Eloise and Abelard, conjunctive and disjunctive judgements.

Eloise and Abelard. We introduce two individuals, Eloise and Abelard we call players. Eloise claims a judgement is true and Abelard claims it is false. We divide judgements in disjunctive and conjunctive judgements, according if they are equivalent disjunction or to the conjunction of their immediate subjudgements (in a few cases, we could consider a judgement both disjunctive and conjunctive, in this case we make some arbitrary choice). We say that disjunctive judgements belongs to Eloise, and conjunctive judgements to Abelard. We draw disjunctive judgements, and everything belonging to Eloise in pink. We draw conjunctive judgements, and everything belonging to Abelard in blue.

When a judgement is "conjunctive", Abelard has to provide an argument against it. When a judgement is disjunctive, Eloise has to provide an argument in favor of it.
An argument by Abelard, against a conjunctive judgement takes the form of a sub-judgement supposed to be false. If a sub-judgement of a conjunctive judgement is false, then the whole judgement, being equivalent to a conjunction including the sub-judgement, is, indeed, false.
  An argument by Eloise, in favor of a disjunctive judgement takes the form of a sub-judgement supposed to be true. If a sub-judgement of a disjunctive judgement is false, then the whole judgement, being equivalent to a disjunction including the sub-judgement, is, indeed, true.

Let  a denote any atom≠T. Disjunctive judgements are:

t.a (atom≠T),        t.A∨B,            t.∃x.A,            t.A→B
            and
f.T,            f.A∧B,            f.∀x.A            f.¬A    

Let  a denote any atom≠T. Conjunctive judgements are:

t.T,            t.A∧B,            t.∀x.A            t.¬A
            and
f.a (atom≠T),        f.A∨B,            f.∃x.A            f.A→B    

Duality. If we switch f, t, then conjunctive judgement become disjunctive, and disjunctive judgement become conjunctive. We call this phenomenon "duality".

The names conjunctive and disjunctive are appropriate. If a judgement is conjunctive, then it is equivalent to the conjunction of its immediate judgments. If a judgement is disjunctive, then it is equivalent to the disjunction of its immediate judgments. For instance, t.A∧B is conjunctive and it is equivalent to the conjunction of t.A, t.B. The judgement t.A∨B is disjunctive and it is equivalent to the disjunction of t.A, t.B.
  We also made some arbitrary choices. The judgment t.¬A has only one subjudgement f.A, therefore it is arbitrary to consider it conjunctive or disjunctive. We consider it conjunctive. If a is atomic (say, an equation x=y+1, or the like), then the judgements t.a and f.a have only one sub-judgement, therefore it is arbitrary to consider it conjunctive or disjunctive. We consider  t.a disjunctive, and f.a conjunctive.

This is the judgement tree of ((A∧B)∨((A∧¬B)∨(¬A∧B))), cut down to the nodes t.A,t.B,f.A, f.B, and coloured in pink and blue.

Created by Mathematica  (October 17, 2006)