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# 数学代写|离散数学代写Discrete Mathematics代考|MAT2520 Valid and Invalid Arguments

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## 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.