Ethical Realism

June 19, 2011

Proving an Argument Is Logically Valid

Filed under: philosophy — JW Gray @ 9:05 am
Tags: , , , , ,

I have discussed the importance of understanding logical form and how to create formal counterexamples. Understanding logic well is a lot easier when we know something about logical validity, and one way to better understand logical validity is to consider an argument that proves an argument to be valid. If we can know why an argument can be valid, then we can know more about logical validity in general. I will now produce a proof of logical validity here. It can take some time to understand the proof, so you might want to take your time to read it carefully.

Consider the following argument:

  1. If dogs are mammals, then they’re animals.
  2. Dogs are mammals.
  3. Therefore, dogs are animals.

This argument is logically valid. If the premises are true, then the conclusion must be true because it has a valid argument form. The argument form is the following:

  1. If A, then B.
  2. A.
  3. Therefore, B.

We can know that this argument is valid simply by knowing what “If A, then B” means. It means “If A is true, then B is true” or “B is true whenever A is true.” Since, A is true, B also has to be true because that’s what “If A, then B” means.

Nonetheless, we could construct a proof for this argument. Consider the following:

We can summarize the proof using the following words:

  1. We can assume the argument form is invalid. In that case the premises can be true and the conclusion can be false at the same time. Let’s assume the premises are true and the conclusion is false.
  2. If this assumption is impossible because it implies a contradiction, then we know the argument can’t be invalid (and must be valid).
  3. In that case “If A, then B” is true because it’s a premise, A is true because it’s a premise, and B is false because it’s a conclusion.
  4. In that case A must also be false because “if A (is true), then B (is true)” is assumed to be true, and B is assumed to be false. B is true whenever A is true, but B is false, so A must be false. (Consider the statement, “If dogs are mammals, then dogs are animals.” If we find out that dogs aren’t animals, then they can’t be mammals either. If the second part of a conditional statement is false, then the first part must be false.)
  5. Therefore, A is true and false.
  6. Therefore, the assumption that the argument form is invalid leads to a contradiction and must be false.
  7. Therefore, the argument must be valid.


We generally demand that our arguments are logically valid and we have an intuitive grasp about what it means for an argument to be logically valid. Validity isn’t sufficient to have a good argument, but it’s generally a very important element for constructing good arguments. Knowing more about logical validity beyond the intuitive level can help us achieve clarity and improve our thinking. Knowing why an argument is valid can help us achieve these goals.

Update (6/21/2011): I updated my proof that the argument form is valid because the other proof I gave was circular and therefore unpersuasive.



  1. I find your steps a little hard to follow, but it appears to me that you are begging the question: In order to arrive at the contradiction, you must prove B, but your proof for B seems to use some variation of the syllogism you originally set out to prove.

    More generally, it is impossible to prove all the rules of logic from scratch: At least some of them must be postulated to begin with, and what you set out to prove is one of the more common postulates.

    Comment by michaeleriksson — June 19, 2011 @ 1:27 pm | Reply

    • I proved that B is true in step 3 on the chart. We know “If A (is true), then B (is true)” and we know A is true. That means B must be true.

      You can prove principles of logic. That’s what logic does all the time. Of course, proofs are only possible with assumptions. I am assuming people know what “If A (is true), then B (is true)” means without a truth table.

      Did you read the other two things I wrote about logic? This piece was not meant to be read in isolation. People should read the other two pieces first.

      Comment by James Gray — June 19, 2011 @ 8:58 pm | Reply

      • > We know “If A (is true), then B (is true)” and we know A is true. That means B must be true.

        But that is exactly the claim you set out to prove, only written slightly less formally. As I said, begging the question.

        The problem with proofs in logic or math is that they must all have some set of assumptions (and not merely concerning meaning, but actual fact) behind them. We cannot start from scratch and develop a system, but can only strive to make the underlying set of assumptions as small and intuitively plausible as possible (while keeping sufficient power and internal consistency). Even a truth table (while immensely useful) implies a certain set of assumptions—including that the values and their rules actually corresponding to logic.

        (I have just skimmed through your other two pieces, but they do not, at least at first glance, change my interpretation.)

        Comment by michaeleriksson — June 19, 2011 @ 9:21 pm

  2. @michaeleriksson

    I am not begging the question. I do assume people know what “If A, then B” means. We have assumptions given every possible argument. My assumption “If A, then B” is intuitively justified and it’s based on what “If A, then B” means for logicians. People really do use the phrase “If A, then B” in ways that are compatible with the way logicians use the phrase quite often. The way the phrase is used here is uncontroversial.

