I didnt understand the truth table for $p \implies q$ (where $p$ and $q$ are statements). Can someone please explain it to me?
It would be better if someone explains with an example.
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Where is the truth table? What are $p$ and $q$? – shardulc Apr 07 '16 at 14:21
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The table should represent "if $p$ then $q$". So every entry in the table is $true$, except for the case of $[p=true]\wedge[q=false]$. – barak manos Apr 07 '16 at 14:26
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If I understand your question correctly, you do not understand why the truth table is as it is. Recall the truth table: $$\begin{array}{c|c|c} P & Q & P\rightarrow Q \\ \hline 1 & 1 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{array}$$ Where $1$ represents True, and $0$ represents False.
The first two lines follow naturally from the "intuitive English language definition", that is $$\text{If }P\text{, Then }Q.$$ So if $P$ holds, and $Q$ holds then the statement $\text{If }P\text{, Then }Q$ is true; if $Q$ did not hold, it would be false.
As for the bottom two lines, we're asking $\text{If }P\text{, Then }Q$ when $P$ is always false. Another way to read this would be $$\text{Whenever }P\text{ is true, Then }Q\text{ is necessarily true.}$$
But if $P$ is not true, this is vacuously true. For example, if $P$ were "There are 25 hours in this day", and $Q$ were "It is sunny today", then we're asking "Whenever there are 25 hours in the day, it is sunny". Well there are never 25 hours in the day, so this is true. Every time there is 25 hours in the day, it is sunny.
See Vacuous truth on Wikipedia for further reading. There are more examples here.
Szmagpie
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1You should explain that you used $1$ and $0$ for "true" and "false", respectively. – GEdgar Apr 07 '16 at 14:41
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The easiest way to handle these types of questions is to ask yourself for every possible combination of the truth values of p and q, "Is q at least as true as p?" If yes, then p -> q is true, if no, then p -> q is false.
cjg123
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