Chapter Five
Also known as propositional calculus（命 题演算） or sentence calculus（句子演 算）, is the study of the truth conditions for propositions（命题的真实条件）: how the truth of a composite proposition is determined by the truth value of its constituent propositions and the
connections between them（复合命题真值 是如何由成分命题真值及成分命题之间的关 系决定的）
", Lyons: A proposition is what is expressed by a declarative sentence when that sentence is
uttered to make a statement. 命题是陈述句被用于叙述事件时 所表达的意义。
", A very important property of the proposition is that it has a truth value. It is either true or false. ", The truth value of a composite proposition is said to be the function(函数)of, or is determined by, the truth values of its component propositions and the logical connectives（逻辑连词） used in it. 复合命 题的真值据称是成分命题真值和所用逻辑连词的函数。(其 真值由成分命题真值和所用逻辑连词决定）
", If a proposition p is true, then its negation ～ p is false. And if p is false, then ～ p is true. ", p stands for a simple proposition（一个简单 命题）.
", ～ is the logical connective negation（逻辑 连词否定）.
", ～ p stands for the negation of a proposition, is a composite proposition（一个命题的否 定，是复合命题）.
", & ,∧ conjunction 合取连词 （相当于英语中 的“and”）
", ∨ disjunction 析取连词 （相当于英语中的 “or”）
", → implication 蕴涵连词, 条件连词 （相当于 英语中的“if?then”如果?那么)
", ≡, equivalence 等值连词, 双条件连词 (相 当于英语中的 “if and only if?then”,有时写作 “ iff?then”当且仅当?则 )
", Two-place connectives 二元连词: & ,∧, ∨, →, ≡,
(because two propositions involved) ",One-place connectives 一元连词: ～
", The truth tables for the two-place connectives: on P121.
", Examples to show the truth tables of the two-place connectives. ", John is an Englishman. (p) John speaks English very well. (q)
", John is an Englishman and he speaks English very well. (p & q)
", John is an Englishman and he doesn’t speak English very well. (p ∨q)
", If John is an Englishman then he speaks English very well. (p → q)
", If and only if John is an Englishman then he speaks English very well. (p ≡q)
", The truth conditions of the logical connectives are not exactly the same as
their counterparts in English--- “not”, “and”, “or”, “if?then”, “if and only if?then” respectively.
", Predicate logic (谓词逻辑) studies the internal structure of simple propositions.
", In this logical system, propositions will be analyzed into two parts: an argument(主目) and a predicate(谓词) .
", An argument refers to some entity about which a
statement is being made.主目是表示实体的项，有关陈述是关于 该实体的。
", A predicate ascribes some property or relation to the entity or entities referred to. 谓词是把一些性质或关系 给予所指实体的项。
", In the proposition Socrates is a man, Socrates is the argument, man is the predicate.
", In logic term, the proposition Socrates is a man is represented as M(s) M=man, s=Socrates
", A simple proposition is seen as a function of its argument. The truth value of a
proposition varies with the argument. 简单命 题可以看作它的主目的函数，命题真值随主 目而改变。
", When Socrates is indeed a man, M(s) is true. On the other hand, as Cupid is an angel, the proposition represented by the logical formula M(c) is false.
", We use the numeral 1 to stand for “true” and 0 for “false”. We can represent these two examples as the formula: M(s) = 1, M(c) = 0.
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Chapter Five
", If we classify predicts in terms of the number of arguments they take, then
", One-place predicate 一元谓词 man (Socrates is a man) M(s)
", Two-place predicate 二元谓词 love (John loves Mary.) L (j, m)
", Three-place predicate 三元谓词 give (John gave Mary a book.) G (j, m, b)
", Propositions with two or more arguments may also be analyzed in the same way as those with one argument. John loves Mary. ( Lm) (j)
a complex predicate (Lm) and a single argument John.
", Predicate logic is applied to the analysis of valid inferences like the syllogism as: All men are rational. Socrates is a man.
Therefore, Socrates is rational.
", All men are rational.
",Quantifier all, known as the universal quantifier（全称量词） and symbolized by an upturned Ａ--- in logic.
", The argument men does not refer to any particular entity, which is known as a variable (变项) and symbolized by x, y.
", The logical structure for All men are rational is x ( M (x) →R (x)). “ For all x, it is the case that, if x is a man, then x is rational”.
",Some men are clever
", The existential quantifier（存在量词）, equivalent to some in English and symbolized by a reversed E--- . ", x( M (x) & C (x))
“There are some x’s that are both men and clever”, or more exactly, “There exists at least one x, such that x is a man and x is clever”.
", The universal and existential quantifiers are related to each other in terms of negation. One is the logical negation of the other. All men are rational = There is no man who is not rational.
", x ( M (x) →R (x)) ≡ ～ x( M (x) & ～ R (x))
", x ( P (x)) ≡ ～ x(～ P (x))
It is the case that all x’s have the property P == There is no x, such that x does not have the property P
", ～ x ( P (x)) ≡ x(～ P (x)) It is not the case that all x’s have the property P == There is at least an x, such that x does not have the property P
", x (P (x)) ≡ ～ x (～ P (x)) There is at least an x, such that x has the property P = It is not the case that all x’s do not have the property P ", ～ x (P (x)) ≡ x (～ P (x))
There is no x, such that x has the property P = It is the case that all x’s do not have the property P
", The logical structure for: All men are rational. Socrates is a man.
Therefore, Socrates is rational. ", x ( M (x) →R (x)) M (s) ∴R (s)
", The following inferences are not valid: ", x ( M (x) →R (x)) R (s) ∴ M (s)
", x ( M (x) &C (x)) M (s) ∴C (s)
",The validity of inferences involving the universal and existential quantifiers may also be shown in terms of set theory (集合论).
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