On Troikas 6: Delta Deduction

Delta Deduction

Delta deduction is a system for determining what propositions in a troika will pass, given which ones have already passed. 

Delta  weakens deduction for positive functions such as “and” and “or”; only half of the usual deduction rules are valid. For instance:

From:              “A and B” passes

Deduce:           “A” passes

AND    “B” passes

From:              “A” passes

OR       “B” passes

Deduce:           “A or  B” passes

The reverse deductions are invalid due to “strong or” and “weak and”.

Majority gets treated like ‘and’ and ‘or’ do in ordinary proof systems.

From:              “A” passes

AND    “B” passes

AND    “C” passes

Deduce:           “Most of (A,B,C)” passes

You need all three; “weak majorities” fail when one of their three components fails.

From:  “Most of (A,B,C)” passes

Deduce:                  “A” passes

OR  “B” passes

OR  “C” passes.

You need all three; “strong majorities” pass when one of their three components pass.

From:              “A” passes

AND    “A = B”  passes unanimously

Deduce:           “B” passes.

            You need unanimity; “equivalence glitches” equate unequals. Unanimous laws include all those of Boolean algebra.

The law of the excluded middle applies, in two ways:

Deduce:           “A or  not A” passes;

“A or else not A” passes

Deduce:           “A” passes  OR  “not A” passes;

                        “A” passes  OR  ELSE “not A” passes

Reductio ad absurdum also applies:

From:              From:  “A” passes

Deduce: “B and not B” passes

Deduce:           “not A” passes.

Conjecture: The above deductive system, recursively applied, is complete for 3-voter elections. That is, it deduces which propositions necessarily pass, given which passed before; and any system which does not yield an explicit contradiction under these rules has a 3-voter model.

Please note that not all inconsistent systems have three-voter models; for some are inconsistent under delta rules. But consider this 5-voter election:

Do you like...

Apples?           Bananas?         Cherries?

Moe:               yes                   no                    no

Larry:              no                    yes                   no

Curly:              no                    no                    yes

Shemp:            yes                   yes                   yes

Curly Joe:       yes                   yes                   yes

 

The following propositions pass by 3/5 each:

I like apples.

I like bananas.

I like cherries.

I do not like most of those three.

Thus 5-voter election deduction is even weaker than 3-voter.