Preliminaries Premise 1 Premise 2 Nontransitive Approaches to Paradox and Compositional Principles of Truth Jonathan Dittrich LMU Munich, MCMP Jonathan.Dittrich@lmu.de 04.05.2017 1 / 22
Preliminaries Premise 1 Premise 2 The argument (1) Non-transitive theories render compositional principles of truth paradoxical in virtue of their notion of paradoxicality. 2 / 22
Preliminaries Premise 1 Premise 2 The argument (1) Non-transitive theories render compositional principles of truth paradoxical in virtue of their notion of paradoxicality. (2) If a theory renders compositional principles of truth paradoxical, it is inadequate. 2 / 22
Preliminaries Premise 1 Premise 2 The argument (1) Non-transitive theories render compositional principles of truth paradoxical in virtue of their notion of paradoxicality. (2) If a theory renders compositional principles of truth paradoxical, it is inadequate. (C) Non-transitive theories of truth are inadequate. 2 / 22
Preliminaries Premise 1 Premise 2 Structure Preliminaries 1 Premise 1 2 Premise 2 3 3 / 22
Preliminaries Premise 1 Premise 2 Preliminaries We work in the language L of FOL = , Q and a unary truth-predicate T 4 / 22
Preliminaries Premise 1 Premise 2 Preliminaries We work in the language L of FOL = , Q and a unary truth-predicate T # φ is the G¨ odel code of φ 4 / 22
Preliminaries Premise 1 Premise 2 Preliminaries We work in the language L of FOL = , Q and a unary truth-predicate T # φ is the G¨ odel code of φ � φ � is the numeral of the G¨ odel code of φ 4 / 22
Preliminaries Premise 1 Premise 2 Preliminaries We work in the language L of FOL = , Q and a unary truth-predicate T # φ is the G¨ odel code of φ � φ � is the numeral of the G¨ odel code of φ We use a suitable proof system in sequent-calculus style that allows for Cut-elimination 4 / 22
Preliminaries Premise 1 Premise 2 The operational rules Γ , φ ⊢ ∆ Γ ⊢ ∆ , φ ¬ R ¬ L Γ ⊢ ∆ , ¬ φ Γ , ¬ φ ⊢ ∆ Γ ′ , ψ ⊢ ∆ ′ Γ ⊢ ∆ , φ, ψ Γ , φ ⊢ ∆ ∨ R ∨ L Γ , Γ ′ , φ ∨ ψ ⊢ ∆ , ∆ ′ Γ ⊢ ∆ , φ ∨ ψ Γ ⊢ ∆ , φ (y) Γ , φ (a) ⊢ ∆ ∀ R ∀ L Γ ⊢ ∆ , ∀ x φ (x) Γ ∀ x φ (x) ⊢ ∆ 5 / 22
Preliminaries Premise 1 Premise 2 Identity rules a=a, Γ ⊢ ∆ =ID Γ ⊢ ∆ a=b, φ (a), φ (b), Γ ⊢ ∆ =repL a=b, φ (b), φ (b) Γ ⊢ ∆ a=b, Γ ⊢ ∆ , φ (a), φ (b) =repR a=b, Γ ⊢ ∆ , φ (b), φ (b) 6 / 22
Preliminaries Premise 1 Premise 2 The structural rules Reflexivity: R φ ⊢ φ 7 / 22
Preliminaries Premise 1 Premise 2 The structural rules Reflexivity: R φ ⊢ φ Weakening: Γ ⊢ ∆ Γ ⊢ ∆ WL WR Γ , φ ⊢ ∆ Γ ⊢ φ , ∆ 7 / 22
Preliminaries Premise 1 Premise 2 The structural rules Reflexivity: R φ ⊢ φ Weakening: Γ ⊢ ∆ Γ ⊢ ∆ WL WR Γ , φ ⊢ ∆ Γ ⊢ φ , ∆ Contraction: Γ , φ , φ ⊢ ∆ Γ ⊢ φ , φ , ∆ CL CR Γ , φ ⊢ ∆ Γ ⊢ φ , ∆ 7 / 22
