logic arguments

Question

If today is Morgan's birthday then today is October 16. Today is October 16. Hence today is Morgan's birthday. I knew that it can be true or false. then
My question is that Today is Morgan's birthday statement is true or false?

Answer 1

$\textit{Today is Morgan's birthday} \implies \textit{Today is October 16}$

It's not an $\textit{if and only if } (\iff)$ so it's false.

If today is October 16, then it may or may not be Morgan's birthday.

Note: this is a weird example, because in real life it should be an $\textit{if and only if.}$

Answer 2

If we are only given that if today is Morgan's birthday then today is October 16 and today is October 16 (in other words, if we forget everything else we know about dates and birthdays), then we cannot conclude that today is Morgan's birthday.

This argument has the format

$$\begin{align}&p\Rightarrow q\\&q\\\therefore\; &p\end{align}$$

which is the fallacy of affirming the consequent.

The problem is that it is difficult to view this particular example without implicitly incorporating additional knowledge of dates and birthdays...

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To say that an argument

$$p_1\\p_2\\\vdots\\p_n\\\therefore q$$

is valid is to say that the proposition $\displaystyle\mathop\land_{k=1}^np_k\Rightarrow q$ is a tautology.

Thus, to demonstrate that an argument is not valid, we just need to show that $\displaystyle\mathop\land_{k=1}^np_k\Rightarrow q$ is not a tautology.

To show that affirming the consequent is not a valid argument, then, we just need to show that the proposition $\left((p\Rightarrow q) \land q\right)\Rightarrow p$ is not a tautology (which is easy enough: consider the case $p$ false, $q$ true).

Compare this to modus ponens:

$$\begin{align}&p\Rightarrow q\\&p\\\therefore\; &q\end{align}$$

Notice that this argument is valid: in other words, the proposition $\left((p\Rightarrow q) \land p\right)\Rightarrow q$ is a tautology.

Answer 3

If the State $Morgens Birthday$ is only true under the condition the Date Oct16 is true is called a Subjunction

$$ Morgans Birthday \, \implies \, Oct16 $$

if $October 16th \; = true $ it does not matter if $Morgans Birthday$ is true or false, the statement in itself is true either way. However you can not backward imply that $Morgans Birthday$ is true or false for the same reason, simply because you can't know. Subjunctions are one way streets.

\begin{array}{|c|c|c|} \hline A & B & A\implies B \\ \hline T & T& T\\ \hline T & F& F\\ \hline F & T& T\\ \hline F & F& T\end{array}