A first-order predicate logic
1, predicate, function, quantifiers
Set a1, a2, ..., an represent individual subject, A represents their attributes, state or relationship, the expression
A(a1, a2, …, an)
In it represents a predicate logic (atom) Proposition. For example,
(1) prime number (2), it means that the proposition "2 is a prime number."
(2) a good friend (John Doe), he says that the proposition "Joe Smith and John Doe are good friends."
In general, expression
P(x1,x2,…,xn)
Referred to as n-ary predicate in predicate logic. Wherein P is a predicate symbols, title words, representative of a determined characteristic or relationship (name). x1, x2, ..., xn variables called predicate or items generally indicates individual.
A collection of individual arguments range called individual domains (Domain or discussed) and taking care of everything called the ACFTU individual domain.
In order to express the correspondence between individuals, we introduce the concepts and notation are usually a function of mathematics. We e.g. Father father of x with (x), is represented by SUM (x, y) and the number of x and y, in general, we use the following form:
f(x1,x2,…,xn)
Argument represents the individual x1, x2, ..., xn corresponding individual y, and n-called individual function, the function referred to (or letter word, letter word naming formula). Where f is a function symbol, with the function of the ability to express concepts and notation, predicate it stronger.
For example, we use the Doctor (father (Li)) said, "Mike's father is a doctor," with E (sq (x), y)) means "square x equals y".
After we agreed with capital letters as predicate symbols, lowercase letters f, g, h, etc. represents a function symbol, lowercase letters x, y, z, etc., as individual argument symbols, lowercase letters a, b, c, etc., as individual Chang yuan symbol.
We "all", "all", "any one", "all", "all" and other words collectively referred to as the universal quantifier , denoted by ∀x; the "existence", "somewhat," "at least one", " some "and other words collectively referred to as the existential quantifier , denoted ∃ x.
Where M (x) denotes "x is human", N (x) indicates "x has a name", the formula can be read as "for any x, if x is a human, then x has a name." Here the individual field is taken as the total subject the whole domain.
If the individual field is taken to be a collection of human beings, the proposition can be expressed as
Similarly, we can put the proposition "there is not an even integer" is expressed as
Wherein G (x) denotes "x is an integer of", E (x) denotes "x is an even number." This formula can be read as "the presence of x, x is an integer and x is not even."
Argument different individuals, individuals may have different domains. In order to convenience and consistency, we express our proposition predicates, general overall take ACFTU individual domains, then take the way to use the qualifying predicate to point out individual domain of each individual argument. Specifically, there are the following two:
(1) universal quantifier, the predicate is defined as a member is added, i.e. ∀x (P (x) → ... ) before implication.
(2) the existence quantifier as a quantifier defining the conjuncts added, i.e. ∃x (P (x) ∧ ... ).
Where P (x) is defined predicate. Let us give a few examples.
Example 5.1 greatest integer not exist, we can translate it to 
There is not a integer x, y are integers better than all the big
or
An arbitrary integer x, there is always an integer greater than x and y
Example 5.2 for all natural numbers, both x + y> x 
Example 5.3 Some people allergic to certain foods
2, predicate formula
Definition 1
(1) and the individual element is often items are individual arguments.
(2) Let f be a function of n-ary symbols, if t1, t2, ..., tn are terms, then f (t1, t2, ..., tn) is a term.
(3) Only limited use (1), (2) a symbol string item is obtained.
Definition 2 Let P n-predicate symbols, t1, t2, ..., tn for the item, then P (t1, t2, ..., tn) called atomic predicate formulas, referred atom or atomic formula.
From the atomic predicate formula, by proposition connectives and quantifiers, you can form a compound predicate formula. Here we give a strict definition of predicate formulas, which generated rule predicate formula.
Definition 3
(1) atomic formula predicate formula.
(2) if A, B are predicate formula, A, A∧B, A∨B, A → B, A↔B, ∀xA, ∃xA formula is the predicate.
(3) only a limited application step (1), (2) the resulting equation is Equation predicate.
