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I don't understand what a sound argument is. And what does it mean for a premise to be false? Why does case 3 (A is false, B is true) not apply in the real world?

Here the author says that the first premise is false. But how can $A \lor B$ be false in case 3 when $B$ is true?

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  • $\begingroup$ Why do you say B is true? $\endgroup$
    – David
    May 21, 2014 at 13:26
  • $\begingroup$ @David: When B is true in the conclusion, it is true in the first premise. $\endgroup$
    – mosceo
    May 21, 2014 at 13:27
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    $\begingroup$ $B$ isn't true. It is false. It is a false conclusion of a valid but unsound argument whose premises are false. $\endgroup$
    – MJD
    May 21, 2014 at 13:29
  • $\begingroup$ But in your comment, "when" is the same as "if": that is, "if B is true in the conclusion". But why do you say that B really is true, in the conclusion or elsewhere? Do you know who Teller is? Do you know that he has never taught logic? $\endgroup$
    – David
    May 21, 2014 at 13:29
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    $\begingroup$ But that is precisely the author's point - he is saying that B is false, therefore the argument has led to a false conclusion because although the logic is correct the facts are not. See my answer for more. $\endgroup$
    – David
    May 21, 2014 at 13:41

5 Answers 5

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The following proof is a valid argument, however, the conclusion of the theorem is clearly false. What went wrong?

Theorem 1 Let $1 = 0$, then all natural numbers are equal to zero.

Proof by induction. Obviously, $0 = 0$. Now, let $k$ be any natural number $\geq 1$. By inductive hypothesis we have $k-1 = 0$. Using our assumption we get $k-1+1 = 0+0$, that is $k = 0$ which concludes the proof.

Some funny examples of this kind happen with loaded questions. For example, if you were to answer the well-known loaded question presented below by "Yes, I have" or "No, I haven't",

                                         Have you stopped beating your wife?

then you would admit that, at some point, you were doing it (and that you actually have a wife). To respond to such a question, one usually points out (in whatever way) that it is based on false premises.

A sound argument is one which is both valid and its premises are true. The above is not sound, because the premise $0 = 1$ is not true. Still, the difference is rather subtle. For example, if the conclusion of the theorem was the whole implication $$(0=1) \implies \forall k\in\mathbb{N}.\ k=0$$ then the only premises are the axioms of logic, natural numbers, etc., and such a theorem would be both valid and sound.

I hope this helps $\ddot\smile$

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    $\begingroup$ @GitGud Clearly thanks. $\endgroup$
    – dtldarek
    May 21, 2014 at 15:36
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    $\begingroup$ These loaded questions always seemed to have a definitively right answer to me. If somebody asks me "Have you stopped beating your wife?", my reply is simply no. If you had a logical statement "Ax: x has stopped beating their wife", A(me) would certainly be false. I have not stopped beating my wife. Of course, the lay person may interpret this to mean that I am currently beating my wife. But that doesn't logically follow. $\endgroup$
    – Cruncher
    May 21, 2014 at 16:45
  • $\begingroup$ I like that inductive proof BTW. The conclusion is intuitively obvious of course(n = 1 + 1 + ...(n times).. + 1), but it's elegant. $\endgroup$
    – Cruncher
    May 21, 2014 at 16:48
  • $\begingroup$ No, the proof is valid but not sound. $\endgroup$
    – cjm
    May 21, 2014 at 17:18
  • $\begingroup$ @Cruncher Propositional calculus deals with declarative statements. Often times one can translate natural language questions into propositional calculus by asking the truth value of $P$. In the case of the question "Have you stopped beating your wife?" one could consider $P:$ "I have at some point beaten on my wife regularly and I've now stopped it". If $P$ is true, then the answer to the question is yes. If $P$ is false, then the answer is no. But I see no reason to consider $P$ instead of $Q:$ "If I have, at some point, beaten on my wife regularly, then I have now stopped". $\endgroup$
    – Git Gud
    May 21, 2014 at 17:32
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Note the very first sentence in the last paragraph: "Viewed as atomic sentences". With this, you're supposed to strip the sentences of their meaning and look at them just as literals (i.e. propositional letters or negations of these).

Thus you can attribute truth values to the propositional letters, (see valuations).

Then the author talks about case 3 which I'm guessing is the valuation $(A,B)\mapsto (F,T)$.
This valuation makes both the premises true and the conclusion true also.

Now the author says that in the real world case 3 can't happen, this means that if you now stop looking at the sentences as propositional letters and give them meaning again, then something impossible happens, namely that Teller has never taught logic (this is somehow known to be false, maybe Teller is the author).

The term false is being used with two different meanings, one of them is the natural language interpretation, the other one is as a valuation.

