数学代写|离散数学代写Discrete Mathematics代考|MA2201/CS2022 Logical Form and Logical Equivalence
数学代写|离散数学代写Discrete Mathematics代考|Logical Form and Logical Equivalence
Logic is a science of the necessary laws of thought, without which no employment of the understanding and the reason takes place. -Immanuel Kant, 1785 An argument is a sequence of statements aimed at demonstrating the truth of an assertion. The assertion at the end of the sequence is called the conclusion, and the preceding statements are called premises. To have confidence in the conclusion that you draw from an argument, you must be sure that the premises are acceptable on their own merits or follow from other statements that are known to be true.
In logic, the form of an argument is distinguished from its content. Logical analysis won’t help you determine the intrinsic merit of an argument’s content, but it will help you analyze an argument’s form to determine whether the truth of the conclusion follows necessarily from the truth of the premises. For this reason logic is sometimes defined as the science of necessary inference or the science of reasoning.
Consider the following two arguments. They have very different content but their logical form is the same. To help make this clear, we use letters like $p, q$, and $r$ to represent component sentences; we let the expression “not $p$ ” refer to the sentence “It is not the case that $p$ “; and we let the symbol $\therefore$ stand for the word “therefore.” Argument 1 If the bell rings or the flag drops, then the race is over. $\therefore$ If the race is not over, then $\overbrace{\text { the bell hasn’t rung and the flag hasn’t dropped. }}^{\text {not } r}$ not $q$.
We now introduce three symbols that are used to build more complicated logical expressions out of simpler ones. The symbol denotes not, $\wedge$ denotes and, and $\vee$ denotes or. Given a statement $p$, the sentence ” $\sim p$ ” is read “not $p$ ” or “It is not the case that $p$ “. In some computer languages the symbol $\neg$ is used in place of $\sim$. Given another statement $q$, the sentence ” $p \wedge q$ ” is read ” $p$ and $q$.” The sentence ” $p \vee q$ ” is read ” $p$ or $q$.”
In expressions that include the symbol $\sim$ as well as $\wedge$ or $\vee$, the order of operations specifies that $\sim$ is performed first. For instance, $\sim p \wedge q=(\sim p) \wedge q$. In logical expressions, as in ordinary algebraic expressions, the order of operations can be overridden through the use of parentheses. Thus $\sim(p \wedge q)$ represents the negation of the conjunction of $p$ and $q$. In this, as in most treatments of logic, the symbols $\wedge$ and $\vee$ are considered coequal in order of operation, and an expression such as $p \wedge q \vee r$ is considered ambiguous. This expression must be written as either $(p \wedge q) \vee r$ or $p \wedge(q \vee r)$ to have meaning.
A variety of English words translate into logic as $\wedge, \vee$, or $\sim$. For instance, the word but translates the same as and when it links two independent clauses, as in “Jim is tall but he is not heavy.” Generally, the word but is used in place of and when the part of the sentence that follows is, in some way, unexpected. Another example involves the words neither-nor. When Shakespeare wrote, “Neither a borrower nor a lender be,” he meant, “Do not be a borrower and do not be a lender.” So if $p$ and $q$ are statements, then \begin{tabular}{|rrl|} \hline$p$ but $q$ & means & $p$ and $q$ \ neither $p$ nor $q$ & means & $\sim p$ and $\sim q$ \ \hline \end{tabular}
数学代写|离散数学代写Discrete Mathematics代考|MA2201/CS2022 Logical Form and Logical Equivalence
现代博弈论始于约翰-冯-诺伊曼（John von Neumann）提出的两人零和博弈中的混合策略均衡的观点及其证明。冯-诺依曼的原始证明使用了关于连续映射到紧凑凸集的布劳威尔定点定理，这成为博弈论和数学经济学的标准方法。在他的论文之后，1944年，他与奥斯卡-莫根斯特恩（Oskar Morgenstern）共同撰写了《游戏和经济行为理论》一书，该书考虑了几个参与者的合作游戏。这本书的第二版提供了预期效用的公理理论，使数理统计学家和经济学家能够处理不确定性下的决策。