Defined by the term, as t1, t2, ..., tn are all individual constants element, the resulting formula is atomic predicates atoms propositional formula (propositional symbol). So, all propositional formulas are also predicate formula. Predicate predicate logic formulas also referred Suitable (type) formula, referred to as Wff.
Immediately after the quantifier is quantifiers (i.e., instructions) called the predicate formula quantifier Scope. For example:
(. 1) ∀xP (X).
(2) ∀x (H (x ) → G (x, y)).
(3) ∃xA (x) ∧B (x).
Wherein (1) P (x) of the Scope ∀x, (2) H (x) → G (x , y) of the Scope ∀x, in (3) A (x) of Scope of ∃x, but the B (x) is not ∃x Scope.
After the variable element such as quantifiers ∀x, ∃y the x, y guidance quantifier called arguments (argument or action), and guide variable domain in a quantifier administer the quantifier same argument, said element argument of the constraint , other arguments (if any) called free argument , such as (2) x is (3) a (x) is constrained argument x, and y arguments of freedom, argument is a constraint, but B (x) is a free argument x. For example (3), an argument can appear in a constraint equation, but also appear freely, but in order to avoid confusion, typically by renaming rule, such a formula in a form occurs in only one argument.
Arguments renamed constraint rules are as follows:
(1) the need for arguments renamed, change the argument should also appear in all its jurisdiction quantifier domain.
(2) must sign the new argument is not originally within the jurisdiction of quantifiers, the best formula is also not appeared. E.g. formula ∀xP (x) ∧Q (x) can be changed ∀yP (y) ∧Q (x) , but both have the same meaning.
Before adding the predicate quantifiers, referred to the respective individual predicate arguments are quantized,E.g. ∀xA x (x) is the quantized, ∃yB (y) where y is quantized. If all the individual arguments of a predicate are quantified, then this becomes a predicate proposition. For example, let P (x) denotes "x is a prime number", the ∀xP (x), ∃xP (x) are on the proposition. So we have two methods predicate propositions from (i.e. propositional functions): one is to the individual predicates becomes Yuan into individual constants element, another individual is to predicate arguments are all quantized.
Incidentally, only the individual arguments quantized first-order predicate verb referred to, if not only the individual arguments are quantized, but also the function symbol is quantized so called second order predicate verb. This book covers only the first-order predicate.
The above concept on quantization can also be extended to predicate formula. So, we can say that if all the individual variables in a formula have been quantified, or all arguments are bound argument (or no free variables), then this formula is a proposition. In particular, we call ∀xA (x) is a universal proposition, ∃xA (x) for the special, said proposition. For both proposition, when the domain is a finite set of individuals (with n elements), we have the following equivalent form:
∀xA (X) ⇔A (A1) ∧A (A2) ... ∧ - ∧A (AN)
∃xA (x) ⇔A (a1) ∨A (a2) ∨ ... ∨A (an)
These two formulas can be extended to an unlimited number of individual field to be set.
Definition 4 Let A be the form of predicate formulas:
B1∧B2∧…∧Bn
Wherein Bi (i = 1,2, ..., n) of the form L1∨L2∨ ... ∨Lm, Lj (j = 1,2, ..., m) of the formula or a negative atom, A is called the conjunctive normal.
For example:
(P (X) ∨Q (Y)) ∧ - (Ya P (x) ∨Q (y) ∨R (x, y)) ∧ ( Ya Q (y) ∨ Ya R (x, y))
is a conjunctive normal.
5 defined set of propositional formulas A is of the form:
B1∨B2∨…∨Bn
Wherein Bi (i = 1,2, ..., n) of the form L1∧L2∧ ... ∧Lm, Lj (j = 1,2, ..., m) of the formula or a negative atom, A is called the disjunctive normal form.
For example:
(P (X) ∧ - Ya Q (y) ∧R (x, y)) ∨ ( Ya P (x) ∧Q (y) ) ∨ ( Ya P (x) ∧R (x, y))
is a disjunctive normal form.