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The original statement is, "If I am in Paris, then I am in France". The inverse statement is, "If I am not in Paris, then I am not in France".

If $\lnot A = \lnot B =$ True, then $\lnot A \implies \lnot B$ = True.

If $\lnot A = \lnot B =$ False, then $\lnot A \implies \lnot B$ = True.

Here is where my misunderstanding occurs:

If $\lnot A =$ True and $\lnot B =$ False, then $\lnot A \implies \lnot B$ = True.

After all, the statement, "If I am not in Paris, then I am in France", is a logical statement: One can be somewhere else in France.

If $\lnot A =$ False and $\lnot B =$ True, then $\lnot A \implies \lnot B$ = False.

After all, the statement, "If I am in Paris, then I am in not France", is an illogical statement: If one is in Paris, then they must necessarily also be in France.

I would greatly appreciate it if someone could please take the time to clarify my misunderstanding.

Thank you.

The Pointer
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3 Answers3

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I would like to show you this in truth tables, hopefully it resolves some of the confusion. Of course, you should read the other answers as well, Patrick Stevens answer I like personally.

The connective "$\Rightarrow$" is defined according to this truth table:

$$ \begin{array}{c|l|c|} P & \text{Q} & \text{P $\Rightarrow$ Q} \\ \hline T & T & T \\ T & F & F \\ F & T & T \\ F & F & T \end{array} $$ This should be in your book. As you can see, $P \Rightarrow Q$ is false only when $P$ is true and $Q$ is false, otherwise $P \Rightarrow Q$ is true.

For the contrapositive, we can draw the following table:

$$ \begin{array}{c|l|c|c} P & Q & \neg Q & \neg P & \neg Q \Rightarrow \neg P \\ \hline T & T & F & F & T \\ T & F & T & F & F\\ F & T & F & T & T\\ F & F & T & T & T \end{array} $$

As you can see, for any valuation of $P, Q$ (for any assignment of $T,F$ to $P,Q$), whenever $ P \Rightarrow Q$ is true, so is $\neg Q \Rightarrow \neg P $.

So as you can see, if $\neg Q$ is false, and $\neg P$ is true, then $\neg Q \Rightarrow \neg P$ is true, by the third line in the above table. This is exactly in line with what is says in your book.

For the converse (or inverse) , we have:

$$ \begin{array}{c|l|c|c} P & \text{Q} & \text{P $\Rightarrow$ Q} & Q \Rightarrow P \\ \hline T & T & T & T \\ T & F & F & T \\ F & T & T & F \\ F & F & T & T \end{array} $$

As you can see, they don't match for all valuations, in particular, if $P$ is true, and $Q$ is false, then one implication is true and the other isn't.

Finally: if $A$ is the statement "you are in Paris" and $B$ is the statement "you are in France" and you know $A \Rightarrow B$ is true, then if you are not in France, then you are not in Paris ($\neg B \Rightarrow \neg A$).

And of course, if you are not in Paris ($\neg A$ is true) then you could be on Mars for all we know. You are not necessarily in France.

EDIT: I would like to encourage you to look at more mathematical examples, similar to other peoples' examples in their answers, natural language is full of vague statements that are not always as clear cut as mathematical ones.

JuliusL33t
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  • Thanks for the response. Unfortunately, despite the many responses, it seems that no one has been able to address my question. I apologise because I must have written the OP badly. I will post it again below. – The Pointer Dec 28 '16 at 14:03
  • If $\lnot A =$ True and $\lnot B =$ False, then $\lnot A \implies \lnot B$ = True.

    After all, the statement, "If I am not in Paris, then I am in France", is a logical statement: One can be somewhere else in France.

    If $\lnot A =$ False and $\lnot B =$ True, then $\lnot A \implies \lnot B$ = False.

    After all, the statement, "If I am in Paris, then I am in not France", is an illogical statement: If one is in Paris, then they must necessarily also be in France.

