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- Thread starter solakis
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- #2

- Jan 26, 2012

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It's an invalid conclusion, obviously. The problem comes as follows:

Let $A$ be the statement that $2>0$, and $B$ be the statement that $2+3=7$. Let $C$ be the statement that $2\le 0$. Let $D$ be the statement that $3+3=8$. Then your premisses are as follows:

\begin{align*}

& \lnot(A \land B) \\

& \lnot A \implies C \\

& B \implies D\\

& \lnot D\\

& \therefore C.

\end{align*}

The first statement can be transformed, via DeMorgan, to

$$\lnot A \lor \lnot B.$$

So your assumption of $\lnot D$ could, via

Last edited:

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why should you not put :It's an invalid conclusion, obviously. The problem comes as follows:

Let $A$ be the statement that $2<0$, and $B$ be the statement that $2+3=7$. Let $C$ be the statement that $2\le 0$. Let $D$ be the statement that $3+3=8$. Then your premisses are as follows:

\begin{align*}

& \lnot(A \land B) \\

& \lnot A \implies C \\

& B \implies D\\

& \lnot D\\

& \therefore C.

\end{align*}

The first statement can be transformed, via DeMorgan, to

$$\lnot A \lor \lnot B.$$

So your assumption of $\lnot D$ could, viamodus tollens, give you $\lnot B$. But then, analyzing the first statement in its DeMorgan form, you are now stating that one of the options of the disjunction is true. That in no way implies that the other disjunct is true. So your reasoning chain ends. You cannot claim that $\lnot A$ is true.

$\neg B$ for $3+2=7$ since $2+3=7$ is false

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- #4

- Jan 26, 2012

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Evgeny can correct me if I'm wrong, but I think if you're in a two-valued logic system, where $\lnot( \lnot B)=B$, then it doesn't matter which you use - just a matter of definition. If you choose $B$ the way I have, it's a false proposition. If you choose your definition, it's a true proposition. You'd have to change your assumptions if you changed your definition, but the logic would work out analogously.why should you not put :

$\neg B$ for $3+2=7$ since $2+3=7$ is false

- Jan 30, 2012

- 2,526

This should say, $2 > 0$.Let $A$ be the statement that $2<0$

One has the right to introduce any notation. Abbreviating some expression by a variable is not a logical step; it does not change a problem in any essential way,why should you not put :

$\neg B$ for $3+2=7$ since $2+3=7$ is false

The premises in the OP are true, say, on integers, and the conclusion is not. So the conclusion cannot be proved in any formal system that is sound with respect to integers. (Regular logic is sound with respect to all models.)

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- #6

- Jan 26, 2012

- 4,198

Thank you! I've corrected that.This should say, $2 > 0$.