1

Modus ponens, that is [(A→B) and A, therefore B], is a valid argument.

If we use the form (1) [(A→B) and B, therefore A], the argument is no longer valid because for the assignment [A = False, B = True], we have the premises (A→B) and B both true but the conclusion A false.

It is said that propositional logic must hold only for the form of the arguments, not for the meaning of the propositions.

But try using A = "Today is Saturday", B = "Tomorrow is Sunday".
The assignment [A = False, B = True], which makes the argument invalid, is not possible, because A and B can only be both true or both false.
So argument (1) becomes valid? It seems like a small question but if you think about it, it is fundamental.

ryang
  • 38,879
  • 14
  • 81
  • 179
Santone Opla
  • 13
  • 2
  • With thoise specific $A$ and $B$, we have more than $A\to B$, we rather have $A\leftrightarrow B$ and can conclude away. – Hagen von Eitzen May 09 '21 at 06:09
  • Thank you, so you confirm that for these two propositions a more appropriate argument requires a double implication ... My problem is that this means that it is not possible to use formalism alone, without a priori knowledge of the meaning of the propositions – Santone Opla May 09 '21 at 06:21
  • An invalid argument is not one that results in a false conclusion; it is one that could result in a false conclusion. – Arturo Magidin May 17 '21 at 19:09

1 Answers1

1

The validity of an argument gauges not the truth of the conclusion, but whether a false conclusion can result from premises that are all true. If yes, the argument is deemed invalid; otherwise—since there is no case whereby all the premises are true yet the conclusion is false—its corresponding conditional must be logically valid (in propositional calculus, this means a tautology).

Argument (1)—its form is called “affirming the consequent”—is invalid, because its corresponding conditional $((A\rightarrow B)\land B)\rightarrow A$ is not a tautology, since it is false for $A$ False and $B$ True.

The fact that when $A$ and $B$ are “Today is Saturday” and “Tomorrow is Sunday”, respectively, the conditional $((A\rightarrow B)\land B)\rightarrow A$ is true (weaker than tautologically true, i.e., true regardless of interpretation of $A$ and $B$) is just a peculiarity of this tautologically-equivalent choice of $A$ and $B.$ However, this doesn't affect argument (1)'s validity, which, as you correctly observe, depends on its form, not its atomic propositions' meanings.

ryang
  • 38,879
  • 14
  • 81
  • 179
  • Thanks so much. My only observation is that if, in their reasoning, someone uses the concept of valid argument as a tool for exclude the consideration of a conclusion of an invalid argument, he/she risks to exclude something that may ultimately be true in reality in certain particular situations. – Santone Opla May 18 '21 at 20:51
  • You’re most welcome @SantoneOpla To be clear, rejecting an argument as invalid is not falsifying its conclusion, but rather, pointing out that the argument is useless for determining its conclusion's truth. In fact, a valid but *unsound* argument is also useless for determining its conclusion's truth. – ryang Apr 10 '22 at 03:33