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$\begingroup$

"If Mike performs well in her examination, he will get a scholarship, if Mike gets a scholarship, he will travel abroad. Mike got a scholarship therefore she performed well in her examination".

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    Is there a significance to the gender changes? Is, say, "Mike" the same person in each clause? – 2017-02-05
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    Ignoring the pronouns, The argument is not valid. Nothing suggests that only those who do well get scholarships. Maybe everybody gets a scholarship. – 2017-02-05
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    The logical form of the argument is : "(if $p$, then $q$) and (if $q$, then $r$) and $q$; therefore $p$". But from $p \to q$ and $q$ we cannot conclude with $p$. You can check it with truth table. – 2017-02-05

1 Answers 1

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A valid argument is one in which for every case where the antecedent is true; the consequent must not be false.

"If Mike performs well in her examination, he will get a scholarship, if Mike gets a scholarship, he will travel abroad. Mike got a scholarship therefore she performed well in her examination".

I have read your expressions as:

$W\rightarrow S\tag{1}\label{1}$

$S\rightarrow T\tag{2}\label{2}$

$S\tag{3}\label{3}$

$((W\rightarrow S)\land S)\rightarrow W\tag{4}\label{4}$

,where W, S and T represent "Mike did well", "Mike got a scholarship", and "Mike Traveled" consecutively.

($\ref{4}$) is an example of affirming the consequence. It is a common formal fallacy and is not a valid argument. This is because the consequent may be false even when the antecedent is true. (https://en.wikipedia.org/wiki/Affirming_the_consequent)

An example of a valid argument that can be formed using ($\ref{1}$), ($\ref{2}$), and ($\ref{3}$) is ($\ref{5}$):

$((S\rightarrow T)\land S)\rightarrow T\tag{5}\label{5}$

($\ref{5}$) takes the form of a valid argument because there is no way for the consequent to be false when the antecedent is true. This can be checked by a truth table. (https://en.wikipedia.org/wiki/Modus_ponens)