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I do not understand how the converse ($B \Rightarrow A$) truth table is logical.

For instance, take the statement, "If I am in Paris, then I am in France".

If I am in Paris, then I am in France. Therefore, $A \Rightarrow B$, since if I am in Paris, then I must also be in France. However, ($B \not \Rightarrow A$), since it can be true that I am in France, but that does not necessarily mean I am in Paris specifically.

I would greatly appreciate it if someone could tell me why my understanding is incorrect.

Thank you.

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    $\begingroup$ Im not sure that I understand what you are asking. Of course just because $A \implies B$, the converse doesn't have to be true. Where is the inconsistency? $\endgroup$ – JuliusL33t Dec 28 '16 at 1:36
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    $\begingroup$ The converse does not follow. You may be thinking of the contrapositive: If I am not in France, then I am not in Paris. $\endgroup$ – John Wayland Bales Dec 28 '16 at 1:37
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    $\begingroup$ If $A$ and $B$ are already known to be true, then $A \Rightarrow B$ and also $B \Rightarrow A$, if it is only known that $A \Rightarrow B$, then $B \Rightarrow A$ doesn't follow necessarily. Also, I've never seen a "converse truth table", could you specify? $\endgroup$ – JuliusL33t Dec 28 '16 at 1:46
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    $\begingroup$ @The Pointer: If you already know that you are in Paris, then whatever $B$ might be, $B$ will still imply that you are in Paris, since you already know that you are in Paris. As you say, if you only know that IF you are in Paris THEN you are in France, then it does not follow that just because you are in France, you must be in Paris. Clearer? $\endgroup$ – JuliusL33t Dec 28 '16 at 1:55
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    $\begingroup$ @The Pointer: Maybe this is the confusion: There are three things here, the truth of $A$, the truth of $B$ and the truth of $A \Rightarrow B$. If you know that $A \Rightarrow B$, then you can neither conclude $A$ nor $B$, all you can say is "if $A$ then $B$", which is different from claiming either $A$ or $B$. Just because you know that IF you are in Paris THEN you are in France doesn't mean that you are either in Paris or in France. $\endgroup$ – JuliusL33t Dec 28 '16 at 2:03
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Let's agree that 'if you are in Paris, then you are in France' ($A \implies B$). (We could get picky, and say maybe you're in Paris, Texas; but let's not!).

But then the converse ($B \implies A$) is not automatically true: for example, we can't then deduce from 'if you are in Paris, then you are in France' that 'if you are in France, then you are in Paris'. In that sense you're right that 'it's not logical' to say the converse is always true.

Now, bear in mind that the converse might be true, or it might not. It must be proven either way based on other information. And it turns out that we know enough other things, about European geography in this case, that we can also prove that the converse is not true.

On the other hand, what we can always deduce is called the contrapositive: once we accept the truth of 'if you are in Paris, then you are in France', then we always automatically can say 'if you are not in France, then you are not in Paris' ($\neg B \implies \neg A$). That will always be true (at least, in the world of mathematical language).

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  • $\begingroup$ So is my textbook incorrect in saying that if $A$ and $B$ are true, then $B \implies A$ is true? That's what it says in the truth table. But if we then substitute the Paris and France statement, it doesn't make sense. $\endgroup$ – The Pointer Dec 28 '16 at 2:29
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    $\begingroup$ Suppose it's raining right now; and I tell you 'if I have an odd number of coins in my pocket, it's raining right now'. Is that a true statement? Does it really matter how many coins I have in my pocket? You're right that it's not particularly helpful of me to make that statement; but it's still a true statement. $\endgroup$ – Chas Brown Dec 28 '16 at 2:35
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    $\begingroup$ Well, that would be a bit silly to put it that way. What matters is that if you ALREADY KNOW the conclusion is true (e.g., that it is raining) it doesn't matter whether the premise itself is actually true or false; in either case it is raining. However if we DON'T KNOW the conclusion is true, what we are being told is that IF we know the premise is true THEN the we know the conclusion is true; and that is useful. $\endgroup$ – Chas Brown Dec 28 '16 at 2:49
  • $\begingroup$ Your comment is illuminating. Thank you. $\endgroup$ – The Pointer Dec 28 '16 at 2:50
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    $\begingroup$ No worries. It takes a bit of practice to wrap you're head around the difference between conversational logic and mathematical logic :). $\endgroup$ – Chas Brown Dec 28 '16 at 2:53
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Let's assume you have a number,for example 5. When you are asked 5 is in which circle you will say it is in B. Then by this statement one can say your number is also in A.

But lets assume you have number 3. And you are asked in which circle number is? You day it is in circle A. So one can't say the number is in B!

This case is same , assume circle B as Paris and fields A as France and see your statement.

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