To impress the understanding, consider the following examples Example 3. Think about the following statements: 1. Figure out what would happen if man can fy like a bird! 2. Everything is possible in folk world. We hope they could help you. If you have any good examples, please sent me an email 4 Connectives From the point of view of function, every connective can be taken as a function with form f 0,1}→{0,1} Definition 4(Truth function). An n-ary connective is truth functional if the truth value for o(Al,., An) is uniquely determined by the truth value of A1,..., An Definition 5. A k-place Boolean function is a function from F, Ti to iT, F]. We let F and T themselves to be 0-place Boolean functions Example 4. We can define- as a Boolean function in Figure 4 x1→x2f→(x1,x2) f→(T,T)=T T FFf,T,,, F)=F FTTf→(F, FF T f→(F,F)=T Let I(a1, T2, .. n)=Ti, which is a projection function of i-th parameter 1. For each n, there are 22 n-place Boolean functions 2. 0-ary connectives: T and F 3. Unary connectives: ,I,T and F 4. Binary connectives: 10 of 16 are real binary functions Definition 6(Adequate connectives). A set S of truth functional connectives is adequate if, given ny truth function connective o, we can find a proposition built up from the connectives is S with the same abbreviated truth table as g In general, we have the following Adequacy theorem Theorem 7(Adequacy)., V, A is adequate(complete)To impress the understanding, consider the following examples: Example 3. Think about the following statements: 1. Figure out what would happen if man can fly like a bird! 2. Everything is possible in folk world. We hope they could help you. If you have any good examples, please sent me an email. 4 Connectives From the point of view of function, every connective can be taken as a function with form f : {0, 1} k → {0, 1}. Definition 4 (Truth function). An n-ary connective is truth functional if the truth value for σ(A1, . . . , An) is uniquely determined by the truth value of A1, . . . , An. Definition 5. A k-place Boolean function is a function from {F, T} k to {T, F}. We let F and T themselves to be 0-place Boolean functions. Example 4. We can define → as a Boolean function in Figure 4. x1 x2 x1 → x2 f→(x1, x2) T T T f→(T, T) = T T F F f→(T, F) = F F T T f→(F, T) = T F F T f→(F, F) = T Figure 4: Boolean Function Let Ii(x1, x2, . . . , xn) = xi , which is a projection function of i-th parameter. 1. For each n, there are 22 n n-place Boolean functions. 2. 0-ary connectives: T and F. 3. Unary connectives: ¬, I , T and F. 4. Binary connectives: 10 of 16 are real binary functions. Definition 6 (Adequate connectives). A set S of truth functional connectives is adequate if, given any truth function connective σ, we can find a proposition built up from the connectives is S with the same abbreviated truth table as σ. In general, we have the following Adequacy theorem. Theorem 7 (Adequacy). {¬, ∨, ∧} is adequate(complete). 4