数学代写|离散数学代写Discrete Mathematics代考|MAT2520 Valid and Invalid Arguments
数学代写|离散数学代写Discrete Mathematics代考|Valid and Invalid Arguments
“Contrariwise,” continued Tweedledee, “if it was so, it might be; and if it were so, it would be; but as it isn’t, it ain’t. That’s logic.” – Lewis Carroll, Through the Looking Glass In mathematics and logic an argument is not a dispute. It is simply a sequence of statements ending in a conclusion. In this section we show how to determine whether an argument is valid-that is, whether the conclusion follows necessarily from the preceding statements. We will show that this determination depends only on the form of an argument, not on its content.
It was shown in Section $2.1$ that the logical form of an argument can be abstracted from its content. For example, the argument If Socrates is a man, then Socrates is mortal. Socrates is a man. $\therefore$ Socrates is mortal. has the abstract form If $p$ then $q$ $\quad p$ $\therefore q$ When considering the abstract form of an argument, think of $p$ and $q$ as variables for which statements may be substituted. An argument form is called valid if, and only if, whenever statements are substituted that make all the premises true, the conclusion is also true.
数学代写|离散数学代写Discrete Mathematics代考|Determining Validity or Invalidity
Determine whether the following argument form is valid or invalid by drawing a truth table, indicating which columns represent the premises and which represent the conclusion, and annotating the table with a sentence of explanation. When you fill in the table, you only need to indicate the truth values for the conclusion in the rows where all the premises are true (the critical rows) because the truth values of the conclusion in the other rows are irrelevant to the validity or invalidity of the argument. $$ \begin{aligned} p & \rightarrow q \vee \sim r \ q & \rightarrow p \wedge r \ \therefore p & \rightarrow r \end{aligned} $$ Solution The truth table shows that even though there are several situations in which the premises and the conclusion are all true (rows 1,7 , and 8), there is one situation (row 4) where the premises are true and the conclusion is false.
数学代写|离散数学代写Discrete Mathematics代考|MAT2520 Valid and Invalid Arguments
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。
