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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism and Axiomatic Theories of Truth Deflationism Peano Proof theoretic and model theoretic conservativities Arithmetic The Compositional Theory of Stella Moon Truth ILLC, UvA 2nd October 2015

Deflationism Deflationism and Axiomatic Theories of Truth Stella Moon Truth is insubstantial so it does not carry any ontological Deflationism weight. Peano Arithmetic The Compositional Theory of Truth

Deflationism Deflationism and Axiomatic Theories of Truth Stella Moon Truth is insubstantial so it does not carry any ontological Deflationism weight. Peano Deflationists are interested how truth works, rather than Arithmetic The what it is. Compositional Theory of Truth

Deflationism Deflationism and Axiomatic Theories of Truth Stella Moon Truth is insubstantial so it does not carry any ontological Deflationism weight. Peano Deflationists are interested how truth works, rather than Arithmetic The what it is. Compositional Theory of It is true that snow is white iff snow is white. Truth

Deflationism Deflationism and Axiomatic Theories of Truth Stella Moon Truth is insubstantial so it does not carry any ontological Deflationism weight. Peano Deflationists are interested how truth works, rather than Arithmetic The what it is. Compositional Theory of It is true that snow is white iff snow is white. Truth Motivated by Tarski’s biconditionals: for any sentence φ T ( φ ) ↔ φ.

Proof theoretic and model theoretic conservativities Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Deflationists desire our extended theory to be conservative Arithmetic over our base theory. The Compositional Theory of Truth

Proof theoretic and model theoretic conservativities Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Deflationists desire our extended theory to be conservative Arithmetic over our base theory. The Compositional There are two notions of conservativities: for models and Theory of Truth for theories.

Proof theoretic and model theoretic conservativities Deflationism and Axiomatic Theories of Truth Stella Moon Definition (Proof theoretic conservativity) Let Γ be a L -theory and Γ ′ be a L ′ -theory extending Γ , that is Deflationism L ′ ⊇ L , such that Γ ′ ⊇ Γ . Γ ′ is proof theoretically conservative Peano Arithmetic over Γ if for any L -setenece θ , Γ ′ ⊢ θ , we have that Γ ⊢ θ . The Compositional Theory of Truth

Proof theoretic and model theoretic conservativities Deflationism and Axiomatic Theories of Truth Stella Moon Definition (Proof theoretic conservativity) Let Γ be a L -theory and Γ ′ be a L ′ -theory extending Γ , that is Deflationism L ′ ⊇ L , such that Γ ′ ⊇ Γ . Γ ′ is proof theoretically conservative Peano Arithmetic over Γ if for any L -setenece θ , Γ ′ ⊢ θ , we have that Γ ⊢ θ . The Compositional Theory of Truth Definition (Model theoretic conservativity) Let Γ be an L -theory and Γ ′ be an L ′ -theory extending Γ . Γ ′ is model-theoretically conservative over Γ if any model of Γ can be expanded to a model of Γ ′ .

Peano Arithmetic Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Axioms of arithmetic (natural numbers) Arithmetic The Compositional Theory of Truth

Peano Arithmetic Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Axioms of arithmetic (natural numbers) – self-reference. Arithmetic The Compositional Theory of Truth

Peano Arithmetic Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Axioms of arithmetic (natural numbers) – self-reference. Arithmetic The L a = { <, + , · , s , 0 } Compositional Theory of Truth

Peano Arithmetic Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Axioms of arithmetic (natural numbers) – self-reference. Arithmetic The L a = { <, + , · , s , 0 } Compositional Theory of G¨ odel’s diagonal Lemma and the incompleteness theorems. Truth

Axioms of Peano Arithmetic (PA) Deflationism ∀ x ( s ( x ) � = 0) and Axiomatic Theories of ∀ x , y ( s ( x ) = s ( y ) → x = y ) Truth Stella Moon ∀ x ( x + 0 = x ) ∀ x , y ( x + s ( y )) = s ( x + y ) Deflationism Peano ∀ x ( x · 0 = 0) Arithmetic ∀ x , y ( x · s ( y ) = ( x · y ) + x ) The Compositional Theory of ∀ x ( ¬ x < 0) Truth ∀ x , y ( x < s ( y ) ↔ ( x < y ∨ x = y )) ∀ x (0 < x ↔ 0 = x ) ∀ x , y ( s ( x ) < y ↔ ( x < y ∧ y � = s ( x ))) For all formulae φ ( x ), �� � �� � φ (0) ∧ ∀ x ( φ ( x ) → φ ( x + 1)) → ∀ x φ ( x ) .

