is not as simple as"I am a student". A propositional letter is something like a word in a sentence. It can't be divide into more basic one A sentence is composed of some words linked by some function word such as verbal word, just like I am a student and I study computer science". To describe a complicated case, we need some word such as and, or, and if. not ., which are called conjunctions. Furthermore, if some more complicated scenario is to be depicted, we may need several sentences, a paragraph, which is separated by a series of punctuation We also need define some symbols in propositional logic. Sometimes we call it language, all complicated strings are constructed based on these symbols with some rules Definition 1. The language of propositional logic consists of the following symbols 1. Connectives:V,∧,-,→,+ 2. Parentheses:),( 3. Propositional Letters: A, Al, A2,..., B, B1, B where we suppose the set of proposition letters is countable, which is reasonable for human can hardly grasp uncountable set There are many words. But we need the grammar to write a sentence, a paragraph, even a book. We also define a inductive approach to construct a proposition Definition 2(Proposition). A proposition is a sequence of many symbols which can be constructed in the following approach 1. Propositional letters are proposit 2. if a and B are propositions, then(avB),(aAB), (a),a,B)and(a+ B)are propositions 3. A string of symbols is a proposition if and only if it can be obtained by starting with proposi- tional letters(1) and repeatedly applying(2) It is obvious that infinite propositions can be generated even if there are only finite propo letters Actually, there are more strings(a sequence of symbols defined in Definition 1)than those generated according to definition 2. Example 1. Given the following strings, check whether it is a proposition 1.(AVB),(A∧B)→C) 2.AV-,(A∧B In practice, we also admit A V B as a valid proposition. But it is not generated according to definition of proposition. So we give a definition of those propositions generated according to definition Definition 3. The proposition constructed according to Definition 2 is well-defined or well-formed We call them well-formed because of their good properties, which are good for us to design some procedure to recognize a proposition. This is a topic in the next lectureis not as simple as ”I am a student”. A propositional letter is something like a word in a sentence. It can’t be divide into more basic one. A sentence is composed of some words linked by some function word such as verbal word, just like ”I am a student and I study computer science”. To describe a complicated case, we need some word such as and, or, and if. . . not . . . , which are called conjunctions. Furthermore, if some more complicated scenario is to be depicted, we may need several sentences, a paragraph, which is separated by a series of punctuation. We also need define some symbols in propositional logic. Sometimes we call it language, all complicated strings are constructed based on these symbols with some rules. Definition 1. The language of propositional logic consists of the following symbols: 1. Connectives: ∨, ∧, ¬, →, ↔ 2. Parentheses: ), ( 3. Propositional Letters: A, A1, A2, · · · , B, B1, B2, · · · . where we suppose the set of proposition letters is countable, which is reasonable for human can hardly grasp uncountable set. There are many words. But we need the grammar to write a sentence, a paragraph, even a book. We also define a inductive approach to construct a proposition. Definition 2 (Proposition). A proposition is a sequence of many symbols which can be constructed in the following approach: 1. Propositional letters are propositions. 2. if α and β are propositions, then (α∨β),(α∧β),(¬α),(α → β) and (α ↔ β) are propositions. 3. A string of symbols is a proposition if and only if it can be obtained by starting with proposi￾tional letters (1) and repeatedly applying (2). It is obvious that infinite propositions can be generated even if there are only finite proposition letters. Actually, there are more strings (a sequence of symbols defined in Definition 1) than those generated according to Definition 2. Example 1. Given the following strings, check whether it is a proposition. 1. (A ∨ B),((A ∧ B) → C) . 2. A ∨ ¬,(A ∧ B In practice, we also admit A ∨ B as a valid proposition. But it is not generated according to definition of proposition. So we give a definition of those propositions generated according to definition. Definition 3. The proposition constructed according to Definition 2 is well-defined or well-formed. We call them well-formed because of their good properties, which are good for us to design some procedure to recognize a proposition. This is a topic in the next lecture. 2