A knockdown argument is one that is indisputable, or virtually so. It has premises and a conclusion. There is no reasonable way to object to the premises’ truth; they are obviously true. And there is no way to argue with the logic of the argument: it’s obvious that if the premises are true, then the conclusion is true too. Hence, we can conclude with all confidence that the conclusion is true too.

That sounds on the right track. But it’s not hard to uncover a paradox at the heart of the notion of a knockdown argument.

Consider the TS1 argument ‘Taylor Swift is or is not human; hence, 1 = 1 or 1 ≠ 1’. I realize that it’s a strange argument, since the premise is weird, the conclusion is weird, and there is no connection between the premise and conclusion. But stick with me while I analyze it.

TS1 has one premise and one conclusion. It’s logically impossible for its conclusion to be false: there is simply no way ‘1 = 1 or 1 ≠ 1’ could be false. It follows that it’s logically impossible for its conclusion to be false and its premise to be true. But that’s the definition of a deductively valid argument. Hence, TS1 is deductively valid. Moreover, it’s as obvious as it can get that its premise is true: surely, Taylor Swift is either human or not. Hence, TS1 is deductively valid with all true premises. But that’s the definition of a deductively sound argument. Thus, TS1 is deductively sound. In addition, it is 100% obvious that it is sound, since everything I just argued is 100% obvious to anyone who knows even the most elementary logic. The premise and conclusion aren’t merely logical truths; they are obviously logical truths.

So far, there is nothing remarkable or controversial going on. Things get interesting when we ask if TS1 is a knockdown argument. There are exactly two options here: hold that that TS1 is/isn’t knockdown.

Suppose that TS1 isn’t knockdown. What follows?

We then have the result that there is a 100% obviously sound argument (TS1) that isn’t knockdown. In fact, we have infinitely many such arguments, since there are infinitely many arguments exactly like TS1, with obviously logical truths as both premises and conclusions. The degree of obviousness is as high as it can ever get.

But that should seem very odd. If it’s perfectly obvious that an argument is deductively valid and has a true premise, then well, that is about as strong as an argument can ever get. It’s impossible to quarrel with the premise’s truth! And it’s impossible to quarrel with the validity of the argument! It’s obvious that it’s literally impossible for the premise to be false, and it’s obvious that it’s literally impossible for the conclusion to be false if the premise is true! Arguments never get any stronger than that. So, it’s weird to think that TS1 isn’t knockdown.

Suppose instead that TS1 is knockdown. What follows?

Either the premise supplies good reason to accept the conclusion or it doesn’t. Let’s look at each of those two options, to see what they come to.

If the premise doesn’t supply good reason to accept the conclusion, then surprisingly enough there are 100% obviously sound knockdown arguments (e.g., TS1) whose premises don’t supply good reason to accept their conclusions. But that’s insane: how on earth can a knockdown argument not supply good reason to believe its conclusion? Knockdown arguments are supposed to be exceedingly excellent ones! How could an exceedingly excellent argument not supply good reason to believe its conclusion?

If the premise does supply good reason to accept the conclusion, then ‘Taylor Swift is or is not human’ amazingly supplies good reason to accept ‘1 = 1 or 1 ≠ 1’, which is bizarre as well. What in God’s name does Taylor Swift have to do with arithmetic? Nothing. She isn’t that talented.

Summing all this up, we have proven that one of the following is true:

• Some 100% obviously sound arguments aren’t knockdown.
• There are 100% obviously sound knockdown arguments whose premises don’t supply good reason to accept their conclusions.
• ‘Taylor Swift is or is not human’ supplies good reason to accept ‘1 = 1 or 1 ≠ 1’.

If we analyze the options for argument TS2, which results when we switch TS1’s premise and conclusion, then we prove that at least one of the following is true (only the third bullet point changes; compare it with the previous one):

• Some 100% obviously sound arguments aren’t knockdown.
• There are 100% obviously sound knockdown arguments whose premises don’t supply good reason to accept their conclusions.
• ‘1 = 1 or 1 ≠ 1’ supplies good reason to accept ‘Taylor Swift is or is not human’.