Models of arithmetic Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Standard model: N = � N ; < N ; + N , · N ; s N ; 0 N � . Peano Arithmetic The Compositional Theory of Truth

Models of arithmetic Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Standard model: N = � N ; < N ; + N , · N ; s N ; 0 N � . Peano Arithmetic Non-standard models: M = � M ; < M ; + M , · M ; s M ;0 M � . The Compositional Theory of Truth

Models of arithmetic Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Standard model: N = � N ; < N ; + N , · N ; s N ; 0 N � . Peano Arithmetic Non-standard models: M = � M ; < M ; + M , · M ; s M ;0 M � . The Compositional There is a c ∈ M such that for any n ∈ N , n < c . Theory of Truth

Models of arithmetic Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Standard model: N = � N ; < N ; + N , · N ; s N ; 0 N � . Peano Arithmetic Non-standard models: M = � M ; < M ; + M , · M ; s M ;0 M � . The Compositional There is a c ∈ M such that for any n ∈ N , n < c . Theory of Truth They are not isomorphic to each other.

G¨ odel coding Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism f : Form L a − → N Peano Arithmetic The Compositional Theory of Truth

G¨ odel coding Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism f : Form L a − → N Peano Arithmetic f is a recursive function The Compositional Theory of Truth

G¨ odel coding Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism f : Form L a − → N Peano Arithmetic f is a recursive function The Compositional im ( f ) ⊆ N is recursive Theory of Truth

G¨ odel coding Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism f : Form L a − → N Peano Arithmetic f is a recursive function The Compositional im ( f ) ⊆ N is recursive Theory of Truth f − 1 is a recursive function

G¨ odel coding — Deflationism and Axiomatic Theories of Truth L a -symbols Natural numbers Stella Moon 0 0 Deflationism s 1 Peano + 2 Arithmetic · 3 The Compositional < 4 Theory of Truth = 5 ∧ 6 ∨ 7 ¬ 8 ∃ 9 ∀ 10 (11 , i ) v i

Theorems Deflationism and Axiomatic Theories of Theorem (Diagonal Lemma) Truth Stella Moon For any formula φ ( x ) , there is a sentence θ such that Deflationism PA ⊢ φ ( � θ � ) ↔ θ. Peano Arithmetic The Compositional Theory of Truth

Theorems Deflationism and Axiomatic Theories of Theorem (Diagonal Lemma) Truth Stella Moon For any formula φ ( x ) , there is a sentence θ such that Deflationism PA ⊢ φ ( � θ � ) ↔ θ. Peano Arithmetic The Compositional Theory of Proof Truth We define a function diag : N → N in the following way: � � ∀ y ( y = n → σ ( y )) � , if n = � σ ( x ) � for some formula diag ( n ) = 0 , otherwise.

Deflationism and Axiomatic Theories of Truth Stella Moon Definition (Provability predicate) Deflationism Peano For any L a formula φ , the provability predicate in PA is defined Arithmetic in the following way. The Compositional If PA ⊢ φ then PA ⊢ Prv ( � φ � ) Theory of Truth PA ⊢ Prv ( � φ → ψ � ) → ( Prv ( � φ � ) → Prv ( � ψ � )) PA ⊢ Prv ( � φ � ) → Prv ( � Prv ( � φ � ) � )

Theorems Deflationism and Axiomatic Theories of Theorem (G¨ odel’s first incompleteness theorem) Truth Stella Moon There is a sentence G such that Deflationism PA ⊢ G ↔ ¬ Prv ( � G � ) . Peano Arithmetic The Compositional Theory of Truth

Theorems Deflationism and Axiomatic Theories of Theorem (G¨ odel’s first incompleteness theorem) Truth Stella Moon There is a sentence G such that Deflationism PA ⊢ G ↔ ¬ Prv ( � G � ) . Peano Arithmetic The Compositional Theory of Theorem (G¨ odel’s second incompleteness theorem) Truth Assume PA is consistent and let Con PA := ¬ Prv ( � ⊥ � ) be the sentence defining the consistency of PA . Then PA � Con PA i.e. PA cannot prove its own consistency.

Liar sentence Deflationism and Axiomatic Theories of Theorem Truth Stella Moon Assume the Tarski-biconditionals for all sentences in PA . Let T Deflationism be a predicate defining truth in PA . T is undefinable in L a . Peano Arithmetic The Compositional Theory of Truth