Preliminaries Premise 1 Premise 2 The structural rules Reflexivity: R φ ⊢ φ Weakening: Γ ⊢ ∆ Γ ⊢ ∆ WL WR Γ , φ ⊢ ∆ Γ ⊢ φ , ∆ Contraction: Γ , φ , φ ⊢ ∆ Γ ⊢ φ , φ , ∆ CL CR Γ , φ ⊢ ∆ Γ ⊢ φ , ∆ Cut: Γ ⊢ φ , Π Σ , φ ⊢ Ω Cut Γ , Σ ⊢ Π , Ω 7 / 22
Preliminaries Premise 1 Premise 2 Q1 Γ , 0=Sa ⊢ ∆ Γ , a=b, Sa=Sb ⊢ ∆ Q2 Γ , Sa=Sb ⊢ ∆ Γ , a=0 ⊢ ∆ Γ , a=Sy ⊢ ∆ Q3 Γ ⊢ ∆ Γ , a+0 = a ⊢ ∆ Q4 Γ ⊢ ∆ 8 / 22
Preliminaries Premise 1 Premise 2 Γ , a+Sb = S(a+b) ⊢ ∆ Q5 Γ ⊢ ∆ Γ ,a × 0=0 ⊢ ∆ Q6 Γ ⊢ ∆ Γ , a × Sb=(a × b) + a ⊢ ∆ Q7 Γ ⊢ ∆ 9 / 22
Preliminaries Premise 1 Premise 2 Fully transparent truth-predicate T expressed by the T-rules: Γ ⊢ φ , ∆ Γ ⊢ T( � φ � ), ∆ Γ ⊢ T( � φ � ), ∆ Γ ⊢ φ , ∆ Γ , φ ⊢ ∆ Γ , T( � φ � ) ⊢ ∆ Γ , T( � φ � ) ⊢ ∆ Γ , φ ⊢ ∆ 10 / 22
Preliminaries Premise 1 Premise 2 The Liar(s): Weak Liar λ There is a recursive diagonalisation function f d ( # φ ( x ) ) = # ∀ x ( x = � φ ( x ) � → φ ( x ) ) for any φ ( x ) with x free. 11 / 22
Preliminaries Premise 1 Premise 2 The Liar(s): Weak Liar λ There is a recursive diagonalisation function f d ( # φ ( x ) ) = # ∀ x ( x = � φ ( x ) � → φ ( x ) ) for any φ ( x ) with x free. We further know that Q includes a predicate D ( x , y ) that strongly represents this function. 11 / 22
Preliminaries Premise 1 Premise 2 The Liar(s): Weak Liar λ There is a recursive diagonalisation function f d ( # φ ( x ) ) = # ∀ x ( x = � φ ( x ) � → φ ( x ) ) for any φ ( x ) with x free. We further know that Q includes a predicate D ( x , y ) that strongly represents this function. This strong representation allows us to prove a weak diagonal lemma: Weak Diagonal Lemma ∀ x ( x = � D ( x , y ) → φ ( y ) � → ∀ y ( D ( x , y ) → φ ( y )) ↔ φ (¯ n ) 11 / 22
Preliminaries Premise 1 Premise 2 The Liar(s): Weak Liar λ By appropriate choice of φ , we get ∀ x ( x = � D ( x , y ) → ¬ T ( y ) � → ∀ y ( D ( x , y ) → ¬ T ( y )) ↔ ¬ T (¯ n )) . 12 / 22
Preliminaries Premise 1 Premise 2 The Liar(s): Weak Liar λ By appropriate choice of φ , we get ∀ x ( x = � D ( x , y ) → ¬ T ( y ) � → ∀ y ( D ( x , y ) → ¬ T ( y )) ↔ ¬ T (¯ n )) . By letting ∀ x ( x = � D ( x , y ) → ¬ T ( y ) � → ∀ y ( D ( x , y ) → ¬ T ( y )) be λ , we get the usual The Liar λ ↔ ¬ T ( � λ � ) 12 / 22
Preliminaries Premise 1 Premise 2 The Liar(s): Weak Liar λ By appropriate choice of φ , we get ∀ x ( x = � D ( x , y ) → ¬ T ( y ) � → ∀ y ( D ( x , y ) → ¬ T ( y )) ↔ ¬ T (¯ n )) . By letting ∀ x ( x = � D ( x , y ) → ¬ T ( y ) � → ∀ y ( D ( x , y ) → ¬ T ( y )) be λ , we get the usual The Liar λ ↔ ¬ T ( � λ � ) By the invertibility of our rules, we know that also ⊢ λ, T ( � λ � ) and λ, T ( � λ � ) ⊢ are provable. 12 / 22
Preliminaries Premise 1 Premise 2 Proving absurdity from λ λ, T ( � λ � ) ⊢ ⊢ λ, T ( � λ � ) T ( � λ � ) , T ( � λ � ) ⊢ ⊢ T ( � λ � ) , T ( � λ � ) CL CR T ( � λ � ) ⊢ ⊢ T ( � λ � ) Cut ⊢ 13 / 22