     – The Pointer Dec 28 '16 at 14:04
  • According to the inverse logic tables, the above statements for $\lnot A \implies \lnot B$ are incorrect. My confusion lies in why this is so, since my reasoning given above seems to be logical. – The Pointer Dec 28 '16 at 14:06
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    "If I am not in Paris, then I am in France" is not a logical statement. Not being in Paris says nothing about whether you are in France or not. You could be in France, or maybe somewhere else. $\neg A \Rightarrow \neg B$ is equivalent to $B \Rightarrow A$, which is different from $A \Rightarrow B$. – JuliusL33t Dec 28 '16 at 14:21
  • Your reasoning for this one makes sense to me. What about the other (more troublesome) one? – The Pointer Dec 28 '16 at 14:22
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    If $\neg B$ is true, i.e. you are not in France, and this is established already to be true, would you agree that whether you are in Paris or not doesn't change the fact that you are in not France? you are ALREADY not in France. Thus "I am in Paris" $\Rightarrow$ "I am not in France" is a true statement, regardless of the truth value of "I am in Paris", under the condition that "I am in not in France" is assigned $T$. And so: $A \Rightarrow T$ is true and $\neg A \Rightarrow T$ is true, for any $A$, whether $A$ is "I am in Paris" or anything else. – JuliusL33t Dec 28 '16 at 14:41
  • If you tell me that you are not in France, then we can have a discussion about whether you are in Paris or not, and whether we agree that you are in Paris or not, we will still arrive at "you are not in France" regardless of whether we say "you are in Paris" or "you are not in Paris". – JuliusL33t Dec 28 '16 at 14:46
  • Note that in the above example, we haven't determined whether "I am in Paris" is true, only that "I am not in France" is true. If we set "I am in Paris" to true, then "I am in France" is necessarily true. I.e setting "I am in France" to false will result in the situation $T \Rightarrow F$ which is false. – JuliusL33t Dec 28 '16 at 14:56
  • Your reasoning is fascinating. For instance, let's take the statement, "If you study hard, you will get good grades". The inverse of this statement would be, "If you do not study hard, you will not get good grades". If $\lnot A =$ false (If you study hard) and $\lnot B =$ true (You will not get good grades), then $\lnot A \implies \lnot B$ must be true (If you study hard, then you will not get good grades). This is because $\lnot A \implies \lnot B$ ONLY makes a claim for the situation where you do not study hard ($\lnot A =$ true). Is this reasoning correct? – The Pointer Dec 28 '16 at 15:17
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    Maybe: Lets take "if you study hard, then you will get good grades" as a true statement. The inverse is "if you dont study hard, you dont get good grades", this is not necessarily true, you could get good grades for other reasons. If we put $\neg B$ as true, then whether you study or not makes no difference, you will still not get good grades, since $\neg B$ is true. $\neg A \Rightarrow \neg B$ does not follow from $A \Rightarrow B$, $A \Rightarrow B$ says nothing about $\neg A \Rightarrow \neg B$, but if $\neg B$ is true, then $\neg A \Rightarrow \neg B$ is automatically true. – JuliusL33t Dec 28 '16 at 15:34
  • Yes, I think I understand now. Thank you for being so gracious with your time. – The Pointer Dec 28 '16 at 15:38
  • From $A \Rightarrow B$, you can claim neither $A$, nor $B$, nor any of their negation, nor $\neg A \Rightarrow \neg B$, but you CAN claim $\neg B \Rightarrow \neg A$ – JuliusL33t Dec 28 '16 at 15:38
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    No sweat, glad I could help. I'm sure this will settle in your mind in a few days, think about various examples for 10 minutes a day, and this will be second nature in no time – JuliusL33t Dec 28 '16 at 15:39
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The negation of $(A\Rightarrow B)$ is not $(\lnot A\Rightarrow\lnot B)$, you have also to reverse the implication.

In fact $(A\Rightarrow B)$ is equivalent to $(\lnot B\Rightarrow \lnot A)$ : "If I'm not in France then I'm not in Paris".

This comes from $(A\Rightarrow B)$ is equivalent to $(\lnot A\lor B)$ and you can clearly see that $(A\Rightarrow B)=(\lnot A\lor B)=(\lnot \lnot B\lor \lnot A)=(\lnot B\Rightarrow \lnot A)$.

Now if you negate the implication you get $\lnot(A\Rightarrow B)=\lnot(\lnot A\lor B)=(A\land \lnot B)$ : "I am in Paris and I am not in France" which is obviously false since you negated a true proposition to start with. But this $(A\land \lnot B)$ does not have a signification with some $\Rightarrow$, it's just something different.