    My discussion of logic is for those who haven’t taken a logic class and it’s based on improving our intuitive idea of logic. I have written a logic book that discusses all of this in much more detail, but most people don’t want to read a logic book.

    You are right that much more could be said about how we use the word “If A, then B” but we don’t need to have that conversation. People already know for the most part what it means. The same is true about the argument involving mammals and animals. Someone giving that argument isn’t begging the question because they assume people know what mammals, animals, and dogs are. Nothing controversial is being said about these things.

    Comment by James Gray — June 19, 2011 @ 9:33 pm | Reply

  3. Your focus on statements like`I do assume people know what “If A, then B” means.’ makes me unsure what your actual agenda is, because that really has nothing to do with the question of whether your syllogism is valid. By all means, if you merely want to make a certain rule means to the layman, this need not be a problem. However, then there are two other issues that confuse me:

    o You explicitly say “How do we know this argument is valid? Consider the following:”, which to me implies that you want to provide at least something akin to a proof. (And if taken as a proof, it does beg the question.)

    o The effort you go through is out of proportion and the explanation likely too hard for most laymen to understand (at least for those who do not intuitively agree with the original syllogism already after seing an example).

    Comment by michaeleriksson — June 19, 2011 @ 10:01 pm | Reply

    • Sorry, make that “if you merely want to explain what a certain rule means”.

      Comment by michaeleriksson — June 19, 2011 @ 10:05 pm | Reply

      • I don’t understand your objection. Yes, I provide a proof. We can prove things using reductio ad absurdum. Perhaps this proof is too complex for laymen, but I have tutored people taking logic classes and taught people logic in the past. I don’t think this is too complex. Of course, the presentation in this piece might be something that can be improved. It’s easier to explain logic in person and be prepared to answer questions.

        The image used in this piece was accidentally put into the counterexample piece somehow. I deleted it, so the counterexample piece should be easier to understand now.

        Comment by James Gray — June 19, 2011 @ 10:51 pm

  4. What I wrote here is not something I expect laymen to skin through quickly and fully understand. It might take them some time to think about what I’m saying before they can understand my proof.

    Comment by James Gray — June 19, 2011 @ 10:53 pm | Reply

  5. I would suggest that you take your proof, write it down more formally and stringently, and then we can see whether it holds up.

    Comment by michaeleriksson — June 20, 2011 @ 5:53 am | Reply

    • The argument as I presented it here is basically the same as it would be presented using symbolic logic. It is a finished proof. I don’t see what more you want from the argument as far as that is concerned. Symbolic logic can be equivalent to ordinary language and can be explained using ordinary language.

      Comment by James Gray — June 20, 2011 @ 9:22 am | Reply

    • Actually I misunderstood your objection and this point helped you get the message across. I was wrong and you were right. The proof doesn’t just have a bad assumption, it’s viciously circular. The argument as I presented it is allowed in a logic class, but it suffers from an informal fallacy and can’t be persuasive to a skeptic for that reason.

      I think I could give a proof almost just like this one for a disjunctive syllogism without the same problem, and a truth table could also be given for that without raising too many counterintuitive problems.

      A better proof for this argument here (modus ponens) wouldn’t require a reductio ad absurdum. It would just require an understanding of what “If P (is true), then Q (is true).”

      Comment by James Gray — June 20, 2011 @ 10:51 am | Reply

  6. Indeed, two parties misunderstanding each other is one of the greatest sources of disagreement in real life. Fortunately, with those inclined to logic there is a good chance that they will overcome the misunderstandings in the end.

    Comment by michaeleriksson — June 20, 2011 @ 5:55 pm | Reply

  7. […] Proving an Argument Is Logically Valid […]

    Pingback by What’s Hot on the Web! (as far as I’m concerned) « Pilant's Business Ethics Blog — June 23, 2011 @ 10:20 pm | Reply

  8. Thank you for clarifying this for me. I’m taking a few introductory philosophy classes and I’ve been having trouble with assumptions/inferences and argument form. So far it is my understanding that valid arguments are much easier to find than sound arguments; was your disagreement about proofs in the comments related to sound arguments? I’m confused.

    Comment by Alexis — September 20, 2011 @ 1:02 am | Reply

    • We were discussing how to prove that an argument form is valid, but such a “proof” needs to be sound.

      Comment by James Gray — September 20, 2011 @ 3:03 am | Reply

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