2. AFmo+ it is not the case that Ahp(We also write as Ap. 3.4=(Vv)分 AFp or y 4.AF(y∧v)分 Fo and F=v. 5.4F(→v)分 Ap orAL 6.AF(9v)兮( AFp andAh yl)or(A≠ p and y) 7. AF3up(u)+ for some ground term t, AFp(t) 8. AF Voo(u)+ for all ground term t, AFp(t) An atomic sentence is a instance of a instantiated atomic formula. a atomic formula can taken as relation. When you determine the truth of an atomic sentence, you can ask whether(a1, a2, .. an) R if R is n-ary We have known that all ground terms just name all element of a domain. So some and all ground term(s)can represent any element of a domain, which are 3 and V expected. Given a little bit complicated sentence like 3zVy R(a, m), its truth can be determined recursively according to our definition. First, we just try to find some ground term ti to make Vy R(t1, y) true. Then, we just verify that R(t1, t2) is true for t2 visits all elements of A This definition just define how to determine the truth of a sentence other than any other else. A sentence is a recursively constructed symbol sequence. When quantifiers occur in a sentence, we need get rid of them from outside to inside one by one as definition defined Example 6. Consider the structure A=< M, P), f(a, y)), c, d>. We have 1. P(c, d)is true. 3.(V)P(c, a) is true 4. Va)P(a, d)is false 5.(Var)(P(,c)->P(, d))is true 6. How about r)(r)P(, ) Determine its truth according to definition Once we can determine the truth of a sentence. We can extend the following concepts defined already in propositional logic Definition 4. Fir some language L 1. A sentence pp of L is valid, Hp, if it is true in all structure for L 2. Given a set of sentences 2=a1,..., we say that a is a logic consequence of E,2ha, if a is true in every structure in which all of the members of are true2. A |= ¬φ ⇔ it is not the case that A |= φ ( We also write as A ̸|= φ.) 3. A |= (φ ∨ ψ) ⇔ A |= φ or A |= ψ. 4. A |= (φ ∧ ψ) ⇔ A |= φ and A |= ψ. 5. A |= (φ → ψ) ⇔ A ̸|= φ or A |= ψ. 6. A |= (φ ↔ ψ) ⇔ (A |= φ and A |= ψ) or (A ̸|= φ and A ̸|= ψ). 7. A |= ∃vφ(v) ⇔ for some ground term t, A |= φ(t). 8. A |= ∀vφ(v) ⇔ for all ground term t, A |= φ(t). An atomic sentence is a instance of a instantiated atomic formula. A atomic formula can taken as a relation. When you determine the truth of an atomic sentence, you can ask whether (a1, a2, . . . , an) ∈ R if R is n-ary. We have known that all ground terms just name all element of a domain. So some and all ground term(s) can represent any element of a domain, which are ∃ and ∀ expected. Given a little bit complicated sentence like ∃x∀yR(x, y), its truth can be determined recursively according to our definition. First, we just try to find some ground term t1 to make ∀yR(t1, y) true. Then, we just verify that R(t1, t2) is true for t2 visits all elements of A. This definition just define how to determine the truth of a sentence other than any other else. A sentence is a recursively constructed symbol sequence. When quantifiers occur in a sentence, we need get rid of them from outside to inside one by one as definition defined. Example 6. Consider the structure A =< N , {P}, {f(x, y)}, {c, d} >. We have 1. P(c, d) is true. 2. (∃x)P(x, d) is true. 3. (∀x)P(c, x) is true. 4. (∀x)P(x, d) is false. 5. (∀x)(P(x, c) → P(x, d)) is true. 6. How about (∃x)(∀x)P(x, y)? Determine its truth according to definition. Once we can determine the truth of a sentence. We can extend the following concepts defined already in propositional logic. Definition 4. Fix some language L. 1. A sentence φ of L is valid, |= φ, if it is true in all structure for L. 2. Given a set of sentences Σ = {α1, . . .}, we say that α is a logic consequence of Σ, Σ |= α, if α is true in every structure in which all of the members of Σ are true. 3