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# 数学代写|离散数学代写Discrete Mathematics代考|MA2201/CS2022 Logical Form and Logical Equivalence

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## 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 $q$.

## 数学代写|离散数学代写Discrete Mathematics代考|Compound Statements

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}
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