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The fallacy of affirming the consequent



The logical fallacy below has been given the name "The fallacy of
affirming the consequent".
A common valid argument is

B -> A, B, therefore A /1/

We have two premises: the first is the conditional B -> A (if B then A)
and the second is the proposition B. These two premises VALIDLY produce
A. In other words, if
B is true and if A is a corollary of B, A is true as well.
The above valid form is typically confused with

B -> A, A, therefore B /2/

which is INVALID. If A is a corollary of B and if A is true, B is NOT
NECESSARILY true. The fact that A, the consequent in the conditional B
-> A, is taken as a
premise, has generated the name "The fallacy of affirming the
consequent".
Look at Einstein's "Relativity: the special and general theory". Let us
assume that A is the postulate of the constancy of the speed of light:

A: x = ct <-> x' = ct'

which is a biconditional and can be read: if and only if x=ct then
x'=ct'. Then Einstein introduces a relation of which A is obviously a
corollary (Einstein
explicitly recognizes this):

B: (x' - ct') = L(x - ct)

If B is true, A is obviously true. However if A is true, there is no
limitation on the values of the variables in B and B is generally
false. So we have B -> A as
a premise. If we write

B -> A, B, therefore A /3/

the argument is valid. However Einstein needs something different:

B -> A, A, therefore B /4/

He proceeds in accordance with /4/ - builds Lorentz equations on B and
so creates the sequence (A therefore B therefore Lorentz equations) -
the illusion is that
Lorentz equations ultimately stem from A. In fact, A CAN be a corollary
of B or Lorentz equations, in accordance with /3/, but Lorentz equations
can BY NO MEANS be
deduced from A.

If there is interest, another logical curiosity - "Deriving from an
inconsistency" - could be discussed. Logicians give examples such as

Logic is easy.
Logic is difficult.
Therefore crocodiles can fly.

The argument is valid(!?!?) so far as no circumstances can be imagined
in which both premises are true and the conclusion false. This approach
is essential in thermodynamics.

Pentcho Valev