1.N,≤,f(x,y)=x·y 2.Q,<,PA(x,y)=x÷y >,f4(x,y) For these simple symbols, we wan assign them many interpretations as possible as you can, which depends only on what you know Term plays a very important role to discuss the truth of a sentence when quantifies are involved Definition 2(The interpretation of ground terms). Term is a recursively defined concept. It can be interpreted also recursively 1 Each constant term c names the element c 2. if the terms ti,. tn of c name the elements tA,..., t a of A and f is an n-ary function symbol of C, then the term f(t1,., t2) names the element f(t1, .. tn)=f(t1,.,tA) of a Example 4. If we add the instance c, d into the language of last example, we can assign to element c4=1/2;d4=2/3 Example 5. Give ground terms f(a, d), f(d, f(d, d)). They name elements of the A as follows 1.f(,d)4=0;f(d,f(d,d)4=1 2.f(c,d)4=3/2;f(d,f(d,d)4=2/ 9.f(c,d)4=2;f(d,f(d,d)4 3 Semantics Given a sentence, its semantics depends on the structure. Here, we should pay attention. For convenience, we expand our language by adding enough constant symbols. After interpretation every element in domain is named by a constant. With the help of these constants, we can"use every element in domain. Specially, when we add a"new"constant into language, before we fix its interpretation or meaning, it can be any element in domain as your wish. However, if you fix its interpretation, you can never change it Definition 3(Truth). The truth of a sentence p of l in a structure A in which every a E A is named by a ground term of L is defined by induction 1.For an atomic sentence R(t1, .. tn), AF R(t1,., tn) if and only if R (t,.,tA)1. N , ≤, f A(x, y) = x · y. 2. Q, <, f A(x, y) = x ÷ y. 3. Z, >, f A(x, y) = x − y. For these simple symbols, we wan assign them many interpretations as possible as you can, which depends only on what you know. Term plays a very important role to discuss the truth of a sentence when quantifies are involved. Definition 2 (The interpretation of ground terms). Term is a recursively defined concept. It can be interpreted also recursively. 1. Each constant term c names the element c A. 2. if the terms t1, . . . , tn of L name the elements t A 1 , . . . , tA 2 of A and f is an n-ary function symbol of L, then the term f(t1, . . . , t2) names the element f(t1, . . . , tn) A = f A(t A 1 , . . . , tA n ) of A. Example 4. If we add the instance c, d into the language of last example, we can assign to element c A, dA as: 1. c A = 0; d A = 1. 2. c A = 1/2; d A = 2/3. 3. c A = 0; d A = −2. Example 5. Give ground terms f(x, d), f(d, f(d, d)). They name elements of the A as follows: 1. f(c, d) A = 0; f(d, f(d, d))A = 1. 2. f(c, d) A = 3/2; f(d, f(d, d))A = 2/3. 3. f(c, d) A = 2; f(d, f(d, d))A = −2. 3 Semantics Given a sentence, its semantics depends on the structure. Here, we should pay attention. For convenience, we expand our language by adding enough constant symbols. After interpretation, every element in domain is named by a constant. With the help of these constants, we can “use” every element in domain. Specially, when we add a “new” constant into language, before we fix its interpretation or meaning, it can be any element in domain as your wish. However, if you fix its interpretation, you can never change it. Definition 3 (Truth). The truth of a sentence φ of L in a structure A in which every a ∈ A is named by a ground term of L is defined by induction. 1. For an atomic sentence R(t1, . . . , tn), A ⊨ R(t1, . . . , tn) if and only if RA(t A 1 , . . . , tA n ). 2