Proof

Proof forms a tree structure, in which a conclusion is reached from axioms through a number of valid steps.


Proofs in Propositional Languages:

There exists an ineffable set C of valuations (Ineffability).
A set S of sentences is satisfiable if and only if every finite subset of S is satisfiable (Compactness).
If there is a deduction of B from S, then S entails B (Soundness).
If S entails B, then there is a deduction of B from S (Completeness).


Proofs in Defeasible Reasoning:

The defeasible entailment relation induced by a finite ranked interpretation is well behaved relative to semantic equivalence,
The defeasible entailment relation induced by a finite ranked interpretation is supraclassical,
The defeasible entailment relation induced by a finite ranked interpretation is reflexive
The defeasible entailment relation induced by a finite ranked interpretation is not necessarily transitive
The defeasible entailment relation induced by a finite ranked interpretation is not necessarily monotonic
The defeasible entailment relation induced by a finite ranked interpretation is cumulative
The defeasible entailment relation induced by a finite ranked interpretation is preferential
The defeasible entailment relation induced by a finite ranked interpretation is rational


Proofs in Belief Change:

Architecture of belief:

[ Ife = Finite-ranked-interpretation result of definite information: fixed (f) and evidence (e) ]
[ Ifed = Finite-ranked-interpretation result of definite information and default rule(d) ]

If alpha is a member of the set Kfe of definite beliefs and beta is a member of the set Kfed of defeasible beliefs, then alpha defeasibly entails beta.

AGM postulates:

alpha is inconsistent with S, if and only if, NOT alpha is a member of the set Common Knowledge(S)


If a set of sentences S is unsatisfiable then Cn(S) = LA. (Agent believes everything)