Laws of Triple Form: 12 of 12

          Chapter 11 : Voter’s Paradox

Voter’s Paradox

Trilemma deduction

Syllogisms by Trilemma

         

11A. Voter’s Paradox

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; z≠x; deduce: x≠y

          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.

         

Some angels read comic books;

Only aliens read comic books;

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.