A New Model for the Liar Some Results Luca Castaldo - PowerPoint PPT Presentation

A New Model for the Liar L. Castaldo What, and Why? The New Model A New Model for the Liar Some Results Luca Castaldo luca.castaldo@bristol.ac.uk University of Bristol PhDs in Logic X 1st-4th May 2018, Prague, Czech Republic

A New Model for Contents the Liar L. Castaldo 1 What, and Why? What, and Why? Preliminaries The New Model Some Results Liars and Truth-Tellers The Gap 2 The New Model 4-valued logic, Truth-Tables 4-valued Logic, Valuation Function 4-valued Logic, Consequence Relation Interpretation of T 3 Some Results Consistency Adequacy Transparency

A New Model for the Liar L. Castaldo What, and Why? Preliminaries Liars and Truth-Tellers The Gap The New Model Section 1 Some Results What, and Why?

A New Model for Preliminaries the Liar L. Castaldo What, and Why? Preliminaries Liars and The formal language I am going to interpret is the language Truth-Tellers The Gap L pa of Peano arithmetic extended by a unary truth predicate The New Model T . Some Results

A New Model for Preliminaries the Liar L. Castaldo What, and Why? Preliminaries Liars and The formal language I am going to interpret is the language Truth-Tellers The Gap L pa of Peano arithmetic extended by a unary truth predicate The New Model T . The addition of a truth predicate T to L pa delivers two Some Results classes of pathological sentences:

A New Model for Preliminaries the Liar L. Castaldo What, and Why? Preliminaries Liars and The formal language I am going to interpret is the language Truth-Tellers The Gap L pa of Peano arithmetic extended by a unary truth predicate The New Model T . The addition of a truth predicate T to L pa delivers two Some Results classes of pathological sentences: • Truth-tellers: Θ := { τ | PA ⊢ τ ↔ T � τ � } • Liars Λ := { λ | PA ⊢ λ ↔ ¬ T � λ � }

A New Model for Questions the Liar L. Castaldo What, and Why? Preliminaries Liars and Truth-Tellers The Gap The New Model Some Results Two diabolical questions: • Are Liars true or untrue? • Are Truth-tellers true or untrue?

A New Model for Liars and Truth-tellers the Liar L. Castaldo What, and Why? Preliminaries Liars and Truth-Tellers The Gap The New Model Some Results Liars are paradoxical, as they cannot be consistently declared true, or untrue. Truth-tellers are not paradoxical, as they can.

A New Model for Kripke’s Idea the Liar L. Castaldo What, and Why? Preliminaries Liars and Truth-Tellers Following Kripke, one defines a “minimal fixed point model” The Gap �M , ( E ∞ , A ∞ ) � for L t The New Model pa , such that: Some Results • M is the standard interpretation of L pa . • The interpretation of T is a pair ( E ∞ , A ∞ ) of disjoint subsets of N .

A New Model for Kripke’s Idea the Liar L. Castaldo What, and Why? Preliminaries Liars and Truth-Tellers Following Kripke, one defines a “minimal fixed point model” The Gap �M , ( E ∞ , A ∞ ) � for L t The New Model pa , such that: Some Results • M is the standard interpretation of L pa . • The interpretation of T is a pair ( E ∞ , A ∞ ) of disjoint subsets of N . • E ∪ A � = N . • Liars and Truth-tellers are outside E ∞ ∪ A ∞ . • Underlying logic 3-valued, e.g. Kleene’s K 3 .

A New Model for Dialetheic Idea the Liar or Liberalisation of Kripke’s construction L. Castaldo What, and Why? Preliminaries Liars and Truth-Tellers The Gap A dialetheic model is a structure �M , ( E , A ) � , such that: The New Model Some Results • M is the standard interpretation of L pa . • The interpretation of T is a pair ( E , A ) of overlapping subsets of N .

A New Model for Dialetheic Idea the Liar or Liberalisation of Kripke’s construction L. Castaldo What, and Why? Preliminaries Liars and Truth-Tellers The Gap A dialetheic model is a structure �M , ( E , A ) � , such that: The New Model Some Results • M is the standard interpretation of L pa . • The interpretation of T is a pair ( E , A ) of overlapping subsets of N . • Liars are element of E ∩ A . • Truth-tellers are outside E ∪ A . • Underlying logic 4-valued, e.g. Belnap-Dunn’s FDE.

A New Model for The Gap the Liar L. Castaldo What, and Why? Preliminaries Liars and Truth-Tellers The Gap The New Model No theory of truth satisfies the two following desiderata: Some Results 1 It differentiates between Liars and Truth-tellers. 2 It does not admit true contradictions. Kripkean theories fall short of 1. Liberalisations of Kripke’s construction admit inconsistent interpretations.

