I understand that we can work forwards using logical implications:

If $A \implies B$ and $B \implies C $, then $A \implies C$.

However, I am unsure of the reasoning that allows us to work backwards.

  1. For instance, If $C \implies D$ and $E \implies C$, then $E \implies D$. Whereas working sequentially forwards with implications seems to make logical sense, I am not sure of the reasoning that makes it logical to work in a 'discontinuous' fashion, as show in the preceding example.
  2. I am also wondering if, in the preceding example, we could also say that $D \implies E$?

I would greatly appreciate it if someone could please take the time to explain the reasoning behind this concept.

Thank you.

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    $\begingroup$ Is this even correct? We cannot derive $E\Rightarrow C$ using $E\Rightarrow D$ and $C\Rightarrow D$. $\endgroup$ – Levent Dec 31 '16 at 9:04
  • $\begingroup$ @Levent You're correct. I made a typing error and have fixed it. I apologise! $\endgroup$ – The Pointer Dec 31 '16 at 9:06
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    $\begingroup$ $E\Rightarrow C$ and $C\Rightarrow D$ is the same thing as $C\Rightarrow D$ and $E\Rightarrow C$. You say that you understand the forwards using logical implications, this is the same thing. $\endgroup$ – Levent Dec 31 '16 at 9:13
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    $\begingroup$ 2) No, you can't. For instance, if this evening I decided to sacrifice a human being to the Sun God while wearing a lioncloth and some ostrich feathers, tomorrow the sun will rise. And, if tomorrow the sun rises, the night won't last forever. On the other hand, the fact that tonight is not going to last forever is not a proof that I murdered a person (in fact, I did not). $\endgroup$ – user228113 Dec 31 '16 at 9:13
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    $\begingroup$ You're saying that you understand "If $A\implies B$ and $B\implies C$ than $A\implies C$" but you don't understand "if $B\implies C$ and $A\implies B$ then $A\implies C$"? What is the difference you see between "$A\implies B$ and $B\implies C$" and "$B\implies C$ and $A\implies B$"? $\endgroup$ – bof Dec 31 '16 at 9:35

In logic we have "And" as a product. And product is commutative.

If $C \implies D$ and $E \implies C$ is same as $E \implies C$ and $C \implies D$ then you can say $E \implies D$.

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    $\begingroup$ The "and" here is a conjunction in the metalanguage, not in the syntax of propositional logic. $\endgroup$ – Hans Hüttel Dec 31 '16 at 12:35

Both situations mentioned in the question arise from the fact that implication is transitive: If $p_1 \Rightarrow p_2$ and $p_2 \Rightarrow p_3$, then $p_1 \Rightarrow p_3$.

Can we conclude that if $p_1 \Rightarrow p_2$ and $p_2 \Rightarrow p_3$, then also $p_3 \rightarrow p_1$? No. A simple counterexample is the case where $p_1$ is false, $p_2$ is false and $p_3$ is true.


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