Evening Note for Wednesday, September 5

Thought for the Evening: The Pseudo-Scotus Validity Paradox

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:



Currently Reading

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

Poem a Day III

Pole and Ice

On arctic snows the central star
is shining like the glinting ice;
around its gleam the heavens spin
until the dawn begins to rise.
Auroral promise will pour out gold
in aureate splendor crowned;
but fair and still are midnight stars
that dance their circumpolar round.
And you, my love, are like the pole,
the pivot on which the heavens turn,
and I am like the ice below,
with ardent starlight graced and burned.
Yet soon the dawn will sear the sky
and every star be lost to view
and I shall blaze with sunlight white
and then shall have forgotten you.

Voyages Extraordinaires #40: Mirifiques Aventures de Maître Antifer

It is September 9th, 1831. The captain left his cabin at six o'clock. The sun is rising, or to speak more exactly, its light is illuminating the lower clouds in the east, for its disk is still below the horizon. A long luminous effluence plays over the surface of the sea, which is broken into gentle waves by the morning breeze.

After a calm night there is every promise of a fine day— one of those September days in which the temperate zone occasionally rejoices at the decline of the hot season.

The captain rests against the skylight on the poop, places the telescope to his right eye, and sweeps the horizon.

Captain Antifer, as it often is titled in English, is a treasure-hunt story. Pierre Antifer, a gruff, sarcastic, and stubborn Breton sailor, is the son of a man who did the wealthy Kamylk Pasha, of Egypt, an extraordinarily important favor. Because of this, the Kamylk Pasha, who wanted to keep his immense fortune from his treacherous family, sent Antifer's father part of the instructions for finding where the Pasha had buried it all -- in particular, the latitude. Another person who had done the Kamylk Pasha a great favor has been given the longitude. And when the two are united, they set off to find a fortune. Dragged along is Antifer's friend Gildas Tregomain, and his nephew, Juhel, who dreads the treasure, because he knows that if Antifer finds it, Juhel will not be allowed to marry the love of his life, Enogate. But the Kamylk Pasha has decided not to make the treasure hunt easy; before the end, they will have had to visit the Gulf of Oman, the Gulf of Guinea, Edinburgh, and the very northern island of Spitzbergen, in the Svalbard archipelago, and finally the mysterious final destination -- where the result of the hunt will not be what anyone expected when they began.

Poem a Day II

Once

I knew you, my darling
when the morning was formed,
when the dawn was a maiden
and the planets just born,
when the courses of ages
had not yet run dry
and the stars were just blooming
in meadows on high.
Then the great orbs, spinning,
had not yet grown old
through the length of their journey
on the Milky Way road.
The waters were fresh,
the airs were still clean,
and you were the splendor
that haunted my dreams
in the yester and once
and the day long ago,
in the yore of the ages
that sleep now below.

Poem a Day I

(This series will be weekdays only.)

Heritage

A tree without roots is overturned by the wind,
unable to battle the storms that descend;
and how, and how can you hold your head high
when your past is nothing but shadows that die?

The building is lost when the ground gives away,
when the base that has crumbled the rest will betray,
and how, and how can you hold your head high
not even remembering the tears that were cried?

When the grain is not stored, the seed will be lost,
no spring will be sown, with harvest its cost;
and how, and how can you hold your head high
with no food for the children save the breezes that sigh?

Servus Servorum Dei

Today is the feast of Pope St. Gregory the Great, also known as the Dialogist, Doctor of the Church. For his Pastoral Rule, Book I, Chapter 2, with salutary words for these days:

Hence it is that the Lord through the prophet complains of the contemptible knowledge of shepherds, saying, When you yourselves had drunk most pure water, you fouled the residue with your feet; and My sheep fed on that which had been trodden by your feet, and drank that which your feet had fouled (Ezekiel 34:18-19). For indeed the shepherds drink most pure water, when with a right understanding they imbibe the streams of truth. But to foul the same water with their feet is to corrupt the studies of holy meditation by evil living. And verily the sheep drink the water fouled by their feet, when any of those subject to them follow not the words which they hear, but only imitate the bad examples which they see. Thirsting for the things said, but perverted by the works observed, they take in mud with their draughts, as from polluted fountains Hence also it is written through the prophet, A snare for the downfall of my people are evil priests (Hosea 5:1; 9:8). Hence again the Lord through the prophet says of the priests, They are made to be for a stumbling-block of iniquity to the house of Israel. For certainly no one does more harm in the Church than one who has the name and rank of sanctity, while he acts perversely. For him, when he transgresses, no one presumes to take to task; and the offense spreads forcibly for example, when out of reverence to his rank the sinner is honoured. But all who are unworthy would fly from the burden of so great guilt, if with the attentive ear of the heart they weighed the sentence of the Truth, Whoever shall offend one of these little ones which believe in me, it were better for him that a millstone were hanged about his neck, and he were drowned in the depth of the sea (Matthew 18:6). By the millstone is expressed the round and labour of worldly life, and by the depth of the sea is denoted final damnation. Whosoever, then, having come to bear the outward show of sanctity, either by word or example destroys others, it had indeed been better for him that earthly deeds in open guise should press him down to death than that sacred offices should point him out to others as imitable in his wrong-doing; because, surely, if he fell alone, the pains of hell would torment him in more tolerable degree.

Fortnightly Book, September 2

Baudolino is the fourth of the novels of Umberto Eco, and also the next fortnightly book. I will, of course, be reading the William Weaver translation. Baudolino of Allessandria has come to Constantinople with singularly bad timing; he has barely arrived in 1204 when the Fourth Crusade sacks the city. He saves the life of the historian, Niketas Choniates, and while they are holed up trying to survive the sack and its aftermath, Baudolino tells Niketas the story of his life.

Of course, it's an Eco novel, so you know that there will be more to it than that. Baudolino is a compulsive liar; it's impossible to tell how much of what he says is true and how much is fiction. Niketas Choniates is a historical figure; he wrote an extraordinarily important history that is one of our sources for what happened during the Sack of Constantinople. So we have a liar telling his fictional history to someone concerned with historical fact; but, as you would expect from Eco, there is some question as to how false the lies are and how true the historical fact actually is. And a recurring theme throughout the work is the notion of a map of regions unknown, the picture of the world we build beyond the borders of the realm for which we can personally vouch. What is fiction and what is fact, especially when we are speaking of fictions about facts and facts about fictions?

In any case, it will be interesting to see Eco handling the Middle Ages again.