# Articular Logic

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```							                                          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
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
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

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