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What do you say to the following way of teaching "if" and "the following are equivalent"? Has somebody ever taught it like this?

An implication A -> B can be viewed as asserting that B is at least as true as A. Thus if A -> B and A -> B then A and B have the same truth value ("at least as true" in both directions). Also, under this interpretation it is easy to see that it suffices to prove a cyclic chain of implication A_1 -> A_2 -> ... -> A_n -> A_1 in order to show that A_1, ..., A_n are equivalent (that is, that they have the same truth value).

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    $\begingroup$ " if A -> B and A -> B" : Typo? $\endgroup$ – Joseph O'Rourke Sep 27 '16 at 22:36
  • $\begingroup$ You can use $A_1 \to A_2 \to \ldots \to A_n \to A_1$ (note the $'s) to get $A_1 \to A_2 \to \ldots \to A_n \to A_1$. $\endgroup$ – dtldarek Sep 28 '16 at 9:53
  • $\begingroup$ Present this as a generalization of $A\Leftrightarrow B$. Your chain is the same as $A_1\Leftrightarrow A_2,\ A_2\Leftrightarrow A_3,\cdots, A_n\Leftrightarrow A_1$ but organized in a way tailored for proving certain "big concepts." $\endgroup$ – user52817 Sep 28 '16 at 19:54
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Maybe Terence Tao's logic section of his text "Analysis 1" is an example. He writes:

One can also think of the statement “if X, then Y ” as “Y is at least as true as X” - if X is true, then Y also has to be true, but if X is false, Y could be as false as X, but it could also be true. This should be compared with “X if and only if Y ”, which asserts that X and Y are equally true.

So there he explains why $(A\implies B)\quad\land\quad (B\implies A)$ is the same as saying that $A$ and $B$ have the same truth value. He also gives the following exercise:

Suppose you know that X is true if and only if Y is true, and you know that Y is true if and only if Z is true. Is this enough to show that X, Y, Z are all logically equivalent? Explain.

Using the information given in the text (that "If A, then B" means "B is at least as true as A"), the solution you have given is obvious.

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I don't know that that's the most common way of teaching it, however, the idea that "if A then B" represents an ordering of truth values such that B is at least as true as A is mathematically and conceptually sound.

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It's an interesting idea. In programmers' terms, if you count 0 as your "truth" value and 1 as your "false" value (which is how it works in some programming languages), then:

$A => B$

means "A is equal to or greater than B."

A >= B

;)

So if A is 0, then B has to be 0 as well (true) because it can't be anything smaller.

Likewise if B is 1 (false), then A can't be anything greater, so must be equal (also false).

But if A is 1, B could satisfy the comparison by being either 0 or 1.


It's a very interesting (and humorous!) thought experiment.

I would NOT recommend it as your first pedagogical entry to the subject of logical implication.

Bring it up after your students are comfortable with the idea of logical implication, and bring it up labeled clearly as an unusual way of looking at it, not generally agreed upon. (My advice.)

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