Now consider argument TS3, which has premise = conclusion = ‘Taylor Swift is or is not human’. In other words, the premise is exactly the same as the conclusion. When we go through the same reasoning as we did for TS1 and TS2, then we prove that one of the following is true, with the new difference showing up in the third bullet point again:

• Some 100% obviously sound arguments aren’t knockdown.
• There are 100% obviously sound knockdown arguments whose premises don’t supply good reason to accept their conclusions.
• ‘Taylor Swift is or is not human’ provides good reason to believe itself.

I have focused on the phrase ‘provides good reason for’, when relating premise and conclusion for TS1, TS2, and TS3. We could use the stronger ‘proves’ instead and get similar results.

Putting all this together, we have uncovered a new philosophical paradox, since we have rigorously proven that at least one of the following highly counterintuitive claims is true, even though we have not said anything about which of them is true: 

1. In some 100% obviously sound knockdown arguments the premises neither prove nor supply good reason to accept the conclusion.
2. Some 100% obviously sound arguments aren’t knockdown.
3. The two claims ‘Taylor Swift is or isn’t human’ and ‘1 = 1 or 1 ≠ 1’ each prove and provide good reason to accept the other. In addition, any obviously logical truth proves itself and supplies good reason to accept itself.

I will articulate and briefly comment on four responses to this paradox. I endorse the second one, but I won’t argue the point.

First response: (1) is true. When one is introduced to logic, one realizes that logicians and philosophers are using ‘argument’ in an odd way so that all you need is any claims at all as premises and conclusions. The premises and conclusions need not be on the same topic at all. Hence, the extension of ‘argument’ as used in logic and philosophy goes far beyond that of ‘typical, real-world argument’. That’s uncontroversial.

Similarly, according to our first response to the paradox, the extension of ‘knockdown argument’ goes far beyond that of ‘typical, real-world knockdown argument’. None of the TS arguments is a typical, real-world one. Since the notion of argument has been revealed to be stretched beyond the typical, we should accept that the notion of knockdown argument is stretched as well. Hence, for some 100% obviously sound knockdown arguments--which are going to be pretty weird--the premises neither prove nor supply good reason to accept the conclusion. That is, (1) is true.

The problem with the first response is that it’s highly counterintuitive to entirely separate ‘knockdown argument’ and ‘supplies good reason for its conclusion’. Sure, we can get used to odd results for almost any notion. But one might think that there is just no way the premises of a knockdown argument can utterly fail to supply good reason for their conclusion. That sounds almost contradictory. One would think it’s bedrock that in a knockdown argument the premises supply very strong reason for the conclusion. If so, then the first response is out.

An advocate of the first response might reply to this criticism by noting that some arguments have no premises at all. Clearly, their premises don’t supply good reason for their conclusions, since there are no premises to do the deed. Let TS4 have no premises but ‘Taylor Swift is or is not human’ as its conclusion. It’s obviously sound. It’s knockdown. It has no premises. Hence, knockdown arguments can fail to have premises that supply good reason for their conclusions.

However, it’s question-begging to assume that TS4 is a knockdown argument. The knockdown status of these peculiar arguments is what is at issue here.

As you might expect, some people will reasonably complain that the logician’s use of ‘argument’ to include zero premise “arguments” is silly, similar to using ‘biological mother’ to include women who have zero children. Perhaps it’s convenient for certain purposes to use ‘argument’ in such a way to apply to premise-less items, but that doesn’t mean that ‘argument’, as it is used in contemporary English, applies to those items.

We need not enter into that fight. Even if there are arguments with no premises, it’s question-begging to assume that TS4 is knockdown, as the advocate of the first response did three paragraphs back.

