## Tuesday, July 24, 2018

### Laws of Triple Form: 12 of 12

Trilemma deduction
Syllogisms by Trilemma

Consider these three orders on the three forms:

6 < 1 < 0
0 < 6 < 1
1 < 0 < 6

The minima on these orders cyclically distribute; so do the maxima. Taken together they form a “troika”, a.k.a. a “voter’s paradox”, which is at the heart of Kenneth Arrow’s Impossibility Theorem. These logic knots have a habit of bollixing political systems. These tiny tangles give politics its notorious perversity.

To simplify presentation of the Troika, I introduce three fictional characters; none other than the Three Stooges.
General Moe rules the Scissors Party with an iron hand. His politics are fascistic; he favors Lies over Truth over Imagination. He prefers monarchy, most preferably if the monarch is himself.
Judge Larry is senior theoretician for the Paper Party. His politics are legalistic: he favors Truth over Imagination over Lies. He prefers to govern by consensus.
Mayor Curly is lead singer for the Rock Party. His politics are populistic: he favors Imagination over Lies over Truth. He prefers to govern by majority rule.

Moe: Imagination  <  Truth  <  Lies
Larry: Lies  <  Imagination  <  Truth
Curly: Truth  <  Lies  <  Imagination

Each single Stooge has a consistent linear ranking of imagination, lies and truth; but when you put them all together, something has got to go. Two-thirds of the Stooges (namely, Moe and Larry) put truth above imagination; Larry and Curly put imagination above lies; and Curly and Moe put lies above truth.

Moe            Larry                    Curly
Lies < Imagination?       no               yes              yes
Truth < Lies?                 yes              no               yes
Imagination < Truth?     yes              yes              no

This gives us a Condorcet Election, or “Voter’s Paradox”:

Truth
<                   <
Imagination        >               Lies

-  by 2/3 majority each; yet they all agree that the ranking is linear!

There are several partial resolutions to this.
If we appoint a single voter as tyrant (Moe, say) then we can decide this consistently; but this is not a fair system.
If we attempt to decide by consensus (as Larry suggests) then that is fair and consistent; but we decide nothing, and that is a weak system.
If we have faith in majority rule (as Curly professes) then we accept the non-linear order, and the linearity of the order. This is fair and decisive, but it is inconsistent.
Finally, we can accept the non-linear ranking, and accept it as non-linear; this goes with every 2/3 majority, but reverses a consensus; and that is perverse.

This political knot is an instance of Arrow’s Impossibility Theorem, which says that no voting system has all four of these virtues:
it is fair: it gives all voters equal power
it is decisive: it decides all questions posed to it
it is logical: it does not believe contradictions
it is responsive: it never defies a voter consensus.

In other words, any government is at least one of:
cruel  ;  weak  ;  absurd  ;  perverse.

Moe prefers cruelty and lies, Larry prefers weakness and truth, and Curly prefers absurdity and imagination; none of them want perversity and paradox, but of course that is what they always get!

11B. Trilemma deduction

A trilemma is a triple of propositions, any two of which can be true but not all three. Any trilemma implies this deductive principle; from any two, derive the negation of the third. For instance, this trilemma, the “disimplication glitch”:

A;
A implies B;
Not B.
implies  these deductive rules:
From: A; A implies B; deduce: B
From: A implies B; Not B; deduce: not A
From: not B; A; deduce: A does not imply B

So modus ponens, modus tollens, and “anti-implication” are encoded by one trilemma. A trilemma is an encapsuled near-absurdity; and near-absurdity is the engine of deduction. Every deduction is a narrow escape, one trilemma wide.
And for each trilemma, there’s a troika that generates it as a voter’s paradox. In it, each of the three voters affirm two-thirds of the trilemma, and reject the third.
In this case:
Moe says: A; A implies B; B
Larry says: A implies B; Not B; not A
Curly says: not B; A; A does not imply B

Take the “strong or glitch”:
Not A;
Not B;
A or B.
It implies  these deductive rules:
From: not A; not B; deduce: not A nor B
From: not B; A or B; deduce: A
From: A or B; not A; deduce: B
Here’s a troika that generates it as a voter’s paradox:
Moe says: not A; not B;  not A nor B
Larry says: not B; A or B; A
Curly says: A or B; not A; B

Here’s the “weak and glitch”:
A;
B;
not(A and B)
It implies  these deductive rules:
From: A; B; deduce: A and B
From: B; not(A and B); deduce: not(A)

This is the “Failed Reductio” glitch:
A implies B;
A implies not B:
A.
It implies these deductive rules:
From: A implies B; A implies not B; deduce: not A
From: A implies not B; A; deduce: A does not imply B
From: A; A implies B; deduce: A does not imply not B

This is the “disequality glitch”:
x = y;
y = z;
z ≠ x.
It implies  these deductive rules:
From: x=y; y=z; deduce: z=x
From: y=z; zx; deduce: xy

Here is “Barbarism”:
All A are B;
All B are C;
Some A are not C.
It implies these deductive rules:
From: All A are B; All B are C; deduce: All A are C
From: All B are C; Some A are not C; deduce: Some A are not B
From: Some A are not C; All A are B; deduce: Some B are not C

This is the “Disinduction Trilemma”:
1 has property P;
For any n, P(n) implies P(n+1);
For some N, not P(N).

It implies these deductive rules:

From: 1 has property P; For any n, P(n) implies P(n+1);
Deduce:       For all N, P(N).
From: For any n, P(n) implies P(n+1); For some N, not P(N);
Deduce:       1 does not have property P.
From: For some N, not P(N); 1 has property P;
Deduce:       For some n, P(n) does not imply P(n+1).

