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Is it impossible from a true statement $P$ to imply a false statement $Q$?

In the language of an implication: $P \Rightarrow Q$, where $P$ is true and $Q$ is false.

In other words is it impossible to deduce from a true statement $P$ a false statement $Q$ ?

Intuitively it must be, since otherwise $Q$ would also be true, which it isn't ? Could one prove this intuively idea ?

Is this the reason why the implication is defined to be false when $P \Rightarrow Q$ with $P$ true and $Q$ false ?

Shuzheng
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    I think the sentence "In the language of an implication..." has the roles of $P$ and $Q$ reversed, in the sense that the rest of your question seems to imply that $P$ be taken to be false, and $Q$ true. – Hayden Jun 02 '14 at 18:53
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    Every multiple of three is even; therefore, six is even. – Shaun Jun 02 '14 at 18:53
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    From the false statement $(0=1)\land \lnot(0=1)$ we can deduce any $Q$, whether true or false. – André Nicolas Jun 02 '14 at 18:54
  • Sorry I've updated: it should say from a true statement $P$ to imply a false statement $Q$. – Shuzheng Jun 02 '14 at 18:59
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    I don't want to be overly philosophical here but what do you mean by "deduce"? – Leo Jun 02 '14 at 19:04
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    And, if philosophy, how are 'true' and 'false' meant? – Berci Jun 02 '14 at 19:07
  • I guess, in mathematics we'd say truth is coherence ;) Which brings me to the following: isn't $ A \vdash B $ but $A \not\models B$ a contradiction since propositional calculus is sound and complete? I'm going to try and work that into a more coherent answer... – Leo Jun 02 '14 at 19:17
  • I mean by deduce that we prove an implication. – Shuzheng Jun 02 '14 at 19:52

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In propositional calculus there are two interesting symbols: $\vdash$ and $\models$. The first one $\vdash$ means that you can deduce something from a given set of premises. For instance from $A$, $A \Rightarrow B$ we can deduce B. More formally you could say: $A, A \Rightarrow B \vdash B$. The second symbol $\models$ means that given certain premises something is actually true. For instance $S \models B$ means that the premise $S$ entails that B is true. A logical system is called sound when $\vdash B$ means $\models B$. It can be shown that propositional calculus is in fact sound. (It even is complete meaning that if something is true, it can also be proven!). From this it follows that it is impossible to "deduce" a false statement from a true statement. Because then that false statement is also true: a blatant contradiction.

amWhy
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Leo
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