set, then when X and Y take any two subsets, the explanation of the above formula does not hold.
Intuitively, logical theorems should hold true in all possible worlds. In a certain sense, predicate calculus satisfies this property. It can be verified that the axioms of predicate calculus are all valid, and if the assumptions of its reasoning rules are established, the conclusion is also established. Therefore, all theorems of predicate calculus are valid. This property is called the validity or reliability of predicate calculus. On the contrary, any effective formula must be a theorem of predicate calculus. This is the famous Godel completeness theorem. Proved by K. Godel in 1930.
Representing a with├a is a formal theorem of predicate calculus, that is, A is a theorem in the system. Reliability and completeness describe the properties of the whole formal system, which is a theorem about the system, also known as meta-theorem. The essence of formal system is one of the main research objects of mathematical logic.
It is easy to infer the reliability and completeness of the first-order theory from the validity and completeness of predicate calculus. The structure that makes all axioms of the first-order theory T true is called the model of T. If a formula A of T is valid in any model of T, it is said that A is valid in T, and the theorem that A is T is written as t ├ a, then the reliability and completeness of T can be expressed as T├ A, and the necessary and sufficient condition is T δ A.
If there is no A to make T├A and ├ A, then T is considered to be harmonious.
If T is harmonic, then T must have a model (generalized completeness theorem).
Shaped like first-order logic
X 1, first-order logic
X2, …, first-order logic
XnB formula is called toe-in formula, in which the first-order logic
Xi stands for first-order logic
Xj or xj, B is a formula without words. Any formula A of the first-order theory T (predicate calculus when T's illogical axiom set is empty) has a formula a', which makes T├A ┠ A┡, where A┡ is the first-order logic of the toe-in formula.
X 1, first-order logic
X2, …, first-order logic
XηB and the illogical symbols in b all appear in A, and A' is also called toe-in normal form of A. This property can be used to classify the formulas of predicate calculus or first-order theory. At this time, we only need to consider the quantifier in the toe-in paradigm as a measure of formula complexity.
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