So all these are true :

  • If I am in Paris then I am in France (proposition)
  • If I am not in France then I am not in Paris (contraposition)
  • If I am not In Paris then either I am not in France either I am in France but not in Paris (negation)

As you can see, the negation doesn't tell much, just that well... you are not in Paris.

zwim
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  • Thanks for the response. My textbook states that the inverse of "A implies B" is "NOT A Implies NOT B"; Is this incorrect, or am I just misunderstanding it? – The Pointer Dec 28 '16 at 12:53
  • If you follow what I wrote (NOT A implies NOT B) = (B implies A). So your textbook just says that the inverse of (A implies B) is (B implies A), this is a semantic inverse but not a logical NOT. – zwim Dec 28 '16 at 12:56
  • I see. So which part of my "if ... , then ... " statement was incorrect? – The Pointer Dec 28 '16 at 13:01
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    @ThePointer What your book calls the "inverse" is often called the "converse"; note that the inverse and the original statement are usually not equivalent. The contrapositive of "A implies B" is "not B implies not A", which is equivalent to the original statement. – Patrick Stevens Dec 28 '16 at 13:01
  • @PatrickStevens What do you mean they're not equivalent? For the inverse statement, don't I just add "not" to the hypothesis and conclusion to make it into a valid inverse statement? – The Pointer Dec 28 '16 at 13:03
  • @ThePointer I've tried to answer this properly in my answer parallel to this one. – Patrick Stevens Dec 28 '16 at 13:14
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Let $P$ be the statement "$A$ implies $B$".

The inverse (or converse) of $P$ is the statement "NOT $A$ implies NOT $B$".

The contrapositive of $P$ is the statement "NOT $B$ implies NOT $A$".

The following is always true: $P$ holds if and only if the contrapositive of $P$ holds. That is, $P$ is equivalent to its contrapositive.

The following need not be true: if $P$ holds, then the inverse of $P$ holds. (Let $A$ be "$x=2$", and $B$ be "$x$ is a natural number", so $P$ is the true statement "if $x=2$, then $x$ is a natural number", and the inverse of $P$ is the false statement "if $x$ is a natural number, then $x=2$".) That is, $P$ is not equivalent to its inverse (indeed, $P$ does not even imply its inverse).

The following need not be true: if the inverse of $P$ holds, then $P$ holds. (Just swap $A$ and $B$ around in the previous case.) This is another reason why $P$ is not equivalent to its inverse: $P$ is not even implied by its inverse.

To address specifically "If $\neg A$ is True, and $\neg B$ is False, then $\neg A \Rightarrow \neg B$ is True": you're saying that something which is True implies something which is False. To see why this can't be right, could you similarly say that $1+1=2$ implies $1=2$?

Notice also that everything is implied by a false statement (this is "vacuous truth"), so if $\neg A$ is false, then $\neg A \Rightarrow X$ for any statement $X$.

Patrick Stevens
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  • Thanks for the elaborate response. I am a novice in logic, so I unfortunately do not understand much of what you are saying. Take my statement in the OP: "If I am in Paris, then I am in not France". This is an illogical statement: If one is in Paris, then they must necessarily also be in France. How does it make sense otherwise? – The Pointer Dec 28 '16 at 13:23
  • "If I am in Paris, then I am not in France" is a false statement, indeed. – Patrick Stevens Dec 28 '16 at 13:32
  • But that's the problem: my textbook states that if $\lnot A =$ false (I am in Paris) and $\lnot B =$ true (I am not in France), then $\lnot A$ implies $\lnot B$ is true. So where is my misunderstanding? – The Pointer Dec 28 '16 at 13:36
  • Oh, I understand now. I misunderstood you the first time. Given that I'm not in Paris, the statement "If I am in Paris, then I am not in France" is true, though it's in general (that is, if I'm allowed to be in Paris) false. You might find it clearer if it's stated as "Suppose I am not in Paris. If I am in Paris, then I am not in France", where it's more obvious that the antecedent "if I am in Paris" is false. – Patrick Stevens Dec 28 '16 at 14:13