This is the “Disintermediation Trilemma”:
f(x) is real and continuous on the interval [a,b];
f(x) does not equal zero anywhere on [a,b];
f(a) and f(b) have different signs.

It implies these deductive rules:

From:
f(x) is real and continuous on the interval [a,b];
f(x) does not equal zero anywhere on [a,b];
Deduce:
f(a) and f(b) have the same sign.

From:
f(a) and f(b) have different signs;
f(x) is real and continuous on the interval [a,b];
Deduce:
f(x) does equal zero somewhere on [a,b].

From:
f(x) does not equal zero anywhere on [a,b];
f(a) and f(b) have different signs.
Deduce:
f(x) is not both real and continuous on the interval [a,b].
Here is the “First Cause Trilemma”:
There is a first cause;
Every cause has a cause;
There are no causal loops.

It implies these deduction rules:
From: there is a first cause; every cause has a cause; deduce:
there are causal loops.
From: every cause has a cause; there are no causal loops; deduce:
there is no first cause.
From: there are no causal loops; there is a first cause; deduce:
not every cause has a cause.

The trilemma can be written as:

There exists A, such that for every B, A causes B.
For every A, there exists B, such that B causes A.
There do not exist A and B such that A causes B and B causes A.

This generalizes to any predicate R:
There exists x, such that for every y, xRy;
For every x, there exists y, such that yRx;
There do not exist x and y such that xRy and yRx.

Call this the “Loop Trilemma”. It’s a variant of the weak-and glitch.

When the predicate is “explains”, you get the “Munchhausen Trilemma”:
Explanation is simple; there is a full explanation.
Explanation is complete; every explanation is explained.
Explanation is not circular; no two explanations explain each other.

Here’s a “Set Loop Trilemma”
There exists an x, such that for every y, yϵx;
For every x, there exists a y, such that xϵy;
There does not exist an x and a y such that xϵy and yϵx.
There’s a universal set; every set’s an element; there are no set loops.

11C. Syllogisms by Trilemma

Consider these modal identities:

Swap:
All A are B                    =       All not-B are not-A
No A are B                    =       No B are A
Some A are B                =       Some B are A
Some A are not B          =       Some not-B are not not-A

Negation:
Not (all A are B)            =       Some A are not-B
Not(no A are B)             =       Some A are B
Not(some A are B)                  =       No A are B
Not(some A are not-B)  =       All A are B

Mode Switch:
All A are B           =       No A are not-B
No A are B           =       All A are not-B
Some A are B       =       Some A are not not-B

From one side of an equation, deduce the other.
To these rules, add one more:

Some-All-None Trilemma: From any two of;
Some A are B
All B are C
No A are C
deduce the negation of the third.

Modal identities plus some-all-none trilemma yields the core of classical syllogism theory. Adding existential import to ‘all’ yields the rest.  For instance, modal identities, substitutions and swap can transform the Some-All-None Trilemma to Barbarism in three steps thus:

Some A are not not-B
All not-C are not-B
All A are not-C                                 (by modal identities)

Some X are not Z
All Y are Z
All X are Y                              (substitute X=A, Y=not-C, Z = not-B)

All X are Y
All Y are Z
Some X are not Z                    (swap)

An anti-syllogism is not a syllogism itself, but it’s always ready to explode into three conflicting syllogisms. For instance; the Barbarism trilemma defies classical logic; yet encodes three classical logic rules:
From: All X are Y, All Y are Z, deduce all X are Z.
From: All Y are Z, some X are not Z, deduce not all X are Y.
From: Some X are not Z, All X are Y, deduce not all Y are Z.

The last two can be changed, by substitutions and swaps, to:
From: Some X are Z, no Y are Z, deduce some X are not Y.
From: Some X are Z, All X are Y, deduce some Y are Z.

Exercise for the student:
Derive deduction rules and troikas from these 24 trilemmas:

Some days are bliss;
All bliss is perfect;
No days are perfect.

All heroes are immortal;
Some men are heroes;
All men are mortal.

No philosophers are liars;
Some philosophers are Cretans;
All Cretans are liars.

Some flattery is stupid;
Stupidity is always boring;
Flattery is never boring.

Some good deeds are wise;
Only foolish deeds are punished;
No good deed goes unpunished.

No aliens are angels.

Only you can prevent forest fires;
Smokey the Bear can prevent forest fires;
You are not Smokey the Bear.

All Scots are canny;
All ghosts are uncanny;
Some Scots are ghosts.

Dragons are wise;
Dragons are brutal;
Dragons are not both wise and brutal.

I am a bum;
All bums are crooks;
I am not a crook.

I love Alice;
I do not love Bob;
I love Alice and Bob equally.

Bob is not a genius;
Bob is not an idiot;
Bob is a genius or an idiot.

If Alice fell down a rabbit hole, then I’ll be a monkey’s uncle;
If Alice did not fall down a rabbit hole, then I’ll be a monkey’s uncle;
I will not be a monkey’s uncle.

There are more saints than sages;
There are more sages than heroes;
There are more heroes than saints.

A crook is better than a fool;
A fool is better than a wimp;
A wimp is better than a crook.

The food is fast;
The food is cheap;
The food is good.

The pundit is honest;
The pundit is intelligent;
The pundit is partisan.

An attorney should investigate zealously;
An attorney should keep client’s secrets;
An attorney should report perjury.

Superman can fly;
Clark Kent can’t fly;
Clark Kent is Superman.

Some frogs are princes;
Some frogs are not princes;
All frogs are equally princes.

All lions are fierce;
Only coffee-drinkers are fierce;
Not all lions drink coffee.

No ducks waltz;
All officers waltz;
Some officers are ducks.

Time is unbounded;
Time is linear;
Time is finite.

There is a final effect;
All effects have effects;
Effectuation does not loop.