Second response: (2) is true. One might argue that although the extension of ‘argument’ is weird/stretched as described in the first response, that’s not so for ‘knockdown argument’. The adjective ‘knockdown’ puts us in the realm of typical, real-world arguments only. For evidence of this claim, just look at what philosophers or anyone else has ever said about knockdown arguments (e.g., see discussion and references in Ballantyne 2014); as far as I know they never, ever, even thought to consider weird arguments like the TS ones. Given that ‘knockdown argument’ has to be restricted to non-bizarre arguments, (2) is true: the bizarre TS arguments aren’t really knockdown. Yes, the TS arguments are perfect in some way--100% obviously sound arguments--which is to their credit, but they aren’t knockdown.

The problem with the second response is that it’s so strange to accept that the best arguments possible aren’t knockdown. If we made one of the TS arguments a little worse, would they then count as knockdown? If the premises are so obviously true that virtually everyone who understands them agrees that they are true, and virtually everyone agrees that it’s impossible for the premise to be true and the conclusion false, that seems to be a pretty damn near perfect argument when it comes to strength. It might not be useful of course, but usefulness is surely different from strength.

One might think that for an argument to be knockdown there has to be some relevance connecting premises and conclusion. We are free to accept that relevance isn’t needed in order to have an argument (cf. relevance logic, Mares 2022), but relevance is needed for an argument to count as knockdown. Since there isn’t any relevance for the TS ones, they aren’t knockdown. Hence, (2) is true.

Well, here’s TS5: Taylor Swift is or is not human; hence, Taylor Swift is or is not human. If you like, replace the second occurrence of ‘human’ with ‘mammal’, ‘female’, ‘physical’, ‘a mother’, ‘a singer’, or anything else you fancy. It’s going to be difficult to insist there is no “relevance”.

Suppose we can spell out ‘relevance’ appropriately so that the TS arguments (as well as others that are similarly problematic) fail to meet some sophisticated relevance condition on knockdown arguments and, hence, don’t qualify as knockdown. Does that really help? We still have to swallow the strange claim that the TS arguments, as strong as they are, aren’t knockdown. That’s the price for embracing this response.

Third response: (3) is true. Well, no. I don’t see any way to make either conjunct of (3) reasonable.

Fourth response: semantics. One might argue that ‘knockdown argument’ is semantically indeterminate enough that neither (1) nor (2) has truth-value (claim (3) can escape because it doesn’t contain ‘knockdown argument’). It’s not implausible to hold that the term ‘electron’ has a precise meaning that is partly determined by what’s out there in nature. But we can’t say the same for ‘knockdown argument’. What the TS arguments do is pressure us into constructing more precise meanings for ‘knockdown argument’. We can go in multiple directions at this point, including either of the first two responses to the paradox. Or so this response says.

One problem with the third response is that it really seems as though each of (1)-(3) is false. No talk about semantic indeterminacy or incompleteness changes that fact. It merely distracts us from the paradox. Even if ‘knockdown argument’ is semantically slippery, we have an apparent proof that at least one of (1)-(3) is true. Semantic theses won’t change that fact. If one wants to argue that ‘TS1 is/isn’t knockdown’ is without truth-value (even though many other sentences of the form ‘Such-and-such argument is/isn’t knockdown’ do have truth-value), then one has a lot of work to do in justifying such a position. Hand-waving remarks about semantic incompleteness or indeterminacy or the like are inadequate.

Moreover, if we take seriously the third response’s point about constructing a more complete/determinate notion of knockdown argument, we end up in the same place as before: once one has constructed something, one has to accept (1), (2), or (3). The paradox doesn’t magically disappear.

The philosophical strength of a paradox lies in the fact that it is unavoidable. Some philosophers are tempted to talk or write endlessly about all sorts of issues related to the paradox and yet they fail to do the one thing that is necessary in any serious response to the paradox: take a stand on the truth-values of each of the pivotal claims. It is not difficult to say reasonable things about the alleged indeterminacy of ‘knockdown argument’, but one doesn’t have a view worth evaluating until one takes a stand on the truth-values of (1)-(3).

References:

Ballantyne, N. 2014. “Knockdown Arguments”. Erkenntnis 79: 525–543.

Mares, E. 2022. “Relevance Logic”. The Stanford Encyclopedia of Philosophy (Fall 2022 Edition), Edward N. Zalta & Uri Nodelman (eds.).