Example 9. All apples are bad Solution. The sentence is simple enough. We first give two possible candidates as following 1.Vx(A(x)→B(x) 2.var(A(x)∧B(x) Here, we suppose the statement is true. However, not both two sentences are correct. You can try to check the truth of them Remark 1. 1. V(A(),B()is correct 2. Var(A(r)A B(r)is wrong. Because it require everything is an apple Example 10. Some apples are bad. Solution. Similarly, we also have two possible candidates 1.彐x(A(x)∧B(x) 2.彐r(A(x)→B(x) In this case, the status exchanges Remark2.1.彐r(A(x)∧B(x) is correct 2. 3c(A(),B(a)) is improper. Because even every apple is good but the statement is still true if a is something other than apple These two examples would be encountered many times. There are some patterns to express sen- tences 1. Va(-,): Everything in a certain category has some property 2. 3x( A: There is some object/objects in the category and having the property 5 Connection between predicate and proposition logic Proposition logic can be embedded into predicate language if we only consider very special category of formula, open formula. All open atomic formula can be thought as a proposition letterExample 9. All apples are bad. Solution. The sentence is simple enough. We first give two possible candidates as following: 1. ∀x(A(x) → B(x)). 2. ∀x(A(x) ∧ B(x)). Here, we suppose the statement is true. However, not both two sentences are correct. You can try to check the truth of them. Remark 1. 1. ∀x(A(x) → B(x) is correct. 2. ∀x(A(x) ∧ B(x) is wrong. Because it require everything is an apple. Example 10. Some apples are bad. Solution. Similarly, we also have two possible candidates: 1. ∃x(A(x) ∧ B(x)). 2. ∃x(A(x) → B(x)). In this case, the status exchanges. Remark 2. 1. ∃x(A(x) ∧ B(x)) is correct. 2. ∃x(A(x) → B(x)) is improper. Because even every apple is good but the statement is still true if x is something other than apple. These two examples would be encountered many times. There are some patterns to express sen￾tences. 1. ∀x( → ): Everything in a certain category has some property. 2. ∃x( ∧ ): There is some object/objects in the category and having the property. 5 Connection between predicate and proposition logic Proposition logic can be embedded into predicate language if we only consider very special category of formula, open formula. All open atomic formula can be thought as a proposition letter. 5