Example 2. Consider the following proposition A1VA2∧A which is definitely not well-defined Solution. We have two possible different propositions 1.(A1VA2)∧A3 2.A1V(A2∧A3) Of course, they have different abbrevation truth tables Mathematics is rigid. One of the responsibilities of mathematical logic is to eliminate ambiguity from mathematics. So we should first eliminate ambiguity in mathematical logic 2.2 Parent Theorem 1. Every well-defined proposition has the same number of left as right parentheses Proof. Consider the symbols without parentheses first And then prove it by induction with more complicated propositions according to the construction of a complicated propositio Here, we actually use the depth of nested connectives, which will be introduced in the next section Theorem 2. Any proper initial segement of a well defined proposition contains an excess of left parenthesis. Thus no proper initial segement of a well defined propositon can itself be a well defined propositions Go back to lecture three if your forget what is a proper initial segment Proof. Prove it by induction from simple to complicated propositions. And we also need Theorem 1 3 Formation tree A proposition is a sequence of symbols. The structure determines the reading of a proposition. In this section, we are to map a proposition to a tree, named formation tree. It will help us to read a proposition Definition 3(Top-down). A formation tree is a finite tree T of binary sequences whose nodes are all labele with propositions. The labeling satisfies the following conditionsExample 2. Consider the following proposition A1 ∨ A2 ∧ A3, which is definitely not well-defined. Solution. We have two possible different propositions 1. (A1 ∨ A2) ∧ A3 2. A1 ∨ (A2 ∧ A3) Of course, they have different abbrevation truth tables. Mathematics is rigid. One of the responsibilities of mathematical logic is to eliminate ambiguity from mathematics. So we should first eliminate ambiguity in mathematical logic. 2.2 Parentheses Theorem 1. Every well-defined proposition has the same number of left as right parentheses. Proof. Consider the symbols without parentheses first. And then prove it by induction with more complicated propositions according to the construction of a complicated proposition. Here, we actually use the depth of nested connectives, which will be introduced in the next section. Theorem 2. Any proper initial segement of a well defined proposition contains an excess of left parenthesiss. Thus no proper initial segement of a well defined propositon can itself be a well defined propositions. Go back to lecture three if your forget what is a proper initial segment. Proof. Prove it by induction from simple to complicated propositions. And we also need Theorem 1. 3 Formation Tree A proposition is a sequence of symbols. The structure determines the reading of a proposition. In this section, we are to map a proposition to a tree, named formation tree. It will help us to read a proposition. Definition 3 (Top-down). A formation tree is a finite tree T of binary sequences whose nodes are all labele with propositions. The labeling satisfies the following conditions: 2