2.(((Va)o(a, y))Av(a) We have observed that some variable r has relation with a quantifier(V r)or r) In order to discuss the relation between variables and quantifiers, we have the following definition Definition 2(Occurrence). An occurrence of a variable v in a formula p is bound if there is a subformula y of p containing that occurrence of v such that w begins with(Vu) or(v). An occurrence of v in p is free if it is not bound Here, we should pay attention to each occurrence in the definition Definition 3. A variable v is said to occur free in p if it has at least one free occurrence there However, if we say a variable is free if it take place free once Consider the following examples Example 3. Given a formula 1.(y(r)p(, u))Av(r)) It is obvious that r is free for the first occurrence is bound and the second is free We have some special formulas. Consider the following example Example 4. Please observe the following formulas 1.((y((Var)p(a, y))A(Va)(a)) 2.((Va)(((yR(, y))v(yT(a, y))) We can find a common feature that there is no free occurrence of any variable. Definition 4. A sentence of predicate logic is a formula with no free occurrences of any variable In the most of cases, we only discuss sentences Contrary to sentence, we have another form of special formulas Definition 5. An open formula is a formula without quantifi Consider the following example Example 5. 1. All atomic formulas: o(a), R(a, y) 2.(R(, y)vo())2. (((∀x)φ(x, y)) ∧ ψ(x)) We have observed that some variable x has relation with a quantifier (∀x) or (∃x). In order to discuss the relation between variables and quantifiers, we have the following definition. Definition 2 (Occurrence). An occurrence of a variable v in a formula φ is bound if there is a subformula ψ of φ containing that occurrence of v such that ψ begins with ((∀v) or ((∃v). An occurrence of v in φ is free if it is not bound. Here, we should pay attention to each occurrence in the definition. Definition 3. A variable v is said to occur free in φ if it has at least one free occurrence there. However, if we say a variable is free if it take place free once. Consider the following examples: Example 3. Given a formula: 1. ((∃y)((∀x)φ(x, y)) ∧ ψ(x)) It is obvious that x is free for the first occurrence is bound and the second is free. We have some special formulas. Consider the following example. Example 4. Please observe the following formulas: 1. ((∃y)((∀x)φ(x, y)) ∧ (∀z)ψ(z)) 2. ((∀x)(((∀y)R(x, y)) ∨ ((∃y)T(x, y))). 3. φ(c0, c1) We can find a common feature that there is no free occurrence of any variable. Definition 4. A sentence of predicate logic is a formula with no free occurrences of any variable. In the most of cases, we only discuss sentences. Contrary to sentence, we have another form of special formulas. Definition 5. An open formula is a formula without quantifiers . Consider the following example. Example 5. 1. All atomic formulas: ϕ(x), R(x, y) .... 2. (R(x, y) ∨ ϕ(x)). 3. R(c0, c1). 2