Why it is not possible to have such argument?

I found some textbook Introduction to Formal Logic and this is from one of the first excercises. This is marked as impossible, but I do not understand why.


The Internet Encyclopaedia of Philosophy has an entry on validity (and soundness, which is often confused). While there are some issues with the entry, as Conifold points out below, the author has the definitions right:

A deductive argument is said to be valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false. Otherwise, a deductive argument is said to be invalid.

A tautology is always true. Therefore, if the conclusion of the argument is a tautology, the conclusion is always true, which means it's impossible for the premises of the argument to be true and the conclusion nevertheless false, which is the definition of the argument's validity.

It's somewhat peculiar that that textbook talks about validity without first defining it. It's a pretty straightforward definition, but usually these books are very precise.

  • Thank you, now I understand clearly. That textbook indeed talks about a basic concepts first. Nevertheless I misunderstood the definition. – Václav Margy Sobotka Jul 7 '15 at 18:26
  • "The author of this article is anonymous. The IEP is actively seeking an author who will write a replacement article", and for a reason. This article adopts model-theoretic notion of logical consequence, confuses it with deductive notion, and then further confuses argument (deduction) with inference, same as the book OP quoted. It is of course perfectly possible to have invalid deduction with a tautological conclusion if it does not follow rules of inference, "deductive argument" in the highlighted passage should be replaced with "inference". iep.utm.edu/logcon/#SSH2b.ii – Conifold Jul 8 '15 at 0:18
  • @Conifold thanks, I added a note about this. – user2953 Jul 8 '15 at 8:13
  • Here is an invalid argument with tautological premise and conclusion: (P) a=a; (S) a=b; (C) b=b. The inference from P to C is valid, because its validity depends on premise and conclusion only, but the argument is not, because S is not inferred by P. For a deduction to be valid every step has to be valid, not just the inference. So either "deductive argument" is not used in its usual sense, or the IEP definition is erroneous. – Conifold Jul 8 '15 at 23:01
  • @Conifold an argument has premises and a conclusion, not something in between. If there is an intermediate step you're actually talking about two arguments that are in some way related. If you think a definition is erroneous, you should provide reference for that. – user2953 Jul 8 '15 at 23:54

If an argument is invalid, then there is an interpretation where all the premises are true and the conclusion is false. So If the conclusion is a tautology, the argument must be valid since the conclusion can't be false under any interpretation.

  • Thank you for your answer. I accepted the second one, because it is more clearer for me. – Václav Margy Sobotka Jul 7 '15 at 18:29

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