The Pseudo-Scotus Validity Paradox gets its name from the fact that it occurs in a commentary to the Prior Analytics that was attributed at one time to Scotus; since it uses technical vocabulary that we have reason to think was only invented some decades after Scotus's death, and have no other serious candidates for authorship, it gets tagged with the 'Pseudo'. The paradox given in that work is:
God exists; therefore this argument is invalid.
The author, of course, takes 'God exists' to be a necessary truth; nothing actually depends on the content of the proposition, so you can substitute any necessary truth that you'd like. A standard definition of validity is that an argument is valid when, if the premises are true, it is impossible for the conclusion to be false. Then it follows that if the above argument is valid, it will have a true conclusion if its premise is true. Its premise is necessarily true. So if it is valid, the conclusion is true, and it is invalid. So, since its validity implies its invalidity, it must be invalid. But we have also just proven that if its premise is true, it must be invalid, and therefore it is impossible for its conclusion to be false. So it is valid.
There are a number of things that could be done here.
(1) Exception. If I recall correctly, Pseudo-Scotus's own solution was to argue that our account of validity is incomplete; that it in fact should have an exception: An argument is valid when it is impossible for the premises to be true and the conclusion false, unless it denies its own inference. A difficulty with this is that it doesn't deal with all kinds of arguments like this. For instance, the following argument is a similar paradox that is not affected by this exception:
This argument is valid; therefore God does not exist
where, of course, the conclusion is taken to be impossible (so you could substitute any impossible proposition; I think Buridan uses "A man is a donkey"). But one could perhaps generalize the idea to saying that an argument is valid when it is impossible for the premises to be true and the conclusion false unless assuming it to be valid implies that it is invalid.
(2) Logical Form. One could perhaps argue that the definition was incomplete in a different way. When we talk about validity we often talk about logical form. So one could argue that an argument is valid only when, if the premises were true, it would be impossible because of its logical form for the conclusion to be false. The validity paradox does not work entirely by virtue of its logical form (the conclusion does not follow strictly formally from the premises and it requires knowing the content of 'This argument is invalid'), so this would work nicely if there were any generally accepted account of logical form. Unfortunately, there are lots of disagreements about the nature of logical form.
(3) Cassation. One could say that the argument is neither valid nor invalid because, despite appearances, it is 'meaningless'. One could blame the self-reference, of course; but as with propositions, there seem to be perfectly unproblematic self-referential arguments. For instance,
This argument is valid; therefore this argument is valid
looks like it is valid by pretty much any standard of validity. But one could very well just refuse to allow arguments to comment on their own validity, or lack thereof; after all, if we were to change the previous argument to
This argument is valid; therefore this argument is invalid
we run into paradoxes again. But on the other hand again, simply eliminating self-reference from propositions to stop the (very similar) Liar paradox for propositions runs into the problem that Liar-style paradoxes don't actually require self-reference; you can create similar paradoxes with pairs or triplets, or any number, of propositions. And it seems that this would be true here, as well.
(4) Dale Jacquette had a nice paper, "The Validity Paradox in Modal S5", in which he argued that (with a significant qualification) the validity paradox depends on a modal fallacy. If the argument is valid, it is invalid. But to get a paradox, you need also to conclude that if it is invalid, it is valid. And this is not as easy to get as the above summary made it sound. You can easily show that if the argument is invalid, the conclusion is true. But we aren't talking about the truth of the conclusion; we'd need to prove that if the conclusion is true, it is impossible for its premise to be true and its conclusion false. But all that follows from the conclusion itself is that it is possible that when the premise is true, the conclusion is false. The conclusion that it is valid comes from assuming that case -- which we have not actually established to be actual, but only possible. Therefore we can't directly get a conclusion stronger than that if it is invalid, it is possibly valid, and in the right modal logical system, there is no problem with something being not-X-but-possibly-X. But Jacquette also argues that this is not a costless solution: there are modal logics in which you can, in fact, get the paradox, most notably the very popular modal system S5 -- so either our concept of validity must change, or we must have more than one kind of validity, or we must reject S5 outright. None of these are especially desirable options.
Various Links of Interest
* Jack Stripling and Megan Zahneis, The Big Lie, about an academic scandal involving a forged job offer.
* Brett Frischman, The Misleading Power of Internet Metaphors
* Eva Del Soldato, Basil (Cardinal) Bessarion, at the SEP
* Ashley Smart, The War Over Supercooled Water
* Martin Chaplin, Water Molecule Structure
* Michael Pakaluk, Capital Punishment and the Sex Abuse Crisis, at "First Things"
* T. Greer, Tradition Is Smarter than You Are; I particularly find interesting the discussion of how augury practices may have often had the effect of breaking up potentially harmful human patterns.
* James Chastek, First Way, visually
* American Catholics have been in an uproar recently because of allegations by Archbishop Carlo Maria Viganò of extensive cover-up in the handling of Cardinal McCarrick; and then in an uproar again because Pope Francis said he would not say a single word about the allegations; and then in uproar again because of the response of Cardinal Cupich Iscariot that Pope Francis had more important things to do than respond to the allegations, selling out the Body of Christ for the thirty pieces of his ecclesiastical career. I've noted before that for the bishops to hunker down in this case is a mistake; that they've chosen to do so is a grave misfortune for us all, since it means that things will get very, very much worse before they get better. But this was always a possibility to begin with. When corrupt underbrush has grown so thick, the fire will burn very fiercely.
It's worth remembering through it all that the betrayed in this case are not merely the laity but also a very large number of priests. There have been a great many priests who have done nothing wrong, and yet must bear the opprobrium of it all, and who have essentially been hung out to dry by all but a fraction of the bishops while they either (in the bad cases) stonewall outright or (in the less bad cases) stall for time until they can come up with something. (There are a few bishops whose response, at least, has been to say reasonable things.) So here's something from Father Mike Schmitz as a reminder of their position:
Umberto Eco, Baudolino
Rosamund Hodge, Endless Water, Starless Sky
Jean-Luc Marion, Believing in Order to See
J. R. R. Tolkien, The Fall of Gondolin
Jules Verne, Michael Strogoff