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    $\begingroup$ Teller is the author; the example is from chapter 4 of Paul Teller, A Modern Formal Logic Primer. $\endgroup$
    – hmakholm left over Monica
    May 21, 2014 at 13:41
  • $\begingroup$ So the whole point is that you can assign truth values that contradict to the real life and get the conclusion true? It is difficult for me to understand why would you assign false to A (~ Teller has never taught logic) knowing that Teller actually taught logic. $\endgroup$
    – mosceo
    May 21, 2014 at 13:56
  • $\begingroup$ @Graduate Regarding your first question, it's not that you get the conclusion as true. What you do get is a valid argument, which says nothing about the conclusion. Regarding the second question (which is missing a question mark, but I'm guessing it is a question), for practical purposes, assigning true to Teller has never taught logic, yields nothing useful, but this is a very simple example. It's not always so obvious what is going on. By assigning truth values you get a mechanical way to deciding whether an argument is valid or not. $\endgroup$
    – Git Gud
    May 21, 2014 at 14:01
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    $\begingroup$ @Graduate When studying the validity of an argument, whether or not the premises of an argument hold true only matters if we have true premises and a false conclusion. So, here Teller assigns false to A to indicate something about validity. $\endgroup$
    – Doug Spoonwood
    May 21, 2014 at 14:05
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    $\begingroup$ The first premise is false because Teller is not ten feet tall and because he has taught logic, that is, because both $A$ and $B$ are false. $\endgroup$
    – Git Gud
    May 21, 2014 at 14:11
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Using the rules of natural deduction or a truth table, you can prove that $$[A\lor B]\land \neg A \implies B$$

As such, it is a valid argument.

It would be a sound argument if and only if both premises $[A\lor B]$ and $\neg A$ are true.

It would be unsound if and only if either premise is false.

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I recently answered this question about valid and invalid arguments, see if it helps.

An argument is sound if it is valid and has true premises. In this case the conclusion is guaranteed true. I don't think your author's example is very good. See if this one helps.

Anyone born outside the USA cannot legally be president.

Barack Obama was born outside the USA.

Therefore, Barack Obama cannot legally be president.

I hope you agree that the logic here is perfectly correct: if the first two statements are true, then the conclusion has to be true. The problem is that (according to most people) the first two statements are not both true - specifically, the second is false - and therefore the conclusion is not guaranteed to be true. This is an example of an argument which is valid but not sound.

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  • $\begingroup$ Conider the premise from my example: Teller is ten feet tall or Teller has never taught logic. Suppose an intelligence agency carries out an investigation and comes up with this exact conclusion. Would this premise be true now? After all we don't know who Teller is and why we are given this premise. There are situations when this premise might be legit. $\endgroup$
    – mosceo
    May 21, 2014 at 13:48
  • $\begingroup$ Of course there are imaginable situations where this statement would be true. But the author's point is that regardless of what any intelligence agency might claim, in the real world the statement is not true - read his third last sentence. $\endgroup$
    – David
    May 21, 2014 at 13:56
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A sound argument comes as one where the premises are true, and the conclusion is true. For example:

(2+2)=4

Therefore,

(2+2)=(4+0)

makes for a sound argument as does:

(4+6)=10

(5+1)=6

Therefore,

(4+(5+1))=10

Case 3 doesn't hold, because neither does it hold that Teller is ten feet tall, nor does it hold that he hasn't taught logic.

One reason the notion of a valid argument becomes important, because we might end up studying or working with some system where we don't know whether the premises hold true, or if they hold false (but we can tell that one of the two must hold... or we know what the possibilities are). For instance, suppose that we know we have some particular binary operation % given to us. We're given the following argument

(3%4)=7

4=(1%3)

Therefore,

(3%(1%3))=7

The argument holds as valid no matter what "%" indicates, because of the structure involved here. The premises could hold true... say if "%" indicates addition. But, the premises could hold false also... say if "%" indicates multiplication. Either way though, the argument holds valid by its form. Consequently, we can tell something about "%" and know that certain consequences will follow even if we can't figure out the truth of the premises.

As perhaps a better example to illustrate why validity can be important, one can argue:

If a set of axioms A is independent, then there does not exist a model of the axioms which has some property P.

There does exist a model of the axioms which has some property P.

Therefore, the set of axiom A is not independent.

This argument holds as valid. But we can't tell a priori whether or not for a given set of axioms there exists a model of the axioms which has some property P. But, the argument holds as valid nonetheless and thus enables a sound argument, in a particular case, if we can find some instance where the premises hold true.

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  • $\begingroup$ Your first sentence is not really accurate. For example, consider "every square is a quadrilateral; every square has four sides; therefore, every quadrilateral has four sides". In this argument the premises and conclusion are all true. However it is not valid because the logic is wrong (converse error). So it is not a sound argument: the conclusion has turned out true "by good luck" , not by logical necessity. $\endgroup$
    – David
    May 22, 2014 at 6:49
  • $\begingroup$ @David Without thinking about the details of what you've written, I'd say that the accuracy of my first sentence depends. What logic did we presuppose to consider that argument? If we work with say a relevant logic, then my sentence won't work as accurate. But, if we work with classical logic, then since (p->(q->r)) is a tautology where r is a tautology, then we can deduce r from "p" and "q" using modus ponens. And with your example, with "p" as the first premise, "q" as the second premise, and "r" as the third premise isn't (p->(q->r)) true? $\endgroup$
    – Doug Spoonwood
    May 22, 2014 at 14:51
  • $\begingroup$ Note that the concept of validity does not fundamentally apply to an argument but to an argument form, that is, an argument composed not of specific statements but of variables like $p,q,r$. The point is to separate the question of whether the logical derivation of the conclusion from the premises is accurate, from the question of whether the premises are actually true. So it is irrelevant to ask if $p\to(q\to r)$ is a tautology when $r$ is a tautology: it only matters whether $p\to(q\to r)$ is a tautology. $\endgroup$
    – David
    May 22, 2014 at 21:04

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