Ba→B na B TT F LF FFI Figure 2: Logically equivalent propositions Consider the following example Example3.a→B≡-VB. Proof. Prove by truth table in Figure 2 Although, they have different formation tree. But they are the same if they are only chan acterized by truth valuation Definition 6. Two proposition a and B such that, for every valuation v, v(a)=v(B)are called logically equivalent. We denote this by a= B 3 Consequence practice,we often mention a pattern that a result can be inferred from some facts.We now consider this pattern from the point of view of semantics Definition 7. Let 2 be a(possibly infinite) set of propositions. We say that o is a conse ence o ∑( and write as∑ha)ij, for any valuation, ((7)= T for all T∈∑)→(a)=T Example4.1.Let∑={A,=AVB}, we have∑hB 2.Let∑={A,=AVB,C},ehae∑hB 3.Let∑={AB}, e have z b. Definition 8. We say that a valuation v is a model ofe ifv(o)=T for every o EE. We denote by M(∑) the set of all models of∑ Example 5. Let 2=A, AVB, we have models 1. Let A(A)=T,A(B)=Tα β α → β T T T T F F F T T F F T α β ¬α ¬α ∨ β T T F T T F T F F T T T F F T T Figure 2: Logically equivalent propositions Consider the following example. Example 3. α → β ≡ ¬α ∨ β. Proof. Prove by truth table in Figure 2. Although, they have different formation tree. But they are the same if they are only char￾acterized by truth valuation. Definition 6. Two proposition α and β such that, for every valuation V, V(α) = V(β) are called logically equivalent. We denote this by α ≡ β. 3 Consequence In practice, we often mention a pattern that a result can be inferred from some facts. We now consider this pattern from the point of view of semantics. Definition 7. Let Σ be a (possibly infinite) set of propositions. We say that σ is a conse￾quence of Σ (and write as Σ |= σ) if, for any valuation V, (V(τ ) = T for all τ ∈ Σ) ⇒ V(σ) = T. Example 4. 1. Let Σ = {A, ¬A ∨ B}, we have Σ |= B. 2. Let Σ = {A, ¬A ∨ B, C}, we have Σ |= B. 3. Let Σ = {¬A ∨ B}, we have Σ 6|= B. Definition 8. We say that a valuation V is a model of Σ if V(σ) = T for every σ ∈ Σ. We denote by M(Σ) the set of all models of Σ. Example 5. Let Σ = {A, ¬A ∨ B}, we have models: 1. Let A(A) = T, A(B) = T 3