What is fallacious about 4 consecutive uses of modus ponens?

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.

[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.

• I imagine this falls into the wide gap between formal logic and practical logic. I don't see anything fallacious in the argument; as given, it's provably correct. The trouble is that for practical usage, these truth values are partial (likelihoods, not certainties), and each statement or premise contains a long string of supporting arguments and assumptions. Those uncertainties are compounded by consecutive usage. Put crudely, if both are 70% likely, the conclusion may only be 50% likely. – Ask About Monica May 28 '15 at 19:13
• A minor technical point, this argument does not use modus tollens four times. It appears to use the transitivity of implication together with one application of modus tollens. – Nick May 28 '15 at 19:53
• @LePressentiment I think the original argument (without premise 3) uses transitivity with modus tollens since, presumably, we do not wish things to be bad (i.e., the negation of the consequent of premise 2, which appears to be assumed as an unstated premise). If we use your suggested premise 3, then we would be applying transitivity and modus ponens to arrive at the conclusion, but again you are basically assuming that we do not wish things to be bad as an unstated premise. – Nick Dec 24 '15 at 2:07
• @NickR +1. Thank you for your explanation which I comprehend now. – AYX.CLDR Jan 3 '16 at 22:43

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).

• If it is valid, it would withstand replacing 'guns' with 'seatbelts'. – user9166 May 29 '15 at 2:25
• @jobermark And you feel it doesn't? You made me doubt myself, so I went ahead and worked it out, see edits above. It's arguably a better argument with "seatbelts" --I believe that makes it both valid AND sound. – Chris Sunami supports Monica May 29 '15 at 3:12
• Sorry, I confused myself. You are right. – user9166 May 29 '15 at 4:30
• @jobermark I'm afraid your argument is entirely incoherent to me. The fact that the premises are context dependent is not structural. And you can't EVER make a formal argument sound or unsound without altering the truth of the premises. What you really mean is that the same premises are true in one situation and false in another. That has nothing to do with validity. – Chris Sunami supports Monica May 30 '15 at 18:49
• Let's simplify the argument. P1 If guns are outlawed, everyone with a gun will be an outlaw. P2 if everyone with a gun is an outlaw, that would be bad. C If guns are outlawed that would be bad. This argument is valid, but unsound --P1 is false, and P2 is questionable. If it were on a plane it doesn't change anything. If you recast the argument to take place on a plane it changes the premises. If P2 is conflating "bad" and "worse" (than the alternative), that is a problem with the argument, but not a structural one. You're abusing the concept of interpretation. – Chris Sunami supports Monica May 30 '15 at 20:25

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.

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.

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.

• It is not a fallacy, it tests validity. Logically valid arguments are independent of their content. (Hence the Lewis Caroll style of logic training exercises about flying pigs and boiling seas.) They may still be sound, but only by having unarticulated premises that fill in the gaps. – user9166 May 29 '15 at 2:10
• One can test the validity of inference by replacing oranges with TV-sets, for example,yes. But one cannot test the validity of a premise such as "it would be bad if criminals had nukes" by replacing "nukes" with "oranges". Possibly however we're talking at cross purposes here. – Cheers and hth. - Alf May 29 '15 at 2:39
• Premises have truth-value, not validity, in the technical language folks now prefer for teaching logic. I am generally not fond of layers of convention fine tuning commonsense words, but if someone is trying to work within this kind of system, I try to stick with the domain's silly conventions. – user9166 May 29 '15 at 3:07
• Thanks for that about truth-value versus validity, yes sorry. – Cheers and hth. - Alf May 29 '15 at 3:22