3. A set of sentence 2=a1,. is satisfiable if there is a structure A in which all the members of 2 are true. Such a structure is called a model of 2. If 2 has no model. If o has no model it s unsatisfiable Given a sentence. it could be true in a structure and be false in another structure. Consider the following example possible structures for L corresponding to two different interpretations of the language e are two Example 7. Consider a language L specified by R(, y) and constants co, C1, c2, ...Her 1. Let domain A consists of the natural numbers, let Ra be usual relation < and co=0, cA 1,..(VayR(, y) is true in this structure 2. Let domain A consists of the rational numbers Q=[g0, q1,. let Ra again be <,and Co=90, C1=q1,..(Va)(Vy)(R(, y)+x)(R(, x)A R(z, y)))is true in this structure 3. The sentence in 2 is not true in 1. So it can not be valid At present, we only discussed how to determine the truth of a sentence. However, formula is more general than sentence according to lecture on syntax. Now, we define the truth of a general formula as follow Definition 5. A formula p of a language L with free variables u1, ... Un is valid in a structure A for L(also written AH p if the universal closure of p, i. e, the sentence Vu1V02, -. Vans gotten by putting Vui in front of y for every free variable vi in p, is true in A The formula of language L is valid if it is valid in every structure of A Example 8. Consider the structure A=<M, (P), f(, y)), c, d>. Give these formula 2.P(x,d) 3.P(d,x) It is obvious that the truth of a formula is determined by the value of a variable. In a structure, sometimes it is true, and sometimes it is false. Once we add universal closure of quantifier before it, its truth is determined. This definition is compatible with definition of truth of a sentence Definition 6. A set> of formulas with free variables is satisfiable if there is a structure in which all of the formulas in 2 are valid Again such a structure is called a model of 2. If 2 has no models it is unsatisfiable 4 Applications In this section, we will use two examples to show how to determine whether your translation from natural language to logic language is proper/right or not3. A set of sentence Σ = {α1, . . .} is satisfiable if there is a structure A in which all the members of Σ are true. Such a structure is called a model of Σ. If Σ has no model. If σ has no model it s unsatisfiable. Given a sentence, it could be true in a structure and be false in another structure. Consider the following example. Example 7. Consider a language L specified by R(x, y) and constants c0, c1, c2, . . .. Here are two possible structures for L corresponding to two different interpretations of the language. 1. Let domain A consists of the natural numbers, let RA be usual relation <, and c A 0 = 0, cA 1 = 1, . . .. (∀x)(∃y)R(x, y) is true in this structure. 2. Let domain A consists of the rational numbers Q = {q0, q1, . . .}, let RA again be <, and c A 0 = q0, cA 1 = q1, . . .. (∀x)(∀y)(R(x, y) → (∃z)(R(x, z) ∧ R(z, y))) is true in this structure. 3. The sentence in 2 is not true in 1. So it can not be valid. At present, we only discussed how to determine the truth of a sentence. However, formula is more general than sentence according to lecture on syntax. Now, we define the truth of a general formula as follow: Definition 5. A formula φ of a language L with free variables v1, . . . , vn is valid in a structure A for L (also written A |= φ ) if the universal closure of φ, i.e., the sentence ∀v1∀v2, . . . , ∀vnφ gotten by putting ∀vi in front of φ for every free variable vi in φ, is true in A. The formula of language L is valid if it is valid in every structure of A. Example 8. Consider the structure A =< N , {P}, {f(x, y)}, {c, d} >. Give these formula: 1. P(c, x). 2. P(x, d). 3. P(d, x). It is obvious that the truth of a formula is determined by the value of a variable. In a structure, sometimes it is true, and sometimes it is false. Once we add universal closure of quantifier before it, its truth is determined. This definition is compatible with definition of truth of a sentence. Definition 6. A set Σ of formulas with free variables is satisfiable if there is a structure in which all of the formulas in Σ are valid. Again such a structure is called a model of Σ. If Σ has no models it is unsatisfiable. 4 Applications In this section, we will use two examples to show how to determine whether your translation from natural language to logic language is proper/right or not. 4