The Liar Paradox

Medieval philosophers categorized several logic puzzles as insolubilia, unsolvable things. Probably the most common of these was (and still is*) the Liar Paradox.

Consider the statement "I am lying." If I am truly lying at that moment, then what I just said was true. If the statement is therefore true, however, then to say "I was lying" would be a lie. So which is it?

One 20th century philosopher used Jean Buridan (c.1300-c.1361, mentioned elsewhere in this blog) to claim that it wasn't really a paradox. Arthur Prior said it wasn't really paradoxical because every statement includes an assertion of its own truth. The statement "I am lying." is therefore taken as true—it carries its own truth independent of other sentences or context— and considering it a paradox is an unnecessary complication.

Buridan actually used the Liar Paradox to prove the existence of God. He put forth two statements:
"God exists."
"None of the sentences in this pair is true."

The only consistent way to assign truth values, that is, to have these two sentences be either true or false, requires making “God exists” be true. In this way, Buridan has “proved” that God does exist. [Internet Encyclopedia of Philosophy]

This particular paradox first appears in the middle of the 4th century BCE. Eubulides of Miletus made a list of seven puzzles, one of which was “A man says that he is lying. Is what he says true or false?” His commentary on whether it is true or false is lost to time.

*Those readers of a certain vintage will remember the Star Trek: The Original Series episode "I, Mudd" in which a controlling super-robot is rendered useless by its inability to process the two statements "Everything Harry Mudd says is a lie." followed by Mudd saying "I am lying."

Anglo-Saxon Riddles

As best we can determine, Symposius was a 4th- or 5th-century author of the Ænigmata, a collection of 100 Latin riddles. The oldest known collection of riddles, they have influenced other riddle-makers through the ages, such as Aldhelm.

Why are we talking about Latin riddles in a post titled "Anglo-Saxon Riddles"? Because without Symposius we might have a more difficult time guessing at some of the Riddles of the Exeter Book. Consider Exeter Book Riddle #61:

#61
A creature came     where many men
sat at council     with wise hearts.
It had one eye     and its ears were two;
it had two feet     and twelve hundred heads,
a back and a belly     and two hands,
arms and shoulders,     one neck,
and two sides.     Say what I’m called.

This might have been more difficult if we did not have Symposius' example #94:

#XCIV
Cernere iam fas est, quod vix tibi credere fas est;
Unus inest oculus, capitum sed milia multa;
Quidquid habet vendit, quod non habet unde parabit?

Now may you see, though not believe, I fear,
One eye and many thousand heads are here,
Whate'er he has, he sells. Whence comes what don't appear?

The answer is the same for both, and I will give it to you in footnotes, along with the answers to the rest. Enjoy.

#18
My garment is darkish.     Bright decorations,
red and radiant,     I have on my raiment.
I mislead the stupid     and stimulate the foolish
toward unwise ways.     Others I restrain
from profitable paths.     But I know not at all
that they, maddened,     robbed of their senses,
astray in their actions     —that they praise to all men
my wicked ways.     Woe to them then
when the Most High holds out     his dearest of gifts
if they do not desist     first from their folly.

#57
I war oft against wave     and fight against wind,
do battle with both,     when I reach to the ground,
covered by the waters.     The land is strange to me.
I am strong in the strife     if I stay at rest.
If I fail at that,     they are stronger than I
and forthwith they wrench me     and put me to rout.
They would carry away     what I ought to defend.
I withstand them then     if my tail endures
and the stones hold me fast.     Ask what my name is. 

#31
I saw a thing     in the dwellings of men
that feeds the cattle;     has many teeth.
The beak is useful to it;     it goes downwards,
ravages faithfully;     pulls homewards;
hunts along walls;     reaches for roots.
Always it finds them,     those which are not fast;
lets them, the beautiful,     when they are fast,
stand in quiet     in their proper places,
brightly shining,     growing, blooming.

*Symposius #94 and Exeter #61: a one-eyed garlic seller
Exeter #18: wine
#57: anchor
#31: rake

Source for Symposius
Source for Exeter riddles

The Exeter Book

"The Wanderer" in the Exeter Book

The Exeter Book was mentioned as the source of two poems about St. Guthlac. It holds much more than that, however. Of the four manuscripts we have of Anglo-Saxon literature, the Exeter Book is the largest collection in existence of Anglo-Saxon poetry, including all the Anglo-Saxon riddles we have (but one), and several poems that survive nowhere else.

The original date of composition is unknown, but it is assumed to have been produced as part of the Benedictine revival in the 10th century, when Benedictine monasteries strove to record and preserve manuscripts of all kinds.

Its existence can reliably be traced to the will of Bishop Leofric (1016 - 1072), who left it to the library of Exeter Cathedral in 1072 along with the rest of his impressive (for the time) collection. Exeter was one of the largest scriptoria in Leofric's lifetime, where manuscripts were created and copied, so it is surprising that this particular manuscript seems to have been so abused.

Several pages at the beginning are believed missing along with the cover. Several pages are scored as if the book was used as a cutting board. One reader of the book clearly set his drink down on the page, leaving a stain, and several pages at the end of the book show burn marks.

The Book contains religious texts; not just the aforementioned Guthlac A and B, but also poems on Christ, Judgment Day, Soul and Body, and The Lord's Prayer. It also has examples of Anglo-Saxon culture in poems such as "The Wanderer," "The Seafarer," "Deor," and "The Wife's Lament." As well it contains over 90 riddles whose answers are usually mundane things, but some of which engage in double entendres, such as the following, whose answer is dough:

I have heard of a something-or-other, growing in its nook, swelling and rising, pushing up its covering. Upon that boneless thing a cocky-minded young woman took a grip with her hands; with her apron a lord's daughter covered the tumescent thing.

Alcuin, Puzzle-master

Is there anyone who hasn't heard the puzzle of the fox, the goose, and the bag of corn? A man has to transport these three things across a river in a boat which can hold him and one other item. His constraints are that he cannot leave the fox and goose together, or the goose and corn together. This is one of several river-crossing puzzles that exist in different cultures. The earliest version we know of in print is in a late 9th century Latin manuscript of the work Propositiones ad Acuendos Juvenes (Problems to Sharpen the Young), attributed to Alcuin of York (c.735-804).*

"Alcuinus abba"=Father Alcuin

Educated in the cathedral school of York, Alcuin became a monk and teacher. On his way back to England from a trip to Rome in 781 he met Charlemagne, King of the Franks, who recognized Alcuin's erudition and invited him to stay and help promote learning to a level unknown since Rome.

Alcuin became head of the palace school at Aachen, where he established a great library, revised the liturgy, wrote treatises and poetry and works on grammar. It is his influence on learning that is said to have vaulted Latin into the position of being the academic language.

Propositiones ad Acuendos Juvenes is attributed to Alcuin, because of its date and because it is the kind of work he would have created for the pupils at Aachen. The 50+ puzzles in it are very mathematical, with three river-crossing problems—although in his early example the items are a wolf, a goat, and a cabbage. Solutions are provided for all the problems.

Or almost all. There is one that has no solution offered, and it goes like this:

A certain man has 300 pigs. He ordered all of them slaughtered in 3 days, but with an uneven number killed each day. What number were to be killed each day? (Problem 43)

There can be no solution to this puzzle, for obvious reasons. (Feel free to post the reason why in the comments to explain it to your fellow readers.) The assumption is that it was given to difficult students to frustrate them.

*Alcuin also recorded the destruction of Lindisfarne.