Use symbols to write the logical form of the following arguments. If valid, iden

tornesasln 2021-11-07 Answered
Use symbols to write the logical form of the following arguments. If valid, iden— tify the rule of inference that guarantees its validity. Otherwise, state whether the converse or the inverse error has been made. If you study hard for your discrete math final you will get an A. Jane got an A on her discrete math final. Therefore, .lane must have studied hard.

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Expert Answer

Pulad1971
Answered 2021-11-08 Author has 1673 answers

Step 1
Let P be the statement where
\(\displaystyle{P}:{S}{t}{u}{\left.{d}{y}\right.}in{g}\ {h}{a}{r}{d}\ {f}{{or}}\ {d}{i}{s}{c}{r}{e}{t}{e}\ {m}{a}{t}{h}\ {f}in{a}{l}\)
Let Q be the statement, where
\(\displaystyle{Q}:{G}{e}t{in}{g}\ {A}\ in\ {d}{i}{s}{c}{r}{e}{t}{e}\ {m}{a}{t}{h}\ {f}in{a}{l}\) Step 2
Definition: "If A is true then B is true." The logical form of this statement is:
\(\displaystyle{A}\rightarrow{B}\)
The given statement is:
\(\displaystyle\text{If you study hard for your discrete math final you will get A}\)
Note that P: Studying hard for discrete math final and Q: Getting A in discrete math final.
The logical form of the given statement is:
\(\displaystyle{P}\rightarrow{Q}\) Step 3
The given statements is:
\(\displaystyle\text{}{A}ne\ {g}{o}{t}\ {a}{n}\ {A}\ {o}{n}\ {h}{e}{r}\ {d}{i}{s}{c}{r}{e}{t}{e}\ {m}{a}{t}{h}\ {f}in{a}{l};\therefore{J}{a}ne\ mu{s}{t}\ {h}{a}{v}{e}\ {s}{t}{u}{d}{i}{e}{d}\ {h}{a}{r}d\) "Jane got an A on her discrete math final." Hence the statement Q is true for Jane.
This statement implies that "Jane must have studied hard." Hence P is true.
\(\displaystyle{I}{f}\ {Q}{i}{s}\ {t}{r}{u}{e}\ {t}{h}{e}{n}\ {P}{i}{s}\ {t}{r}{u}{e}.\)
The logical form of the statement is:
\(\displaystyle{Q}\rightarrow{P}\) Step 4
From the first statement,
\(\displaystyle{P}\rightarrow{Q}\)
From the second and third statements,
\(\displaystyle{Q}\rightarrow{P}\)
which is not always true.
For example, If the fruit is banana then it is yellow in color; if the fruit is yellow in color one can not assure whether the fruit is a banana.
Hence, converse or inverse error has been made.

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