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In my course book we have something called implication arrows $\Rightarrow$ and equivalence arrows $\Leftrightarrow$ and I have never managed to understand them.

When do I know which to use and how do I know that I'm correct when I use them?

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  • $\begingroup$ @Theo Buehler: Thanks, updated! $\endgroup$ – Zolomon Jul 5 '11 at 10:18
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    $\begingroup$ I replaced the arrows by their LaTeX equivalent and changed the (induction) tag to (logic). $\endgroup$ – t.b. Jul 5 '11 at 10:20
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    $\begingroup$ This question deals with the "equivalence arrow". $\endgroup$ – Américo Tavares Jul 5 '11 at 11:48
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Suppose you have two propositions $P$ and $Q$.

  • $P\Rightarrow Q$ means that $P$ implies $Q$ (or if $P$, then $Q$).
  • $P\Leftrightarrow Q$ means that $P$ implies $Q$ and $Q$ implies $P$ (or $P$ if and only if $Q$).

Let me add a few simple examples.

  1. Since $\frac{2p}{2q}=\frac{p}{q}$ ($q\ne 0$) we can write $$\frac{2p}{2q}=5\Leftrightarrow \frac{p}{q}=5.\qquad (1)$$

    Its meaning is that $\frac{2p}{2q}=5$ is true if and only if $\frac{p}{q}=5$.

  2. a) If you have $y=x$, then you can say that $y^2=x^2$ and write $$y=x\Rightarrow y^2=x^2.\qquad (2)$$

    Its meaning is that if $y=x$ is true so is $y^2=x^2.$

    b) If you have $y=-x$, then you can say that $y^2=x^2$ and write

    $$y=-x\Rightarrow y^2=x^2.\qquad (3)$$

    Its meaning is that if $y=-x$ is true so is $y^2=x^2.$

    c) If you know that $y^2=x^2$ you can guarantee that $y=x$ or $y=-x$ and write

    $$y^2=x^2\Rightarrow y=x\quad \text{or}\quad y=-x.\qquad (4)$$

    Its meanig is that if $y^2=x^2$, then $y=x$ or $y=-x$.

    d) Combining $(2),(3)$ and $(4)$ you have the following equivalence

    $$y^2=x^2\Leftrightarrow y=x\quad \text{or}\quad y=-x.\qquad (5)$$

    Its meanig is that $y^2=x^2$ if and only if $y=x\quad \text{or}\quad y=-x$.

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$A \Rightarrow B$ has the meaning $B$ follows from $A,$ but this does not necessarily hold the other way around. For instance, from $x = 13$, it follows that $x$ is prime. But you can't argue the other way around. So if $\mathbb P$ is the set of prime numbers, you can write $x = 13 \Rightarrow x\in\mathbb P$, but not $x\in\mathbb P\Rightarrow x=13.$

$A\Leftrightarrow B$ has the meaning that $A$ and $B$ are the same statement, just transformed. If $A\Leftrightarrow B$, then both $A\Rightarrow B$ and $B\Rightarrow A.$ For instance, consider an integer $x$. You can say that if $x$ is even, then $x$ is not odd. This certainly also holds the other way around, so you can write $\mathrm{even}(x)\Leftrightarrow\lnot\;\mathrm{odd}(x).$

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Well, as far as I know the $\Rightarrow$ which you call the implication arrow can be used to for implying statements.

  • Example : $x^{2}-1 = 0 \Rightarrow (x+1)(x-1)=0$.

The second arrow that is $\Leftrightarrow$ I have seen it often being used for if and only if statements. That is $A \Leftrightarrow B$, means If you assume $A$ then $B$ is true and if you assume $B$ then $A$ should be true (or can be deduced.)

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There is a significant subtlety about $A \Rightarrow B$ which it is useful to clear up right at the beginning and which has been the subject of other threads. It is based on the use of implication in logic.

It is that $A \Rightarrow B$ is taken to be True except in the case that A is True and B is False.

Stated like that it can seem obvious, but this means that if A is false, then the implication is taken to be true whether B is true or false. This is a convention, which can seem counterintuitive at times, but which is very useful (which is why the convention is the way it is). Intuitively the convention encodes the fact that if A is false, the implication tells us nothing about B.

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I find the implication of logical implication really interesting. Starting off with the truth tables for the two, note only one subtle difference:

+---+---+-------+
| P | Q | P ⇒ Q |    
+---+---+-------+
| T | T |     T |
| T | F |     F |
| F | T |     T |
| F | F |     T |
+---+---+-------+

+---+---+-------+
| P | Q | P ⇔ Q |    
+---+---+-------+
| T | T |     T |
| T | F |     F |
| F | T |     F |
| F | F |     T |
+---+---+-------+

There is a common misconception that P ⇒ Q means that if P is true, Q is true. What the truth table shows is that P ⇒ Q is always true when P is false, which I think has profoundly interesting implications.

In some ways the truth table is simply a formal version of the old computer cliche "garbage in, garbage out": P actually tells us nothing about Q.

Q, on the other hand, if we know it is false, tells us if P is true or not provided we know if P ⇒ Q is true or false. But if Q is true, it gives us no clue as to whether P is true or false.

It is a lesson that we can only advance knowledge by working back from known wrong conclusions. There is no way to link known correct conclusions to what we think causes them with logical certainty.

This may seem obvious to lawyers and detectives, but it is very counterintuitive to us computer programmers who think of P as a pattern to be matched by an if statement, and Q as the resulting action.

My view is logical implication creeps up in other fields such as the old statistics cliche "correlation does not imply causation" and Karl Popper's scientific method of falsifiability. But I've never really seen this articulated anywhere.

Through deduction we can whittle down the choice of possible causes, Ps, for a given Q until the answer seems obvious, but even then it is never 100% certain.

Logical implication encourages an open mind which remembers “there are more things in heaven and earth, Horatio, than are dreamt of in your philosophy”. Many zealots out there could have been cured if our basic education system taught logical implication at an early age.

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  • $\begingroup$ Somewhat at a loss why my answer got voted down, but nevermind. Just stumbled on this quote from Bertrand Russel who said what I was trying to say above better: “Causation, like much that passes muster among philosophers, is a relic of a bygone age, surviving, like the monarchy, only because it is erroneously supposed to do no harm.” $\endgroup$ – joeblog Oct 18 '19 at 13:01

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