[Math] “If P, then Q; If P, then R; Therefore: If Q, then R.” Fallacy and Transitivity

first-order-logiclogicmetalogicpropositional-calculus

Two fallacious arguments:

  1. If P, then Q
  2. If P, then R
  3. Therefore: If Q, then R

And

  1. If P, then Q
  2. If R, then Q
  3. Therefore: If P, then R

However, if these particular propositions were interpreted as being connected not by a conditional sign but by an "=" (identity) sign, wouldn't we have examples of the transitive property (i.e. P=Q, P=R, so Q=R)?

Best Answer

Why are your statements indented? It does not seem to serve a clear purpose.

If you have both:

  If P then Q.

  If P then R.

It does not at all imply that:

  If Q then R.

Why? Because if P is false, the first two would be (vacuously) true, but it might be that also Q is true and R is false, which would make the last one false! (See this post for an explanation of the conditional.)

Even if you have:

  If ( P implies Q ) then ( P implies R ).

You cannot infer that:

  If Q then R.

Again this is because the first is always true when P is false, but choosing Q and R appropriately makes the second false.

Also, you are very confused about logic. An implication of the form "P implies Q" has absolutely nothing to do with "P iff Q", so it makes no sense to talk about equality. For now, make sure you understand the truth tables of all the logical symbols, and then when you want to write a logical statement you simply ensure that in all situations its truth value matches the value you wish it to have. Do not try to blindly translate from an English sentence into logical form; it is bound to fail in too many ways to count.

Related Question