Preliminaries Premise 1 Premise 2 Proving absurdity from λ λ, T ( � λ � ) ⊢ ⊢ λ, T ( � λ � ) T ( � λ � ) , T ( � λ � ) ⊢ ⊢ T ( � λ � ) , T ( � λ � ) CL CR T ( � λ � ) ⊢ ⊢ T ( � λ � ) Cut ⊢ But the empty sequent is not derivable without Cut! This motivates non-transitive solutions which give up Cut (see e.g. Ripley, Tennant). 13 / 22
Preliminaries Premise 1 Premise 2 Non-transitive theories: Fully transparent, classical (yet inconsistent but non-trivial) theory of truth. 14 / 22
Preliminaries Premise 1 Premise 2 Non-transitive theories: Fully transparent, classical (yet inconsistent but non-trivial) theory of truth. Thesis: Although non-transitive theories can deal with a fully transparent truth-predicate, they are inadequate with respect to stronger demands to the truth-predicate: The compositional principles of truth. 14 / 22
Preliminaries Premise 1 Premise 2 The argument (1) Non-transitive theories render compositional principles of truth paradoxical in virtue of their notion of paradoxicality. 15 / 22
Preliminaries Premise 1 Premise 2 The argument (1) Non-transitive theories render compositional principles of truth paradoxical in virtue of their notion of paradoxicality. (2) If a theory renders compositional principles of truth paradoxical, it is inadequate. 15 / 22
Preliminaries Premise 1 Premise 2 The argument (1) Non-transitive theories render compositional principles of truth paradoxical in virtue of their notion of paradoxicality. (2) If a theory renders compositional principles of truth paradoxical, it is inadequate. (C) Non-transitive theories of truth are inadequate. 15 / 22
Preliminaries Premise 1 Premise 2 The compositional principles of truth (Neg) ∀ x ( Sent ( x ) → ( T ( ¬ . x ) ↔ ¬ T ( x ))) (Con) ∀ x ∀ y ( Sent ( x ) ∧ Sent ( y ) → ( T ( x ∧ . y ) ↔ T ( x ) ∧ T ( y ))) (Dis) ∀ x ∀ y ( Sent ( x ) ∧ Sent ( y ) → ( T ( x ∨ . y ) ↔ T ( x ) ∨ T ( y ))) (Cond) ∀ x ∀ y ( Sent ( x ) ∧ Sent ( y ) → ( T ( x → . y ) ↔ T ( x ) → T ( y ))) 16 / 22
Preliminaries Premise 1 Premise 2 The compositional principles of truth (Neg) ∀ x ( Sent ( x ) → ( T ( ¬ . x ) ↔ ¬ T ( x ))) (Con) ∀ x ∀ y ( Sent ( x ) ∧ Sent ( y ) → ( T ( x ∧ . y ) ↔ T ( x ) ∧ T ( y ))) (Dis) ∀ x ∀ y ( Sent ( x ) ∧ Sent ( y ) → ( T ( x ∨ . y ) ↔ T ( x ) ∨ T ( y ))) (Cond) ∀ x ∀ y ( Sent ( x ) ∧ Sent ( y ) → ( T ( x → . y ) ↔ T ( x ) → T ( y ))) Note that although their instances are provable, the universally quantified principles are not. However, they are typically added to formal theories of truth. 16 / 22
Preliminaries Premise 1 Premise 2 In support of (1) By its solution to the paradoxes in terms of Cut-inadmissibility, NT is committed to the following notion of paradoxicality: 17 / 22
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