What is the logical reasoning that allows us to work backwards with implications? If $C \implies D$ and $E \implies C$, then $E \implies D$

Question

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.

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.
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.

Is this even correct? We cannot derive $E\Rightarrow C$ using $E\Rightarrow D$ and $C\Rightarrow D$. — Levent, Dec 31 '16 at 09:04
@Levent You're correct. I made a typing error and have fixed it. I apologise! — The Pointer, Dec 31 '16 at 09:06
$E\Rightarrow C$ 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. — Levent, Dec 31 '16 at 09:13
@Levent But notice that we did not work forwards to $D \implies E$: that's what I meant by sequential. — The Pointer, Dec 31 '16 at 09:14
@G.Sassatelli That makes sense. Thank you. What about the reasoning behind (1)? — The Pointer, Dec 31 '16 at 09:15
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 $A\implies B$"? — bof, Dec 31 '16 at 09:35

score 4 · Accepted Answer · answered Dec 31 '16 at 09:48

4

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 $E \implies D$.

answered Dec 31 '16 at 09:48

Kanwaljit Singh

8,610

1

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

score 1 · Answer 2 · answered Dec 31 '16 at 10:10

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.

What is the logical reasoning that allows us to work backwards with implications? If $C \implies D$ and $E \implies C$, then $E \implies D$

2 Answers2