A New Model for Purpose the Liar L. Castaldo What, and Why? Preliminaries Liars and Truth-Tellers The Gap The New Model Some Results Without arguing for or against dialetheism, I propose a theory which fills the gap: a consistent interpretation of L t pa that sees the difference between Θ and Λ.

A New Model for the Liar L. Castaldo What, and Why? The New Model 4-valued logic, Truth-Tables 4-valued Logic, Valuation Function Section 2 4-valued Logic, Consequence Relation Interpretation of T Some Results The New Model

A New Model for Model Outline the Liar L. Castaldo What, and Why? The New Model 4-valued logic, Truth-Tables �M , ( E , A , X ) � will be the interpretation of L t pa with: 4-valued Logic, Valuation Function 4-valued Logic, Consequence Relation • M is the standard interpretation of L pa . Interpretation of T Some Results

A New Model for Model Outline the Liar L. Castaldo What, and Why? The New Model 4-valued logic, Truth-Tables �M , ( E , A , X ) � will be the interpretation of L t pa with: 4-valued Logic, Valuation Function 4-valued Logic, Consequence Relation • M is the standard interpretation of L pa . Interpretation of T Some Results • The interpretation of T is a triple ( E , A , X ) of pairwise disjoint subsets of N .

A New Model for Model Outline the Liar L. Castaldo What, and Why? The New Model 4-valued logic, Truth-Tables �M , ( E , A , X ) � will be the interpretation of L t pa with: 4-valued Logic, Valuation Function 4-valued Logic, Consequence Relation • M is the standard interpretation of L pa . Interpretation of T Some Results • The interpretation of T is a triple ( E , A , X ) of pairwise disjoint subsets of N . • Liars are element of X . • Truth-tellers are outside E ∪ A ∪ X . • Underlying logic 4-valued, but not FDE.

A New Model for The new Logic the Liar Truth Values L. Castaldo What, and Why? The New Model 4-valued logic, The new 4-valued logic is defined by the structure: Truth-Tables 4-valued Logic, Valuation Function 4-valued Logic, W = { 1 , 0 , p , u } Consequence Relation Interpretation of T D = { 1 } Some Results C = { f ¬ , f ∨ , f ∧ } where: • W = set of truth values (true, false, paradoxical, undefined); • D = set of the sole designated value; • C = set of truth functions.

A New Model for The new Logic the Liar Truth Tables L. Castaldo What, and Why? The New Model 4-valued logic, Truth-Tables 4-valued Logic, ¬ ∨ 1 0 Valuation Function p u 4-valued Logic, Consequence Relation 1 0 1 1 1 1 1 Interpretation of T 0 1 0 1 0 p u Some Results 1 p p p p p u 1 u u u u u u ∧ 1 0 p u 1 1 0 p u 0 0 0 0 0 0 p p p u 0 u u u u

A New Model for Valuation Function the Liar Part 1 L. Castaldo The valuation function V �M , ( E , A , X ) � : L t pa − → { 1 , 0 , p , u } , What, and Why? assigning to each sentence ϕ ∈ L t pa a truth value in the The New Model 4-valued logic, model �M , ( E , A , X ) � , is defined as follows: Truth-Tables 4-valued Logic, Valuation Function (a) For atomic L pa -sentences: 4-valued Logic, Consequence Relation Interpretation of T Some Results (b) For atomic L t pa -sentences T ( n ):

A New Model for Valuation Function the Liar Part 1 L. Castaldo The valuation function V �M , ( E , A , X ) � : L t pa − → { 1 , 0 , p , u } , What, and Why? assigning to each sentence ϕ ∈ L t pa a truth value in the The New Model 4-valued logic, model �M , ( E , A , X ) � , is defined as follows: Truth-Tables 4-valued Logic, Valuation Function (a) For atomic L pa -sentences: 4-valued Logic, Consequence Relation Interpretation of T � 1 if M | = ϕ Some Results V ( ϕ ) = 0 if M | = ¬ ϕ (b) For atomic L t pa -sentences T ( n ):

A New Model for Valuation Function the Liar Part 1 L. Castaldo The valuation function V �M , ( E , A , X ) � : L t pa − → { 1 , 0 , p , u } , What, and Why? assigning to each sentence ϕ ∈ L t pa a truth value in the The New Model 4-valued logic, model �M , ( E , A , X ) � , is defined as follows: Truth-Tables 4-valued Logic, Valuation Function (a) For atomic L pa -sentences: 4-valued Logic, Consequence Relation Interpretation of T � 1 if M | = ϕ Some Results V ( ϕ ) = 0 if M | = ¬ ϕ (b) For atomic L t pa -sentences T ( n ):  1 if n ∈ E     0 if n ∈ A  V ( T ( n )) = if n ∈ X p     if n / ∈ E ∪ A ∪ X u 