Articular Logic

							                                          R. E. Jennings
                                                 Y. Chen




Laboratory for Logic and Experimental Philosophy
             http://www.sfu.ca/llep/
             Simon Fraser University
Inarticulation
        What is truth
        said doughty Pilate.
        But snappy answer came there none
        and he made good his escape.
        Francis Bacon: Truth is noble.
        Immanuel Jenkins: Whoop-te-doo!*
        (*Quoted in Tessa-Lou Thomas. Immanuel Jenkins: the myth and
        the man.)
Theory and Observation
 Conversational understanding of truth will do for
  observation sentences.
 Theoretical sentences (causality, necessity, implication
  and so on) require something more.
Articulation
 G. W. Leibniz: All truths are analytic.
 Contingent truths are infinitely so.
 Only God can articulate the analysis.
Leibniz realized
 Every wff of classical propositional logic has a finite
  analysis into articulated form:
 Viz. its CNF (A conjunction of disjunctions of literals).
Protecting the analysis
 Classical Semantic representation of CNF’s:
 the intersection of a set of unions of truth-sets of
  literals. (Propositions are single sets.)
 Taking intersections of unions masks the articulation.
 Instead, we suggest, make use of it.
 An analysed proposition is a set of sets of sets.
Hypergraphs
 Hypergraphs provide a natural way of thinking about
  Normal Forms.
 We use hypergraphs instead of sets to represent wffs.
 Classically, inference relations are represented by
  subset relations between sets.
Hypergraphic Representation
 Inference relations are represented by relations
 between hypergraphs.
   α entails β iff the α-hypergraph, Hα is in the relation,
    Bob Loblaw, to the β-hypergraph, Hβ .
   What the inference relation is is determined by how we
    characterize Bob Loblaw.
Hypergraphic Models (h-models)




   Each atom is assigned a hypergraph on the
     power set of the universe .
H-models cont’d
Definition 1




Definition 2
H-models cont’d
Definition 3



Definition 4
Contradictions and Tautologies
H-models cont’d
 We are now in a position to define Bob Loblaw.
 We consider four definitions.
Definition one
FDE (Anderson & Belnap)
 α├ β iff CNF(α) ≤ DNF(β)
 Definition 5:
Definition 0ne
Subsumption




 In the class of h-models, the relation of
 subsumption corresponds to FDE.
First-degree
entailment (FDE)
A ^ B├ B                           A. R. Anderson & N. Belnap,
 A├AvB
A ^ (B v C) ├ (A ^ B) v (A v C)     Tautological entailments, 1961.
~~A ├ A
A ├ ~~A
                                   FDE is determined by a
~(A ^ B) ├ ~A v ~B                  subsumption in the class of h-
~(A v B) ├ ~A ^ ~B
[Mon] Σ ├ A / Σ, Δ ├ A
                                    models.
[Ref] A  Σ / Σ ├ A                FD entailment preserves the
[Trans] Σ, A ├ B, Σ ├ A / Σ ├ B
                                    cardinality of a set of
                                    contradictions.
Definition two
Definition two
First-degree analytic entailment (FDAE):
RFDAE: subsumption + prescriptive principle



In the class of h-models, RFDAE corresponds to
  FDAE.
Analytic Implication
 Kit Fine: analytic implication
 Strict implication + prescriptive principle
 Arthur Prior
First degree analytic
entailment (FDAE)
A ^ B├ B                          First-Degree fragment of Parry’s original system
A├AvB
A^B├AvB                           A├A^A
A ^ (B v C) ├ (A ^ B) v (A v C)   A^B├B^A
~~A ├ A                           ~~A ├ A
A ├ ~~A                           A ├ ~~A
~(A ^ B) ├ ~A v ~B                A ^ (B v C) ├ (A ^ B) v (A v C)
~(A v B) ├ ~A ^ ~B                A├ B^C/A├ B
[Mon] Σ ├ A / Σ, Δ ├ A            A ├ B, C ├ D / A ^ B ├ C ^ D
[Ref] A  Σ / Σ ├ A               A ├ B, C ├ D / A v B ├ C v D
[Trans] Σ, A ├ B, Σ ├ A / Σ ├ B   A v (B ^ ~B) ├ A
                                  A ├ B, B ├ C / A ├ C
FDAE preserves classical          f (A) / A ├ A
contingency and colourability.    A ├ B, B ├ A / f (A) ├ f (B), f (B) ├ f(A)
                                  A, B ├ A ^ B
                                  ~ A ├ A, A ├ B / ~ B ├ B
Definition three
Definition Three
First-degree Parry entailment (FDPE)
First degree Parry
entailment (FDPE)
A ^ B├ B                          While the prescriptive principle in FDAE preserves
A├AvB                               vertices of hypergraphs that semantically
A^B├AvB                             represent wffs, that in FDPE preserves atoms of
A ├ A v ~A                          wffs.
A ^ (B v C) ├ (A ^ B) v (A v C)
~~A ├ A
A ├ ~~A
~(A ^ B) ├ ~A v ~B
~(A v B) ├ ~A ^ ~B
[Mon] Σ ├ A / Σ, Δ ├ A
[Ref] A  Σ / Σ ├ A
[Trans] Σ, A ├ B, Σ ├ A / Σ ├ B
Definition four
Definition Four
 First-degree sub-entailment (FDSE)
FDSE
A ^ B├ B
                                   Comparing with FDAE
A├AvB
A ^ (B v C) ├ (A ^ B) v (A v C)    and FDPE:
~~A ├ A
A ├ ~~A
                                  A^B├AvB
~(A ^ B) ├ ~A v ~B                A ├ A v ~A
~(A v B) ├ ~A ^ ~B
[Mon] Σ ├ A / Σ, Δ ├ A
[Ref] A  Σ / Σ ├ A
[Trans] Σ, A ├ B, Σ ├ A / Σ ├ B
FDSE Lattice
Future Research

 First-degree modal logics
 Higher-degree systems
 Other non-Boolean algebras