Presentation on theme: "R. E. Jennings Y. Chen Laboratory for Logic and Experimental Philosophy Simon Fraser University."— Presentation transcript:
R. E. Jennings Y. Chen Laboratory for Logic and Experimental Philosophy Simon Fraser University
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.)
Conversational understanding of truth will do for observation sentences. Theoretical sentences (causality, necessity, implication and so on) require something more.
G. W. Leibniz: All truths are analytic. Contingent truths are infinitely so. Only God can articulate the analysis.
Every wff of classical propositional logic has a finite analysis into articulated form: Viz. its CNF (A conjunction of disjunctions of literals).
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 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.
Inference relations are represented by relations between hypergraphs. α entails β iff the α -hypergraph, H α is in the relation, Bob Loblaw, to the β -hypergraph, H β. What the inference relation is is determined by how we characterize Bob Loblaw.
Each atom is assigned a hypergraph on the power set of the universe.
Definition 2 Definition 1
Definition 3 Definition 4
We are now in a position to define Bob Loblaw. We consider four definitions.
α├ β iff DNF(α) ≤ CNF(β) Definition 5 :
Subsumption In the class of a-models, the relation of subsumption corresponds to FDE.
First-degree entailment (FDE) FDE A ^ B├ B A ├ A v B 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 A. R. Anderson & N. Belnap, Tautological entailments, FDE is determined by a subsumption in the class of a- models. FD entailment preserves the cardinality of a set of contradictions.
A&B ((A → A) → B) → B; (A → B) → ((B → C) → (A → C)); (A → (A → B)) → (A → B); (A → B) ∧ (A → C) ├ A → B ∧ C ; (A → C) ∧ (B → C) ├ AVB → C ; (A →~ A) →~ A; (A →~ B) →( B →~ A); NA ∧ NB → N(A ∧ B ). NA=def (A → A) → A R&C (A → B) ∧ (A → C) ├ A → B ∧ C ; (A → C) ∧ (B → C) ├ AVB → C ; A → C ├ A ∧ B → C ; (A → B) ├ AVC → BVC ; A → B ∧ C ├ A → C ;
First-degree analytic entailment (FDAE): R FDAE : subsumption + prescriptive principle In the class of h-models, R FDAE corresponds to FDAE.
First degree analytic entailment (FDAE) FDAE A ^ B├ B A ├ A v B A ^ B ├ A v B 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 FDAE preserves classical contingency and colourability. First-Degree fragment of Parry’s original system A ├ A ^ A A ^ B ├ B ^ A ~~A ├ A A ├ ~~A A ^ (B v C) ├ (A ^ B) v (A v C) A ├ B ^ C / A ├ B A ├ B, C ├ D / A ^ B ├ C ^ D A ├ B, C ├ D / A v B ├ C v D A v (B ^ ~B) ├ A A ├ B, B ├ C / A ├ C f (A) / A ├ A A ├ B, B ├ A / f (A) ├ f (B), f (B) ├ f(A) A, B ├ A ^ B ~ A ├ A, A ├ B / ~ B ├ B
First-degree Parry entailment (FDPE)
First degree Parry entailment (FDPE) FDPE A ^ B├ B A ├ A v B A ^ B ├ A v B A ├ A v ~A 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 While the prescriptive principle in FDAE preserves vertices of hypergraphs that semantically represent wffs, that in FDPE preserves atoms of wffs.
First-degree sub-entailment (FDSE)
A ^ B├ B A ├ A v B 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 Comparing with FDAE and FDPE: A ^ B ├ A v B A ├ A v ~A
First-degree modal logics Higher-degree systems Other non-Boolean algebras