P → Q ⇔ ¬Q  → ¬P

Read as so: P implies Q, if and only if, not Q implies not P. Implies in this context means “leads to", “implies", or “means". What the above equation means is in layman terms is

if P is true, then Q is also true if and only if Q is false means P is false. 

In other words, transposition logic have the following properties:

1. When we know P is true, then we know Q is also true;
2. When we know P Is false, then we know Q is also false; and finally
3. When we know Q is false, then we know P is also false
4. When we know Q is true, we don’t actually know what P is (P is unknown)

Combined it gives us a proper cause and effect relationship.

Knowing P always gives us Q, but knowing what Q is don’t always gives us the answer to P.

Knowing the “cause", we can always predict what the “effect” will be. The reverse is not always true - knowing the “effect” do not identify the “cause".

If knowing Q gives us the answer to P, then effectively P is the same as Q (a contradiction because P and Q are supposed to be different). Therefore, a claim will pass this logic test when”P always leads to Q”, “if and only if” (this is the bounding predicate condition, ⇔) “not Q always means not P“.  You can refer to Wikipedia for a more in depth article on this, as well as the proof.

Now we look at the left hand side (LHS) of the equation (P → Q). If we let P be Audible and Q be Measurable), the left hand side of the equation now reads:

If something is audible, then it is measurable.

Exactly the same assertion claimed on the Internet. The LHS is quoted on a regular basis by the people in the second camp. The implication that there exists a very strong “one to one” relationship between measurable and audible. 

However, for this statement to be true when applied under the transposition logic test, the right hand side (RHS) of the equation (¬Q  → ¬P) must hold. In other words if we can show ¬Q don’t always imply ¬P, then it means P don’t always imply Q. To test this, with the same P and Q terms, the RHS now reads:

If something is not measurable, then it’s not audible.

The RHS appears on surface to be a close resemblance to claim #2 from the previous page. That is not the case. The two claims in the previous page still follows the P → Q equation (just different P and Q terms). They are in fact not mathematically related, unlike the transposition logic equation shown here.

This RHS equation is hardly ever seen uttered on the Internet, and for good reason! Because there are several examples where something is not measurable but audible. By presenting only the LHS part of the equation, the second more militant camp is applying a passive aggressive form of dismissing a view point that do not agree with their world view - i.e. he is indirectly making his argument that anything you have claim cannot be real because measurements have showed no differences.

The RHS is often omitted because the latter is impossible to argue for because it can be shown to be false readily. And as far as transposition logic is concerned, both sides needs to be true for the claim to be true. If RHS is shown to be false, the LHS is too.

If we can prove that “if something is not measurable, then something is not audible” to be false, then the “If something is audible, then that something is measurable” is illogical.

Or as I prefer to call it - complete and utter bullshit.

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