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• The syntax语法学 and semantics语义学 of propositional calculus (命题演算的语法学、语义学) • Tautologies and contradictions (重言式、矛盾式) • Truth tables (真值表)
Propositional logic: connectives
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conjunction连接 of p and q); “p∨q” is the proposition “p or q or both”, (the disjunction 析取 of p and q.） “p → q” is the proposition “if p then q”, (the implication蕴含 of p and q.) “p ↔ q” is the proposition “if p then q, and vice替代 versa反之亦然”, (the equivalence of p and q.) “¬p” is the proposition “not p”.
Propositional logic
• The meaning of the logical connectives Conjunction ∧, & Difference between ∧, & vs. and () Run a mile every day and you will feel like a new man. Not conjunction but implication蕴涵: () If you run a mile every day then you will feel like a new man. p→q
Propositional logic
• The meaning of the logical connectives Conjunction ∧, & Difference between ∧, & vs. and “And” often expresses sequences 后果 of events. () He lay down on the bed and died. () He died and lay down on the bed. In logic, p ∧ q is always equivalent to q ∧ p. p ∧ q ≡q ∧ p (commutative 交换的) ∧ is atemporal不受时间影响的.
Propositional logic
• The meaning of the logical connectives Disjunction ∨
The disjunction析取 is used to create a compound sentence并列（复合）句 which is false only if both the simple sentences in it are false. It will be enough that one disjunct 不相连的is true for the whole disjunction to be true.
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Contents: • Connectives 联接词 • The meaning of the logical connectives (negation否定,conjunction连接, disjunction分离, implication蕴含, equivalence相等) • How to indicate constitute structure(表示组成成分的
2. The possibility of focusing different constituents 构成部分 in a negated sentence by stress and intonation语调 is also lost in propositional logic. () Mary didn’t break the window. Mary didn’t break the window. The difference between () and () cannot be captured 捕获in propositional logic; both would be translated as ～p.
Propositional logic
• The meaning of the logical connectives Conjunction ∧, & Difference between ∧, & vs. and “∧, &” can only be used to combine sentences “And” can be used to combine constituents below the sentence level. John and Bill (×John ∧ Bill) () John and Bill own a car. () John owns a car and Bill owns a car. (p ∧ q)
Propositional logic命题逻辑
• Propositional logic largely 大部分地 involves studying logical connectives 连接词 such as the words “and” and “or” and the rules determining 确定 the truth-values of the propositions they are used to join连接, as well as what these rules mean for the validity正确性 of arguments.论
• Four of the propositional connectives
“and”, “or” , “if... then”, “if and only if当且 仅当” plus “not” If p and q are propositions, then
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• “p∧q” is the proposition “p and q ”, (the
Propositional logic命题逻辑
• Propositional logic (sentential logic句子
逻辑), is that branch of logic that studies ways of combining联合 or altering改 变 statements声明 or propositions命题 to form more complicated结构复杂的 statements or propositions. Joining two simpler propositions with the word “and” is one common way of combining statements. When two statements are joined together with “and”, the complex statement formed by them is true if and only if both the component成分的 statements are true.
• Bill is a vegetarian even though he eats pork. • Bill is a vegetarian and he reads eats pork. • Bill is a vegetarian or he eats pork.
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p even though q p and q p or q The logical form provided by the sentential connectives therefore determines the logical consequences 逻辑结论of the sentences that have been related by the connectives.
3. The hope of capturing complexity捕捉的复杂性 in the formal representation of propositional logic is lost. () Harold did not think Alfred was fond of cakes. () is ambiguous for there being a choice between interpreting理解 the subordinate次要的 or the main clause as negated.
连接词 • We will use letters such as p,q,r,s,... or A,B,C,D,... to represent propositions. The letters are called logical variables逻辑变量/数. We combine simple propositions to form compound复合 propositions using connectives (logical constants逻辑常数). • logical variables逻辑变量/数: signs that can stand for any declarative sentence 陈述句 (represent the content) • Constants常数，常量: signs that have a permanent non-variable 恒定的meaning (represent the structure) • Logical constants逻辑常数[常项]: signs that through their permanent meanings and functions determine the logical structure of the sentences they occur发生 in.