What is fallacious about 4 consecutive uses of modus ponens?

Question

_{Source: 8 minutes 56 seconds juncture, Lecture 12-2 (transcription), ... How to Reason and Argue,
by Prof Ram Neta PhD (U Pittsbugh; in Philosophy)}

Okay, now refutation by parallel reasoning doesn't always work. Sometimes, we get results that are unclear or don't show what we were trying to show. [...] consider the following argument.

[Premise 1:] If guns are outlawed, only outlaws will have guns.

[Premise 2:] If only outlaws had guns, it would be bad.
[Caution: I reordered this premise; strangely, Neta preposed the apodosis before the protasis].

[Conclusion:] Therefore, guns should not be outlawed.

[...] that's an argument. Is it a fallacy or not? Well suppose we try to refute it by parallel reasoning ... [by replacing 'guNs' with 'guMs'.] Now notice, this parallel argument has exactly the same form as this earlier argument about guns. The two arguments have the same form. So if one of them is a fallacy then the other one is a fallacy. But is this argument a fallacy. That's not clear.
[...] So we can't tell by looking at the parallel argument whether the earlier argument about guns was a fallacy. So this is a case of refutation by parallel reasoning that doesn't succeed.

Where is the fallacy? The argument appears to lack only the following Suppressed Premise:

[Premise 3:] If it is bad that only outlaws had guns, then guns shouldn't be outlawed.

Answer 1

The argument is not fallacious, it is a valid argument (although, as you noted, an elliptical one with some suppressed premises).

If we include your premise 3, it is this form:

P1: If A then B
P2: If B then C
P3: If C then NOT A

Indirect Proof; Proof by Reductio Ad Absurdum:
1. Assume A.
2. B        by P1, 1, Modus Ponens
3. C        by P2, 2, Modus Ponens.
4. NOT A      by P3, 3, Modus Ponens.
5. A and NOT A   by 1, 4, Contradiction!

Conclusion: Therefore NOT A    by 1-5, Indirect Proof.

However it is arguably unsound. A valid argument is correct structurally, but the conclusion is only as sure as the premises. If the premises are false, the conclusion may or may not be true; the conclusion's truth is no longer certified by the argument, regardless of structural validity.

It's also worth nothing that Premise 3 is arguably begging-the-question (assuming the conclusion) so it may be illegitimate to include it in the first place (the argument isn't valid without it).

Answer 2

Regarding the lecturer's “replacing 'guNs' with 'guMs'”, that's a fallacy. It ignores the connection between guns and outlaws, a connection that's not there for oranges or TV sets. That said, that fallacy in the attempted disproof does not make the original argument less fallacious.

A main problem with the original argument is that “outlaws” is used in two different senses: before the ban on guns, and after the ban of guns (where it's a much larger set). I.e. the original argument uses equivocation to deceive. That's not the only problem, but it's enough to dismiss it as a concrete argument.

However, on the third and gripping hand, we presently have a very important case of the general form of that argument, to deal with:

• If secure cryptography is outlawed, then only the state and outlaws will have secure cryptography.

• If only the state and outlaws have secure cryptography, it would be bad (we do want to be able to look the state in the cards, we don't want criminals to have a guaranteed strong advantage over us).

• Therefore, secure cryptography should not be outlawed.

I happen to believe that this concrete instance of the argument form is valid. But I may be wrong – the conclusion is after all in line with my views on the matter, so I may tend to only see that which supports my views.

Answer 3

The fallacy is that these two propositions are not the same kind of fact, in a way that does not allow them to be straightforwardly combined.

The form of the second premise is simply too broad to argue from. It is in effect a modal statement in disguise, containing an unarticulated prediction and a moral posture. Such statements escape logic by the magic of modal ambiguity, like counterfactuals. Things can 'be bad' in too many unrelated ways. You need a set of additional constraining understandings to make sense of 'bad'.

Consider replacing 'guns' with something less reciprocally empowering, like 'tools specifically intended for starting cars you don't own'. (Yes, these exist, and automatic start buttons made them less easy to control -- no key blank in a radio key.) Pretty much only criminals do have those, and I might wish I had such a thing in an emergency, but by and large normal people would never use them. So we happily outlaw them.

For the gun argument to hold water, the 'badness' has to be a kind of 'badness' that is not overly general or absolute. It seems obvious that, say, on a plane, the danger a gun creates is pretty absolute and balance would not help. You could hardly have a standoff in the aisle or the cockpit, productively. And if anyone shot, even in self-defense, others would be unreasonably endangered -- you might lose cabin pressure. So they are banned. If the argument were truly valid, this should not matter.

I am sure there are other examples, but to my mind they must all depend upon the changeability of the level of badness and how contingent its application might be, because those are the parameters of the modal aspect of the assertion "If only criminals had guns, that would be bad."

But both premises are also pretty weak on their face.

First, as a matter of practice premise one is false. The police and the army would still have guns. But assuming premise one is true, premise two is highly questionable, because the police and army have far worse things to unleash on outlaws than mere guns, and so do civilians, really.

Answer 4

Premise 2 is a gusty one. If you're not a fan of arguments favouring legalized gun ownership, that'd be the interesting one to question.

For example, one could argue along the lines of "if only outlaws have guns, it would be bad, but it would be worse if we had even more guns to go around," which is a very popular argument. One could also argue that it might not be bad. What makes it so bad for outlaws to be the only ones with guns? There's arguments for either side of that question, dealing with the ability to force others to do as one wishes. The point is: premise 2 is arguable.

For many reasons, premise 2 is sufficiently suspect that you can challenge the argument without having to resort to calling people out on fallacies.

Another interesting argument: going from premise 2 to premise 3 goes from a statement about the state of the world to a recommended action. No premise like the following is written down:

[Premise 2a:] if a state is bad, we should avoid actions which lead to that state"

While people would not generally disagree with 2a at first, once we start discussing the definition of "bad," we find that arguments like this are... tricky at best. See religious situations, where the religions clearly dictate what we should do, and yet we do otherwise (and often defend our actions using logic!).

Heck, if you felt like it, you may argue against premise 1 as leading the question. Premise 1 explicitly focuses only on the negative aspects of making guns illegal. You may argue that, without including a complete realistic view of the effects of gun ownership, the logic may be valid but the premises are insufficient to act on the results. For example, it's trivial to make a solid case against vaticcinating your kid, by ignoring all the good things about vaccines, and showing there is at least one downside to it.