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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)?

David Marlowe
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  • I would tend to agree that you cannot necessarily translate a verb of being into an equal sign. For instance, we could agree that "Alice is smart" and "Bob is smart"; but we could not then conclude (by the transitive and symmetric properties of 'is' (equals)) that "Alice is Bob". Here though, I would say that "Alice is smart" really means "Alice is an element of the set of smart people". – paw88789 Sep 01 '16 at 20:30
  • In first order logic, I don't know if I've seen "=" used. We can say P is equivalent to Q by $\equiv$, in which case we expect transitivity. The main problem is that the conditional only works one direction, while equivalence allows both directions, so you can logically conclude R from Q or P from R. –  Sep 01 '16 at 20:31
  • The conditional connective $\to$ is not the verb "is". – Mauro ALLEGRANZA Sep 02 '16 at 05:50
  • By the way, the conditionasl is transitive : "if $P \to Q$ and $Q \to R$, then $P \to R$". It is not reflexive : not "if $P \to Q$, then $Q \to P$". Thus, it isnot an equivalence relation (refl, symm, trans), that is the "minimal requirement" to be interpreted as "equality". – Mauro ALLEGRANZA Sep 02 '16 at 05:52

1 Answers1

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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.

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